+------------------------------------------------------------------+ | | | Transcriber's Note: In the parts containing mathematical | | notation, superscript characters have been used where possible. | | The remaining few are denoted using underscores and curly | | brackets, for example e_{n+1}. Italicised words are marked using | | underscores _like this_, but the letters used in the mathematics | | (which were all in italic font) have not been marked, to aid | | legibility and to avoid confusion with the subscripts. | | | | The special characters used in this file are: | | α β γ δ ε η θ ι λ μ ν ο π ρ ς σ τ υ | | à é ê ï ô ➝ ′ ″ ² ₁ ₂ ₃ ₄ ⁸ | | If you can only display the first few on the 2nd line, you may | | prefer the ISO-8859-1 version of this ebook; if you can see all | | apart from the last six, then either the ISO-8859-1 text or the | | HTML versions should be suitable. If you can see none of the | | special characters, use the ASCII version. | | | | Two printer errors have been corrected. These are: | | | | "... Relating the entity discriminated by sight with that | | discriminated by [sight]... " sight has been changed to touch as | | suggested by the sense. | | | | "... [universely] proportional to... " universely has been changed | | to inversely. | | | +------------------------------------------------------------------+

 The Concept of

 NATURE

 THE TARNER LECTURES DELIVERED IN TRINITY COLLEGE NOVEMBER 1919

 Alfred North Whitehead

PREFACE

The contents of this book were originally delivered at Trinity Collegein the autumn of 1919 as the inaugural course of Tarner lectures. TheTarner lectureship is an occasional office founded by the liberality ofMr Edward Tarner. The duty of each of the successive holders of the postwill be to deliver a course on 'the Philosophy of the Sciences and theRelations or Want of Relations between the different Departments ofKnowledge. ' The present book embodies the endeavour of the firstlecturer of the series to fulfil his task. 

The chapters retain their original lecture form and remain as deliveredwith the exception of minor changes designed to remove obscurities ofexpression. The lecture form has the advantage of suggesting an audiencewith a definite mental background which it is the purpose of the lectureto modify in a specific way. In the presentation of a novel outlook withwide ramifications a single line of communications from premises toconclusions is not sufficient for intelligibility. Your audience willconstrue whatever you say into conformity with their pre-existingoutlook. For this reason the first two chapters and the last twochapters are essential for intelligibility though they hardly add to theformal completeness of the exposition. Their function is to prevent thereader from bolting up side tracks in pursuit of misunderstandings. Thesame reason dictates my avoidance of the existing technical terminologyof philosophy. The modern natural philosophy is shot through andthrough with the fallacy of bifurcation which is discussed in the secondchapter of this work. Accordingly all its technical terms in some subtleway presuppose a misunderstanding of my thesis. It is perhaps as well tostate explicitly that if the reader indulges in the facile vice ofbifurcation not a word of what I have here written will be intelligible. 

The last two chapters do not properly belong to the special course. Chapter VIII is a lecture delivered in the spring of 1920 before theChemical Society of the students of the Imperial College of Science andTechnology. It has been appended here as conveniently summing up andapplying the doctrine of the book for an audience with one definite typeof outlook. 

This volume on 'the Concept of Nature' forms a companion book to myprevious work _An Enquiry concerning the Principles of NaturalKnowledge_. Either book can be read independently, but they supplementeach other. In part the present book supplies points of view which wereomitted from its predecessor; in part it traverses the same ground withan alternative exposition. For one thing, mathematical notation has beencarefully avoided, and the results of mathematical deductions areassumed. Some of the explanations have been improved and others havebeen set in a new light. On the other hand important points of theprevious work have been omitted where I have had nothing fresh to sayabout them. On the whole, whereas the former work based itself chieflyon ideas directly drawn from mathematical physics, the present bookkeeps closer to certain fields of philosophy and physics to theexclusion of mathematics. The two works meet in their discussions ofsome details of space and time. 

I am not conscious that I have in any way altered my views. Somedevelopments have been made. Those that are capable of anon-mathematical exposition have been incorporated in the text. Themathematical developments are alluded to in the last two chapters. Theyconcern the adaptation of the principles of mathematical physics to theform of the relativity principle which is here maintained. Einstein'smethod of using the theory of tensors is adopted, but the application isworked out on different lines and from different assumptions. Those ofhis results which have been verified by experience are obtained also bymy methods. The divergence chiefly arises from the fact that I do notaccept his theory of non-uniform space or his assumption as to thepeculiar fundamental character of light-signals. I would not however bemisunderstood to be lacking in appreciation of the value of his recentwork on general relativity which has the high merit of first disclosingthe way in which mathematical physics should proceed in the light of theprinciple of relativity. But in my judgment he has cramped thedevelopment of his brilliant mathematical method in the narrow bounds ofa very doubtful philosophy. 

The object of the present volume and of its predecessor is to lay thebasis of a natural philosophy which is the necessary presupposition of areorganised speculative physics. The general assimilation of space andtime which dominates the constructive thought can claim the independentsupport of Minkowski from the side of science and also of succeedingrelativists, while on the side of philosophers it was, I believe, onetheme of Prof. Alexander's Gifford lectures delivered some few years agobut not yet published. He also summarised his conclusions on thisquestion in a lecture to the Aristotelian Society in the July of 1918. Since the publication of _An Enquiry concerning the Principles ofNatural Knowledge_ I have had the advantage of reading Mr C.  D. Broad's_Perception, Physics, and Reality_ [Camb. Univ. Press, 1914]. Thisvaluable book has assisted me in my discussion in Chapter II, though Iam unaware as to how far Mr Broad would assent to any of my arguments asthere stated. 

It remains for me to thank the staff of the University Press, itscompositors, its proof-readers, its clerks, and its managing officials, not only for the technical excellence of their work, but for the waythey have co-operated so as to secure my convenience. 

A.  N.  W. 

 IMPERIAL COLLEGE OF SCIENCE AND TECHNOLOGY. _April_, 1920. 

CONTENTS

 CHAP. PAGE

 I NATURE AND THOUGHT 1

 II THEORIES OF THE BIFURCATION OF NATURE 26

 III TIME 49

 IV THE METHOD OF EXTENSIVE ABSTRACTION 74

 V SPACE AND MOTION 99

 VI CONGRUENCE 120

 VII OBJECTS 143

 VIII SUMMARY 164

 IX THE ULTIMATE PHYSICAL CONCEPTS 185

 NOTE: ON THE GREEK CONCEPT OF A POINT 197

 NOTE: ON SIGNIFICANCE AND INFINITE EVENTS 197

 INDEX 199

THE CONCEPT OF NATURE

CHAPTER I

NATURE AND THOUGHT

The subject-matter of the Tarner lectures is defined by the founder tobe 'the Philosophy of the Sciences and the Relations or Want ofRelations between the different Departments of Knowledge. ' It is fittingat the first lecture of this new foundation to dwell for a few momentson the intentions of the donor as expressed in this definition; and I doso the more willingly as I shall thereby be enabled to introduce thetopics to which the present course is to be devoted. 

We are justified, I think, in taking the second clause of the definitionas in part explanatory of the earlier clause. What is the philosophy ofthe sciences? It is not a bad answer to say that it is the study of therelations between the different departments of knowledge. Then withadmirable solicitude for the freedom of learning there is inserted inthe definition after the word 'relations' the phrase 'or want ofrelations. ' A disproof of relations between sciences would in itselfconstitute a philosophy of the sciences. But we could not dispenseeither with the earlier or the later clause. It is not every relationbetween sciences which enters into their philosophy. For example biologyand physics are connected by the use of the microscope. Still, I maysafely assert that a technical description of the uses of the microscopein biology is not part of the philosophy of the sciences. Again, youcannot abandon the later clause of the definition; namely thatreferring to the relations between the sciences, without abandoning theexplicit reference to an ideal in the absence of which philosophy mustlanguish from lack of intrinsic interest. That ideal is the attainmentof some unifying concept which will set in assigned relationships withinitself all that there is for knowledge, for feeling, and for emotion. That far off ideal is the motive power of philosophic research; andclaims allegiance even as you expel it. The philosophic pluralist is astrict logician; the Hegelian thrives on contradictions by the help ofhis absolute; the Mohammedan divine bows before the creative will ofAllah; and the pragmatist will swallow anything so long as it 'works. '

The mention of these vast systems and of the age-long controversies fromwhich they spring, warns us to concentrate. Our task is the simpler oneof the philosophy of the sciences. Now a science has already a certainunity which is the very reason why that body of knowledge has beeninstinctively recognised as forming a science. The philosophy of ascience is the endeavour to express explicitly those unifyingcharacteristics which pervade that complex of thoughts and make it to bea science. The philosophy of the sciences--conceived as one subject--isthe endeavour to exhibit all sciences as one science, or--in case ofdefeat--the disproof of such a possibility. 

Again I will make a further simplification, and confine attention to thenatural sciences, that is, to the sciences whose subject-matter isnature. By postulating a common subject-matter for this group ofsciences a unifying philosophy of natural science has been therebypresupposed. 

What do we mean by nature? We have to discuss the philosophy of naturalscience. Natural science is the science of nature. But--What is nature?

Nature is that which we observe in perception through the senses. Inthis sense-perception we are aware of something which is not thought andwhich is self-contained for thought. This property of beingself-contained for thought lies at the base of natural science. It meansthat nature can be thought of as a closed system whose mutual relationsdo not require the expression of the fact that they are thought about. 

Thus in a sense nature is independent of thought. By this statement nometaphysical pronouncement is intended. What I mean is that we can thinkabout nature without thinking about thought. I shall say that then weare thinking 'homogeneously' about nature. 

Of course it is possible to think of nature in conjunction with thoughtabout the fact that nature is thought about. In such a case I shall saythat we are thinking 'heterogeneously' about nature. In fact during thelast few minutes we have been thinking heterogeneously about nature. Natural science is exclusively concerned with homogeneous thoughts aboutnature. 

But sense-perception has in it an element which is not thought. It is adifficult psychological question whether sense-perception involvesthought; and if it does involve thought, what is the kind of thoughtwhich it necessarily involves. Note that it has been stated above thatsense-perception is an awareness of something which is not thought. Namely, nature is not thought. But this is a different question, namelythat the fact of sense-perception has a factor which is not thought. Icall this factor 'sense-awareness. ' Accordingly the doctrine thatnatural science is exclusively concerned with homogeneous thoughts aboutnature does not immediately carry with it the conclusion that naturalscience is not concerned with sense-awareness. 

However, I do assert this further statement; namely, that though naturalscience is concerned with nature which is the terminus ofsense-perception, it is not concerned with the sense-awareness itself. 

I repeat the main line of this argument, and expand it in certaindirections. 

Thought about nature is different from the sense-perception of nature. Hence the fact of sense-perception has an ingredient or factor which isnot thought. I call this ingredient sense-awareness. It is indifferentto my argument whether sense-perception has or has not thought asanother ingredient. If sense-perception does not involve thought, thensense-awareness and sense-perception are identical. But the somethingperceived is perceived as an entity which is the terminus of thesense-awareness, something which for thought is beyond the fact of thatsense-awareness. Also the something perceived certainly does not containother sense-awarenesses which are different from the sense-awarenesswhich is an ingredient in that perception. Accordingly nature asdisclosed in sense-perception is self-contained as againstsense-awareness, in addition to being self-contained as against thought. I will also express this self-containedness of nature by saying thatnature is closed to mind. 

This closure of nature does not carry with it any metaphysical doctrineof the disjunction of nature and mind. It means that in sense-perceptionnature is disclosed as a complex of entities whose mutual relations areexpressible in thought without reference to mind, that is, withoutreference either to sense-awareness or to thought. Furthermore, I do notwish to be understood as implying that sense-awareness and thought arethe only activities which are to be ascribed to mind. Also I am notdenying that there are relations of natural entities to mind or mindsother than being the termini of the sense-awarenesses of minds. Accordingly I will extend the meaning of the terms 'homogeneousthoughts' and 'heterogeneous thoughts' which have already beenintroduced. We are thinking 'homogeneously' about nature when we arethinking about it without thinking about thought or aboutsense-awareness, and we are thinking 'heterogeneously' about nature whenwe are thinking about it in conjunction with thinking either aboutthought or about sense-awareness or about both. 

I also take the homogeneity of thought about nature as excluding anyreference to moral or aesthetic values whose apprehension is vivid inproportion to self-conscious activity. The values of nature are perhapsthe key to the metaphysical synthesis of existence. But such a synthesisis exactly what I am not attempting. I am concerned exclusively with thegeneralisations of widest scope which can be effected respecting thatwhich is known to us as the direct deliverance of sense-awareness. 

I have said that nature is disclosed in sense-perception as a complex ofentities. It is worth considering what we mean by an entity in thisconnexion. 'Entity' is simply the Latin equivalent for 'thing' unlesssome arbitrary distinction is drawn between the words for technicalpurposes. All thought has to be about things. We can gain some idea ofthis necessity of things for thought by examination of the structure ofa proposition. 

Let us suppose that a proposition is being communicated by an expositorto a recipient. Such a proposition is composed of phrases; some of thesephrases may be demonstrative and others may be descriptive. 

By a demonstrative phrase I mean a phrase which makes the recipientaware of an entity in a way which is independent of the particulardemonstrative phrase. You will understand that I am here using'demonstration' in the non-logical sense, namely in the sense in which alecturer demonstrates by the aid of a frog and a microscope thecirculation of the blood for an elementary class of medical students. Iwill call such demonstration 'speculative' demonstration, rememberingHamlet's use of the word 'speculation' when he says, There is no speculation in those eyes. 

Thus a demonstrative phrase demonstrates an entity speculatively. It mayhappen that the expositor has meant some other entity--namely, thephrase demonstrates to him an entity which is diverse from the entitywhich it demonstrates to the recipient. In that case there is confusion;for there are two diverse propositions, namely the proposition for theexpositor and the proposition for the recipient. I put this possibilityaside as irrelevant for our discussion, though in practice it may bedifficult for two persons to concur in the consideration of exactly thesame proposition, or even for one person to have determined exactly theproposition which he is considering. 

Again the demonstrative phrase may fail to demonstrate any entity. Inthat case there is no proposition for the recipient. I think that wemay assume (perhaps rashly) that the expositor knows what he means. 

A demonstrative phrase is a gesture. It is not itself a constituent ofthe proposition, but the entity which it demonstrates is such aconstituent. You may quarrel with a demonstrative phrase as in some wayobnoxious to you; but if it demonstrates the right entity, theproposition is unaffected though your taste may be offended. Thissuggestiveness of the phraseology is part of the literary quality of thesentence which conveys the proposition. This is because a sentencedirectly conveys one proposition, while in its phraseology it suggests apenumbra of other propositions charged with emotional value. We are nowtalking of the one proposition directly conveyed in any phraseology. 

This doctrine is obscured by the fact that in most cases what is in forma mere part of the demonstrative gesture is in fact a part of theproposition which it is desired directly to convey. In such a case wewill call the phraseology of the proposition elliptical. In ordinaryintercourse the phraseology of nearly all propositions is elliptical. 

Let us take some examples. Suppose that the expositor is in London, sayin Regent's Park and in Bedford College, the great women's college whichis situated in that park. He is speaking in the college hall and hesays, 'This college building is commodious. '

The phrase 'this college building' is a demonstrative phrase. Nowsuppose the recipient answers, 'This is not a college building, it is the lion-house in the Zoo. '

Then, provided that the expositor's original proposition has not beencouched in elliptical phraseology, the expositor sticks to his originalproposition when he replies, 'Anyhow, _it_ is commodious. '

Note that the recipient's answer accepts the speculative demonstrationof the phrase 'This college building. ' He does not say, 'What do youmean?' He accepts the phrase as demonstrating an entity, but declaresthat same entity to be the lion-house in the Zoo. In his reply, theexpositor in his turn recognises the success of his original gesture asa speculative demonstration, and waives the question of the suitabilityof its mode of suggestiveness with an 'anyhow. ' But he is now in aposition to repeat the original proposition with the aid of ademonstrative gesture robbed of any suggestiveness, suitable orunsuitable, by saying, '_It_ is commodious. '

The '_it_' of this final statement presupposes that thought has seizedon the entity as a bare objective for consideration. 

We confine ourselves to entities disclosed in sense-awareness. Theentity is so disclosed as a relatum in the complex which is nature. Itdawns on an observer because of its relations; but it is an objectivefor thought in its own bare individuality. Thought cannot proceedotherwise; namely, it cannot proceed without the ideal bare 'it' whichis speculatively demonstrated. This setting up of the entity as a bareobjective does not ascribe to it an existence apart from the complex inwhich it has been found by sense-perception. The 'it' for thought isessentially a relatum for sense-awareness. 

The chances are that the dialogue as to the college building takesanother form. Whatever the expositor originally meant, he almostcertainly now takes his former statement as couched in ellipticalphraseology, and assumes that he was meaning, 'This is a college building and is commodious. '

Here the demonstrative phrase or the gesture, which demonstrates the'it' which is commodious, has now been reduced to 'this'; and theattenuated phrase, under the circumstances in which it is uttered, issufficient for the purpose of correct demonstration. This brings out thepoint that the verbal form is never the whole phraseology of theproposition; this phraseology also includes the general circumstances ofits production. Thus the aim of a demonstrative phrase is to exhibit adefinite 'it' as a bare objective for thought; but the _modus operandi_of a demonstrative phrase is to produce an awareness of the entity as aparticular relatum in an auxiliary complex, chosen merely for the sakeof the speculative demonstration and irrelevant to the proposition. Forexample, in the above dialogue, colleges and buildings, as related tothe 'it' speculatively demonstrated by the phrase 'this collegebuilding, ' set that 'it' in an auxiliary complex which is irrelevant tothe proposition

 'It is commodious. '

Of course in language every phrase is invariably highly elliptical. Accordingly the sentence

 'This college building is commodious'

means probably

 'This college building is commodious as a college building. '

But it will be found that in the above discussion we can replace'commodious' by 'commodious as a college building' without altering ourconclusion; though we can guess that the recipient, who thought he wasin the lion-house of the Zoo, would be less likely to assent to. 

 'Anyhow, it is commodious as a college building. '

A more obvious instance of elliptical phraseology arises if theexpositor should address the recipient with the remark, 'That criminal is your friend. '

The recipient might answer, 'He is my friend and you are insulting. '

Here the recipient assumes that the phrase 'That criminal' is ellipticaland not merely demonstrative. In fact, pure demonstration is impossiblethough it is the ideal of thought. This practical impossibility of puredemonstration is a difficulty which arises in the communication ofthought and in the retention of thought. Namely, a proposition about aparticular factor in nature can neither be expressed to others norretained for repeated consideration without the aid of auxiliarycomplexes which are irrelevant to it. 

I now pass to descriptive phrases. The expositor says, 'A college in Regent's Park is commodious. '

The recipient knows Regent's Park well. The phrase 'A college inRegent's Park' is descriptive for him. If its phraseology is notelliptical, which in ordinary life it certainly will be in some way orother, this proposition simply means, 'There is an entity which is a college building in Regent's Park and is commodious. '

If the recipient rejoins, 'The lion-house in the Zoo is the only commodious building in Regent's Park, '

he now contradicts the expositor, on the assumption that a lion-house ina Zoo is not a college building. 

Thus whereas in the first dialogue the recipient merely quarrelled withthe expositor without contradicting him, in this dialogue he contradictshim. Thus a descriptive phrase is part of the proposition which it helpsto express, whereas a demonstrative phrase is not part of theproposition which it helps to express. 

Again the expositor might be standing in Green Park--where there are nocollege buildings--and say, 'This college building is commodious. '

Probably no proposition will be received by the recipient because thedemonstrative phrase, 'This college building'

has failed to demonstrate owing to the absence of the background ofsense-awareness which it presupposes. 

But if the expositor had said, 'A college building in Green Park is commodious, '

the recipient would have received a proposition, but a false one. 

Language is usually ambiguous and it is rash to make general assertionsas to its meanings. But phrases which commence with 'this' or 'that' areusually demonstrative, whereas phrases which commence with 'the' or 'a'are often descriptive. In studying the theory of propositionalexpression it is important to remember the wide difference between theanalogous modest words 'this' and 'that' on the one hand and 'a' and'the' on the other hand. The sentence

 'The college building in Regent's Park is commodious'

means, according to the analysis first made by Bertrand Russell, theproposition, 'There is an entity which (i) is a college building in Regent's Park and (ii) is commodious and (iii) is such that any college building in Regent's Park is identical with it. '

The descriptive character of the phrase 'The college building inRegent's Park' is thus evident. Also the proposition is denied by thedenial of any one of its three component clauses or by the denial of anycombination of the component clauses. If we had substituted 'Green Park'for 'Regent's Park' a false proposition would have resulted. Also theerection of a second college in Regent's Park would make the propositionfalse, though in ordinary life common sense would politely treat it asmerely ambiguous. 

'The Iliad' for a classical scholar is usually a demonstrative phrase;for it demonstrates to him a well-known poem. But for the majority ofmankind the phrase is descriptive, namely, it is synonymous with 'Thepoem named "the Iliad". '

Names may be either demonstrative or descriptive phrases. For example'Homer' is for us a descriptive phrase, namely, the word with someslight difference in suggestiveness means 'The man who wrote the Iliad. '

This discussion illustrates that thought places before itself bareobjectives, entities as we call them, which the thinking clothes byexpressing their mutual relations. Sense-awareness discloses fact withfactors which are the entities for thought. The separate distinction ofan entity in thought is not a metaphysical assertion, but a method ofprocedure necessary for the finite expression of individualpropositions. Apart from entities there could be no finite truths; theyare the means by which the infinitude of irrelevance is kept out ofthought. 

To sum up: the termini for thought are entities, primarily with bareindividuality, secondarily with properties and relations ascribed tothem in the procedure of thought; the termini for sense-awareness arefactors in the fact of nature, primarily relata and only secondarilydiscriminated as distinct individualities. 

No characteristic of nature which is immediately posited for knowledgeby sense-awareness can be explained. It is impenetrable by thought, inthe sense that its peculiar essential character which enters intoexperience by sense-awareness is for thought merely the guardian of itsindividuality as a bare entity. Thus for thought 'red' is merely adefinite entity, though for awareness 'red' has the content of itsindividuality. The transition from the 'red' of awareness to the 'red'of thought is accompanied by a definite loss of content, namely by thetransition from the factor 'red' to the entity 'red. ' This loss in thetransition to thought is compensated by the fact that thought iscommunicable whereas sense-awareness is incommunicable. 

Thus there are three components in our knowledge of nature, namely, fact, factors, and entities. Fact is the undifferentiated terminus ofsense-awareness; factors are termini of sense-awareness, differentiatedas elements of fact; entities are factors in their function as thetermini of thought. The entities thus spoken of are natural entities. Thought is wider than nature, so that there are entities for thoughtwhich are not natural entities. 

When we speak of nature as a complex of related entities, the 'complex'is fact as an entity for thought, to whose bare individuality isascribed the property of embracing in its complexity the naturalentities. It is our business to analyse this conception and in thecourse of the analysis space and time should appear. Evidently therelations holding between natural entities are themselves naturalentities, namely they are also factors of fact, there forsense-awareness. Accordingly the structure of the natural complex cannever be completed in thought, just as the factors of fact can never beexhausted in sense-awareness. Unexhaustiveness is an essential characterof our knowledge of nature. Also nature does not exhaust the matter forthought, namely there are thoughts which would not occur in anyhomogeneous thinking about nature. 

The question as to whether sense-perception involves thought is largelyverbal. If sense-perception involves a cognition of individualityabstracted from the actual position of the entity as a factor in fact, then it undoubtedly does involve thought. But if it is conceived assense-awareness of a factor in fact competent to evoke emotion andpurposeful action without further cognition, then it does not involvethought. In such a case the terminus of the sense-awareness is somethingfor mind, but nothing for thought. The sense-perception of some lowerforms of life may be conjectured to approximate to this characterhabitually. Also occasionally our own sense-perception in moments whenthought-activity has been lulled to quiescence is not far off theattainment of this ideal limit. 

The process of discrimination in sense-awareness has two distinct sides. There is the discrimination of fact into parts, and the discriminationof any part of fact as exhibiting relations to entities which are notparts of fact though they are ingredients in it. Namely the immediatefact for awareness is the whole occurrence of nature. It is nature as anevent present for sense-awareness, and essentially passing. There is noholding nature still and looking at it. We cannot redouble our effortsto improve our knowledge of the terminus of our present sense-awareness;it is our subsequent opportunity in subsequent sense-awareness whichgains the benefit of our good resolution. Thus the ultimate fact forsense-awareness is an event. This whole event is discriminated by usinto partial events. We are aware of an event which is our bodily life, of an event which is the course of nature within this room, and of avaguely perceived aggregate of other partial events. This is thediscrimination in sense-awareness of fact into parts. 

I shall use the term 'part' in the arbitrarily limited sense of an eventwhich is part of the whole fact disclosed in awareness. 

Sense-awareness also yields to us other factors in nature which are notevents. For example, sky-blue is seen as situated in a certain event. This relation of situation requires further discussion which ispostponed to a later lecture. My present point is that sky-blue is foundin nature with a definite implication in events, but is not an eventitself. Accordingly in addition to events, there are other factors innature directly disclosed to us in sense-awareness. The conception inthought of all the factors in nature as distinct entities with definitenatural relations is what I have in another place[1] called the'diversification of nature. '

[1] Cf. _Enquiry_. 

There is one general conclusion to be drawn from the foregoingdiscussion. It is that the first task of a philosophy of science shouldbe some general classification of the entities disclosed to us insense-perception. 

Among the examples of entities in addition to 'events' which we haveused for the purpose of illustration are the buildings of BedfordCollege, Homer, and sky-blue. Evidently these are very different sortsof things; and it is likely that statements which are made about onekind of entity will not be true about other kinds. If human thoughtproceeded with the orderly method which abstract logic would suggest toit, we might go further and say that a classification of naturalentities should be the first step in science itself. Perhaps you will beinclined to reply that this classification has already been effected, and that science is concerned with the adventures of material entitiesin space and time. 

The history of the doctrine of matter has yet to be written. It is thehistory of the influence of Greek philosophy on science. That influencehas issued in one long misconception of the metaphysical status ofnatural entities. The entity has been separated from the factor which isthe terminus of sense-awareness. It has become the substratum for thatfactor, and the factor has been degraded into an attribute of theentity. In this way a distinction has been imported into nature which isin truth no distinction at all. A natural entity is merely a factor offact, considered in itself. Its disconnexion from the complex of fact isa mere abstraction. It is not the substratum of the factor, but the veryfactor itself as bared in thought. Thus what is a mere procedure of mindin the translation of sense-awareness into discursive knowledge has beentransmuted into a fundamental character of nature. In this way matterhas emerged as being the metaphysical substratum of its properties, andthe course of nature is interpreted as the history of matter. 

Plato and Aristotle found Greek thought preoccupied with the quest forthe simple substances in terms of which the course of events could beexpressed. We may formulate this state of mind in the question, What isnature made of? The answers which their genius gave to this question, and more particularly the concepts which underlay the terms in whichthey framed their answers, have determined the unquestionedpresuppositions as to time, space and matter which have reigned inscience. 

In Plato the forms of thought are more fluid than in Aristotle, andtherefore, as I venture to think, the more valuable. Their importanceconsists in the evidence they yield of cultivated thought about naturebefore it had been forced into a uniform mould by the long tradition ofscientific philosophy. For example in the _Timaeus_ there is apresupposition, somewhat vaguely expressed, of a distinction between thegeneral becoming of nature and the measurable time of nature. In a laterlecture I have to distinguish between what I call the passage of natureand particular time-systems which exhibit certain characteristics ofthat passage. I will not go so far as to claim Plato in direct supportof this doctrine, but I do think that the sections of the _Timaeus_which deal with time become clearer if my distinction is admitted. 

This is however a digression. I am now concerned with the origin of thescientific doctrine of matter in Greek thought. In the _Timaeus_ Platoasserts that nature is made of fire and earth with air and water asintermediate between them, so that 'as fire is to air so is air towater, and as air is to water so is water to earth. ' He also suggests amolecular hypothesis for these four elements. In this hypothesiseverything depends on the shape of the atoms; for earth it is cubicaland for fire it is pyramidal. To-day physicists are again discussingthe structure of the atom, and its shape is no slight factor in thatstructure. Plato's guesses read much more fantastically than doesAristotle's systematic analysis; but in some ways they are morevaluable. The main outline of his ideas is comparable with that ofmodern science. It embodies concepts which any theory of naturalphilosophy must retain and in some sense must explain. Aristotle askedthe fundamental question, What do we mean by 'substance'? Here thereaction between his philosophy and his logic worked very unfortunately. In his logic, the fundamental type of affirmative proposition is theattribution of a predicate to a subject. Accordingly, amid the manycurrent uses of the term 'substance' which he analyses, he emphasisesits meaning as 'the ultimate substratum which is no longer predicated ofanything else. '

The unquestioned acceptance of the Aristotelian logic has led to aningrained tendency to postulate a substratum for whatever is disclosedin sense-awareness, namely, to look below what we are aware of for thesubstance in the sense of the 'concrete thing. ' This is the origin ofthe modern scientific concept of matter and of ether, namely they arethe outcome of this insistent habit of postulation. 

Accordingly ether has been invented by modern science as the substratumof the events which are spread through space and time beyond the reachof ordinary ponderable matter. Personally, I think that predication is amuddled notion confusing many different relations under a convenientcommon form of speech. For example, I hold that the relation of green toa blade of grass is entirely different from the relation of green tothe event which is the life history of that blade for some short period, and is different from the relation of the blade to that event. In asense I call the event the situation of the green, and in another senseit is the situation of the blade. Thus in one sense the blade is acharacter or property which can be predicated of the situation, and inanother sense the green is a character or property of the same eventwhich is also its situation. In this way the predication of propertiesveils radically different relations between entities. 

Accordingly 'substance, ' which is a correlative term to 'predication, 'shares in the ambiguity. If we are to look for substance anywhere, Ishould find it in events which are in some sense the ultimate substanceof nature. 

Matter, in its modern scientific sense, is a return to the Ionian effortto find in space and time some stuff which composes nature. It has amore refined signification than the early guesses at earth and water byreason of a certain vague association with the Aristotelian idea ofsubstance. 

Earth, water, air, fire, and matter, and finally ether are related indirect succession so far as concerns their postulated characters ofultimate substrata of nature. They bear witness to the undying vitalityof Greek philosophy in its search for the ultimate entities which arethe factors of the fact disclosed in sense-awareness. This search is theorigin of science. 

The succession of ideas starting from the crude guesses of the earlyIonian thinkers and ending in the nineteenth century ether reminds usthat the scientific doctrine of matter is really a hybrid through whichphilosophy passed on its way to the refined Aristotelian concept ofsubstance and to which science returned as it reacted againstphilosophic abstractions. Earth, fire, and water in the Ionic philosophyand the shaped elements in the _Timaeus_ are comparable to the matterand ether of modern scientific doctrine. But substance represents thefinal philosophic concept of the substratum which underlies anyattribute. Matter (in the scientific sense) is already in space andtime. Thus matter represents the refusal to think away spatial andtemporal characteristics and to arrive at the bare concept of anindividual entity. It is this refusal which has caused the muddle ofimporting the mere procedure of thought into the fact of nature. Theentity, bared of all characteristics except those of space and time, hasacquired a physical status as the ultimate texture of nature; so thatthe course of nature is conceived as being merely the fortunes of matterin its adventure through space. 

Thus the origin of the doctrine of matter is the outcome of uncriticalacceptance of space and time as external conditions for naturalexistence. By this I do not mean that any doubt should be thrown onfacts of space and time as ingredients in nature. What I do mean is 'theunconscious presupposition of space and time as being that within whichnature is set. ' This is exactly the sort of presupposition which tingesthought in any reaction against the subtlety of philosophical criticism. My theory of the formation of the scientific doctrine of matter is thatfirst philosophy illegitimately transformed the bare entity, which issimply an abstraction necessary for the method of thought, into themetaphysical substratum of these factors in nature which in varioussenses are assigned to entities as their attributes; and that, as asecond step, scientists (including philosophers who were scientists) inconscious or unconscious ignoration of philosophy presupposed thissubstratum, _qua_ substratum for attributes, as nevertheless in time andspace. 

This is surely a muddle. The whole being of substance is as a substratumfor attributes. Thus time and space should be attributes of thesubstance. This they palpably are not, if the matter be the substance ofnature, since it is impossible to express spatio-temporal truths withouthaving recourse to relations involving relata other than bits of matter. I waive this point however, and come to another. It is not the substancewhich is in space, but the attributes. What we find in space are the redof the rose and the smell of the jasmine and the noise of cannon. Wehave all told our dentists where our toothache is. Thus space is not arelation between substances, but between attributes. 

Thus even if you admit that the adherents of substance can be allowed toconceive substance as matter, it is a fraud to slip substance into spaceon the plea that space expresses relations between substances. On theface of it space has nothing to do with substances, but only with theirattributes. What I mean is, that if you choose--as I think wrongly--toconstrue our experience of nature as an awareness of the attributes ofsubstances, we are by this theory precluded from finding any analogousdirect relations between substances as disclosed in our experience. Whatwe do find are relations between the attributes of substances. Thus ifmatter is looked on as substance in space, the space in which it findsitself has very little to do with the space of our experience. 

The above argument has been expressed in terms of the relational theoryof space. But if space be absolute--namely, if it have a beingindependent of things in it--the course of the argument is hardlychanged. For things in space must have a certain fundamental relation tospace which we will call occupation. Thus the objection that it is theattributes which are observed as related to space, still holds. 

The scientific doctrine of matter is held in conjunction with anabsolute theory of time. The same arguments apply to the relationsbetween matter and time as apply to the relations between space andmatter. There is however (in the current philosophy) a difference in theconnexions of space with matter from those of time with matter, which Iwill proceed to explain. 

Space is not merely an ordering of material entities so that any oneentity bears certain relations to other material entities. Theoccupation of space impresses a certain character on each materialentity in itself. By reason of its occupation of space matter hasextension. By reason of its extension each bit of matter is divisibleinto parts, and each part is a numerically distinct entity from everyother such part. Accordingly it would seem that every material entity isnot really one entity. It is an essential multiplicity of entities. There seems to be no stopping this dissociation of matter intomultiplicities short of finding each ultimate entity occupying oneindividual point. This essential multiplicity of material entities iscertainly not what is meant by science, nor does it correspond toanything disclosed in sense-awareness. It is absolutely necessary thatat a certain stage in this dissociation of matter a halt should becalled, and that the material entities thus obtained should be treatedas units. The stage of arrest may be arbitrary or may be set by thecharacteristics of nature; but all reasoning in science ultimately dropsits space-analysis and poses to itself the problem, 'Here is onematerial entity, what is happening to it as a unit entity?' Yet thismaterial entity is still retaining its extension, and as thus extendedis a mere multiplicity. Thus there is an essential atomic property innature which is independent of the dissociation of extension. There issomething which in itself is one, and which is more than the logicalaggregate of entities occupying points within the volume which the unitoccupies. Indeed we may well be sceptical as to these ultimate entitiesat points, and doubt whether there are any such entities at all. Theyhave the suspicious character that we are driven to accept them byabstract logic and not by observed fact. 

Time (in the current philosophy) does not exert the same disintegratingeffect on matter which occupies it. If matter occupies a duration oftime, the whole matter occupies every part of that duration. Thus theconnexion between matter and time differs from the connexion betweenmatter and space as expressed in current scientific philosophy. There isobviously a greater difficulty in conceiving time as the outcome ofrelations between different bits of matter than there is in theanalogous conception of space. At an instant distinct volumes of spaceare occupied by distinct bits of matter. Accordingly there is so far nointrinsic difficulty in conceiving that space is merely the resultant ofrelations between the bits of matter. But in the one-dimensional timethe same bit of matter occupies different portions of time. Accordinglytime would have to be expressible in terms of the relations of a bit ofmatter with itself. My own view is a belief in the relational theoryboth of space and of time, and of disbelief in the current form of therelational theory of space which exhibits bits of matter as the relatafor spatial relations. The true relata are events. The distinction whichI have just pointed out between time and space in their connexion withmatter makes it evident that any assimilation of time and space cannotproceed along the traditional line of taking matter as a fundamentalelement in space-formation. 

The philosophy of nature took a wrong turn during its development byGreek thought. This erroneous presupposition is vague and fluid inPlato's _Timaeus_. The general groundwork of the thought is stilluncommitted and can be construed as merely lacking due explanation andthe guarding emphasis. But in Aristotle's exposition the currentconceptions were hardened and made definite so as to produce a faultyanalysis of the relation between the matter and the form of nature asdisclosed in sense-awareness. In this phrase the term 'matter' is notused in its scientific sense. 

I will conclude by guarding myself against a misapprehension. It isevident that the current doctrine of matter enshrines some fundamentallaw of nature. Any simple illustration will exemplify what I mean. Forexample, in a museum some specimen is locked securely in a glass case. It stays there for years: it loses its colour, and perhaps falls topieces. But it is the same specimen; and the same chemical elements andthe same quantities of those elements are present within the case at theend as were present at the beginning. Again the engineer and theastronomer deal with the motions of real permanences in nature. Anytheory of nature which for one moment loses sight of these great basicfacts of experience is simply silly. But it is permissible to point outthat the scientific expression of these facts has become entangled in amaze of doubtful metaphysics; and that, when we remove the metaphysicsand start afresh on an unprejudiced survey of nature, a new light isthrown on many fundamental concepts which dominate science and guide theprogress of research. 

CHAPTER II

THEORIES OF THE BIFURCATION OF NATURE

In my previous lecture I criticised the concept of matter as thesubstance whose attributes we perceive. This way of thinking of matteris, I think, the historical reason for its introduction into science, and is still the vague view of it at the background of our thoughtswhich makes the current scientific doctrine appear so obvious. Namely weconceive ourselves as perceiving attributes of things, and bits ofmatter are the things whose attributes we perceive. 

In the seventeenth century the sweet simplicity of this aspect of matterreceived a rude shock. The transmission doctrines of science were thenin process of elaboration and by the end of the century wereunquestioned, though their particular forms have since been modified. The establishment of these transmission theories marks a turning pointin the relation between science and philosophy. The doctrines to which Iam especially alluding are the theories of light and sound. I have nodoubt that the theories had been vaguely floating about before asobvious suggestions of common sense; for nothing in thought is evercompletely new. But at that epoch they were systematised and made exact, and their complete consequences were ruthlessly deduced. It is theestablishment of this procedure of taking the consequences seriouslywhich marks the real discovery of a theory. Systematic doctrines oflight and sound as being something proceeding from the emitting bodieswere definitely established, and in particular the connexion of lightwith colour was laid bare by Newton. 

The result completely destroyed the simplicity of the 'substance andattribute' theory of perception. What we see depends on the lightentering the eye. Furthermore we do not even perceive what enters theeye. The things transmitted are waves or--as Newton thought--minuteparticles, and the things seen are colours. Locke met this difficulty bya theory of primary and secondary qualities. Namely, there are someattributes of the matter which we do perceive. These are the primaryqualities, and there are other things which we perceive, such ascolours, which are not attributes of matter, but are perceived by us asif they were such attributes. These are the secondary qualities ofmatter. 

Why should we perceive secondary qualities? It seems an extremelyunfortunate arrangement that we should perceive a lot of things that arenot there. Yet this is what the theory of secondary qualities in factcomes to. There is now reigning in philosophy and in science anapathetic acquiescence in the conclusion that no coherent account can begiven of nature as it is disclosed to us in sense-awareness, withoutdragging in its relations to mind. The modern account of nature is not, as it should be, merely an account of what the mind knows of nature; butit is also confused with an account of what nature does to the mind. Theresult has been disastrous both to science and to philosophy, butchiefly to philosophy. It has transformed the grand question of therelations between nature and mind into the petty form of the interactionbetween the human body and mind. 

Berkeley's polemic against matter was based on this confusion introducedby the transmission theory of light. He advocated, rightly as I think, the abandonment of the doctrine of matter in its present form. He hadhowever nothing to put in its place except a theory of the relation offinite minds to the divine mind. 

But we are endeavouring in these lectures to limit ourselves to natureitself and not to travel beyond entities which are disclosed insense-awareness. 

Percipience in itself is taken for granted. We consider indeedconditions for percipience, but only so far as those conditions areamong the disclosures of perception. We leave to metaphysics thesynthesis of the knower and the known. Some further explanation anddefence of this position is necessary, if the line of argument of theselectures is to be comprehensible. 

The immediate thesis for discussion is that any metaphysicalinterpretation is an illegitimate importation into the philosophy ofnatural science. By a metaphysical interpretation I mean any discussionof the how (beyond nature) and of the why (beyond nature) of thought andsense-awareness. In the philosophy of science we seek the generalnotions which apply to nature, namely, to what we are aware of inperception. It is the philosophy of the thing perceived, and it shouldnot be confused with the metaphysics of reality of which the scopeembraces both perceiver and perceived. No perplexity concerning theobject of knowledge can be solved by saying that there is a mind knowingit[2]. 

[2] Cf. _Enquiry_, preface. 

In other words, the ground taken is this: sense-awareness is anawareness of something. What then is the general character of thatsomething of which we are aware? We do not ask about the percipient orabout the process, but about the perceived. I emphasise this pointbecause discussions on the philosophy of science are usually extremelymetaphysical--in my opinion, to the great detriment of the subject. 

The recourse to metaphysics is like throwing a match into the powdermagazine. It blows up the whole arena. This is exactly what scientificphilosophers do when they are driven into a corner and convicted ofincoherence. They at once drag in the mind and talk of entities in themind or out of the mind as the case may be. For natural philosophyeverything perceived is in nature. We may not pick and choose. For usthe red glow of the sunset should be as much part of nature as are themolecules and electric waves by which men of science would explain thephenomenon. It is for natural philosophy to analyse how these variouselements of nature are connected. 

In making this demand I conceive myself as adopting our immediateinstinctive attitude towards perceptual knowledge which is onlyabandoned under the influence of theory. We are instinctively willing tobelieve that by due attention, more can be found in nature than thatwhich is observed at first sight. But we will not be content with less. What we ask from the philosophy of science is some account of thecoherence of things perceptively known. 

This means a refusal to countenance any theory of psychic additions tothe object known in perception. For example, what is given in perceptionis the green grass. This is an object which we know as an ingredient innature. The theory of psychic additions would treat the greenness as apsychic addition furnished by the perceiving mind, and would leave tonature merely the molecules and the radiant energy which influence themind towards that perception. My argument is that this dragging in ofthe mind as making additions of its own to the thing posited forknowledge by sense-awareness is merely a way of shirking the problem ofnatural philosophy. That problem is to discuss the relations _inter se_of things known, abstracted from the bare fact that they are known. Natural philosophy should never ask, what is in the mind and what is innature. To do so is a confession that it has failed to express relationsbetween things perceptively known, namely to express those naturalrelations whose expression is natural philosophy. It may be that thetask is too hard for us, that the relations are too complex and toovarious for our apprehension, or are too trivial to be worth the troubleof exposition. It is indeed true that we have gone but a very small wayin the adequate formulation of such relations. But at least do not letus endeavour to conceal failure under a theory of the byplay of theperceiving mind. 

What I am essentially protesting against is the bifurcation of natureinto two systems of reality, which, in so far as they are real, are realin different senses. One reality would be the entities such as electronswhich are the study of speculative physics. This would be the realitywhich is there for knowledge; although on this theory it is never known. For what is known is the other sort of reality, which is the byplay ofthe mind. Thus there would be two natures, one is the conjecture and theother is the dream. 

Another way of phrasing this theory which I am arguing against is tobifurcate nature into two divisions, namely into the nature apprehendedin awareness and the nature which is the cause of awareness. The naturewhich is the fact apprehended in awareness holds within it the greennessof the trees, the song of the birds, the warmth of the sun, the hardnessof the chairs, and the feel of the velvet. The nature which is the causeof awareness is the conjectured system of molecules and electrons whichso affects the mind as to produce the awareness of apparent nature. Themeeting point of these two natures is the mind, the causal nature beinginfluent and the apparent nature being effluent. 

There are four questions which at once suggest themselves for discussionin connexion with this bifurcation theory of nature. They concern (i)causality, (ii) time, (iii) space, and (iv) delusions. These questionsare not really separable. They merely constitute four distinct startingpoints from which to enter upon the discussion of the theory. 

Causal nature is the influence on the mind which is the cause of theeffluence of apparent nature from the mind. This conception of causalnature is not to be confused with the distinct conception of one part ofnature as being the cause of another part. For example, the burning ofthe fire and the passage of heat from it through intervening space isthe cause of the body, its nerves and its brain, functioning in certainways. But this is not an action of nature on the mind. It is aninteraction within nature. The causation involved in this interaction iscausation in a different sense from the influence of this system ofbodily interactions within nature on the alien mind which thereuponperceives redness and warmth. 

The bifurcation theory is an attempt to exhibit natural science as aninvestigation of the cause of the fact of knowledge. Namely, it is anattempt to exhibit apparent nature as an effluent from the mind becauseof causal nature. The whole notion is partly based on the implicitassumption that the mind can only know that which it has itself producedand retains in some sense within itself, though it requires an exteriorreason both as originating and as determining the character of itsactivity. But in considering knowledge we should wipe out all thesespatial metaphors, such as 'within the mind' and 'without the mind. 'Knowledge is ultimate. There can be no explanation of the 'why' ofknowledge; we can only describe the 'what' of knowledge. Namely we cananalyse the content and its internal relations, but we cannot explainwhy there is knowledge. Thus causal nature is a metaphysical chimera;though there is need of a metaphysics whose scope transcends thelimitation to nature. The object of such a metaphysical science is notto explain knowledge, but exhibit in its utmost completeness our conceptof reality. 

However, we must admit that the causality theory of nature has itsstrong suit. The reason why the bifurcation of nature is always creepingback into scientific philosophy is the extreme difficulty of exhibitingthe perceived redness and warmth of the fire in one system of relationswith the agitated molecules of carbon and oxygen, with the radiantenergy from them, and with the various functionings of the materialbody. Unless we produce the all-embracing relations, we are faced with abifurcated nature; namely, warmth and redness on one side, andmolecules, electrons and ether on the other side. Then the two factorsare explained as being respectively the cause and the mind's reaction tothe cause. 

Time and space would appear to provide these all-embracing relationswhich the advocates of the philosophy of the unity of nature require. The perceived redness of the fire and the warmth are definitely relatedin time and in space to the molecules of the fire and the molecules ofthe body. 

It is hardly more than a pardonable exaggeration to say that thedetermination of the meaning of nature reduces itself principally to thediscussion of the character of time and the character of space. Insucceeding lectures I shall explain my own view of time and space. Ishall endeavour to show that they are abstractions from more concreteelements of nature, namely, from events. The discussion of the detailsof the process of abstraction will exhibit time and space asinterconnected, and will finally lead us to the sort of connexionsbetween their measurements which occur in the modern theory ofelectromagnetic relativity. But this is anticipating our subsequent lineof development. At present I wish to consider how the ordinary views oftime and space help, or fail to help, in unifying our conception ofnature. 

First, consider the absolute theories of time and space. We are toconsider each, namely both time and space, to be a separate andindependent system of entities, each system known to us in itself andfor itself concurrently with our knowledge of the events of nature. Timeis the ordered succession of durationless instants; and these instantsare known to us merely as the relata in the serial relation which is thetime-ordering relation, and the time-ordering relation is merely knownto us as relating the instants. Namely, the relation and the instantsare jointly known to us in our apprehension of time, each implying theother. 

This is the absolute theory of time. Frankly, I confess that it seems tome to be very unplausible. I cannot in my own knowledge find anythingcorresponding to the bare time of the absolute theory. Time is known tome as an abstraction from the passage of events. The fundamental factwhich renders this abstraction possible is the passing of nature, itsdevelopment, its creative advance, and combined with this fact isanother characteristic of nature, namely the extensive relation betweenevents. These two facts, namely the passage of events and the extensionof events over each other, are in my opinion the qualities from whichtime and space originate as abstractions. But this is anticipating myown later speculations. 

Meanwhile, returning to the absolute theory, we are to suppose that timeis known to us independently of any events in time. What happens in timeoccupies time. This relation of events to the time occupied, namely thisrelation of occupation, is a fundamental relation of nature to time. Thus the theory requires that we are aware of two fundamental relations, the time-ordering relation between instants, and the time-occupationrelation between instants of time and states of nature which happen atthose instants. 

There are two considerations which lend powerful support to the reigningtheory of absolute time. In the first place time extends beyond nature. Our thoughts are in time. Accordingly it seems impossible to derive timemerely from relations between elements of nature. For in that casetemporal relations could not relate thoughts. Thus, to use a metaphor, time would apparently have deeper roots in reality than has nature. Forwe can imagine thoughts related in time without any perception ofnature. For example we can imagine one of Milton's angels with thoughtssucceeding each other in time, who does not happen to have noticed thatthe Almighty has created space and set therein a material universe. As amatter of fact I think that Milton set space on the same absolute levelas time. But that need not disturb the illustration. In the second placeit is difficult to derive the true serial character of time from therelative theory. Each instant is irrevocable. It can never recur by thevery character of time. But if on the relative theory an instant of timeis simply the state of nature at that time, and the time-orderingrelation is simply the relation between such states, then theirrevocableness of time would seem to mean that an actual state of allnature can never return. I admit it seems unlikely that there shouldever be such a recurrence down to the smallest particular. But extremeunlikeliness is not the point. Our ignorance is so abysmal that ourjudgments of likeliness and unlikeliness of future events hardly count. The real point is that the exact recurrence of a state of nature seemsmerely unlikely, while the recurrence of an instant of time violates ourwhole concept of time-order. The instants of time which have passed, arepassed, and can never be again. 

Any alternative theory of time must reckon with these two considerationswhich are buttresses of the absolute theory. But I will not now continuetheir discussion. 

The absolute theory of space is analogous to the corresponding theory oftime, but the reasons for its maintenance are weaker. Space, on thistheory, is a system of extensionless points which are the relata inspace-ordering relations which can technically be combined into onerelation. This relation does not arrange the points in one linear seriesanalogously to the simple method of the time-ordering relation forinstants. The essential logical characteristics of this relation fromwhich all the properties of space spring are expressed by mathematiciansin the axioms of geometry. From these axioms[3] as framed by modernmathematicians the whole science of geometry can be deduced by thestrictest logical reasoning. The details of these axioms do not nowconcern us. The points and the relations are jointly known to us in ourapprehension of space, each implying the other. What happens in space, occupies space. This relation of occupation is not usually stated forevents but for objects. For example, Pompey's statue would be said tooccupy space, but not the event which was the assassination of JuliusCaesar. In this I think that ordinary usage is unfortunate, and I holdthat the relations of events to space and to time are in all respectsanalogous. But here I am intruding my own opinions which are to bediscussed in subsequent lectures. Thus the theory of absolute spacerequires that we are aware of two fundamental relations, thespace-ordering relation, which holds between points, and thespace-occupation relation between points of space and material objects. 

[3] Cf. (for example) _Projective Geometry_ by Veblen and Young, vol.  i. 1910, vol.  ii. 1917, Ginn and Company, Boston, U. S. A. 

This theory lacks the two main supports of the corresponding theory ofabsolute time. In the first place space does not extend beyond nature inthe sense that time seems to do. Our thoughts do not seem to occupyspace in quite the same intimate way in which they occupy time. Forexample, I have been thinking in a room, and to that extent my thoughtsare in space. But it seems nonsense to ask how much volume of the roomthey occupied, whether it was a cubic foot or a cubic inch; whereas thesame thoughts occupy a determinate duration of time, say, from eleven totwelve on a certain date. 

Thus whereas the relations of a relative theory of time are required torelate thoughts, it does not seem so obvious that the relations of arelative theory of space are required to relate them. The connexion ofthought with space seems to have a certain character of indirectnesswhich appears to be lacking in the connexion of thought with time. 

Again the irrevocableness of time does not seem to have any parallel forspace. Space, on the relative theory, is the outcome of certainrelations between objects commonly said to be in space; and wheneverthere are the objects, so related, there is the space. No difficultyseems to arise like that of the inconvenient instants of time whichmight conceivably turn up again when we thought that we had done withthem. 

The absolute theory of space is not now generally popular. The knowledgeof bare space, as a system of entities known to us in itself and foritself independently of our knowledge of the events in nature, does notseem to correspond to anything in our experience. Space, like time, would appear to be an abstraction from events. According to my owntheory it only differentiates itself from time at a somewhat developedstage of the abstractive process. The more usual way of expressing therelational theory of space would be to consider space as an abstractionfrom the relations between material objects. 

Suppose now we assume absolute time and absolute space. What bearinghas this assumption on the concept of nature as bifurcated into causalnature and apparent nature? Undoubtedly the separation between the twonatures is now greatly mitigated. We can provide them with two systemsof relations in common; for both natures can be presumed to occupy thesame space and the same time. The theory now is this: Causal eventsoccupy certain periods of the absolute time and occupy certain positionsof the absolute space. These events influence a mind which thereuponperceives certain apparent events which occupy certain periods in theabsolute time and occupy certain positions of the absolute space; andthe periods and positions occupied by the apparent events bear adeterminate relation to the periods and positions occupied by the causalevents. 

Furthermore definite causal events produce for the mind definiteapparent events. Delusions are apparent events which appear in temporalperiods and spatial positions without the intervention of these causalevents which are proper for influencing of the mind to their perception. 

The whole theory is perfectly logical. In these discussions we cannothope to drive an unsound theory to a logical contradiction. A reasoner, apart from mere slips, only involves himself in a contradiction when heis shying at a _reductio ad absurdum_. The substantial reason forrejecting a philosophical theory is the 'absurdum' to which it reducesus. In the case of the philosophy of natural science the 'absurdum' canonly be that our perceptual knowledge has not the character assigned toit by the theory. If our opponent affirms that his knowledge has thatcharacter, we can only--after making doubly sure that we understandeach other--agree to differ. Accordingly the first duty of an expositorin stating a theory in which he disbelieves is to exhibit it as logical. It is not there where his trouble lies. 

Let me summarise the previously stated objections to this theory ofnature. In the first place it seeks for the cause of the knowledge ofthe thing known instead of seeking for the character of the thing known:secondly it assumes a knowledge of time in itself apart from eventsrelated in time: thirdly it assumes a knowledge of space in itself apartfrom events related in space. There are in addition to these objectionsother flaws in the theory. 

Some light is thrown on the artificial status of causal nature in thistheory by asking, why causal nature is presumed to occupy time andspace. This really raises the fundamental question as to whatcharacteristics causal nature should have in common with apparentnature. Why--on this theory--should the cause which influences the mindto perception have any characteristics in common with the effluentapparent nature? In particular, why should it be in space? Why should itbe in time? And more generally, What do we know about mind which wouldallow us to infer any particular characteristics of a cause which shouldinfluence mind to particular effects?

The transcendence of time beyond nature gives some slight reason forpresuming that causal nature should occupy time. For if the mindoccupies periods of time, there would seem to be some vague reason forassuming that influencing causes occupy the same periods of time, or atleast, occupy periods which are strictly related to the mental periods. But if the mind does not occupy volumes of space, there seems to be noreason why causal nature should occupy any volumes of space. Thus spacewould seem to be merely apparent in the same sense as apparent nature ismerely apparent. Accordingly if science is really investigating causeswhich operate on the mind, it would seem to be entirely on the wrongtack in presuming that the causes which it is seeking for have spatialrelations. Furthermore there is nothing else in our knowledge analogousto these causes which influence the mind to perception. Accordingly, beyond the rashly presumed fact that they occupy time, there is reallyno ground by which we can determine any point of their character. Theymust remain for ever unknown. 

Now I assume as an axiom that science is not a fairy tale. It is notengaged in decking out unknowable entities with arbitrary and fantasticproperties. What then is it that science is doing, granting that it iseffecting something of importance? My answer is that it is determiningthe character of things known, namely the character of apparent nature. But we may drop the term 'apparent'; for there is but one nature, namelythe nature which is before us in perceptual knowledge. The characterswhich science discerns in nature are subtle characters, not obvious atfirst sight. They are relations of relations and characters ofcharacters. But for all their subtlety they are stamped with a certainsimplicity which makes their consideration essential in unravelling thecomplex relations between characters of more perceptive insistence. 

The fact that the bifurcation of nature into causal and apparentcomponents does not express what we mean by our knowledge is broughtbefore us when we realise our thoughts in any discussion of the causesof our perceptions. For example, the fire is burning and we see a redcoal. This is explained in science by radiant energy from the coalentering our eyes. But in seeking for such an explanation we are notasking what are the sort of occurrences which are fitted to cause a mindto see red. The chain of causation is entirely different. The mind iscut out altogether. The real question is, When red is found in nature, what else is found there also? Namely we are asking for an analysis ofthe accompaniments in nature of the discovery of red in nature. In asubsequent lecture I shall expand this line of thought. I simply drawattention to it here in order to point out that the wave-theory of lighthas not been adopted because waves are just the sort of things whichought to make a mind perceive colours. This is no part of the evidencewhich has ever been adduced for the wave-theory, yet on the causaltheory of perception, it is really the only relevant part. In otherwords, science is not discussing the causes of knowledge, but thecoherence of knowledge. The understanding which is sought by science isan understanding of relations within nature. 

So far I have discussed the bifurcation of nature in connexion with thetheories of absolute time and of absolute space. My reason has been thatthe introduction of the relational theories only weakens the case forbifurcation, and I wished to discuss this case on its strongest grounds. 

For instance, suppose we adopt the relational theory of space. Then thespace in which apparent nature is set is the expression of certainrelations between the apparent objects. It is a set of apparentrelations between apparent relata. Apparent nature is the dream, andthe apparent relations of space are dream relations, and the space isthe dream space. Similarly the space in which causal nature is set isthe expression of certain relations between the causal objects. It isthe expression of certain facts about the causal activity which is goingon behind the scenes. Accordingly causal space belongs to a differentorder of reality to apparent space. Hence there is no pointwiseconnexion between the two and it is meaningless to say that themolecules of the grass are in any place which has a determinate spatialrelation to the place occupied by the grass which we see. Thisconclusion is very paradoxical and makes nonsense of all scientificphraseology. The case is even worse if we admit the relativity of time. For the same arguments apply, and break up time into the dream time andcausal time which belong to different orders of reality. 

I have however been discussing an extreme form of the bifurcationtheory. It is, as I think, the most defensible form. But its verydefiniteness makes it the more evidently obnoxious to criticism. Theintermediate form allows that the nature we are discussing is always thenature directly known, and so far it rejects the bifurcation theory. Butit holds that there are psychic additions to nature as thus known, andthat these additions are in no proper sense part of nature. For example, we perceive the red billiard ball at its proper time, in its properplace, with its proper motion, with its proper hardness, and with itsproper inertia. But its redness and its warmth, and the sound of theclick as a cannon is made off it are psychic additions, namely, secondary qualities which are only the mind's way of perceiving nature. This is not only the vaguely prevalent theory, but is, I believe, thehistorical form of the bifurcation theory in so far as it is derivedfrom philosophy. I shall call it the theory of psychic additions. 

This theory of psychic additions is a sound common-sense theory whichlays immense stress on the obvious reality of time, space, solidity andinertia, but distrusts the minor artistic additions of colour, warmthand sound. 

The theory is the outcome of common-sense in retreat. It arose in anepoch when the transmission theories of science were being elaborated. For example, colour is the result of a transmission from the materialobject to the perceiver's eye; and what is thus transmitted is notcolour. Thus colour is not part of the reality of the material object. Similarly for the same reason sounds evaporate from nature. Also warmthis due to the transfer of something which is not temperature. Thus weare left with spatio-temporal positions, and what I may term the'pushiness' of the body. This lands us to eighteenth and nineteenthcentury materialism, namely, the belief that what is real in nature ismatter, in time and in space and with inertia. 

Evidently a distinction in quality has been presupposed separating offsome perceptions due to touch from other perceptions. Thesetouch-perceptions are perceptions of the real inertia, whereas the otherperceptions are psychic additions which must be explained on the causaltheory. This distinction is the product of an epoch in which physicalscience has got ahead of medical pathology and of physiology. Perceptions of push are just as much the outcome of transmission as areperceptions of colour. When colour is perceived the nerves of the bodyare excited in one way and transmit their message towards the brain, andwhen push is perceived other nerves of the body are excited in anotherway and transmit their message towards the brain. The message of the oneset is not the conveyance of colour, and the message of the other set isnot the conveyance of push. But in one case colour is perceived and inthe other case the push due to the object. If you snip certain nerves, there is an end to the perception of colour; and if you snip certainother nerves, there is an end to the perception of push. It would appeartherefore that any reasons which should remove colour from the realityof nature should also operate to remove inertia. 

Thus the attempted bifurcation of apparent nature into two parts ofwhich one part is both causal for its own appearance and for theappearance of the other part, which is purely apparent, fails owing tothe failure to establish any fundamental distinction between our ways ofknowing about the two parts of nature as thus partitioned. I am notdenying that the feeling of muscular effort historically led to theformulation of the concept of force. But this historical fact does notwarrant us in assigning a superior reality in nature to material inertiaover colour or sound. So far as reality is concerned all oursense-perceptions are in the same boat, and must be treated on the sameprinciple. The evenness of treatment is exactly what this compromisetheory fails to achieve. 

The bifurcation theory however dies hard. The reason is that therereally is a difficulty to be faced in relating within the same system ofentities the redness of the fire with the agitation of the molecules. Inanother lecture I will give my own explanation of the origin of thedifficulty and of its solution. 

Another favourite solution, the most attenuated form which thebifurcation theory assumes, is to maintain that the molecules and etherof science are purely conceptual. Thus there is but one nature, namelyapparent nature, and atoms and ether are merely names for logical termsin conceptual formulae of calculation. 

But what is a formula of calculation? It is presumably a statement thatsomething or other is true for natural occurrences. Take the simplest ofall formulae, Two and two make four. This--so far as it applies tonature--asserts that if you take two natural entities, and then againtwo other natural entities, the combined class contains four naturalentities. Such formulae which are true for any entities cannot result inthe production of the concepts of atoms. Then again there are formulaewhich assert that there are entities in nature with such and suchspecial properties, say, for example, with the properties of the atomsof hydrogen. Now if there are no such entities, I fail to see how anystatements about them can apply to nature. For example, the assertionthat there is green cheese in the moon cannot be a premiss in anydeduction of scientific importance, unless indeed the presence of greencheese in the moon has been verified by experiment. The current answerto these objections is that, though atoms are merely conceptual, yetthey are an interesting and picturesque way of saying something elsewhich is true of nature. But surely if it is something else that youmean, for heaven's sake say it. Do away with this elaborate machinery ofa conceptual nature which consists of assertions about things whichdon't exist in order to convey truths about things which do exist. I ammaintaining the obvious position that scientific laws, if they are true, are statements about entities which we obtain knowledge of as being innature; and that, if the entities to which the statements refer are notto be found in nature, the statements about them have no relevance toany purely natural occurrence. Thus the molecules and electrons ofscientific theory are, so far as science has correctly formulated itslaws, each of them factors to be found in nature. The electrons are onlyhypothetical in so far as we are not quite certain that the electrontheory is true. But their hypothetical character does not arise from theessential nature of the theory in itself after its truth has beengranted. 

Thus at the end of this somewhat complex discussion, we return to theposition which was affirmed at its beginning. The primary task of aphilosophy of natural science is to elucidate the concept of nature, considered as one complex fact for knowledge, to exhibit the fundamentalentities and the fundamental relations between entities in terms ofwhich all laws of nature have to be stated, and to secure that theentities and relations thus exhibited are adequate for the expression ofall the relations between entities which occur in nature. 

The third requisite, namely that of adequacy, is the one over which allthe difficulty occurs. The ultimate data of science are commonly assumedto be time, space, material, qualities of material, and relationsbetween material objects. But data as they occur in the scientific lawsdo not relate all the entities which present themselves in ourperception of nature. For example, the wave-theory of light is anexcellent well-established theory; but unfortunately it leaves outcolour as perceived. Thus the perceived redness--or, other colour--hasto be cut out of nature and made into the reaction of the mind under theimpulse of the actual events of nature. In other words this concept ofthe fundamental relations within nature is inadequate. Thus we have tobend our energies to the enunciation of adequate concepts. 

But in so doing, are we not in fact endeavouring to solve a metaphysicalproblem? I do not think so. We are merely endeavouring to exhibit thetype of relations which hold between the entities which we in factperceive as in nature. We are not called on to make any pronouncement asto the psychological relation of subjects to objects or as to the statusof either in the realm of reality. It is true that the issue of ourendeavour may provide material which is relevant evidence for adiscussion on that question. It can hardly fail to do so. But it is onlyevidence, and is not itself the metaphysical discussion. In order tomake clear the character of this further discussion which is out of ourken, I will set before you two quotations. One is from Schelling and Iextract the quotation from the work of the Russian philosopher Losskywhich has recently been so excellently translated into English[4]--'Inthe "Philosophy of Nature" I considered the subject-object called naturein its activity of self-constructing. In order to understand it, we mustrise to an intellectual intuition of nature. The empiricist does notrise thereto, and for this reason in all his explanations it is always_he himself_ that proves to be constructing nature. It is no wonder, then, that his construction and that which was to be constructed soseldom coincide. A _Natur-philosoph_ raises nature to independence, andmakes it construct itself, and he never feels, therefore, the necessityof opposing nature as constructed (_i. E. _ as experience) to realnature, or of correcting the one by means of the other. '

[4] _The Intuitive Basis of Knowledge_, by N.  O. Lossky, transl. By MrsDuddington, Macmillan and Co. , 1919. 

The other quotation is from a paper read by the Dean of St Paul's beforethe Aristotelian Society in May of 1919. Dr Inge's paper is entitled'Platonism and Human Immortality, ' and in it there occurs the followingstatement: 'To sum up. The Platonic doctrine of immortality rests on the_independence_ of the spiritual world. The spiritual world is not aworld of unrealised ideals, over against a real world of unspiritualfact. It is, on the contrary, the real world, of which we have a truethough very incomplete knowledge, over against a world of commonexperience which, as a complete whole, is not real, since it iscompacted out of miscellaneous data, not all on the same level, by thehelp of the imagination. There is no world corresponding to the world ofour common experience. Nature makes abstractions for us, deciding whatrange of vibrations we are to see and hear, what things we are to noticeand remember. '

I have cited these statements because both of them deal with topicswhich, though they lie outside the range of our discussion, are alwaysbeing confused with it. The reason is that they lie proximate to ourfield of thought, and are topics which are of burning interest to themetaphysically minded. It is difficult for a philosopher to realise thatanyone really is confining his discussion within the limits that I haveset before you. The boundary is set up just where he is beginning to getexcited. But I submit to you that among the necessary prolegomena forphilosophy and for natural science is a thorough understanding of thetypes of entities, and types of relations among those entities, whichare disclosed to us in our perceptions of nature. 

CHAPTER III

TIME

The two previous lectures of this course have been mainly critical. Inthe present lecture I propose to enter upon a survey of the kinds ofentities which are posited for knowledge in sense-awareness. My purposeis to investigate the sorts of relations which these entities of variouskinds can bear to each other. A classification of natural entities isthe beginning of natural philosophy. To-day we commence with theconsideration of Time. 

In the first place there is posited for us a general fact: namely, something is going on; there is an occurrence for definition. 

This general fact at once yields for our apprehension two factors, whichI will name, the 'discerned' and the 'discernible. ' The discerned iscomprised of those elements of the general fact which are discriminatedwith their own individual peculiarities. It is the field directlyperceived. But the entities of this field have relations to otherentities which are not particularly discriminated in this individualway. These other entities are known merely as the relata in relation tothe entities of the discerned field. Such an entity is merely a'something' which has such-and-such definite relations to some definiteentity or entities in the discerned field. As being thus related, theyare--owing to the particular character of these relations--known aselements of the general fact which is going on. But we are not aware ofthem except as entities fulfilling the functions of relata in theserelations. 

Thus the complete general fact, posited as occurring, comprises bothsets of entities, namely the entities perceived in their ownindividuality and other entities merely apprehended as relata withoutfurther definition. This complete general fact is the discernible and itcomprises the discerned. The discernible is all nature as disclosed inthat sense-awareness, and extends beyond and comprises all of nature asactually discriminated or discerned in that sense-awareness. Thediscerning or discrimination of nature is a peculiar awareness ofspecial factors in nature in respect to their peculiar characters. Butthe factors in nature of which we have this peculiar sense-awareness areknown as not comprising all the factors which together form the wholecomplex of related entities within the general fact there fordiscernment. This peculiarity of knowledge is what I call itsunexhaustive character. This character may be metaphorically describedby the statement that nature as perceived always has a ragged edge. Forexample, there is a world beyond the room to which our sight is confinedknown to us as completing the space-relations of the entities discernedwithin the room. The junction of the interior world of the room with theexterior world beyond is never sharp. Sounds and subtler factorsdisclosed in sense-awareness float in from the outside. Every type ofsense has its own set of discriminated entities which are known to berelata in relation with entities not discriminated by that sense. Forexample we see something which we do not touch and we touch somethingwhich we do not see, and we have a general sense of the space-relationsbetween the entity disclosed in sight and the entity disclosed in touch. Thus in the first place each of these two entities is known as a relatumin a general system of space-relations and in the second place theparticular mutual relation of these two entities as related to eachother in this general system is determined. But the general system ofspace-relations relating the entity discriminated by sight with thatdiscriminated by sight is not dependent on the peculiar character of theother entity as reported by the alternative sense. For example, thespace-relations of the thing seen would have necessitated an entity as arelatum in the place of the thing touched even although certain elementsof its character had not been disclosed by touch. Thus apart from thetouch an entity with a certain specific relation to the thing seen wouldhave been disclosed by sense-awareness but not otherwise discriminatedin respect to its individual character. An entity merely known asspatially related to some discerned entity is what we mean by the bareidea of 'place. ' The concept of place marks the disclosure insense-awareness of entities in nature known merely by their spatialrelations to discerned entities. It is the disclosure of the discernibleby means of its relations to the discerned. 

This disclosure of an entity as a relatum without further specificdiscrimination of quality is the basis of our concept of significance. In the above example the thing seen was significant, in that itdisclosed its spatial relations to other entities not necessarilyotherwise entering into consciousness. Thus significance is relatedness, but it is relatedness with the emphasis on one end only of the relation. 

For the sake of simplicity I have confined the argument to spatialrelations; but the same considerations apply to temporal relations. Theconcept of 'period of time' marks the disclosure in sense-awareness ofentities in nature known merely by their temporal relations todiscerned entities. Still further, this separation of the ideas of spaceand time has merely been adopted for the sake of gaining simplicity ofexposition by conformity to current language. What we discern is thespecific character of a place through a period of time. This is what Imean by an 'event. ' We discern some specific character of an event. Butin discerning an event we are also aware of its significance as arelatum in the structure of events. This structure of events is thecomplex of events as related by the two relations of extension andcogredience. The most simple expression of the properties of thisstructure are to be found in our spatial and temporal relations. Adiscerned event is known as related in this structure to other eventswhose specific characters are otherwise not disclosed in that immediateawareness except so far as that they are relata within the structure. 

The disclosure in sense-awareness of the structure of events classifiesevents into those which are discerned in respect to some furtherindividual character and those which are not otherwise disclosed exceptas elements of the structure. These signified events must include eventsin the remote past as well as events in the future. We are aware ofthese as the far off periods of unbounded time. But there is anotherclassification of events which is also inherent in sense-awareness. These are the events which share the immediacy of the immediatelypresent discerned events. These are the events whose characters togetherwith those of the discerned events comprise all nature present fordiscernment. They form the complete general fact which is all nature nowpresent as disclosed in that sense-awareness. It is in this secondclassification of events that the differentiation of space from timetakes its origin. The germ of space is to be found in the mutualrelations of events within the immediate general fact which is allnature now discernible, namely within the one event which is thetotality of present nature. The relations of other events to thistotality of nature form the texture of time. 

The unity of this general present fact is expressed by the concept ofsimultaneity. The general fact is the whole simultaneous occurrence ofnature which is now for sense-awareness. This general fact is what Ihave called the discernible. But in future I will call it a 'duration, 'meaning thereby a certain whole of nature which is limited only by theproperty of being a simultaneity. Further in obedience to the principleof comprising within nature the whole terminus of sense-awareness, simultaneity must not be conceived as an irrelevant mental conceptimposed upon nature. Our sense-awareness posits for immediatediscernment a certain whole, here called a 'duration'; thus a durationis a definite natural entity. A duration is discriminated as a complexof partial events, and the natural entities which are components of thiscomplex are thereby said to be 'simultaneous with this duration. ' Alsoin a derivative sense they are simultaneous with each other in respectto this duration. Thus simultaneity is a definite natural relation. Theword 'duration' is perhaps unfortunate in so far as it suggests a mereabstract stretch of time. This is not what I mean. A duration is aconcrete slab of nature limited by simultaneity which is an essentialfactor disclosed in sense-awareness. 

Nature is a process. As in the case of everything directly exhibited insense-awareness, there can be no explanation of this characteristic ofnature. All that can be done is to use language which may speculativelydemonstrate it, and also to express the relation of this factor innature to other factors. 

It is an exhibition of the process of nature that each duration happensand passes. The process of nature can also be termed the passage ofnature. I definitely refrain at this stage from using the word 'time, 'since the measurable time of science and of civilised life generallymerely exhibits some aspects of the more fundamental fact of the passageof nature. I believe that in this doctrine I am in full accord withBergson, though he uses 'time' for the fundamental fact which I call the'passage of nature. ' Also the passage of nature is exhibited equally inspatial transition as well as in temporal transition. It is in virtue ofits passage that nature is always moving on. It is involved in themeaning of this property of 'moving on' that not only is any act ofsense-awareness just that act and no other, but the terminus of each actis also unique and is the terminus of no other act. Sense-awarenessseizes its only chance and presents for knowledge something which is forit alone. 

There are two senses in which the terminus of sense-awareness is unique. It is unique for the sense-awareness of an individual mind and it isunique for the sense-awareness of all minds which are operating undernatural conditions. There is an important distinction between the twocases. (i) For one mind not only is the discerned component of thegeneral fact exhibited in any act of sense-awareness distinct from thediscerned component of the general fact exhibited in any other act ofsense-awareness of that mind, but the two corresponding durations whichare respectively related by simultaneity to the two discerned componentsare necessarily distinct. This is an exhibition of the temporal passageof nature; namely, one duration has passed into the other. Thus not onlyis the passage of nature an essential character of nature in its _rôle_of the terminus of sense-awareness, but it is also essential forsense-awareness in itself. It is this truth which makes time appear toextend beyond nature. But what extends beyond nature to mind is not theserial and measurable time, which exhibits merely the character ofpassage in nature, but the quality of passage itself which is in no waymeasurable except so far as it obtains in nature. That is to say, 'passage' is not measurable except as it occurs in nature in connexionwith extension. In passage we reach a connexion of nature with theultimate metaphysical reality. The quality of passage in durations is aparticular exhibition in nature of a quality which extends beyondnature. For example passage is a quality not only of nature, which isthe thing known, but also of sense-awareness which is the procedure ofknowing. Durations have all the reality that nature has, though whatthat may be we need not now determine. The measurableness of time isderivative from the properties of durations. So also is the serialcharacter of time. We shall find that there are in nature competingserial time-systems derived from different families of durations. Theseare a peculiarity of the character of passage as it is found in nature. This character has the reality of nature, but we must not necessarilytransfer natural time to extra-natural entities. (ii) For two minds, thediscerned components of the general facts exhibited in their respectiveacts of sense-awareness must be different. For each mind, in itsawareness of nature is aware of a certain complex of related naturalentities in their relations to the living body as a focus. But theassociated durations may be identical. Here we are touching on thatcharacter of the passage nature which issues in the spatial relations ofsimultaneous bodies. This possible identity of the durations in the caseof the sense-awareness of distinct minds is what binds into one naturethe private experiences of sentient beings. We are here considering thespatial side of the passage of nature. Passage in this aspect of it alsoseems to extend beyond nature to mind. 

It is important to distinguish simultaneity from instantaneousness. Ilay no stress on the mere current usage of the two terms. There are twoconcepts which I want to distinguish, and one I call simultaneity andthe other instantaneousness. I hope that the words are judiciouslychosen; but it really does not matter so long as I succeed in explainingmy meaning. Simultaneity is the property of a group of natural elementswhich in some sense are components of a duration. A duration can be allnature present as the immediate fact posited by sense-awareness. Aduration retains within itself the passage of nature. There are withinit antecedents and consequents which are also durations which may be thecomplete specious presents of quicker consciousnesses. In other words aduration retains temporal thickness. Any concept of all nature asimmediately known is always a concept of some duration though it may beenlarged in its temporal thickness beyond the possible specious presentof any being known to us as existing within nature. Thus simultaneity isan ultimate factor in nature, immediate for sense-awareness. 

Instantaneousness is a complex logical concept of a procedure in thoughtby which constructed logical entities are produced for the sake of thesimple expression in thought of properties of nature. Instantaneousnessis the concept of all nature at an instant, where an instant isconceived as deprived of all temporal extension. For example we conceiveof the distribution of matter in space at an instant. This is a veryuseful concept in science especially in applied mathematics; but it is avery complex idea so far as concerns its connexions with the immediatefacts of sense-awareness. There is no such thing as nature at an instantposited by sense-awareness. What sense-awareness delivers over forknowledge is nature through a period. Accordingly nature at an instant, since it is not itself a natural entity, must be defined in terms ofgenuine natural entities. Unless we do so, our science, which employsthe concept of instantaneous nature, must abandon all claim to befounded upon observation. 

I will use the term 'moment' to mean 'all nature at an instant. ' Amoment, in the sense in which the term is here used, has no temporalextension, and is in this respect to be contrasted with a duration whichhas such extension. What is directly yielded to our knowledge bysense-awareness is a duration. Accordingly we have now to explain howmoments are derived from durations, and also to explain the purposeserved by their introduction. 

A moment is a limit to which we approach as we confine attention todurations of minimum extension. Natural relations among the ingredientsof a duration gain in complexity as we consider durations of increasingtemporal extension. Accordingly there is an approach to ideal simplicityas we approach an ideal diminution of extension. 

The word 'limit' has a precise signification in the logic of number andeven in the logic of non-numerical one-dimensional series. As used hereit is so far a mere metaphor, and it is necessary to explain directlythe concept which it is meant to indicate. 

Durations can have the two-termed relational property of extending oneover the other. Thus the duration which is all nature during a certainminute extends over the duration which is all nature during the30th second of that minute. This relation of 'extendingover'--'extension' as I shall call it--is a fundamental natural relationwhose field comprises more than durations. It is a relation which twolimited events can have to each other. Furthermore as holding betweendurations the relation appears to refer to the purely temporalextension. I shall however maintain that the same relation of extensionlies at the base both of temporal and spatial extension. This discussioncan be postponed; and for the present we are simply concerned with therelation of extension as it occurs in its temporal aspect for thelimited field of durations. 

The concept of extension exhibits in thought one side of the ultimatepassage of nature. This relation holds because of the special characterwhich passage assumes in nature; it is the relation which in the case ofdurations expresses the properties of 'passing over. ' Thus the durationwhich was one definite minute passed over the duration which was its30th second. The duration of the 30th second was part of the duration ofthe minute. I shall use the terms 'whole' and 'part' exclusively in thissense, that the 'part' is an event which is extended over by the otherevent which is the 'whole. ' Thus in my nomenclature 'whole' and 'part'refer exclusively to this fundamental relation of extension; andaccordingly in this technical usage only events can be either wholes orparts. 

The continuity of nature arises from extension. Every event extends overother events, and every event is extended over by other events. Thus inthe special case of durations which are now the only events directlyunder consideration, every duration is part of other durations; andevery duration has other durations which are parts of it. Accordinglythere are no maximum durations and no minimum durations. Thus there isno atomic structure of durations, and the perfect definition of aduration, so as to mark out its individuality and distinguish it fromhighly analogous durations over which it is passing, or which arepassing over it, is an arbitrary postulate of thought. Sense-awarenessposits durations as factors in nature but does not clearly enablethought to use it as distinguishing the separate individualities of theentities of an allied group of slightly differing durations. This is oneinstance of the indeterminateness of sense-awareness. Exactness is anideal of thought, and is only realised in experience by the selection ofa route of approximation. 

The absence of maximum and minimum durations does not exhaust theproperties of nature which make up its continuity. The passage of natureinvolves the existence of a family of durations. When two durationsbelong to the same family either one contains the other, or they overlapeach other in a subordinate duration without either containing theother; or they are completely separate. The excluded case is that ofdurations overlapping in finite events but not containing a thirdduration as a common part. 

It is evident that the relation of extension is transitive; namely asapplied to durations, if duration A is part of duration B, andduration B is part of duration C, then A is part of C. Thus thefirst two cases may be combined into one and we can say that twodurations which belong to the same family _either_ are such that thereare durations which are parts of both _or_ are completely separate. 

Furthermore the converse of this proposition holds; namely, if twodurations have other durations which are parts of both _or_ if the twodurations are completely separate, then they belong to the same family. 

The further characteristics of the continuity of nature--so far asdurations are concerned--which has not yet been formulated arises inconnexion with a family of durations. It can be stated in this way:There are durations which contain as parts any two durations of the samefamily. For example a week contains as parts any two of its days. It isevident that a containing duration satisfies the conditions forbelonging to the same family as the two contained durations. 

We are now prepared to proceed to the definition of a moment of time. Consider a set of durations all taken from the same family. Let it havethe following properties: (i) of any two members of the set one containsthe other as a part, and (ii) there is no duration which is a commonpart of every member of the set. 

Now the relation of whole and part is asymmetrical; and by this I meanthat if A is part of B, then B is not part of A. Also we havealready noted that the relation is transitive. Accordingly we can easilysee that the durations of any set with the properties just enumeratedmust be arranged in a one-dimensional serial order in which as wedescend the series we progressively reach durations of smaller andsmaller temporal extension. The series may start with any arbitrarilyassumed duration of any temporal extension, but in descending theseries the temporal extension progressively contracts and the successivedurations are packed one within the other like the nest of boxes of aChinese toy. But the set differs from the toy in this particular: thetoy has a smallest box which forms the end box of its series; but theset of durations can have no smallest duration nor can it convergetowards a duration as its limit. For the parts either of the endduration or of the limit would be parts of all the durations of the setand thus the second condition for the set would be violated. 

I will call such a set of durations an 'abstractive set' of durations. It is evident that an abstractive set as we pass along it converges tothe ideal of all nature with no temporal extension, namely, to the idealof all nature at an instant. But this ideal is in fact the ideal of anonentity. What the abstractive set is in fact doing is to guide thoughtto the consideration of the progressive simplicity of natural relationsas we progressively diminish the temporal extension of the durationconsidered. Now the whole point of the procedure is that thequantitative expressions of these natural properties do converge tolimits though the abstractive set does not converge to any limitingduration. The laws relating these quantitative limits are the laws ofnature 'at an instant, ' although in truth there is no nature at aninstant and there is only the abstractive set. Thus an abstractive setis effectively the entity meant when we consider an instant of timewithout temporal extension. It subserves all the necessary purposes ofgiving a definite meaning to the concept of the properties of nature atan instant. I fully agree that this concept is fundamental in theexpression of physical science. The difficulty is to express ourmeaning in terms of the immediate deliverances of sense-awareness, and Ioffer the above explanation as a complete solution of the problem. 

In this explanation a moment is the set of natural properties reached bya route of approximation. An abstractive series is a route ofapproximation. There are different routes of approximation to the samelimiting set of the properties of nature. In other words there aredifferent abstractive sets which are to be regarded as routes ofapproximation to the same moment. Accordingly there is a certain amountof technical detail necessary in explaining the relations of suchabstractive sets with the same convergence and in guarding againstpossible exceptional cases. Such details are not suitable for expositionin these lectures, and I have dealt with them fully elsewhere[5]. 

[5] Cf. _An Enquiry concerning the Principles of Natural Knowledge_, Cambridge University Press, 1919. 

It is more convenient for technical purposes to look on a moment asbeing the class of all abstractive sets of durations with the sameconvergence. With this definition (provided that we can successfullyexplain what we mean by the 'same convergence' apart from a detailedknowledge of the set of natural properties arrived at by approximation)a moment is merely a class of sets of durations whose relations ofextension in respect to each other have certain definite peculiarities. We may term these connexions of the component durations the 'extrinsic'properties of a moment; the 'intrinsic' properties of the moment are theproperties of nature arrived at as a limit as we proceed along any oneof its abstractive sets. These are the properties of nature 'at thatmoment, ' or 'at that instant. '

The durations which enter into the composition of a moment all belong toone family. Thus there is one family of moments corresponding to onefamily of durations. Also if we take two moments of the same family, among the durations which enter into the composition of one moment thesmaller durations are completely separated from the smaller durationswhich enter into the composition of the other moment. Thus the twomoments in their intrinsic properties must exhibit the limits ofcompletely different states of nature. In this sense the two moments arecompletely separated. I will call two moments of the same family'parallel. '

Corresponding to each duration there are two moments of the associatedfamily of moments which are the boundary moments of that duration. A'boundary moment' of a duration can be defined in this way. There aredurations of the same family as the given duration which overlap it butare not contained in it. Consider an abstractive set of such durations. Such a set defines a moment which is just as much without the durationas within it. Such a moment is a boundary moment of the duration. Alsowe call upon our sense-awareness of the passage of nature to inform usthat there are two such boundary moments, namely the earlier one and thelater one. We will call them the initial and the final boundaries. 

There are also moments of the same family such that the shorterdurations in their composition are entirely separated from the givenduration. Such moments will be said to lie 'outside' the given duration. Again other moments of the family are such that the shorter durations intheir composition are parts of the given duration. Such moments are saidto lie 'within' the given duration or to 'inhere' in it. The wholefamily of parallel moments is accounted for in this way by reference toany given duration of the associated family of durations. Namely, thereare moments of the family which lie without the given duration, thereare the two moments which are the boundary moments of the givenduration, and the moments which lie within the given duration. Furthermore any two moments of the same family are the boundary momentsof some one duration of the associated family of durations. 

It is now possible to define the serial relation of temporal order amongthe moments of a family. For let A and C be any two moments of thefamily, these moments are the boundary moments of one duration d ofthe associated family, and any moment B which lies within the durationd will be said to lie between the moments A and C. Thus thethree-termed relation of 'lying-between' as relating three moments A, B, and C is completely defined. Also our knowledge of the passage ofnature assures us that this relation distributes the moments of thefamily into a serial order. I abstain from enumerating the definiteproperties which secure this result, I have enumerated them in myrecently published book[6] to which I have already referred. Furthermorethe passage of nature enables us to know that one direction along theseries corresponds to passage into the future and the other directioncorresponds to retrogression towards the past. 

[6] Cf. _Enquiry_. 

Such an ordered series of moments is what we mean by time defined as aseries. Each element of the series exhibits an instantaneous state ofnature. Evidently this serial time is the result of an intellectualprocess of abstraction. What I have done is to give precise definitionsof the procedure by which the abstraction is effected. This procedure ismerely a particular case of the general method which in my book I namethe 'method of extensive abstraction. ' This serial time is evidently notthe very passage of nature itself. It exhibits some of the naturalproperties which flow from it. The state of nature 'at a moment' hasevidently lost this ultimate quality of passage. Also the temporalseries of moments only retains it as an extrinsic relation of entitiesand not as the outcome of the essential being of the terms of theseries. 

Nothing has yet been said as to the measurement of time. Suchmeasurement does not follow from the mere serial property of time; itrequires a theory of congruence which will be considered in a laterlecture. 

In estimating the adequacy of this definition of the temporal series asa formulation of experience it is necessary to discriminate between thecrude deliverance of sense-awareness and our intellectual theories. Thelapse of time is a measurable serial quantity. The whole of scientifictheory depends on this assumption and any theory of time which fails toprovide such a measurable series stands self-condemned as unable toaccount for the most salient fact in experience. Our difficulties onlybegin when we ask what it is that is measured. It is evidently somethingso fundamental in experience that we can hardly stand back from it andhold it apart so as to view it in its own proportions. 

We have first to make up our minds whether time is to be found in natureor nature is to be found in time. The difficulty of the latteralternative--namely of making time prior to nature--is that time thenbecomes a metaphysical enigma. What sort of entities are its instantsor its periods? The dissociation of time from events discloses to ourimmediate inspection that the attempt to set up time as an independentterminus for knowledge is like the effort to find substance in a shadow. There is time because there are happenings, and apart from happeningsthere is nothing. 

It is necessary however to make a distinction. In some sense timeextends beyond nature. It is not true that a timeless sense-awarenessand a timeless thought combine to contemplate a timeful nature. Sense-awareness and thought are themselves processes as well as theirtermini in nature. In other words there is a passage of sense-awarenessand a passage of thought. Thus the reign of the quality of passageextends beyond nature. But now the distinction arises between passagewhich is fundamental and the temporal series which is a logicalabstraction representing some of the properties of nature. A temporalseries, as we have defined it, represents merely certain properties of afamily of durations--properties indeed which durations only possessbecause of their partaking of the character of passage, but on the otherhand properties which only durations do possess. Accordingly time in thesense of a measurable temporal series is a character of nature only, anddoes not extend to the processes of thought and of sense-awarenessexcept by a correlation of these processes with the temporal seriesimplicated in their procedures. 

So far the passage of nature has been considered in connexion with thepassage of durations; and in this connexion it is peculiarly associatedwith temporal series. We must remember however that the character ofpassage is peculiarly associated with the extension of events, and thatfrom this extension spatial transition arises just as much as temporaltransition. The discussion of this point is reserved for a later lecturebut it is necessary to remember it now that we are proceeding to discussthe application of the concept of passage beyond nature, otherwise weshall have too narrow an idea of the essence of passage. 

It is necessary to dwell on the subject of sense-awareness in thisconnexion as an example of the way in which time concerns mind, althoughmeasurable time is a mere abstract from nature and nature is closed tomind. 

Consider sense-awareness--not its terminus which is nature, butsense-awareness in itself as a procedure of mind. Sense-awareness is arelation of mind to nature. Accordingly we are now considering mind as arelatum in sense-awareness. For mind there is the immediatesense-awareness and there is memory. The distinction between memory andthe present immediacy has a double bearing. On the one hand it disclosesthat mind is not impartially aware of all those natural durations towhich it is related by awareness. Its awareness shares in the passage ofnature. We can imagine a being whose awareness, conceived as his privatepossession, suffers no transition, although the terminus of hisawareness is our own transient nature. There is no essential reason whymemory should not be raised to the vividness of the present fact; andthen from the side of mind, What is the difference between the presentand the past? Yet with this hypothesis we can also suppose that thevivid remembrance and the present fact are posited in awareness as intheir temporal serial order. Accordingly we must admit that though wecan imagine that mind in the operation of sense-awareness might be freefrom any character of passage, yet in point of fact our experience ofsense-awareness exhibits our minds as partaking in this character. 

On the other hand the mere fact of memory is an escape from transience. In memory the past is present. It is not present as overleaping thetemporal succession of nature, but it is present as an immediate factfor the mind. Accordingly memory is a disengagement of the mind from themere passage of nature; for what has passed for nature has not passedfor mind. 

Furthermore the distinction between memory and the immediate present isnot so clear as it is conventional to suppose. There is an intellectualtheory of time as a moving knife-edge, exhibiting a present fact withouttemporal extension. This theory arises from the concept of an idealexactitude of observation. Astronomical observations are successivelyrefined to be exact to tenths, to hundredths, and to thousandths ofseconds. But the final refinements are arrived at by a system ofaveraging, and even then present us with a stretch of time as a marginof error. Here error is merely a conventional term to express the factthat the character of experience does not accord with the ideal ofthought. I have already explained how the concept of a momentconciliates the observed fact with this ideal; namely, there is alimiting simplicity in the quantitative expression of the properties ofdurations, which is arrived at by considering any one of the abstractivesets included in the moment. In other words the extrinsic character ofthe moment as an aggregate of durations has associated with it theintrinsic character of the moment which is the limiting expression ofnatural properties. 

Thus the character of a moment and the ideal of exactness which itenshrines do not in any way weaken the position that the ultimateterminus of awareness is a duration with temporal thickness. Thisimmediate duration is not clearly marked out for our apprehension. Itsearlier boundary is blurred by a fading into memory, and its laterboundary is blurred by an emergence from anticipation. There is no sharpdistinction either between memory and the present immediacy or betweenthe present immediacy and anticipation. The present is a waveringbreadth of boundary between the two extremes. Thus our ownsense-awareness with its extended present has some of the character ofthe sense-awareness of the imaginary being whose mind was free frompassage and who contemplated all nature as an immediate fact. Our ownpresent has its antecedents and its consequents, and for the imaginarybeing all nature has its antecedent and its consequent durations. Thusthe only difference in this respect between us and the imaginary beingis that for him all nature shares in the immediacy of our presentduration. 

The conclusion of this discussion is that so far as sense-awareness isconcerned there is a passage of mind which is distinguishable from thepassage of nature though closely allied with it. We may speculate, if welike, that this alliance of the passage of mind with the passage ofnature arises from their both sharing in some ultimate character ofpassage which dominates all being. But this is a speculation in which wehave no concern. The immediate deduction which is sufficient for us isthat--so far as sense-awareness is concerned--mind is not in time or inspace in the same sense in which the events of nature are in time, butthat it is derivatively in time and in space by reason of the peculiaralliance of its passage with the passage of nature. Thus mind is in timeand in space in a sense peculiar to itself. This has been a longdiscussion to arrive at a very simple and obvious conclusion. We allfeel that in some sense our minds are here in this room and at thistime. But it is not quite in the same sense as that in which the eventsof nature which are the existences of our brains have their spatial andtemporal positions. The fundamental distinction to remember is thatimmediacy for sense-awareness is not the same as instantaneousness fornature. This last conclusion bears on the next discussion with which Iwill terminate this lecture. This question can be formulated thus, Canalternative temporal series be found in nature?

A few years ago such a suggestion would have been put aside as beingfantastically impossible. It would have had no bearing on the sciencethen current, and was akin to no ideas which had ever entered into thedreams of philosophy. The eighteenth and nineteenth centuries acceptedas their natural philosophy a certain circle of concepts which were asrigid and definite as those of the philosophy of the middle ages, andwere accepted with as little critical research. I will call this naturalphilosophy 'materialism. ' Not only were men of science materialists, butalso adherents of all schools of philosophy. The idealists only differedfrom the philosophic materialists on question of the alignment of naturein reference to mind. But no one had any doubt that the philosophy ofnature considered in itself was of the type which I have calledmaterialism. It is the philosophy which I have already examined in mytwo lectures of this course preceding the present one. It can besummarised as the belief that nature is an aggregate of material andthat this material exists in some sense _at_ each successive member of aone-dimensional series of extensionless instants of time. Furthermorethe mutual relations of the material entities at each instant formedthese entities into a spatial configuration in an unbounded space. Itwould seem that space--on this theory--would be as instantaneous as theinstants, and that some explanation is required of the relations betweenthe successive instantaneous spaces. The materialistic theory is howeversilent on this point; and the succession of instantaneous spaces istacitly combined into one persistent space. This theory is a purelyintellectual rendering of experience which has had the luck to getitself formulated at the dawn of scientific thought. It has dominatedthe language and the imagination of science since science flourished inAlexandria, with the result that it is now hardly possible to speakwithout appearing to assume its immediate obviousness. 

But when it is distinctly formulated in the abstract terms in which Ihave just stated it, the theory is very far from obvious. The passingcomplex of factors which compose the fact which is the terminus ofsense-awareness places before us nothing corresponding to the trinity ofthis natural materialism. This trinity is composed (i) of the temporalseries of extensionless instants, (ii) of the aggregate of materialentities, and (iii) of space which is the outcome of relations ofmatter. 

There is a wide gap between these presuppositions of the intellectualtheory of materialism and the immediate deliverances of sense-awareness. I do not question that this materialistic trinity embodies importantcharacters of nature. But it is necessary to express these characters interms of the facts of experience. This is exactly what in this lecture Ihave been endeavouring to do so far as time is concerned; and we havenow come up against the question, Is there only one temporal series? Theuniqueness of the temporal series is presupposed in the materialistphilosophy of nature. But that philosophy is merely a theory, like theAristotelian scientific theories so firmly believed in the middle ages. If in this lecture I have in any way succeeded in getting behind thetheory to the immediate facts, the answer is not nearly so certain. Thequestion can be transformed into this alternative form, Is there onlyone family of durations? In this question the meaning of a 'family ofdurations' has been defined earlier in this lecture. The answer is nownot at all obvious. On the materialistic theory the instantaneouspresent is the only field for the creative activity of nature. The pastis gone and the future is not yet. Thus (on this theory) the immediacyof perception is of an instantaneous present, and this unique present isthe outcome of the past and the promise of the future. But we deny thisimmediately given instantaneous present. There is no such thing to befound in nature. As an ultimate fact it is a nonentity. What isimmediate for sense-awareness is a duration. Now a duration has withinitself a past and a future; and the temporal breadths of the immediatedurations of sense-awareness are very indeterminate and dependent on theindividual percipient. Accordingly there is no unique factor in naturewhich for every percipient is pre-eminently and necessarily the present. The passage of nature leaves nothing between the past and the future. What we perceive as present is the vivid fringe of memory tinged withanticipation. This vividness lights up the discriminated field within aduration. But no assurance can thereby be given that the happenings ofnature cannot be assorted into other durations of alternative families. We cannot even know that the series of immediate durations posited bythe sense-awareness of one individual mind all necessarily belong to thesame family of durations. There is not the slightest reason to believethat this is so. Indeed if my theory of nature be correct, it will notbe the case. 

The materialistic theory has all the completeness of the thought of themiddle ages, which had a complete answer to everything, be it in heavenor in hell or in nature. There is a trimness about it, with itsinstantaneous present, its vanished past, its non-existent future, andits inert matter. This trimness is very medieval and ill accords withbrute fact. 

The theory which I am urging admits a greater ultimate mystery and adeeper ignorance. The past and the future meet and mingle in theill-defined present. The passage of nature which is only another namefor the creative force of existence has no narrow ledge of definiteinstantaneous present within which to operate. Its operative presencewhich is now urging nature forward must be sought for throughout thewhole, in the remotest past as well as in the narrowest breadth of anypresent duration. Perhaps also in the unrealised future. Perhaps also inthe future which might be as well as the actual future which will be. Itis impossible to meditate on time and the mystery of the creativepassage of nature without an overwhelming emotion at the limitations ofhuman intelligence. 

CHAPTER IV

THE METHOD OF EXTENSIVE ABSTRACTION

To-day's lecture must commence with the consideration of limited events. We shall then be in a position to enter upon an investigation of thefactors in nature which are represented by our conception of space. 

The duration which is the immediate disclosure of our sense-awareness isdiscriminated into parts. There is the part which is the life of allnature within a room, and there is the part which is the life of allnature within a table in the room. These parts are limited events. Theyhave the endurance of the present duration, and they are parts of it. But whereas a duration is an unlimited whole and in a certain limitedsense is all that there is, a limited event possesses a completelydefined limitation of extent which is expressed for us inspatio-temporal terms. 

We are accustomed to associate an event with a certain melodramaticquality. If a man is run over, that is an event comprised within certainspatio-temporal limits. We are not accustomed to consider the enduranceof the Great Pyramid throughout any definite day as an event. But thenatural fact which is the Great Pyramid throughout a day, meaningthereby all nature within it, is an event of the same character as theman's accident, meaning thereby all nature with spatio-temporallimitations so as to include the man and the motor during the periodwhen they were in contact. 

We are accustomed to analyse these events into three factors, time, space, and material. In fact, we at once apply to them the concepts ofthe materialistic theory of nature. I do not deny the utility of thisanalysis for the purpose of expressing important laws of nature. What Iam denying is that anyone of these factors is posited for us insense-awareness in concrete independence. We perceive one unit factor innature; and this factor is that something is going on then--there. Forexample, we perceive the going-on of the Great Pyramid in its relationsto the goings-on of the surrounding Egyptian events. We are so trained, both by language and by formal teaching and by the resultingconvenience, to express our thoughts in terms of this materialisticanalysis that intellectually we tend to ignore the true unity of thefactor really exhibited in sense-awareness. It is this unit factor, retaining in itself the passage of nature, which is the primary concreteelement discriminated in nature. These primary factors are what I meanby events. 

Events are the field of a two-termed relation, namely the relation ofextension which was considered in the last lecture. Events are thethings related by the relation of extension. If an event A extendsover an event B, then B is 'part of' A, and A is a 'whole' ofwhich B is a part. Whole and part are invariably used in theselectures in this definite sense. It follows that in reference to thisrelation any two events A and B may have any one of four relationsto each other, namely (i) A may extend over B, or (ii) B mayextend over A, or (iii) A and B may both extend over some thirdevent C, but neither over the other, or (iv) A and B may beentirely separate. These alternatives can obviously be illustrated byEuler's diagrams as they appear in logical textbooks. 

The continuity of nature is the continuity of events. This continuity ismerely the name for the aggregate of a variety of properties of eventsin connexion with the relation of extension. 

In the first place, this relation is transitive; secondly, every eventcontains other events as parts of itself; thirdly every event is a partof other events; fourthly given any two finite events there are eventseach of which contains both of them as parts; and fifthly there is aspecial relation between events which I term 'junction. '

Two events have junction when there is a third event of which bothevents are parts, and which is such that no part of it is separated fromboth of the two given events. Thus two events with junction make upexactly one event which is in a sense their sum. 

Only certain pairs of events have this property. In general any eventcontaining two events also contains parts which are separated from bothevents. 

There is an alternative definition of the junction of two events which Ihave adopted in my recent book[7]. Two events have junction when thereis a third event such that (i) it overlaps both events and (ii) it hasno part which is separated from both the given events. If either ofthese alternative definitions is adopted as the definition of junction, the other definition appears as an axiom respecting the character ofjunction as we know it in nature. But we are not thinking of logicaldefinition so much as the formulation of the results of directobservation. There is a certain continuity inherent in the observedunity of an event, and these two definitions of junction are reallyaxioms based on observation respecting the character of this continuity. 

[7] Cf. _Enquiry_. 

The relations of whole and part and of overlapping are particular casesof the junction of events. But it is possible for events to havejunction when they are separate from each other; for example, the upperand the lower part of the Great Pyramid are divided by some imaginaryhorizontal plane. 

The continuity which nature derives from events has been obscured by theillustrations which I have been obliged to give. For example I havetaken the existence of the Great Pyramid as a fairly well-known fact towhich I could safely appeal as an illustration. This is a type of eventwhich exhibits itself to us as the situation of a recognisable object;and in the example chosen the object is so widely recognised that it hasreceived a name. An object is an entity of a different type from anevent. For example, the event which is the life of nature within theGreat Pyramid yesterday and to-day is divisible into two parts, namelythe Great Pyramid yesterday and the Great Pyramid to-day. But therecognisable object which is also called the Great Pyramid is the sameobject to-day as it was yesterday. I shall have to consider the theoryof objects in another lecture. 

The whole subject is invested with an unmerited air of subtlety by thefact that when the event is the situation of a well-marked object, wehave no language to distinguish the event from the object. In the caseof the Great Pyramid, the object is the perceived unit entity which asperceived remains self-identical throughout the ages; while the wholedance of molecules and the shifting play of the electromagnetic fieldare ingredients of the event. An object is in a sense out of time. It isonly derivatively in time by reason of its having the relation to eventswhich I term 'situation. ' This relation of situation will requirediscussion in a subsequent lecture. 

The point which I want to make now is that being the situation of awell-marked object is not an inherent necessity for an event. Whereverand whenever something is going on, there is an event. Furthermore'wherever and whenever' in themselves presuppose an event, for space andtime in themselves are abstractions from events. It is therefore aconsequence of this doctrine that something is always going oneverywhere, even in so-called empty space. This conclusion is in accordwith modern physical science which presupposes the play of anelectromagnetic field throughout space and time. This doctrine ofscience has been thrown into the materialistic form of an all-pervadingether. But the ether is evidently a mere idle concept--in thephraseology which Bacon applied to the doctrine of final causes, it is abarren virgin. Nothing is deduced from it; and the ether merelysubserves the purpose of satisfying the demands of the materialistictheory. The important concept is that of the shifting facts of thefields of force. This is the concept of an ether of events which shouldbe substituted for that of a material ether. 

It requires no illustration to assure you that an event is a complexfact, and the relations between two events form an almost impenetrablemaze. The clue discovered by the common sense of mankind andsystematically utilised in science is what I have elsewhere[8] calledthe law of convergence to simplicity by diminution of extent. 

[8] Cf. _Organisation of Thought_, pp.  146 et seq. Williams and Norgate, 1917. 

If A and B are two events, and A′ is part of A and B′ is partof B, then in many respects the relations between the parts A′ andB′ will be simpler than the relations between A and B. This is theprinciple which presides over all attempts at exact observation. 

The first outcome of the systematic use of this law has been theformulation of the abstract concepts of Time and Space. In the previouslecture I sketched how the principle was applied to obtain thetime-series. I now proceed to consider how the spatial entities areobtained by the same method. The systematic procedure is identical inprinciple in both cases, and I have called the general type of procedurethe 'method of extensive abstraction. '

You will remember that in my last lecture I defined the concept of anabstractive set of durations. This definition can be extended so as toapply to any events, limited events as well as durations. The onlychange that is required is the substitution of the word 'event' for theword 'duration. ' Accordingly an abstractive set of events is any set ofevents which possesses the two properties, (i) of any two members of theset one contains the other as a part, and (ii) there is no event whichis a common part of every member of the set. Such a set, as you willremember, has the properties of the Chinese toy which is a nest ofboxes, one within the other, with the difference that the toy has asmallest box, while the abstractive class has neither a smallest eventnor does it converge to a limiting event which is not a member of theset. 

Thus, so far as the abstractive sets of events are concerned, anabstractive set converges to nothing. There is the set with its membersgrowing indefinitely smaller and smaller as we proceed in thoughttowards the smaller end of the series; but there is no absolute minimumof any sort which is finally reached. In fact the set is just itself andindicates nothing else in the way of events, except itself. But eachevent has an intrinsic character in the way of being a situation ofobjects and of having parts which are situations of objects and--tostate the matter more generally--in the way of being a field of the lifeof nature. This character can be defined by quantitative expressionsexpressing relations between various quantities intrinsic to the eventor between such quantities and other quantities intrinsic to otherevents. In the case of events of considerable spatio-temporal extensionthis set of quantitative expressions is of bewildering complexity. Ife be an event, let us denote by q(e) the set of quantitative expressionsdefining its character including its connexions with the rest of nature. Let e₁, e₂, e₃, etc. Be an abstractive set, the members being soarranged that each member such as e_{n} extends over all the succeedingmembers such as e_{n+1}, e_{n+2} and so on. Then corresponding to theseries

 e₁, e₂, e₃, ... , e_{n}, e_{n+1}, ... , there is the series

 q(e₁), q(e₂), q(e₃), ... , q(e_{n}), q(e_{n+1}), .... 

Call the series of events s and the series of quantitative expressionsq(s). The series s has no last term and no events which are containedin every member of the series. Accordingly the series of eventsconverges to nothing. It is just itself. Also the series q(s) has nolast term. But the sets of homologous quantities running through thevarious terms of the series do converge to definite limits. For exampleif Q₁ be a quantitative measurement found in q(e₁), and Q₂ the homologueto Q₁ to be found in q(e₂), and Q₃ the homologue to Q₁ and Q₂ to befound in q(e₃), and so on, then the series

 Q₁, Q₂, Q₃, ... , Q_{n}, Q_{n+1}, ... , though it has no last term, does in general converge to a definitelimit. Accordingly there is a class of limits l(s) which is theclass of the limits of those members of q(e_{n}) which havehomologues throughout the series q(s) as n indefinitely increases. We can represent this statement diagrammatically by using an arrow (➝)to mean 'converges to. ' Then

 e₁, e₂, e₃, ... , e_{n}, e_{n+1}, ... ➝ nothing, and

 q(e₁), q(e₂), q(e₃), ... , q(e_{n}), q(e_{n+1}), ... ➝ l(s). 

The mutual relations between the limits in the set l(s), and alsobetween these limits and the limits in other sets l(s′), l(s″), ... , which arise from other abstractive sets s′, s″, etc. , have a peculiarsimplicity. 

Thus the set s does indicate an ideal simplicity of natural relations, though this simplicity is not the character of any actual event in s. We can make an approximation to such a simplicity which, as estimatednumerically, is as close as we like by considering an event which is farenough down the series towards the small end. It will be noted that itis the infinite series, as it stretches away in unending successiontowards the small end, which is of importance. The arbitrarily largeevent with which the series starts has no importance at all. We canarbitrarily exclude any set of events at the big end of an abstractiveset without the loss of any important property to the set as thusmodified. 

I call the limiting character of natural relations which is indicated byan abstractive set, the 'intrinsic character' of the set; also theproperties, connected with the relation of whole and part as concerningits members, by which an abstractive set is defined together form what Icall its 'extrinsic character. ' The fact that the extrinsic character ofan abstractive set determines a definite intrinsic character is thereason of the importance of the precise concepts of space and time. Thisemergence of a definite intrinsic character from an abstractive set isthe precise meaning of the law of convergence. 

For example, we see a train approaching during a minute. The event whichis the life of nature within that train during the minute is of greatcomplexity and the expression of its relations and of the ingredients ofits character baffles us. If we take one second of that minute, the morelimited event which is thus obtained is simpler in respect to itsingredients, and shorter and shorter times such as a tenth of thatsecond, or a hundredth, or a thousandth--so long as we have a definiterule giving a definite succession of diminishing events--give eventswhose ingredient characters converge to the ideal simplicity of thecharacter of the train at a definite instant. Furthermore there aredifferent types of such convergence to simplicity. For example, we canconverge as above to the limiting character expressing nature at aninstant within the whole volume of the train at that instant, or tonature at an instant within some portion of that volume--for examplewithin the boiler of the engine--or to nature at an instant on some areaof surface, or to nature at an instant on some line within the train, orto nature at an instant at some point of the train. In the last case thesimple limiting characters arrived at will be expressed as densities, specific gravities, and types of material. Furthermore we need notnecessarily converge to an abstraction which involves nature at aninstant. We may converge to the physical ingredients of a certain pointtrack throughout the whole minute. Accordingly there are different typesof extrinsic character of convergence which lead to the approximation todifferent types of intrinsic characters as limits. 

We now pass to the investigation of possible connexions betweenabstractive sets. One set may 'cover' another. I define 'covering' asfollows: An abstractive set p covers an abstractive set q when everymember of p contains as its parts some members of q. It is evidentthat if any event e contains as a part any member of the set q, thenowing to the transitive property of extension every succeeding member ofthe small end of q is part of e. In such a case I will say that theabstractive set q 'inheres in' the event e. Thus when an abstractiveset p covers an abstractive set q, the abstractive set q inheresin every member of p. 

Two abstractive sets may each cover the other. When this is the case Ishall call the two sets 'equal in abstractive force. ' When there is nodanger of misunderstanding I shall shorten this phrase by simply sayingthat the two abstractive sets are 'equal. ' The possibility of thisequality of abstractive sets arises from the fact that both sets, pand q, are infinite series towards their small ends. Thus the equalitymeans, that given any event x belonging to p, we can always byproceeding far enough towards the small end of q find an event ywhich is part of x, and that then by proceeding far enough towards thesmall end of p we can find an event z which is part of y, and soon indefinitely. 

The importance of the equality of abstractive sets arises from theassumption that the intrinsic characters of the two sets are identical. If this were not the case exact observation would be at an end. 

It is evident that any two abstractive sets which are equal to a thirdabstractive set are equal to each other. An 'abstractive element' is thewhole group of abstractive sets which are equal to any one ofthemselves. Thus all abstractive sets belonging to the same element areequal and converge to the same intrinsic character. Thus an abstractiveelement is the group of routes of approximation to a definite intrinsiccharacter of ideal simplicity to be found as a limit among naturalfacts. 

If an abstractive set p covers an abstractive set q, then anyabstractive set belonging to the abstractive element of which p is amember will cover any abstractive set belonging to the element of whichq is a member. Accordingly it is useful to stretch the meaning of theterm 'covering, ' and to speak of one abstractive element 'covering'another abstractive element. If we attempt in like manner to stretch theterm 'equal' in the sense of 'equal in abstractive force, ' it is obviousthat an abstractive element can only be equal to itself. Thus anabstractive element has a unique abstractive force and is the constructfrom events which represents one definite intrinsic character which isarrived at as a limit by the use of the principle of convergence tosimplicity by diminution of extent. 

When an abstractive element A covers an abstractive element B, theintrinsic character of A in a sense includes the intrinsic characterof B. It results that statements about the intrinsic character of Bare in a sense statements about the intrinsic character of A; but theintrinsic character of A is more complex than that of B. 

The abstractive elements form the fundamental elements of space andtime, and we now turn to the consideration of the properties involved inthe formation of special classes of such elements. In my last lecture Ihave already investigated one class of abstractive elements, namelymoments. Each moment is a group of abstractive sets, and the eventswhich are members of these sets are all members of one family ofdurations. The moments of one family form a temporal series; and, allowing the existence of different families of moments, there will bealternative temporal series in nature. Thus the method of extensiveabstraction explains the origin of temporal series in terms of theimmediate facts of experience and at the same time allows for theexistence of the alternative temporal series which are demanded by themodern theory of electromagnetic relativity. 

We now turn to space. The first thing to do is to get hold of the classof abstractive elements which are in some sense the points of space. Such an abstractive element must in some sense exhibit a convergence toan absolute minimum of intrinsic character. Euclid has expressed for alltime the general idea of a point, as being without parts and withoutmagnitude. It is this character of being an absolute minimum which wewant to get at and to express in terms of the extrinsic characters ofthe abstractive sets which make up a point. Furthermore, points whichare thus arrived at represent the ideal of events without any extension, though there are in fact no such entities as these ideal events. Thesepoints will not be the points of an external timeless space but ofinstantaneous spaces. We ultimately want to arrive at the timeless spaceof physical science, and also of common thought which is now tinged withthe concepts of science. It will be convenient to reserve the term'point' for these spaces when we get to them. I will therefore use thename 'event-particles' for the ideal minimum limits to events. Thus anevent-particle is an abstractive element and as such is a group ofabstractive sets; and a point--namely a point of timeless space--will bea class of event-particles. 

Furthermore there is a separate timeless space corresponding to eachseparate temporal series, that is to each separate family of durations. We will come back to points in timeless spaces later. I merely mentionthem now that we may understand the stages of our investigation. Thetotality of event-particles will form a four-dimensional manifold, theextra dimension arising from time--in other words--arising from thepoints of a timeless space being each a class of event-particles. 

The required character of the abstractive sets which formevent-particles would be secured if we could define them as having theproperty of being covered by any abstractive set which they cover. Forthen any other abstractive set which an abstractive set of anevent-particle covered, would be equal to it, and would therefore be amember of the same event-particle. Accordingly an event-particle couldcover no other abstractive element. This is the definition which Ioriginally proposed at a congress in Paris in 1914[9]. There is howevera difficulty involved in this definition if adopted without some furtheraddition, and I am now not satisfied with the way in which I attemptedto get over that difficulty in the paper referred to. 

[9] Cf. 'La Théorie Relationniste de l'Espace, ' _Rev. De Métaphysique etde Morale_, vol.  XXIII, 1916. 

The difficulty is this: When event-particles have once been defined itis easy to define the aggregate of event-particles forming the boundaryof an event; and thence to define the point-contact at their boundariespossible for a pair of events of which one is part of the other. We canthen conceive all the intricacies of tangency. In particular we canconceive an abstractive set of which all the members have point-contactat the same event-particle. It is then easy to prove that there will beno abstractive set with the property of being covered by everyabstractive set which it covers. I state this difficulty at some lengthbecause its existence guides the development of our line of argument. Wehave got to annex some condition to the root property of being coveredby any abstractive set which it covers. When we look into this questionof suitable conditions we find that in addition to event-particles allthe other relevant spatial and spatio-temporal abstractive elements canbe defined in the same way by suitably varying the conditions. Accordingly we proceed in a general way suitable for employment beyondevent-particles. 

Let σ be the name of any condition which some abstractive sets fulfil. Isay that an abstractive set is 'σ-prime' when it has the twoproperties, (i) that it satisfies the condition σ and (ii) that it iscovered by every abstractive set which both is covered by it andsatisfies the condition σ. 

In other words you cannot get any abstractive set satisfying thecondition σ which exhibits intrinsic character more simple than that ofa σ-prime. 

There are also the correlative abstractive sets which I call the sets ofσ-antiprimes. An abstractive set is a σ-antiprime when it has the twoproperties, (i) that it satisfies the condition σ and (ii) that itcovers every abstractive set which both covers it and satisfies thecondition σ. In other words you cannot get any abstractive setsatisfying the condition σ which exhibits an intrinsic character morecomplex than that of a σ-antiprime. 

The intrinsic character of a σ-prime has a certain minimum of fullnessamong those abstractive sets which are subject to the condition ofsatisfying σ; whereas the intrinsic character of a σ-antiprime has acorresponding maximum of fullness, and includes all it can in thecircumstances. 

Let us first consider what help the notion of antiprimes could give usin the definition of moments which we gave in the last lecture. Let thecondition σ be the property of being a class whose members are alldurations. An abstractive set which satisfies this condition is thus anabstractive set composed wholly of durations. It is convenient then todefine a moment as the group of abstractive sets which are equal to someσ-antiprime, where the condition σ has this special meaning. It will befound on consideration (i) that each abstractive set forming a moment isa σ-antiprime, where σ has this special meaning, and (ii) that we haveexcluded from membership of moments abstractive sets of durations whichall have one common boundary, either the initial boundary or the finalboundary. We thus exclude special cases which are apt to confuse generalreasoning. The new definition of a moment, which supersedes our previousdefinition, is (by the aid of the notion of antiprimes) the moreprecisely drawn of the two, and the more useful. 

The particular condition which 'σ' stood for in the definition ofmoments included something additional to anything which can be derivedfrom the bare notion of extension. A duration exhibits for thought atotality. The notion of totality is something beyond that of extension, though the two are interwoven in the notion of a duration. 

In the same way the particular condition 'σ' required for the definitionof an event-particle must be looked for beyond the mere notion ofextension. The same remark is also true of the particular conditionsrequisite for the other spatial elements. This additional notion isobtained by distinguishing between the notion of 'position' and thenotion of convergence to an ideal zero of extension as exhibited by anabstractive set of events. 

In order to understand this distinction consider a point of theinstantaneous space which we conceive as apparent to us in an almostinstantaneous glance. This point is an event-particle. It has twoaspects. In one aspect it is there, where it is. This is its position inthe space. In another aspect it is got at by ignoring the circumambientspace, and by concentrating attention on the smaller and smaller set ofevents which approximate to it. This is its extrinsic character. Thus apoint has three characters, namely, its position in the wholeinstantaneous space, its extrinsic character, and its intrinsiccharacter. The same is true of any other spatial element. For example aninstantaneous volume in instantaneous space has three characters, namely, its position, its extrinsic character as a group of abstractivesets, and its intrinsic character which is the limit of naturalproperties which is indicated by any one of these abstractive sets. 

Before we can talk about position in instantaneous space, we mustevidently be quite clear as to what we mean by instantaneous space initself. Instantaneous space must be looked for as a character of amoment. For a moment is all nature at an instant. It cannot be theintrinsic character of the moment. For the intrinsic character tells usthe limiting character of nature in space at that instant. Instantaneousspace must be an assemblage of abstractive elements considered in theirmutual relations. Thus an instantaneous space is the assemblage ofabstractive elements covered by some one moment, and it is theinstantaneous space of that moment. 

We have now to ask what character we have found in nature which iscapable of according to the elements of an instantaneous space differentqualities of position. This question at once brings us to theintersection of moments, which is a topic not as yet considered in theselectures. 

The locus of intersection of two moments is the assemblage ofabstractive elements covered by both of them. Now two moments of thesame temporal series cannot intersect. Two moments respectively ofdifferent families necessarily intersect. Accordingly in theinstantaneous space of a moment we should expect the fundamentalproperties to be marked by the intersections with moments of otherfamilies. If M be a given moment, the intersection of M with anothermoment A is an instantaneous plane in the instantaneous space of M; andif B be a third moment intersecting both M and A, the intersection of Mand B is another plane in the space M. Also the common intersection ofA, B, and M is the intersection of the two planes in the space M, namelyit is a straight line in the space M. An exceptional case arises if Band M intersect in the same plane as A and M. Furthermore if C be afourth moment, then apart from special cases which we need not consider, it intersects M in a plane which the straight line (A, B, M) meets. Thusthere is in general a common intersection of four moments of differentfamilies. This common intersection is an assemblage of abstractiveelements which are each covered (or 'lie in') all four moments. Thethree-dimensional property of instantaneous space comes to this, that(apart from special relations between the four moments) any fifth momenteither contains the whole of their common intersection or none of it. Nofurther subdivision of the common intersection is possible by means ofmoments. The 'all or none' principle holds. This is not an _à priori_truth but an empirical fact of nature. 

It will be convenient to reserve the ordinary spatial terms 'plane, ''straight line, ' 'point' for the elements of the timeless space of atime-system. Accordingly an instantaneous plane in the instantaneousspace of a moment will be called a 'level, ' an instantaneous straightline will be called a 'rect, ' and an instantaneous point will be calleda 'punct. ' Thus a punct is the assemblage of abstractive elements whichlie in each of four moments whose families have no special relations toeach other. Also if P be any moment, either every abstractive elementbelonging to a given punct lies in P, or no abstractive element ofthat punct lies in P. 

Position is the quality which an abstractive element possesses in virtueof the moments in which it lies. The abstractive elements which lie inthe instantaneous space of a given moment M are differentiated fromeach other by the various other moments which intersect M so as tocontain various selections of these abstractive elements. It is thisdifferentiation of the elements which constitutes their differentiationof position. An abstractive element which belongs to a punct has thesimplest type of position in M, an abstractive element which belongsto a rect but not to a punct has a more complex quality of position, anabstractive element which belongs to a level and not to a rect has astill more complex quality of position, and finally the most complexquality of position belongs to an abstractive element which belongs to avolume and not to a level. A volume however has not yet been defined. This definition will be given in the next lecture. 

Evidently levels, rects, and puncts in their capacity as infiniteaggregates cannot be the termini of sense-awareness, nor can they belimits which are approximated to in sense-awareness. Any one member of alevel has a certain quality arising from its character as also belongingto a certain set of moments, but the level as a whole is a mere logicalnotion without any route of approximation along entities posited insense-awareness. 

On the other hand an event-particle is defined so as to exhibit thischaracter of being a route of approximation marked out by entitiesposited in sense-awareness. A definite event-particle is defined inreference to a definite punct in the following manner: Let the conditionσ mean the property of covering all the abstractive elements which aremembers of that punct; so that an abstractive set which satisfies thecondition σ is an abstractive set which covers every abstractive elementbelonging to the punct. Then the definition of the event-particleassociated with the punct is that it is the group of all the σ-primes, where σ has this particular meaning. 

It is evident that--with this meaning of σ--every abstractive set equalto a σ-prime is itself a σ-prime. Accordingly an event-particle as thusdefined is an abstractive element, namely it is the group of thoseabstractive sets which are each equal to some given abstractive set. Ifwe write out the definition of the event-particle associated with somegiven punct, which we will call π, it is as follows: The event-particleassociated with π is the group of abstractive classes each of which hasthe two properties (i) that it covers every abstractive set in π and(ii) that all the abstractive sets which also satisfy the formercondition as to π and which it covers, also cover it. 

An event-particle has position by reason of its association with apunct, and conversely the punct gains its derived character as a routeof approximation from its association with the event-particle. These twocharacters of a point are always recurring in any treatment of thederivation of a point from the observed facts of nature, but in generalthere is no clear recognition of their distinction. 

The peculiar simplicity of an instantaneous point has a twofold origin, one connected with position, that is to say with its character as apunct, and the other connected with its character as an event-particle. The simplicity of the punct arises from its indivisibility by a moment. 

The simplicity of an event-particle arises from the indivisibility ofits intrinsic character. The intrinsic character of an event-particle isindivisible in the sense that every abstractive set covered by itexhibits the same intrinsic character. It follows that, though there arediverse abstractive elements covered by event-particles, there is noadvantage to be gained by considering them since we gain no additionalsimplicity in the expression of natural properties. 

These two characters of simplicity enjoyed respectively byevent-particles and puncts define a meaning for Euclid's phrase, 'without parts and without magnitude. '

It is obviously convenient to sweep away out of our thoughts all thesestray abstractive sets which are covered by event-particles withoutthemselves being members of them. They give us nothing new in the way ofintrinsic character. Accordingly we can think of rects and levels asmerely loci of event-particles. In so doing we are also cutting outthose abstractive elements which cover sets of event-particles, withoutthese elements being event-particles themselves. There are classes ofthese abstractive elements which are of great importance. I willconsider them later on in this and in other lectures. Meanwhile we willignore them. Also I will always speak of 'event-particles' in preferenceto 'puncts, ' the latter being an artificial word for which I have nogreat affection. 

Parallelism among rects and levels is now explicable. 

Consider the instantaneous space belonging to a moment A, and let Abelong to the temporal series of moments which I will call α. Considerany other temporal series of moments which I will call β. The moments ofβ do not intersect each other and they intersect the moment A in afamily of levels. None of these levels can intersect, and they form afamily of parallel instantaneous planes in the instantaneous space ofmoment A. Thus the parallelism of moments in a temporal series begetsthe parallelism of levels in an instantaneous space, and thence--as itis easy to see--the parallelism of rects. Accordingly the Euclideanproperty of space arises from the parabolic property of time. It may bethat there is reason to adopt a hyperbolic theory of time and acorresponding hyperbolic theory of space. Such a theory has not beenworked out, so it is not possible to judge as to the character of theevidence which could be brought forward in its favour. 

The theory of order in an instantaneous space is immediately derivedfrom time-order. For consider the space of a moment M. Let α be the nameof a time-system to which M does not belong. Let A₁, A₂, A₃ etc. Bemoments of α in the order of their occurrences. Then A₁, A₂, A₃, etc. Intersect M in parallel levels l₁, l₂, l₃, etc. Then the relative orderof the parallel levels in the space of M is the same as the relativeorder of the corresponding moments in the time-system α. Any rect in Mwhich intersects all these levels in its set of puncts, thereby receivesfor its puncts an order of position on it. So spatial order isderivative from temporal order. Furthermore there are alternativetime-systems, but there is only one definite spatial order in eachinstantaneous space. Accordingly the various modes of deriving spatialorder from diverse time-systems must harmonise with one spatial order ineach instantaneous space. In this way also diverse time-orders arecomparable. 

We have two great questions still on hand to be settled before ourtheory of space is fully adjusted. One of these is the question of thedetermination of the methods of measurement within the space, in otherwords, the congruence-theory of the space. The measurement of space willbe found to be closely connected with the measurement of time, withrespect to which no principles have as yet been determined. Thus ourcongruence-theory will be a theory both for space and for time. Secondlythere is the determination of the timeless space which corresponds toany particular time-system with its infinite set of instantaneous spacesin its successive moments. This is the space--or rather, these are thespaces--of physical science. It is very usual to dismiss this space bysaying that this is conceptual. I do not understand the virtue of thesephrases. I suppose that it is meant that the space is the conception ofsomething in nature. Accordingly if the space of physical science is tobe called conceptual, I ask, What in nature is it the conception of? Forexample, when we speak of a point in the timeless space of physicalscience, I suppose that we are speaking of something in nature. If weare not so speaking, our scientists are exercising their wits in therealms of pure fantasy, and this is palpably not the case. This demandfor a definite Habeas Corpus Act for the production of the relevantentities in nature applies whether space be relative or absolute. On thetheory of relative space, it may perhaps be argued that there is notimeless space for physical science, and that there is only themomentary series of instantaneous spaces. 

An explanation must then be asked for the meaning of the very commonstatement that such and such a man walked four miles in some definitehour. How can you measure distance from one space into another space? Iunderstand walking out of the sheet of an ordnance map. But the meaningof saying that Cambridge at 10 o'clock this morning in the appropriateinstantaneous space for that instant is 52 miles from London at11 o'clock this morning in the appropriate instantaneous space for thatinstant beats me entirely. I think that, by the time a meaning has beenproduced for this statement, you will find that you have constructedwhat is in fact a timeless space. What I cannot understand is how toproduce an explanation of meaning without in effect making some suchconstruction. Also I may add that I do not know how the instantaneousspaces are thus correlated into one space by any method which isavailable on the current theories of space. 

You will have noticed that by the aid of the assumption of alternativetime-systems, we are arriving at an explanation of the character ofspace. In natural science 'to explain' means merely to discover'interconnexions. ' For example, in one sense there is no explanation ofthe red which you see. It is red, and there is nothing else to be saidabout it. Either it is posited before you in sense-awareness or you areignorant of the entity red. But science has explained red. Namely it hasdiscovered interconnexions between red as a factor in nature and otherfactors in nature, for example waves of light which are waves ofelectromagnetic disturbances. There are also various pathologicalstates of the body which lead to the seeing of red without theoccurrence of light waves. Thus connexions have been discovered betweenred as posited in sense-awareness and various other factors in nature. The discovery of these connexions constitutes the scientific explanationof our vision of colour. In like manner the dependence of the characterof space on the character of time constitutes an explanation in thesense in which science seeks to explain. The systematising intellectabhors bare facts. The character of space has hitherto been presented asa collection of bare facts, ultimate and disconnected. The theory whichI am expounding sweeps away this disconnexion of the facts of space. 

CHAPTER V

SPACE AND MOTION

The topic for this lecture is the continuation of the task of explainingthe construction of spaces as abstracts from the facts of nature. It wasnoted at the close of the previous lecture that the question ofcongruence had not been considered, nor had the construction of atimeless space which should correlate the successive momentary spaces ofa given time-system. Furthermore it was also noted that there were manyspatial abstractive elements which we had not yet defined. We will firstconsider the definition of some of these abstractive elements, namelythe definitions of solids, of areas, and of routes. By a 'route' I meana linear segment, whether straight or curved. The exposition of thesedefinitions and the preliminary explanations necessary will, I hope, serve as a general explanation of the function of event-particles in theanalysis of nature. 

We note that event-particles have 'position' in respect to each other. In the last lecture I explained that 'position' was quality gained by aspatial element in virtue of the intersecting moments which covered it. Thus each event-particle has position in this sense. The simplest modeof expressing the position in nature of an event-particle is by firstfixing on any definite time-system. Call it α. There will be one momentof the temporal series of α which covers the given event-particle. Thusthe position of the event-particle in the temporal series α is definedby this moment, which we will call M. The position of the particle inthe space of M is then fixed in the ordinary way by three levels whichintersect in it and in it only. This procedure of fixing the position ofan event-particle shows that the aggregate of event-particles forms afour-dimensional manifold. A finite event occupies a limited chunk ofthis manifold in a sense which I now proceed to explain. 

Let e be any given event. The manifold of event-particles falls intothree sets in reference to e. Each event-particle is a group of equalabstractive sets and each abstractive set towards its small-end iscomposed of smaller and smaller finite events. When we select from thesefinite events which enter into the make-up of a given event-particlethose which are small enough, one of three cases must occur. Either (i)all of these small events are entirely separate from the given evente, or (ii) all of these small events are parts of the event e, or(iii) all of these small events overlap the event e but are not partsof it. In the first case the event-particle will be said to 'lieoutside' the event e, in the second case the event-particle will besaid to 'lie inside' the event e, and in the third case theevent-particle will be said to be a 'boundary-particle' of the evente. Thus there are three sets of particles, namely the set of thosewhich lie outside the event e, the set of those which lie inside theevent e, and the boundary of the event e which is the set ofboundary-particles of e. Since an event is four-dimensional, theboundary of an event is a three-dimensional manifold. For a finite eventthere is a continuity of boundary; for a duration the boundary consistsof those event-particles which are covered by either of the two boundingmoments. Thus the boundary of a duration consists of two momentarythree-dimensional spaces. An event will be said to 'occupy' theaggregate of event-particles which lie within it. 

Two events which have 'junction' in the sense in which junction wasdescribed in my last lecture, and yet are separated so that neitherevent either overlaps or is part of the other event, are said to be'adjoined. '

This relation of adjunction issues in a peculiar relation between theboundaries of the two events. The two boundaries must have a commonportion which is in fact a continuous three-dimensional locus ofevent-particles in the four-dimensional manifold. 

A three-dimensional locus of event-particles which is the common portionof the boundary of two adjoined events will be called a 'solid. ' A solidmay or may not lie completely in one moment. A solid which does not liein one moment will be called 'vagrant. ' A solid which does lie in onemoment will be called a volume. A volume may be defined as the locus ofthe event-particles in which a moment intersects an event, provided thatthe two do intersect. The intersection of a moment and an event willevidently consist of those event-particles which are covered by themoment and lie in the event. The identity of the two definitions of avolume is evident when we remember that an intersecting moment dividesthe event into two adjoined events. 

A solid as thus defined, whether it be vagrant or be a volume, is a mereaggregate of event-particles illustrating a certain quality of position. We can also define a solid as an abstractive element. In order to do sowe recur to the theory of primes explained in the preceding lecture. Letthe condition named σ stand for the fact that each of the events of anyabstractive set satisfying it has all the event-particles of someparticular solid lying in it. Then the group of all the σ-primes is theabstractive element which is associated with the given solid. I willcall this abstractive element the solid as an abstractive element, and Iwill call the aggregate of event-particles the solid as a locus. Theinstantaneous volumes in instantaneous space which are the ideals of oursense-perception are volumes as abstractive elements. What we reallyperceive with all our efforts after exactness are small events farenough down some abstractive set belonging to the volume as anabstractive element. 

It is difficult to know how far we approximate to any perception ofvagrant solids. We certainly do not think that we make any suchapproximation. But then our thoughts--in the case of people who do thinkabout such topics--are so much under the control of the materialistictheory of nature that they hardly count for evidence. If Einstein'stheory of gravitation has any truth in it, vagrant solids are of greatimportance in science. The whole boundary of a finite event may belooked on as a particular example of a vagrant solid as a locus. Itsparticular property of being closed prevents it from being definable asan abstractive element. 

When a moment intersects an event, it also intersects the boundary ofthat event. This locus, which is the portion of the boundary containedin the moment, is the bounding surface of the corresponding volume ofthat event contained in the moment. It is a two-dimensional locus. 

The fact that every volume has a bounding surface is the origin of theDedekindian continuity of space. 

Another event may be cut by the same moment in another volume and thisvolume will also have its boundary. These two volumes in theinstantaneous space of one moment may mutually overlap in the familiarway which I need not describe in detail and thus cut off portions fromeach other's surfaces. These portions of surfaces are 'momental areas. '

It is unnecessary at this stage to enter into the complexity of adefinition of vagrant areas. Their definition is simple enough when thefour-dimensional manifold of event-particles has been more fullyexplored as to its properties. 

Momental areas can evidently be defined as abstractive elements byexactly the same method as applied to solids. We have merely tosubstitute 'area' for a 'solid' in the words of the definition alreadygiven. Also, exactly as in the analogous case of a solid, what weperceive as an approximation to our ideal of an area is a small eventfar enough down towards the small end of one of the equal abstractivesets which belongs to the area as an abstractive element. 

Two momental areas lying in the same moment can cut each other in amomental segment which is not necessarily rectilinear. Such a segmentcan also be defined as an abstractive element. It is then called a'momental route. ' We will not delay over any general consideration ofthese momental routes, nor is it important for us to proceed to thestill wider investigation of vagrant routes in general. There arehowever two simple sets of routes which are of vital importance. One isa set of momental routes and the other of vagrant routes. Both sets canbe classed together as straight routes. We proceed to define themwithout any reference to the definitions of volumes and surfaces. 

The two types of straight routes will be called rectilinear routes andstations. Rectilinear routes are momental routes and stations arevagrant routes. Rectilinear routes are routes which in a sense lie inrects. Any two event-particles on a rect define the set ofevent-particles which lie between them on that rect. Let thesatisfaction of the condition σ by an abstractive set mean that the twogiven event-particles and the event-particles lying between them on therect all lie in every event belonging to the abstractive set. The groupof σ-primes, where σ has this meaning, form an abstractive element. Suchabstractive elements are rectilinear routes. They are the segments ofinstantaneous straight lines which are the ideals of exact perception. Our actual perception, however exact, will be the perception of a smallevent sufficiently far down one of the abstractive sets of theabstractive element. 

A station is a vagrant route and no moment can intersect any station inmore than one event-particle. Thus a station carries with it acomparison of the positions in their respective moments of theevent-particles covered by it. Rects arise from the intersection ofmoments. But as yet no properties of events have been mentioned by whichany analogous vagrant loci can be found out. 

The general problem for our investigation is to determine a method ofcomparison of position in one instantaneous space with positions inother instantaneous spaces. We may limit ourselves to the spaces of theparallel moments of one time-system. How are positions in these variousspaces to be compared? In other words, What do we mean by motion? It isthe fundamental question to be asked of any theory of relative space, and like many other fundamental questions it is apt to be leftunanswered. It is not an answer to reply, that we all know what we meanby motion. Of course we do, so far as sense-awareness is concerned. I amasking that your theory of space should provide nature with something tobe observed. You have not settled the question by bringing forward atheory according to which there is nothing to be observed, and by thenreiterating that nevertheless we do observe this non-existent fact. Unless motion is something as a fact in nature, kinetic energy andmomentum and all that depends on these physical concepts evaporate fromour list of physical realities. Even in this revolutionary age myconservatism resolutely opposes the identification of momentum andmoonshine. 

Accordingly I assume it as an axiom, that motion is a physical fact. Itis something that we perceive as in nature. Motion presupposes rest. Until theory arose to vitiate immediate intuition, that is to say tovitiate the uncriticised judgments which immediately arise fromsense-awareness, no one doubted that in motion you leave behind thatwhich is at rest. Abraham in his wanderings left his birthplace where ithad ever been. A theory of motion and a theory of rest are the samething viewed from different aspects with altered emphasis. 

Now you cannot have a theory of rest without in some sense admitting atheory of absolute position. It is usually assumed that relative spaceimplies that there is no absolute position. This is, according to mycreed, a mistake. The assumption arises from the failure to make anotherdistinction; namely, that there may be alternative definitions ofabsolute position. This possibility enters with the admission ofalternative time-systems. Thus the series of spaces in the parallelmoments of one temporal series may have their own definition of absoluteposition correlating sets of event-particles in these successive spaces, so that each set consists of event-particles, one from each space, allwith the property of possessing the same absolute position in thatseries of spaces. Such a set of event-particles will form a point in thetimeless space of that time-system. Thus a point is really an absoluteposition in the timeless space of a given time-system. 

But there are alternative time-systems, and each time-system has its ownpeculiar group of points--that is to say, its own peculiar definition ofabsolute position. This is exactly the theory which I will elaborate. 

In looking to nature for evidence of absolute position it is of no useto recur to the four-dimensional manifold of event-particles. Thismanifold has been obtained by the extension of thought beyond theimmediacy of observation. We shall find nothing in it except what wehave put there to represent the ideas in thought which arise from ourdirect sense-awareness of nature. To find evidence of the propertieswhich are to be found in the manifold of event-particles we must alwaysrecur to the observation of relations between events. Our problem is todetermine those relations between events which issue in the property ofabsolute position in a timeless space. This is in fact the problem ofthe determination of the very meaning of the timeless spaces ofphysical science. 

In reviewing the factors of nature as immediately disclosed insense-awareness, we should note the fundamental character of the perceptof 'being here. ' We discern an event merely as a factor in a determinatecomplex in which each factor has its own peculiar share. 

There are two factors which are always ingredient in this complex, oneis the duration which is represented in thought by the concept of allnature that is present now, and the other is the peculiar _locus standi_for mind involved in the sense-awareness. This _locus standi_ in natureis what is represented in thought by the concept of 'here, ' namely of an'event here. '

This is the concept of a definite factor in nature. This factor is anevent in nature which is the focus in nature for that act of awareness, and the other events are perceived as referred to it. This event is partof the associated duration. I call it the 'percipient event. ' This eventis not the mind, that is to say, not the percipient. It is that innature from which the mind perceives. The complete foothold of the mindin nature is represented by the pair of events, namely, the presentduration which marks the 'when' of awareness and the percipient eventwhich marks the 'where' of awareness and the 'how' of awareness. Thispercipient event is roughly speaking the bodily life of the incarnatemind. But this identification is only a rough one. For the functions ofthe body shade off into those of other events in nature; so that forsome purposes the percipient event is to be reckoned as merely part ofthe bodily life and for other purposes it may even be reckoned as morethan the bodily life. In many respects the demarcation is purelyarbitrary, depending upon where in a sliding scale you choose to drawthe line. 

I have already in my previous lecture on Time discussed the associationof mind with nature. The difficulty of the discussion lies in theliability of constant factors to be overlooked. We never note them bycontrast with their absences. The purpose of a discussion of suchfactors may be described as being to make obvious things look odd. Wecannot envisage them unless we manage to invest them with some of thefreshness which is due to strangeness. 

It is because of this habit of letting constant factors slip fromconsciousness that we constantly fall into the error of thinking of thesense-awareness of a particular factor in nature as being a two-termedrelation between the mind and the factor. For example, I perceive agreen leaf. Language in this statement suppresses all reference to anyfactors other than the percipient mind and the green leaf and therelation of sense-awareness. It discards the obvious inevitable factorswhich are essential elements in the perception. I am here, the leaf isthere; and the event here and the event which is the life of the leafthere are both embedded in a totality of nature which is now, and withinthis totality there are other discriminated factors which it isirrelevant to mention. Thus language habitually sets before the mind amisleading abstract of the indefinite complexity of the fact ofsense-awareness. 

What I now want to discuss is the special relation of the percipientevent which is 'here' to the duration which is 'now. ' This relation is afact in nature, namely the mind is aware of nature as being with thesetwo factors in this relation. 

Within the short present duration the 'here' of the percipient event hasa definite meaning of some sort. This meaning of 'here' is the contentof the special relation of the percipient event to its associatedduration. I will call this relation 'cogredience. ' Accordingly I ask fora description of the character of the relation of cogredience. Thepresent snaps into a past and a present when the 'here' of cogredienceloses its single determinate meaning. There has been a passage of naturefrom the 'here' of perception within the past duration to the different'here' of perception within the present duration. But the two 'heres' ofsense-awareness within neighbouring durations may be indistinguishable. In this case there has been a passage from the past to the present, buta more retentive perceptive force might have retained the passing natureas one complete present instead of letting the earlier duration slipinto the past. Namely, the sense of rest helps the integration ofdurations into a prolonged present, and the sense of motiondifferentiates nature into a succession of shortened durations. As welook out of a railway carriage in an express train, the present is pastbefore reflexion can seize it. We live in snippits too quick forthought. On the other hand the immediate present is prolonged accordingas nature presents itself to us in an aspect of unbroken rest. Anychange in nature provides ground for a differentiation among durationsso as to shorten the present. But there is a great distinction betweenself-change in nature and change in external nature. Self-change innature is change in the quality of the standpoint of the percipientevent. It is the break up of the 'here' which necessitates the break upof the present duration. Change in external nature is compatible with aprolongation of the present of contemplation rooted in a givenstandpoint. What I want to bring out is that the preservation of apeculiar relation to a duration is a necessary condition for thefunction of that duration as a present duration for sense-awareness. This peculiar relation is the relation of cogredience between thepercipient event and the duration. Cogredience is the preservation ofunbroken quality of standpoint within the duration. It is thecontinuance of identity of station within the whole of nature which isthe terminus of sense-awareness. The duration may comprise change withinitself, but cannot--so far as it is one present duration--comprisechange in the quality of its peculiar relation to the containedpercipient event. 

In other words, perception is always 'here, ' and a duration can only beposited as present for sense-awareness on condition that it affords oneunbroken meaning of 'here' in its relation to the percipient event. Itis only in the past that you can have been 'there' with a standpointdistinct from your present 'here. '

Events there and events here are facts of nature, and the qualities ofbeing 'there' and 'here' are not merely qualities of awareness as arelation between nature and mind. The quality of determinate station inthe duration which belongs to an event which is 'here' in onedeterminate sense of 'here' is the same kind of quality of station whichbelongs to an event which is 'there' in one determinate sense of'there. ' Thus cogredience has nothing to do with any biologicalcharacter of the event which is related by it to the associatedduration. This biological character is apparently a further conditionfor the peculiar connexion of a percipient event with the percipience ofmind; but it has nothing to do with the relation of the percipient eventto the duration which is the present whole of nature posited as thedisclosure of the percipience. 

Given the requisite biological character, the event in its character ofa percipient event selects that duration with which the operative pastof the event is practically cogredient within the limits of theexactitude of observation. Namely, amid the alternative time-systemswhich nature offers there will be one with a duration giving the bestaverage of cogredience for all the subordinate parts of the percipientevent. This duration will be the whole of nature which is the terminusposited by sense-awareness. Thus the character of the percipient eventdetermines the time-system immediately evident in nature. As thecharacter of the percipient event changes with the passage ofnature--or, in other words, as the percipient mind in its passagecorrelates itself with the passage of the percipient event into anotherpercipient event--the time-system correlated with the percipience ofthat mind may change. When the bulk of the events perceived arecogredient in a duration other than that of the percipient event, thepercipience may include a double consciousness of cogredience, namelythe consciousness of the whole within which the observer in the train is'here, ' and the consciousness of the whole within which the trees andbridges and telegraph posts are definitely 'there. ' Thus in perceptionsunder certain circumstances the events discriminated assert their ownrelations of cogredience. This assertion of cogredience is peculiarlyevident when the duration to which the perceived event is cogredient isthe same as the duration which is the present whole of nature--in otherwords, when the event and the percipient event are both cogredient tothe same duration. 

We are now prepared to consider the meaning of stations in a duration, where stations are a peculiar kind of routes, which define absoluteposition in the associated timeless space. 

There are however some preliminary explanations. A finite event will besaid to extend throughout a duration when it is part of the durationand is intersected by any moment which lies in the duration. Such anevent begins with the duration and ends with it. Furthermore every eventwhich begins with the duration and ends with it, extends throughout theduration. This is an axiom based on the continuity of events. Bybeginning with a duration and ending with it, I mean (i) that the eventis part of the duration, and (ii) that both the initial and finalboundary moments of the duration cover some event-particles on theboundary of the event. 

Every event which is cogredient with a duration extends throughout thatduration. 

It is not true that all the parts of an event cogredient with a durationare also cogredient with the duration. The relation of cogredience mayfail in either of two ways. One reason for failure may be that the partdoes not extend throughout the duration. In this case the part may becogredient with another duration which is part of the given duration, though it is not cogredient with the given duration itself. Such a partwould be cogredient if its existence were sufficiently prolonged in thattime-system. The other reason for failure arises from thefour-dimensional extension of events so that there is no determinateroute of transition of events in linear series. For example, the tunnelof a tube railway is an event at rest in a certain time-system, that isto say, it is cogredient with a certain duration. A train travelling init is part of that tunnel, but is not itself at rest. 

If an event e be cogredient with a duration d, and d′ be anyduration which is part of d. Then d′ belongs to the same time-systemas d. Also d′ intersects e in an event e′ which is part of eand is cogredient with d′. 

Let P be any event-particle lying in a given duration d. Considerthe aggregate of events in which P lies and which are also cogredientwith d. Each of these events occupies its own aggregate ofevent-particles. These aggregates will have a common portion, namely theclass of event-particle lying in all of them. This class ofevent-particles is what I call the 'station' of the event-particle Pin the duration d. This is the station in the character of a locus. Astation can also be defined in the character of an abstractive element. Let the property σ be the name of the property which an abstractive setpossesses when (i) each of its events is cogredient with the durationd and (ii) the event-particle P lies in each of its events. Then thegroup of σ-primes, where σ has this meaning, is an abstractive elementand is the station of P in d as an abstractive element. The locus ofevent-particles covered by the station of P in d as an abstractiveelement is the station of P in d as a locus. A station hasaccordingly the usual three characters, namely, its character ofposition, its extrinsic character as an abstractive element, and itsintrinsic character. 

It follows from the peculiar properties of rest that two stationsbelonging to the same duration cannot intersect. Accordingly everyevent-particle on a station of a duration has that station as itsstation in the duration. Also every duration which is part of a givenduration intersects the stations of the given duration in loci which areits own stations. By means of these properties we can utilise theoverlappings of the durations of one family--that is, of onetime-system--to prolong stations indefinitely backwards and forwards. Such a prolonged station will be called a point-track. A point-track isa locus of event-particles. It is defined by reference to oneparticular time-system, α say. Corresponding to any other time-systemthese will be a different group of point-tracks. Every event-particlewill lie on one and only one point-track of the group belonging to anyone time-system. The group of point-tracks of the time-system α is thegroup of points of the timeless space of α. Each such point indicates acertain quality of absolute position in reference to the durations ofthe family associated with α, and thence in reference to the successiveinstantaneous spaces lying in the successive moments of α. Each momentof α will intersect a point-track in one and only one event-particle. 

This property of the unique intersection of a moment and a point-trackis not confined to the case when the moment and the point-track belongto the same time-system. Any two event-particles on a point-track aresequential, so that they cannot lie in the same moment. Accordingly nomoment can intersect a point-track more than once, and every momentintersects a point-track in one event-particle. 

Anyone who at the successive moments of α should be at theevent-particles where those moments intersect a given point of α will beat rest in the timeless space of time-system α. But in any othertimeless space belonging to another time-system he will be at adifferent point at each succeeding moment of that time-system. In otherwords he will be moving. He will be moving in a straight line withuniform velocity. We might take this as the definition of a straightline. Namely, a straight line in the space of time-system β is the locusof those points of β which all intersect some one point-track which is apoint in the space of some other time-system. Thus each point in thespace of a time-system α is associated with one and only one straightline of the space of any other time-system β. Furthermore the set ofstraight lines in space β which are thus associated with points in spaceα form a complete family of parallel straight lines in space β. Thusthere is a one-to-one correlation of points in space α with the straightlines of a certain definite family of parallel straight lines in spaceβ. Conversely there is an analogous one-to-one correlation of the pointsin space β with the straight lines of a certain family of parallelstraight lines in space α. These families will be called respectivelythe family of parallels in β associated with α, and the family ofparallels in α associated with β. The direction in the space of βindicated by the family of parallels in β will be called the directionof α in space β, and the family of parallels in α is the direction of βin space α. Thus a being at rest at a point of space α will be movinguniformly along a line in space β which is in the direction of α inspace β, and a being at rest at a point of space β will be movinguniformly along a line in space α which is in the direction of β inspace α. 

I have been speaking of the timeless spaces which are associated withtime-systems. These are the spaces of physical science and of anyconcept of space as eternal and unchanging. But what we actuallyperceive is an approximation to the instantaneous space indicated byevent-particles which lie within some moment of the time-systemassociated with our awareness. The points of such an instantaneous spaceare event-particles and the straight lines are rects. Let thetime-system be named α, and let the moment of time-system α to which ourquick perception of nature approximates be called M. Any straight liner in space α is a locus of points and each point is a point-track whichis a locus of event-particles. Thus in the four-dimensional geometry ofall event-particles there is a two-dimensional locus which is the locusof all event-particles on points lying on the straight line r. I willcall this locus of event-particles the matrix of the straight line r. Amatrix intersects any moment in a rect. Thus the matrix of r intersectsthe moment M in a rect ρ. Thus ρ is the instantaneous rect in M whichoccupies at the moment M the straight line r in the space of α. Accordingly when one sees instantaneously a moving being and its pathahead of it, what one really sees is the being at some event-particle Alying in the rect ρ which is the apparent path on the assumption ofuniform motion. But the actual rect ρ which is a locus ofevent-particles is never traversed by the being. These event-particlesare the instantaneous facts which pass with the instantaneous moment. What is really traversed are other event-particles which at succeedinginstants occupy the same points of space α as those occupied by theevent-particles of the rect ρ. For example, we see a stretch of road anda lorry moving along it. The instantaneously seen road is a portion ofthe rect ρ--of course only an approximation to it. The lorry is themoving object. But the road as seen is never traversed. It is thought ofas being traversed because the intrinsic characters of the later eventsare in general so similar to those of the instantaneous road that we donot trouble to discriminate. But suppose a land mine under the road hasbeen exploded before the lorry gets there. Then it is fairly obviousthat the lorry does not traverse what we saw at first. Suppose the lorryis at rest in space β. Then the straight line r of space α is in thedirection of β in space α, and the rect ρ is the representative in themoment M of the line r of space α. The direction of ρ in theinstantaneous space of the moment M is the direction of β in M, where Mis a moment of time-system α. Again the matrix of the line r of space αwill also be the matrix of some line s of space β which will be in thedirection of α in space β. Thus if the lorry halts at some point P ofspace α which lies on the line r, it is now moving along the line s ofspace β. This is the theory of relative motion; the common matrix is thebond which connects the motion of β in space α with the motions of α inspace β. 

Motion is essentially a relation between some object of nature and theone timeless space of a time-system. An instantaneous space is static, being related to the static nature at an instant. In perception when wesee things moving in an approximation to an instantaneous space, thefuture lines of motion as immediately perceived are rects which arenever traversed. These approximate rects are composed of small events, namely approximate routes and event-particles, which are passed awaybefore the moving objects reach them. Assuming that our forecasts ofrectilinear motion are correct, these rects occupy the straight lines intimeless space which are traversed. Thus the rects are symbols inimmediate sense-awareness of a future which can only be expressed interms of timeless space. 

We are now in a position to explore the fundamental character ofperpendicularity. Consider the two time-systems α and β, each with itsown timeless space and its own family of instantaneous moments withtheir instantaneous spaces. Let M and N be respectively a moment ofα and a moment of β. In M there is the direction of β and in N thereis the direction of α. But M and N, being moments of differenttime-systems, intersect in a level. Call this level λ. Then λ is aninstantaneous plane in the instantaneous space of M and also in theinstantaneous space of N. It is the locus of all the event-particleswhich lie both in M and in N. 

In the instantaneous space of M the level λ is perpendicular to thedirection of β in M, and in the instantaneous space of N the level λis perpendicular to the direction of α in N. This is the fundamentalproperty which forms the definition of perpendicularity. The symmetry ofperpendicularity is a particular instance of the symmetry of the mutualrelations between two time-systems. We shall find in the next lecturethat it is from this symmetry that the theory of congruence is deduced. 

The theory of perpendicularity in the timeless space of any time-systemα follows immediately from this theory of perpendicularity in each ofits instantaneous spaces. Let ρ be any rect in the moment M of α andlet λ be a level in M which is perpendicular to ρ. The locus of thosepoints of the space of α which intersect M in event-particles on ρ isthe straight line r of space α, and the locus of those points of thespace of α which intersect M in event-particles on λ is the plane lof space α. Then the plane l is perpendicular to the line r. 

In this way we have pointed out unique and definite properties in naturewhich correspond to perpendicularity. We shall find that this discoveryof definite unique properties defining perpendicularity is of criticalimportance in the theory of congruence which is the topic for the nextlecture. 

I regret that it has been necessary for me in this lecture to administersuch a large dose of four-dimensional geometry. I do not apologise, because I am really not responsible for the fact that nature in its mostfundamental aspect is four-dimensional. Things are what they are; and itis useless to disguise the fact that 'what things are' is often verydifficult for our intellects to follow. It is a mere evasion of theultimate problems to shirk such obstacles. 

CHAPTER VI

CONGRUENCE

The aim of this lecture is to establish a theory of congruence. You mustunderstand at once that congruence is a controversial question. It isthe theory of measurement in space and in time. The question seemssimple. In fact it is simple enough for a standard procedure to havebeen settled by act of parliament; and devotion to metaphysicalsubtleties is almost the only crime which has never been imputed to anyEnglish parliament. But the procedure is one thing and its meaning isanother. 

First let us fix attention on the purely mathematical question. When thesegment between two points A and B is congruent to that between thetwo points C and D, the quantitative measurements of the twosegments are equal. The equality of the numerical measures and thecongruence of the two segments are not always clearly discriminated, andare lumped together under the term equality. But the procedure ofmeasurement presupposes congruence. For example, a yard measure isapplied successively to measure two distances between two pairs ofpoints on the floor of a room. It is of the essence of the procedure ofmeasurement that the yard measure remains unaltered as it is transferredfrom one position to another. Some objects can palpably alter as theymove--for example, an elastic thread; but a yard measure does not alterif made of the proper material. What is this but a judgment ofcongruence applied to the train of successive positions of the yardmeasure? We know that it does not alter because we judge it to becongruent to itself in various positions. In the case of the thread wecan observe the loss of self-congruence. Thus immediate judgments ofcongruence are presupposed in measurement, and the process ofmeasurement is merely a procedure to extend the recognition ofcongruence to cases where these immediate judgments are not available. Thus we cannot define congruence by measurement. 

In modern expositions of the axioms of geometry certain conditions arelaid down which the relation of congruence between segments is tosatisfy. It is supposed that we have a complete theory of points, straight lines, planes, and the order of points on planes--in fact, acomplete theory of non-metrical geometry. We then enquire aboutcongruence and lay down the set of conditions--or axioms as they arecalled--which this relation satisfies. It has then been proved thatthere are alternative relations which satisfy these conditions equallywell and that there is nothing intrinsic in the theory of space to leadus to adopt any one of these relations in preference to any other as therelation of congruence which we adopt. In other words there arealternative metrical geometries which all exist by an equal right so faras the intrinsic theory of space is concerned. 

Poincaré, the great French mathematician, held that our actual choiceamong these geometries is guided purely by convention, and that theeffect of a change of choice would be simply to alter our expression ofthe physical laws of nature. By 'convention' I understand Poincaré tomean that there is nothing inherent in nature itself giving any peculiar_rôle_ to one of these congruence relations, and that the choice of oneparticular relation is guided by the volitions of the mind at the otherend of the sense-awareness. The principle of guidance is intellectualconvenience and not natural fact. 

This position has been misunderstood by many of Poincaré's expositors. They have muddled it up with another question, namely that owing to theinexactitude of observation it is impossible to make an exact statementin the comparison of measures. It follows that a certain subset ofclosely allied congruence relations can be assigned of which each memberequally well agrees with that statement of observed congruence when thestatement is properly qualified with its limits of error. 

This is an entirely different question and it presupposes a rejection ofPoincaré's position. The absolute indetermination of nature in respectof all the relations of congruence is replaced by the indetermination ofobservation with respect to a small subgroup of these relations. 

Poincaré's position is a strong one. He in effect challenges anyone topoint out any factor in nature which gives a preeminent status to thecongruence relation which mankind has actually adopted. But undeniablythe position is very paradoxical. Bertrand Russell had a controversywith him on this question, and pointed out that on Poincaré's principlesthere was nothing in nature to determine whether the earth is larger orsmaller than some assigned billiard ball. Poincaré replied that theattempt to find reasons in nature for the selection of a definitecongruence relation in space is like trying to determine the position ofa ship in the ocean by counting the crew and observing the colour ofthe captain's eyes. 

In my opinion both disputants were right, assuming the grounds on whichthe discussion was based. Russell in effect pointed out that apart fromminor inexactitudes a determinate congruence relation is among thefactors in nature which our sense-awareness posits for us. Poincaré asksfor information as to the factor in nature which might lead anyparticular congruence relation to play a preeminent _rôle_ among thefactors posited in sense-awareness. I cannot see the answer to either ofthese contentions provided that you admit the materialistic theory ofnature. With this theory nature at an instant in space is an independentfact. Thus we have to look for our preeminent congruence relation amidnature in instantaneous space; and Poincaré is undoubtedly right insaying that nature on this hypothesis gives us no help in finding it. 

On the other hand Russell is in an equally strong position when heasserts that, as a fact of observation, we do find it, and what is moreagree in finding the same congruence relation. On this basis it is oneof the most extraordinary facts of human experience that all mankindwithout any assignable reason should agree in fixing attention on justone congruence relation amid the indefinite number of indistinguishablecompetitors for notice. One would have expected disagreement on thisfundamental choice to have divided nations and to have rent families. But the difficulty was not even discovered till the close of thenineteenth century by a few mathematical philosophers and philosophicmathematicians. The case is not like that of our agreement on somefundamental fact of nature such as the three dimensions of space. Ifspace has only three dimensions we should expect all mankind to be awareof the fact, as they are aware of it. But in the case of congruence, mankind agree in an arbitrary interpretation of sense-awareness whenthere is nothing in nature to guide it. 

I look on it as no slight recommendation of the theory of nature which Iam expounding to you that it gives a solution of this difficulty bypointing out the factor in nature which issues in the preeminence of onecongruence relation over the indefinite herd of other such relations. 

The reason for this result is that nature is no longer confined withinspace at an instant. Space and time are now interconnected; and thispeculiar factor of time which is so immediately distinguished among thedeliverances of our sense-awareness, relates itself to one particularcongruence relation in space. 

Congruence is a particular example of the fundamental fact ofrecognition. In perception we recognise. This recognition does notmerely concern the comparison of a factor of nature posited by memorywith a factor posited by immediate sense-awareness. Recognition takesplace within the present without any intervention of pure memory. Forthe present fact is a duration with its antecedent and consequentdurations which are parts of itself. The discrimination insense-awareness of a finite event with its quality of passage is alsoaccompanied by the discrimination of other factors of nature which donot share in the passage of events. Whatever passes is an event. But wefind entities in nature which do not pass; namely we recognisesamenesses in nature. Recognition is not primarily an intellectual actof comparison; it is in its essence merely sense-awareness in itscapacity of positing before us factors in nature which do not pass. Forexample, green is perceived as situated in a certain finite event withinthe present duration. This green preserves its self-identity throughout, whereas the event passes and thereby obtains the property of breakinginto parts. The green patch has parts. But in talking of the green patchwe are speaking of the event in its sole capacity of being for us thesituation of green. The green itself is numerically one self-identicalentity, without parts because it is without passage. 

Factors in nature which are without passage will be called objects. There are radically different kinds of objects which will be consideredin the succeeding lecture. 

Recognition is reflected into the intellect as comparison. Therecognised objects of one event are compared with the recognised objectsof another event. The comparison may be between two events in thepresent, or it may be between two events of which one is posited bymemory-awareness and the other by immediate sense-awareness. But it isnot the events which are compared. For each event is essentially uniqueand incomparable. What are compared are the objects and relations ofobjects situated in events. The event considered as a relation betweenobjects has lost its passage and in this aspect is itself an object. This object is not the event but only an intellectual abstraction. Thesame object can be situated in many events; and in this sense even thewhole event, viewed as an object, can recur, though not the very eventitself with its passage and its relations to other events. 

Objects which are not posited by sense-awareness may be known to theintellect. For example, relations between objects and relations betweenrelations may be factors in nature not disclosed in sense-awareness butknown by logical inference as necessarily in being. Thus objects for ourknowledge may be merely logical abstractions. For example, a completeevent is never disclosed in sense-awareness, and thus the object whichis the sum total of objects situated in an event as thus inter-relatedis a mere abstract concept. Again a right-angle is a perceived objectwhich can be situated in many events; but, though rectangularity isposited by sense-awareness, the majority of geometrical relations arenot so posited. Also rectangularity is in fact often not perceived whenit can be proved to have been there for perception. Thus an object isoften known merely as an abstract relation not directly posited insense-awareness although it is there in nature. 

The identity of quality between congruent segments is generally of thischaracter. In certain special cases this identity of quality can bedirectly perceived. But in general it is inferred by a process ofmeasurement depending on our direct sense-awareness of selected casesand a logical inference from the transitive character of congruence. 

Congruence depends on motion, and thereby is generated the connexionbetween spatial congruence and temporal congruence. Motion along astraight line has a symmetry round that line. This symmetry is expressedby the symmetrical geometrical relations of the line to the family ofplanes normal to it. 

Also another symmetry in the theory of motion arises from the fact thatrest in the points of β corresponds to uniform motion along a definitefamily of parallel straight lines in the space of α. We must note thethree characteristics, (i) of the uniformity of the motioncorresponding to any point of β along its correlated straight line in α, and (ii) of the equality in magnitude of the velocities along thevarious lines of α correlated to rest in the various points of β, and(iii) of the parallelism of the lines of this family. 

We are now in possession of a theory of parallels and a theory ofperpendiculars and a theory of motion, and from these theories thetheory of congruence can be constructed. It will be remembered that afamily of parallel levels in any moment is the family of levels in whichthat moment is intersected by the family of moments of some othertime-system. Also a family of parallel moments is the family of momentsof some one time-system. Thus we can enlarge our concept of a family ofparallel levels so as to include levels in different moments of onetime-system. With this enlarged concept we say that a complete family ofparallel levels in a time-system α is the complete family of levels inwhich the moments of α intersect the moments of β. This complete familyof parallel levels is also evidently a family lying in the moments ofthe time-system β. By introducing a third time-system γ, parallel rectsare obtained. Also all the points of any one time-system form a familyof parallel point-tracks. Thus there are three types of parallelogramsin the four-dimensional manifold of event-particles. 

In parallelograms of the first type the two pairs of parallel sides areboth of them pairs of rects. In parallelograms of the second type onepair of parallel sides is a pair of rects and the other pair is a pairof point-tracks. In parallelograms of the third type the two pairs ofparallel sides are both of them pairs of point-tracks. 

The first axiom of congruence is that the opposite sides of anyparallelogram are congruent. This axiom enables us to compare thelengths of any two segments either respectively on parallel rects or onthe same rect. Also it enables us to compare the lengths of any twosegments either respectively on parallel point-tracks or on the samepoint-track. It follows from this axiom that two objects at rest in anytwo points of a time-system β are moving with equal velocities in anyother time-system α along parallel lines. Thus we can speak of thevelocity in α due to the time-system β without specifying any particularpoint in β. The axiom also enables us to measure time in anytime-system; but does not enable us to compare times in differenttime-systems. 

The second axiom of congruence concerns parallelograms on congruentbases and between the same parallels, which have also their other pairsof sides parallel. The axiom asserts that the rect joining the twoevent-particles of intersection of the diagonals is parallel to the recton which the bases lie. By the aid of this axiom it easily follows thatthe diagonals of a parallelogram bisect each other. 

Congruence is extended in any space beyond parallel rects to all rectsby two axioms depending on perpendicularity. The first of these axioms, which is the third axiom of congruence, is that if ABC is a triangleof rects in any moment and D is the middle event-particle of the baseBC, then the level through D perpendicular to BC contains A whenand only when AB is congruent to AC. This axiom evidently expressesthe symmetry of perpendicularity, and is the essence of the famous ponsasinorum expressed as an axiom. 

The second axiom depending on perpendicularity, and the fourth axiom ofcongruence, is that if r and A be a rect and an event-particle in thesame moment and AB and AC be a pair of rectangular rects intersecting rin B and C, and AD and AE be another pair of rectangular rectsintersecting r in D and E, then either D or E lies in the segment BC andthe other one of the two does not lie in this segment. Also as aparticular case of this axiom, if AB be perpendicular to r and inconsequence AC be parallel to r, then D and E lie on opposite sides of Brespectively. By the aid of these two axioms the theory of congruencecan be extended so as to compare lengths of segments on any two rects. Accordingly Euclidean metrical geometry in space is completelyestablished and lengths in the spaces of different time-systems arecomparable as the result of definite properties of nature which indicatejust that particular method of comparison. 

The comparison of time-measurements in diverse time-systems requires twoother axioms. The first of these axioms, forming the fifth axiom ofcongruence, will be called the axiom of 'kinetic symmetry. ' It expressesthe symmetry of the quantitative relations between two time-systems whenthe times and lengths in the two systems are measured in congruentunits. 

The axiom can be explained as follows: Let α and β be the names of twotime-systems. The directions of motion in the space of α due to rest ina point of β is called the 'β-direction in α' and the direction ofmotion in the space of β due to rest in a point of α is called the'α-direction in β. ' Consider a motion in the space of α consisting of acertain velocity in the β-direction of α and a certain velocity atright-angles to it. This motion represents rest in the space of anothertime-system--call it π. Rest in π will also be represented in the spaceof β by a certain velocity in the α-direction in β and a certainvelocity at right-angles to this α-direction. Thus a certain motion inthe space of α is correlated to a certain motion in the space of β, asboth representing the same fact which can also be represented by rest inπ. Now another time-system, which I will name σ, can be found which issuch that rest in its space is represented by the same magnitudes ofvelocities along and perpendicular to the α-direction in β as thosevelocities in α, along and perpendicular to the β-direction, whichrepresent rest in π. The required axiom of kinetic symmetry is that restin σ will be represented in α by the same velocities along andperpendicular to the β-direction in α as those velocities in β along andperpendicular to the α-direction which represent rest in π. 

A particular case of this axiom is that relative velocities are equaland opposite. Namely rest in α is represented in β by a velocity alongthe α-direction which is equal to the velocity along the β-direction inα which represents rest in β. 

Finally the sixth axiom of congruence is that the relation of congruenceis transitive. So far as this axiom applies to space, it is superfluous. For the property follows from our previous axioms. It is howevernecessary for time as a supplement to the axiom of kinetic symmetry. Themeaning of the axiom is that if the time-unit of system α is congruentto the time-unit of system β, and the time-unit of system β is congruentto the time-unit of system γ, then the time-units of α and γ are alsocongruent. 

By means of these axioms formulae for the transformation ofmeasurements made in one time-system to measurements of the same factsof nature made in another time-system can be deduced. These formulaewill be found to involve one arbitrary constant which I will call k. 

It is of the dimensions of the square of a velocity. Accordingly fourcases arise. In the first case k is zero. This case producesnonsensical results in opposition to the elementary deliverances ofexperience. We put this case aside. 

In the second case k is infinite. This case yields the ordinaryformulae for transformation in relative motion, namely those formulaewhich are to be found in every elementary book on dynamics. 

In the third case, k is negative. Let us call it -c², where cwill be of the dimensions of a velocity. This case yields the formulaeof transformation which Larmor discovered for the transformation ofMaxwell's equations of the electromagnetic field. These formulae wereextended by H.  A. Lorentz, and used by Einstein and Minkowski as thebasis of their novel theory of relativity. I am not now speaking ofEinstein's more recent theory of general relativity by which he deduceshis modification of the law of gravitation. If this be the case whichapplies to nature, then c must be a close approximation to thevelocity of light _in vacuo_. Perhaps it is this actual velocity. Inthis connexion '_in vacuo_' must not mean an absence of events, namelythe absence of the all-pervading ether of events. It must mean theabsence of certain types of objects. 

In the fourth case, k is positive. Let us call it h², where hwill be of the dimensions of a velocity. This gives a perfectly possibletype of transformation formulae, but not one which explains any factsof experience. It has also another disadvantage. With the assumption ofthis fourth case the distinction between space and time becomes undulyblurred. The whole object of these lectures has been to enforce thedoctrine that space and time spring from a common root, and that theultimate fact of experience is a space-time fact. But after all mankinddoes distinguish very sharply between space and time, and it is owing tothis sharpness of distinction that the doctrine of these lectures issomewhat of a paradox. Now in the third assumption this sharpness ofdistinction is adequately preserved. There is a fundamental distinctionbetween the metrical properties of point-tracks and rects. But in thefourth assumption this fundamental distinction vanishes. 

Neither the third nor the fourth assumption can agree with experienceunless we assume that the velocity c of the third assumption, and thevelocity h of the fourth assumption, are extremely large compared tothe velocities of ordinary experience. If this be the case the formulaeof both assumptions will obviously reduce to a close approximation tothe formulae of the second assumption which are the ordinary formulae ofdynamical textbooks. For the sake of a name, I will call these textbookformulae the 'orthodox' formulae. 

There can be no question as to the general approximate correctness ofthe orthodox formulae. It would be merely silly to raise doubts on thispoint. But the determination of the status of these formulae is by nomeans settled by this admission. The independence of time and space isan unquestioned presupposition of the orthodox thought which hasproduced the orthodox formulae. With this presupposition and given theabsolute points of one absolute space, the orthodox formulae areimmediate deductions. Accordingly, these formulae are presented to ourimaginations as facts which cannot be otherwise, time and space beingwhat they are. The orthodox formulae have therefore attained to thestatus of necessities which cannot be questioned in science. Any attemptto replace these formulae by others was to abandon the _rôle_ ofphysical explanation and to have recourse to mere mathematical formulae. 

But even in physical science difficulties have accumulated round theorthodox formulae. In the first place Maxwell's equations of theelectromagnetic field are not invariant for the transformations of theorthodox formulae; whereas they are invariant for the transformations ofthe formulae arising from the third of the four cases mentioned above, provided that the velocity c is identified with a famouselectromagnetic constant quantity. 

Again the null results of the delicate experiments to detect the earth'svariations of motion through the ether in its orbital path are explainedimmediately by the formulae of the third case. But if we assume theorthodox formulae we have to make a special and arbitrary assumption asto the contraction of matter during motion. I mean the Fitzgerald-Lorentzassumption. 

Lastly Fresnel's coefficient of drag which represents the variation ofthe velocity of light in a moving medium is explained by the formulae ofthe third case, and requires another arbitrary assumption if we use theorthodox formulae. 

It appears therefore that on the mere basis of physical explanationthere are advantages in the formulae of the third case as compared withthe orthodox formulae. But the way is blocked by the ingrained beliefthat these latter formulae possess a character of necessity. It istherefore an urgent requisite for physical science and for philosophy toexamine critically the grounds for this supposed necessity. The onlysatisfactory method of scrutiny is to recur to the first principles ofour knowledge of nature. This is exactly what I am endeavouring to do inthese lectures. I ask what it is that we are aware of in oursense-perception of nature. I then proceed to examine those factors innature which lead us to conceive nature as occupying space andpersisting through time. This procedure has led us to an investigationof the characters of space and time. It results from theseinvestigations that the formulae of the third case and the orthodoxformulae are on a level as possible formulae resulting from the basiccharacter of our knowledge of nature. The orthodox formulae have thuslost any advantage as to necessity which they enjoyed over the serialgroup. The way is thus open to adopt whichever of the two groups bestaccords with observation. 

I take this opportunity of pausing for a moment from the course of myargument, and of reflecting on the general character which my doctrineascribes to some familiar concepts of science. I have no doubt that someof you have felt that in certain aspects this character is veryparadoxical. 

This vein of paradox is partly due to the fact that educated languagehas been made to conform to the prevalent orthodox theory. We are thus, in expounding an alternative doctrine, driven to the use of eitherstrange terms or of familiar words with unusual meanings. This victoryof the orthodox theory over language is very natural. Events are namedafter the prominent objects situated in them, and thus both in languageand in thought the event sinks behind the object, and becomes the mereplay of its relations. The theory of space is then converted into atheory of the relations of objects instead of a theory of the relationsof events. But objects have not the passage of events. Accordingly spaceas a relation between objects is devoid of any connexion with time. Itis space at an instant without any determinate relations between thespaces at successive instants. It cannot be one timeless space becausethe relations between objects change. 

A few minutes ago in speaking of the deduction of the orthodox formulaefor relative motion I said that they followed as an immediate deductionfrom the assumption of absolute points in absolute space. This referenceto absolute space was not an oversight. I know that the doctrine of therelativity of space at present holds the field both in science andphilosophy. But I do not think that its inevitable consequences areunderstood. When we really face them the paradox of the presentation ofthe character of space which I have elaborated is greatly mitigated. Ifthere is no absolute position, a point must cease to be a simple entity. What is a point to one man in a balloon with his eyes fixed on aninstrument is a track of points to an observer on the earth who iswatching the balloon through a telescope, and is another track of pointsto an observer in the sun who is watching the balloon through someinstrument suited to such a being. Accordingly if I am reproached withthe paradox of my theory of points as classes of event-particles, and ofmy theory of event-particles as groups of abstractive sets, I ask mycritic to explain exactly what he means by a point. While you explainyour meaning about anything, however simple, it is always apt to looksubtle and fine spun. I have at least explained exactly what I do meanby a point, what relations it involves and what entities are the relata. If you admit the relativity of space, you also must admit that pointsare complex entities, logical constructs involving other entities andtheir relations. Produce your theory, not in a few vague phrases ofindefinite meaning, but explain it step by step in definite termsreferring to assigned relations and assigned relata. Also show that yourtheory of points issues in a theory of space. Furthermore note that theexample of the man in the balloon, the observer on earth, and theobserver in the sun, shows that every assumption of relative restrequires a timeless space with radically different points from thosewhich issue from every other such assumption. The theory of therelativity of space is inconsistent with any doctrine of one unique setof points of one timeless space. 

The fact is that there is no paradox in my doctrine of the nature ofspace which is not in essence inherent in the theory of the relativityof space. But this doctrine has never really been accepted in science, whatever people say. What appears in our dynamical treatises is Newton'sdoctrine of relative motion based on the doctrine of differential motionin absolute space. When you once admit that the points are radicallydifferent entities for differing assumptions of rest, then the orthodoxformulae lose all their obviousness. They were only obvious because youwere really thinking of something else. When discussing this topic youcan only avoid paradox by taking refuge from the flood of criticism inthe comfortable ark of no meaning. 

The new theory provides a definition of the congruence of periods oftime. The prevalent view provides no such definition. Its position isthat if we take such time-measurements so that certain familiarvelocities which seem to us to be uniform are uniform, then the laws ofmotion are true. Now in the first place no change could appear either asuniform or non-uniform without involving a definite determination of thecongruence for time-periods. So in appealing to familiar phenomena itallows that there is some factor in nature which we can intellectuallyconstruct as a congruence theory. It does not however say anything aboutit except that the laws of motion are then true. Suppose that with someexpositors we cut out the reference to familiar velocities such as therate of rotation of the earth. We are then driven to admit that there isno meaning in temporal congruence except that certain assumptions makethe laws of motion true. Such a statement is historically false. KingAlfred the Great was ignorant of the laws of motion, but knew very wellwhat he meant by the measurement of time, and achieved his purpose bymeans of burning candles. Also no one in past ages justified the use ofsand in hour-glasses by saying that some centuries later interestinglaws of motion would be discovered which would give a meaning to thestatement that the sand was emptied from the bulbs in equal times. Uniformity in change is directly perceived, and it follows that mankindperceives in nature factors from which a theory of temporal congruencecan be formed. The prevalent theory entirely fails to produce suchfactors. 

The mention of the laws of motion raises another point where theprevalent theory has nothing to say and the new theory gives a completeexplanation. It is well known that the laws of motion are not valid forany axes of reference which you may choose to take fixed in any rigidbody. You must choose a body which is not rotating and has noacceleration. For example they do not really apply to axes fixed in theearth because of the diurnal rotation of that body. The law which failswhen you assume the wrong axes as at rest is the third law, that actionand reaction are equal and opposite. With the wrong axes uncompensatedcentrifugal forces and uncompensated composite centrifugal forcesappear, due to rotation. The influence of these forces can bedemonstrated by many facts on the earth's surface, Foucault's pendulum, the shape of the earth, the fixed directions of the rotations ofcyclones and anticyclones. It is difficult to take seriously thesuggestion that these domestic phenomena on the earth are due to theinfluence of the fixed stars. I cannot persuade myself to believe that alittle star in its twinkling turned round Foucault's pendulum in theParis Exhibition of 1861. Of course anything is believable when adefinite physical connexion has been demonstrated, for example theinfluence of sunspots. Here all demonstration is lacking in the form ofany coherent theory. According to the theory of these lectures the axesto which motion is to be referred are axes at rest in the space of sometime-system. For example, consider the space of a time-system α. Thereare sets of axes at rest in the space of α. These are suitable dynamicalaxes. Also a set of axes in this space which is moving with uniformvelocity without rotation is another suitable set. All the movingpoints fixed in these moving axes are really tracing out parallel lineswith one uniform velocity. In other words they are the reflections inthe space of α of a set of fixed axes in the space of some othertime-system β. Accordingly the group of dynamical axes required forNewton's Laws of Motion is the outcome of the necessity of referringmotion to a body at rest in the space of some one time-system in orderto obtain a coherent account of physical properties. If we do not do sothe meaning of the motion of one portion of our physical configurationis different from the meaning of the motion of another portion of thesame configuration. Thus the meaning of motion being what it is, inorder to describe the motion of any system of objects without changingthe meaning of your terms as you proceed with your description, you arebound to take one of these sets of axes as axes of reference; though youmay choose their reflections into the space of any time-system which youwish to adopt. A definite physical reason is thereby assigned for thepeculiar property of the dynamical group of axes. 

On the orthodox theory the position of the equations of motion is mostambiguous. The space to which they refer is completely undetermined andso is the measurement of the lapse of time. Science is simply settingout on a fishing expedition to see whether it cannot find some procedurewhich it can call the measurement of space and some procedure which itcan call the measurement of time, and something which it can call asystem of forces, and something which it can call masses, so that theseformulae may be satisfied. The only reason--on this theory--why anyoneshould want to satisfy these formulae is a sentimental regard forGalileo, Newton, Euler and Lagrange. The theory, so far from foundingscience on a sound observational basis, forces everything to conform toa mere mathematical preference for certain simple formulae. 

I do not for a moment believe that this is a true account of the realstatus of the Laws of Motion. These equations want some slightadjustment for the new formulae of relativity. But with theseadjustments, imperceptible in ordinary use, the laws deal withfundamental physical quantities which we know very well and wish tocorrelate. 

The measurement of time was known to all civilised nations long beforethe laws were thought of. It is this time as thus measured that the lawsare concerned with. Also they deal with the space of our daily life. When we approach to an accuracy of measurement beyond that ofobservation, adjustment is allowable. But within the limits ofobservation we know what we mean when we speak of measurements of spaceand measurements of time and uniformity of change. It is for science togive an intellectual account of what is so evident in sense-awareness. It is to me thoroughly incredible that the ultimate fact beyond whichthere is no deeper explanation is that mankind has really been swayed byan unconscious desire to satisfy the mathematical formulae which we callthe Laws of Motion, formulae completely unknown till the seventeenthcentury of our epoch. 

The correlation of the facts of sense-experience effected by thealternative account of nature extends beyond the physical properties ofmotion and the properties of congruence. It gives an account of themeaning of the geometrical entities such as points, straight lines, andvolumes, and connects the kindred ideas of extension in time andextension in space. The theory satisfies the true purpose of anintellectual explanation in the sphere of natural philosophy. Thispurpose is to exhibit the interconnexions of nature, and to show thatone set of ingredients in nature requires for the exhibition of itscharacter the presence of the other sets of ingredients. 

The false idea which we have to get rid of is that of nature as a mereaggregate of independent entities, each capable of isolation. Accordingto this conception these entities, whose characters are capable ofisolated definition, come together and by their accidental relationsform the system of nature. This system is thus thoroughly accidental;and, even if it be subject to a mechanical fate, it is only accidentallyso subject. 

With this theory space might be without time, and time might be withoutspace. The theory admittedly breaks down when we come to the relationsof matter and space. The relational theory of space is an admission thatwe cannot know space without matter or matter without space. But theseclusion of both from time is still jealously guarded. The relationsbetween portions of matter in space are accidental facts owing to theabsence of any coherent account of how space springs from matter or howmatter springs from space. Also what we really observe in nature, itscolours and its sounds and its touches are secondary qualities; in otherwords, they are not in nature at all but are accidental products of therelations between nature and mind. 

The explanation of nature which I urge as an alternative ideal to thisaccidental view of nature, is that nothing in nature could be what it isexcept as an ingredient in nature as it is. The whole which is presentfor discrimination is posited in sense-awareness as necessary for thediscriminated parts. An isolated event is not an event, because everyevent is a factor in a larger whole and is significant of that whole. There can be no time apart from space; and no space apart from time; andno space and no time apart from the passage of the events of nature. Theisolation of an entity in thought, when we think of it as a bare 'it, 'has no counterpart in any corresponding isolation in nature. Suchisolation is merely part of the procedure of intellectual knowledge. 

The laws of nature are the outcome of the characters of the entitieswhich we find in nature. The entities being what they are, the laws mustbe what they are; and conversely the entities follow from the laws. Weare a long way from the attainment of such an ideal; but it remains asthe abiding goal of theoretical science. 

CHAPTER VII

OBJECTS

The ensuing lecture is concerned with the theory of objects. Objects areelements in nature which do not pass. The awareness of an object as somefactor not sharing in the passage of nature is what I call'recognition. ' It is impossible to recognise an event, because an eventis essentially distinct from every other event. Recognition is anawareness of sameness. But to call recognition an awareness of samenessimplies an intellectual act of comparison accompanied with judgment. Iuse recognition for the non-intellectual relation of sense-awarenesswhich connects the mind with a factor of nature without passage. On theintellectual side of the mind's experience there are comparisons ofthings recognised and consequent judgments of sameness or diversity. Probably 'sense-recognition' would be a better term for what I mean by'recognition. ' I have chosen the simpler term because I think that Ishall be able to avoid the use of 'recognition' in any other meaningthan that of 'sense-recognition. ' I am quite willing to believe thatrecognition, in my sense of the term, is merely an ideal limit, and thatthere is in fact no recognition without intellectual accompaniments ofcomparison and judgment. But recognition is that relation of the mind tonature which provides the material for the intellectual activity. 

An object is an ingredient in the character of some event. In fact thecharacter of an event is nothing but the objects which are ingredient init and the ways in which those objects make their ingression into theevent. Thus the theory of objects is the theory of the comparison ofevents. Events are only comparable because they body forth permanences. We are comparing objects in events whenever we can say, 'There it isagain. ' Objects are the elements in nature which can 'be again. '

Sometimes permanences can be proved to exist which evade recognition inthe sense in which I am using that term. The permanences which evaderecognition appear to us as abstract properties either of events or ofobjects. All the same they are there for recognition althoughundiscriminated in our sense-awareness. The demarcation of events, thesplitting of nature up into parts is effected by the objects which werecognise as their ingredients. The discrimination of nature is therecognition of objects amid passing events. It is a compound of theawareness of the passage of nature, of the consequent partition ofnature, and of the definition of certain parts of nature by the modes ofthe ingression of objects into them. 

You may have noticed that I am using the term 'ingression' to denote thegeneral relation of objects to events. The ingression of an object intoan event is the way the character of the event shapes itself in virtueof the being of the object. Namely the event is what it is, because theobject is what it is; and when I am thinking of this modification of theevent by the object, I call the relation between the two 'the ingressionof the object into the event. ' It is equally true to say that objectsare what they are because events are what they are. Nature is such thatthere can be no events and no objects without the ingression of objectsinto events. Although there are events such that the ingredient objectsevade our recognition. These are the events in empty space. Such eventsare only analysed for us by the intellectual probing of science. 

Ingression is a relation which has various modes. There are obviouslyvery various kinds of objects; and no one kind of object can have thesame sort of relations to events as objects of another kind can have. Weshall have to analyse out some of the different modes of ingressionwhich different kinds of objects have into events. 

But even if we stick to one and the same kind of objects, an object ofthat kind has different modes of ingression into different events. Science and philosophy have been apt to entangle themselves in asimple-minded theory that an object is at one place at any definitetime, and is in no sense anywhere else. This is in fact the attitude ofcommon sense thought, though it is not the attitude of language which isnaïvely expressing the facts of experience. Every other sentence in awork of literature which is endeavouring truly to interpret the facts ofexperience expresses differences in surrounding events due to thepresence of some object. An object is ingredient throughout itsneighbourhood, and its neighbourhood is indefinite. Also themodification of events by ingression is susceptible of quantitativedifferences. Finally therefore we are driven to admit that each objectis in some sense ingredient throughout nature; though its ingression maybe quantitatively irrelevant in the expression of our individualexperiences. 

This admission is not new either in philosophy or science. It isobviously a necessary axiom for those philosophers who insist thatreality is a system. In these lectures we are keeping off the profoundand vexed question as to what we mean by 'reality. ' I am maintaining thehumbler thesis that nature is a system. But I suppose that in this casethe less follows from the greater, and that I may claim the support ofthese philosophers. The same doctrine is essentially interwoven in allmodern physical speculation. As long ago as 1847 Faraday in a paper inthe _Philosophical Magazine_ remarked that his theory of tubes of forceimplies that in a sense an electric charge is everywhere. Themodification of the electromagnetic field at every point of space ateach instant owing to the past history of each electron is another wayof stating the same fact. We can however illustrate the doctrine by themore familiar facts of life without recourse to the abstrusespeculations of theoretical physics. 

The waves as they roll on to the Cornish coast tell of a gale inmid-Atlantic; and our dinner witnesses to the ingression of the cookinto the dining room. It is evident that the ingression of objects intoevents includes the theory of causation. I prefer to neglect this aspectof ingression, because causation raises the memory of discussions basedupon theories of nature which are alien to my own. Also I think thatsome new light may be thrown on the subject by viewing it in this freshaspect. 

The examples which I have given of the ingression of objects into eventsremind us that ingression takes a peculiar form in the case of someevents; in a sense, it is a more concentrated form. For example, theelectron has a certain position in space and a certain shape. Perhaps itis an extremely small sphere in a certain test-tube. The storm is agale situated in mid-Atlantic with a certain latitude and longitude, andthe cook is in the kitchen. I will call this special form of ingressionthe 'relation of situation'; also, by a double use of the word'situation, ' I will call the event in which an object is situated 'thesituation of the object. ' Thus a situation is an event which is arelatum in the relation of situation. Now our first impression is thatat last we have come to the simple plain fact of where the object reallyis; and that the vaguer relation which I call ingression should not bemuddled up with the relation of situation, as if including it as aparticular case. It seems so obvious that any object is in such and sucha position, and that it is influencing other events in a totallydifferent sense. Namely, in a sense an object is the character of theevent which is its situation, but it only influences the character ofother events. Accordingly the relations of situation and influencing arenot generally the same sort of relation, and should not be subsumedunder the same term 'ingression. ' I believe that this notion is amistake, and that it is impossible to draw a clear distinction betweenthe two relations. 

For example, Where was your toothache? You went to a dentist and pointedout the tooth to him. He pronounced it perfectly sound, and cured you bystopping another tooth. Which tooth was the situation of the toothache?Again, a man has an arm amputated, and experiences sensations in thehand which he has lost. The situation of the imaginary hand is in factmerely thin air. You look into a mirror and see a fire. The flames thatyou see are situated behind the mirror. Again at night you watch thesky; if some of the stars had vanished from existence hours ago, youwould not be any the wiser. Even the situations of the planets differfrom those which science would assign to them. 

Anyhow you are tempted to exclaim, the cook is in the kitchen. If youmean her mind, I will not agree with you on the point; for I am onlytalking of nature. Let us think only of her bodily presence. What do youmean by this notion? We confine ourselves to typical manifestations ofit. You can see her, touch her, and hear her. But the examples which Ihave given you show that the notions of the situations of what you see, what you touch, and what you hear are not so sharply separated out as todefy further questioning. You cannot cling to the idea that we have twosets of experiences of nature, one of primary qualities which belong tothe objects perceived, and one of secondary qualities which are theproducts of our mental excitements. All we know of nature is in the sameboat, to sink or swim together. The constructions of science are merelyexpositions of the characters of things perceived. Accordingly to affirmthat the cook is a certain dance of molecules and electrons is merely toaffirm that the things about her which are perceivable have certaincharacters. The situations of the perceived manifestations of her bodilypresence have only a very general relation to the situations of themolecules, to be determined by discussion of the circumstances ofperception. 

In discussing the relations of situation in particular and of ingressionin general, the first requisite is to note that objects are of radicallydifferent types. For each type 'situation' and 'ingression' have theirown special meanings which are different from their meanings for othertypes, though connexions can be pointed out. It is necessary thereforein discussing them to determine what type of objects are underconsideration. There are, I think, an indefinite number of types ofobjects. Happily we need not think of them all. The idea of situationhas its peculiar importance in reference to three types of objects whichI call sense-objects, perceptual objects and scientific objects. Thesuitability of these names for the three types is of minor importance, so long as I can succeed in explaining what I mean by them. 

These three types form an ascending hierarchy, of which each memberpresupposes the type below. The base of the hierarchy is formed by thesense-objects. These objects do not presuppose any other type ofobjects. A sense-object is a factor of nature posited by sense-awarenesswhich (i), in that it is an object, does not share in the passage ofnature and (ii) is not a relation between other factors of nature. Itwill of course be a relatum in relations which also implicate otherfactors of nature. But it is always a relatum and never the relationitself. Examples of sense-objects are a particular sort of colour, sayCambridge blue, or a particular sort of sound, or a particular sort ofsmell, or a particular sort of feeling. I am not talking of a particularpatch of blue as seen during a particular second of time at a definitedate. Such a patch is an event where Cambridge blue is situated. Similarly I am not talking of any particular concert-room as filled withthe note. I mean the note itself and not the patch of volume filled bythe sound for a tenth of a second. It is natural for us to think of thenote in itself, but in the case of colour we are apt to think of itmerely as a property of the patch. No one thinks of the note as aproperty of the concert-room. We see the blue and we hear the note. Boththe blue and the note are immediately posited by the discrimination ofsense-awareness which relates the mind to nature. The blue is posited asin nature related to other factors in nature. In particular it isposited as in the relation of being situated in the event which is itssituation. 

The difficulties which cluster around the relation of situation arisefrom the obstinate refusal of philosophers to take seriously theultimate fact of multiple relations. By a multiple relation I mean arelation which in any concrete instance of its occurrence necessarilyinvolves more than two relata. For example, when John likes Thomas thereare only two relata, John and Thomas. But when John gives that book toThomas there are three relata, John, that book, and Thomas. 

Some schools of philosophy, under the influence of the Aristotelianlogic and the Aristotelian philosophy, endeavour to get on withoutadmitting any relations at all except that of substance and attribute. Namely all apparent relations are to be resolvable into the concurrentexistence of substances with contrasted attributes. It is fairly obviousthat the Leibnizian monadology is the necessary outcome of any suchphilosophy. If you dislike pluralism, there will be only one monad. 

Other schools of philosophy admit relations but obstinately refuse tocontemplate relations with more than two relata. I do not think thatthis limitation is based on any set purpose or theory. It merely arisesfrom the fact that more complicated relations are a bother to peoplewithout adequate mathematical training, when they are admitted into thereasoning. 

I must repeat that we have nothing to do in these lectures with theultimate character of reality. It is quite possible that in the truephilosophy of reality there are only individual substances withattributes, or that there are only relations with pairs of relata. I donot believe that such is the case; but I am not concerned to argue aboutit now. Our theme is Nature. So long as we confine ourselves to thefactors posited in the sense-awareness of nature, it seems to me thatthere certainly are instances of multiple relations between thesefactors, and that the relation of situation for sense-objects is oneexample of such multiple relations. 

Consider a blue coat, a flannel coat of Cambridge blue belonging to someathlete. The coat itself is a perceptual object and its situation is notwhat I am talking about. We are talking of someone's definitesense-awareness of Cambridge blue as situated in some event of nature. He may be looking at the coat directly. He then sees Cambridge blue assituated practically in the same event as the coat at that instant. Itis true that the blue which he sees is due to light which left the coatsome inconceivably small fraction of a second before. This differencewould be important if he were looking at a star whose colour wasCambridge blue. The star might have ceased to exist days ago, or evenyears ago. The situation of the blue will not then be very intimatelyconnected with the situation (in another sense of 'situation') of anyperceptual object. This disconnexion of the situation of the blue andthe situation of some associated perceptual object does not require astar for its exemplification. Any looking glass will suffice. Look atthe coat through a looking glass. Then blue is seen as situated behindthe mirror. The event which is its situation depends upon the positionof the observer. 

The sense-awareness of the blue as situated in a certain event which Icall the situation, is thus exhibited as the sense-awareness of arelation between the blue, the percipient event of the observer, thesituation, and intervening events. All nature is in fact required, though only certain intervening events require their characters to be ofcertain definite sorts. The ingression of blue into the events of natureis thus exhibited as systematically correlated. The awareness of theobserver depends on the position of the percipient event in thissystematic correlation. I will use the term 'ingression into nature' forthis systematic correlation of the blue with nature. Thus the ingressionof blue into any definite event is a part statement of the fact of theingression of blue into nature. 

In respect to the ingression of blue into nature events may be roughlyput into four classes which overlap and are not very clearly separated. These classes are (i) the percipient events, (ii) the situations, (iii)the active conditioning events, (iv) the passive conditioning events. Tounderstand this classification of events in the general fact of theingression of blue into nature, let us confine attention to onesituation for one percipient event and to the consequent _rôles_ of theconditioning events for the ingression as thus limited. The percipientevent is the relevant bodily state of the observer. The situation iswhere he sees the blue, say, behind the mirror. The active conditioningevents are the events whose characters are particularly relevant for theevent (which is the situation) to be the situation for that percipientevent, namely the coat, the mirror, and the state of the room as tolight and atmosphere. The passive conditioning events are the events ofthe rest of nature. 

In general the situation is an active conditioning event; namely thecoat itself, when there is no mirror or other such contrivance toproduce abnormal effects. But the example of the mirror shows us thatthe situation may be one of the passive conditioning events. We are thenapt to say that our senses have been cheated, because we demand as aright that the situation should be an active condition in theingression. 

This demand is not so baseless as it may seem when presented as I haveput it. All we know of the characters of the events of nature is basedon the analysis of the relations of situations to percipient events. Ifsituations were not in general active conditions, this analysis wouldtell us nothing. Nature would be an unfathomable enigma to us and therecould be no science. Accordingly the incipient discontent when asituation is found to be a passive condition is in a sense justifiable;because if that sort of thing went on too often, the _rôle_ of theintellect would be ended. 

Furthermore the mirror is itself the situation of other sense-objectseither for the same observer with the same percipient event, or forother observers with other percipient events. Thus the fact that anevent is a situation in the ingression of one set of sense-objects intonature is presumptive evidence that that event is an active condition inthe ingression of other sense-objects into nature which may have othersituations. 

This is a fundamental principle of science which it has derived fromcommon sense. 

I now turn to perceptual objects. When we look at the coat, we do not ingeneral say, There is a patch of Cambridge blue; what naturally occursto us is, There is a coat. Also the judgment that what we have seen isa garment of man's attire is a detail. What we perceive is an objectother than a mere sense-object. It is not a mere patch of colour, butsomething more; and it is that something more which we judge to be acoat. I will use the word 'coat' as the name for that crude object whichis more than a patch of colour, and without any allusion to thejudgments as to its usefulness as an article of attire either in thepast or the future. The coat which is perceived--in this sense of theword 'coat'--is what I call a perceptual object. We have to investigatethe general character of these perceptual objects. 

It is a law of nature that in general the situation of a sense-object isnot only the situation of that sense-object for one definite percipientevent, but is the situation of a variety of sense-objects for a varietyof percipient events. For example, for any one percipient event, thesituation of a sense-object of sight is apt also to be the situations ofsense-objects of sight, of touch, of smell, and of sound. Furthermorethis concurrence in the situations of sense-objects has led to thebody--_i. E. _ the percipient event--so adapting itself that theperception of one sense-object in a certain situation leads to asubconscious sense-awareness of other sense-objects in the samesituation. This interplay is especially the case between touch andsight. There is a certain correlation between the ingressions ofsense-objects of touch and sense-objects of sight into nature, and in aslighter degree between the ingressions of other pairs of sense-objects. I call this sort of correlation the 'conveyance' of one sense-object byanother. When you see the blue flannel coat you subconsciously feelyourself wearing it or otherwise touching it. If you are a smoker, youmay also subconsciously be aware of the faint aroma of tobacco. Thepeculiar fact, posited by this sense-awareness of the concurrence ofsubconscious sense-objects along with one or more dominatingsense-objects in the same situation, is the sense-awareness of theperceptual object. The perceptual object is not primarily the issue of ajudgment. It is a factor of nature directly posited in sense-awareness. The element of judgment comes in when we proceed to classify theparticular perceptual object. For example, we say, That is flannel, andwe think of the properties of flannel and the uses of athletes' coats. But that all takes place after we have got hold of the perceptualobject. Anticipatory judgments affect the perceptual object perceived byfocussing and diverting attention. 

The perceptual object is the outcome of the habit of experience. Anything which conflicts with this habit hinders the sense-awareness ofsuch an object. A sense-object is not the product of the association ofintellectual ideas; it is the product of the association ofsense-objects in the same situation. This outcome is not intellectual;it is an object of peculiar type with its own particular ingression intonature. 

There are two kinds of perceptual objects, namely, 'delusive perceptualobjects' and 'physical objects. ' The situation of a delusive perceptualobject is a passive condition in the ingression of that object intonature. Also the event which is the situation will have the relation ofsituation to the object only for one particular percipient event. Forexample, an observer sees the image of the blue coat in a mirror. It isa blue coat that he sees and not a mere patch of colour. This shows thatthe active conditions for the conveyance of a group of subconscioussense-objects by a dominating sense-object are to be found in thepercipient event. Namely we are to look for them in the investigationsof medical psychologists. The ingression into nature of the delusivesense-object is conditioned by the adaptation of bodily events to themore normal occurrence, which is the ingression of the physical object. 

A perceptual object is a physical object when (i) its situation is anactive conditioning event for the ingression of any of its componentsense-objects, and (ii) the same event can be the situation of theperceptual object for an indefinite number of possible percipientevents. Physical objects are the ordinary objects which we perceive whenour senses are not cheated, such as chairs, tables and trees. In a wayphysical objects have more insistent perceptive power thansense-objects. Attention to the fact of their occurrence in nature isthe first condition for the survival of complex living organisms. Theresult of this high perceptive power of physical objects is thescholastic philosophy of nature which looks on the sense-objects as mereattributes of the physical objects. This scholastic point of view isdirectly contradicted by the wealth of sense-objects which enter intoour experience as situated in events without any connexion with physicalobjects. For example, stray smells, sounds, colours and more subtlenameless sense-objects. There is no perception of physical objectswithout perception of sense-objects. But the converse does not hold:namely, there is abundant perception of sense-objects unaccompanied byany perception of physical objects. This lack of reciprocity in therelations between sense-objects and physical objects is fatal to thescholastic natural philosophy. 

There is a great difference in the _rôles_ of the situations ofsense-objects and physical objects. The situations of a physical objectare conditioned by uniqueness and continuity. The uniqueness is an ideallimit to which we approximate as we proceed in thought along anabstractive set of durations, considering smaller and smaller durationsin the approach to the ideal limit of the moment of time. In otherwords, when the duration is small enough, the situation of the physicalobject within that duration is practically unique. 

The identification of the same physical object as being situated indistinct events in distinct durations is effected by the condition ofcontinuity. This condition of continuity is the condition that acontinuity of passage of events, each event being a situation of theobject in its corresponding duration, can be found from the earlier tothe later of the two given events. So far as the two events arepractically adjacent in one specious present, this continuity of passagemay be directly perceived. Otherwise it is a matter of judgment andinference. 

The situations of a sense-object are not conditioned by any suchconditions either of uniqueness or of continuity. In any durationshowever small a sense-object may have any number of situations separatedfrom each other. Thus two situations of a sense-object, either in thesame duration or in different durations, are not necessarily connectedby any continuous passage of events which are also situations of thatsense-object. 

The characters of the conditioning events involved in the ingression ofa sense-object into nature can be largely expressed in terms of thephysical objects which are situated in those events. In one respect thisis also a tautology. For the physical object is nothing else than thehabitual concurrence of a certain set of sense-objects in one situation. Accordingly when we know all about the physical object, we thereby knowits component sense-objects. But a physical object is a condition forthe occurrence of sense-objects other than those which are itscomponents. For example, the atmosphere causes the events which are itssituations to be active conditioning events in the transmission ofsound. A mirror which is itself a physical object is an active conditionfor the situation of a patch of colour behind it, due to the reflectionof light in it. 

Thus the origin of scientific knowledge is the endeavour to express interms of physical objects the various _rôles_ of events as activeconditions in the ingression of sense-objects into nature. It is in theprogress of this investigation that scientific objects emerge. Theyembody those aspects of the character of the situations of the physicalobjects which are most permanent and are expressible without referenceto a multiple relation including a percipient event. Their relations toeach other are also characterised by a certain simplicity anduniformity. Finally the characters of the observed physical objects andsense-objects can be expressed in terms of these scientific objects. Infact the whole point of the search for scientific objects is theendeavour to obtain this simple expression of the characters of events. These scientific objects are not themselves merely formulae forcalculation; because formulae must refer to things in nature, and thescientific objects are the things in nature to which the formulae refer. 

A scientific object such as a definite electron is a systematiccorrelation of the characters of all events throughout all nature. It isan aspect of the systematic character of nature. The electron is notmerely where its charge is. The charge is the quantitative character ofcertain events due to the ingression of the electron into nature. Theelectron is its whole field of force. Namely the electron is thesystematic way in which all events are modified as the expression of itsingression. The situation of an electron in any small duration may bedefined as that event which has the quantitative character which is thecharge of the electron. We may if we please term the mere charge theelectron. But then another name is required for the scientific objectwhich is the full entity which concerns science, and which I have calledthe electron. 

According to this conception of scientific objects, the rival theoriesof action at a distance and action by transmission through a medium areboth incomplete expressions of the true process of nature. The stream ofevents which form the continuous series of situations of the electron isentirely self-determined, both as regards having the intrinsic characterof being the series of situations of that electron and as regards thetime-systems with which its various members are cogredient, and the fluxof their positions in their corresponding durations. This is thefoundation of the denial of action at a distance; namely the progress ofthe stream of the situations of a scientific object can be determined byan analysis of the stream itself. 

On the other hand the ingression of every electron into nature modifiesto some extent the character of every event. Thus the character of thestream of events which we are considering bears marks of the existenceof every other electron throughout the universe. If we like to think ofthe electrons as being merely what I call their charges, then thecharges act at a distance. But this action consists in the modificationof the situation of the other electron under consideration. Thisconception of a charge acting at a distance is a wholly artificial one. The conception which most fully expresses the character of nature isthat of each event as modified by the ingression of each electron intonature. The ether is the expression of this systematic modification ofevents throughout space and throughout time. The best expression of thecharacter of this modification is for physicists to find out. My theoryhas nothing to do with that and is ready to accept any outcome ofphysical research. 

The connexion of objects with space requires elucidation. Objects aresituated in events. The relation of situation is a different relationfor each type of object, and in the case of sense-objects it cannot beexpressed as a two-termed relation. It would perhaps be better to use adifferent word for these different types of the relation of situation. It has not however been necessary to do so for our purposes in theselectures. It must be understood however that, when situation is spokenof, some one definite type is under discussion, and it may happen thatthe argument may not apply to situation of another type. In all caseshowever I use situation to express a relation between objects and eventsand not between objects and abstractive elements. There is a derivativerelation between objects and spatial elements which I call the relationof location; and when this relation holds, I say that the object islocated in the abstractive element. In this sense, an object may belocated in a moment of time, in a volume of space, an area, a line, or apoint. There will be a peculiar type of location corresponding to eachtype of situation; and location is in each case derivative from thecorresponding relation of situation in a way which I will proceed toexplain. 

Also location in the timeless space of some time-system is a relationderivative from location in instantaneous spaces of the sametime-system. Accordingly location in an instantaneous space is theprimary idea which we have to explain. Great confusion has beenoccasioned in natural philosophy by the neglect to distinguish betweenthe different types of objects, the different types of situation, thedifferent types of location, and the difference between location andsituation. It is impossible to reason accurately in the vague concerningobjects and their positions without keeping these distinctions in view. An object is located in an abstractive element, when an abstractive setbelonging to that element can be found such that each event belonging tothat set is a situation of the object. It will be remembered that anabstractive element is a certain group of abstractive sets, and thateach abstractive set is a set of events. This definition defines thelocation of an element in any type of abstractive element. In this sensewe can talk of the existence of an object at an instant, meaning therebyits location in some definite moment. It may also be located in somespatial element of the instantaneous space of that moment. 

A quantity can be said to be located in an abstractive element when anabstractive set belonging to the element can be found such that thequantitative expressions of the corresponding characters of its eventsconverge to the measure of the given quantity as a limit when we passalong the abstractive set towards its converging end. 

By these definitions location in elements of instantaneous spaces isdefined. These elements occupy corresponding elements of timelessspaces. An object located in an element of an instantaneous space willalso be said to be located at that moment in the timeless element of thetimeless space which is occupied by that instantaneous element. 

It is not every object which can be located in a moment. An object whichcan be located in every moment of some duration will be called a'uniform' object throughout that duration. Ordinary physical objectsappear to us to be uniform objects, and we habitually assume thatscientific objects such as electrons are uniform. But some sense-objectscertainly are not uniform. A tune is an example of a non-uniform object. We have perceived it as a whole in a certain duration; but the tune as atune is not at any moment of that duration though one of the individualnotes may be located there. 

It is possible therefore that for the existence of certain sorts ofobjects, _e. G. _ electrons, minimum quanta of time are requisite. Somesuch postulate is apparently indicated by the modern quantum theory andit is perfectly consistent with the doctrine of objects maintained inthese lectures. 

Also the instance of the distinction between the electron as the merequantitative electric charge of its situation and the electron asstanding for the ingression of an object throughout nature illustratesthe indefinite number of types of objects which exist in nature. We canintellectually distinguish even subtler and subtler types of objects. Here I reckon subtlety as meaning seclusion from the immediateapprehension of sense-awareness. Evolution in the complexity of lifemeans an increase in the types of objects directly sensed. Delicacy ofsense-apprehension means perceptions of objects as distinct entitieswhich are mere subtle ideas to cruder sensibilities. The phrasing ofmusic is a mere abstract subtlety to the unmusical; it is a directsense-apprehension to the initiated. For example, if we could imaginesome lowly type of organic being thinking and aware of our thoughts, itwould wonder at the abstract subtleties in which we indulge as we thinkof stones and bricks and drops of water and plants. It only knows ofvague undifferentiated feelings in nature. It would consider us as givenover to the play of excessively abstract intellects. But then if itcould think, it would anticipate; and if it anticipated, it would soonperceive for itself. 

In these lectures we have been scrutinising the foundations of naturalphilosophy. We are stopping at the very point where a boundless ocean ofenquiries opens out for our questioning. 

I agree that the view of Nature which I have maintained in theselectures is not a simple one. Nature appears as a complex system whosefactors are dimly discerned by us. But, as I ask you, Is not this thevery truth? Should we not distrust the jaunty assurance with which everyage prides itself that it at last has hit upon the ultimate concepts inwhich all that happens can be formulated? The aim of science is to seekthe simplest explanations of complex facts. We are apt to fall into theerror of thinking that the facts are simple because simplicity is thegoal of our quest. The guiding motto in the life of every naturalphilosopher should be, Seek simplicity and distrust it. 

CHAPTER VIII

SUMMARY

There is a general agreement that Einstein's investigations have onefundamental merit irrespective of any criticisms which we may feelinclined to pass on them. They have made us think. But when we haveadmitted so far, we are most of us faced with a distressing perplexity. What is it that we ought to think about? The purport of my lecture thisafternoon will be to meet this difficulty and, so far as I am able, toset in a clear light the changes in the background of our scientificthought which are necessitated by any acceptance, however qualified, ofEinstein's main positions. I remember that I am lecturing to the membersof a chemical society who are not for the most part versed in advancedmathematics. The first point that I would urge upon you is that whatimmediately concerns you is not so much the detailed deductions of thenew theory as this general change in the background of scientificconceptions which will follow from its acceptance. Of course, thedetailed deductions are important, because unless our colleagues theastronomers and the physicists find these predictions to be verified wecan neglect the theory altogether. But we may now take it as grantedthat in many striking particulars these deductions have been found to bein agreement with observation. Accordingly the theory has to be takenseriously and we are anxious to know what will be the consequences ofits final acceptance. Furthermore during the last few weeks thescientific journals and the lay press have been filled with articles asto the nature of the crucial experiments which have been made and as tosome of the more striking expressions of the outcome of the new theory. 'Space caught bending' appeared on the news-sheet of a well-knownevening paper. This rendering is a terse but not inapt translation ofEinstein's own way of interpreting his results. I should say at oncethat I am a heretic as to this explanation and that I shall expound toyou another explanation based upon some work of my own, an explanationwhich seems to me to be more in accordance with our scientific ideas andwith the whole body of facts which have to be explained. We have toremember that a new theory must take account of the old well-attestedfacts of science just as much as of the very latest experimental resultswhich have led to its production. 

To put ourselves in the position to assimilate and to criticise anychange in ultimate scientific conceptions we must begin at thebeginning. So you must bear with me if I commence by making some simpleand obvious reflections. Let us consider three statements, (i)'Yesterday a man was run over on the Chelsea Embankment, ' (ii)'Cleopatra's Needle is on the Charing Cross Embankment, ' and (iii)'There are dark lines in the Solar Spectrum. ' The first statement aboutthe accident to the man is about what we may term an 'occurrence, ' a'happening, ' or an 'event. ' I will use the term 'event' because it isthe shortest. In order to specify an observed event, the place, thetime, and character of the event are necessary. In specifying the placeand the time you are really stating the relation of the assigned eventto the general structure of other observed events. For example, the manwas run over between your tea and your dinner and adjacently to apassing barge in the river and the traffic in the Strand. The pointwhich I want to make is this: Nature is known to us in our experience asa complex of passing events. In this complex we discern definite mutualrelations between component events, which we may call their relativepositions, and these positions we express partly in terms of space andpartly in terms of time. Also in addition to its mere relative positionto other events, each particular event has its own peculiar character. In other words, nature is a structure of events and each event has itsposition in this structure and its own peculiar character or quality. 

Let us now examine the other two statements in the light of this generalprinciple as to the meaning of nature. Take the second statement, 'Cleopatra's Needle is on the Charing Cross Embankment. ' At first sightwe should hardly call this an event. It seems to lack the element oftime or transitoriness. But does it? If an angel had made the remarksome hundreds of millions of years ago, the earth was not in existence, twenty millions of years ago there was no Thames, eighty years ago therewas no Thames Embankment, and when I was a small boy Cleopatra's Needlewas not there. And now that it is there, we none of us expect it to beeternal. The static timeless element in the relation of Cleopatra'sNeedle to the Embankment is a pure illusion generated by the fact thatfor purposes of daily intercourse its emphasis is needless. What itcomes to is this: Amidst the structure of events which form the mediumwithin which the daily life of Londoners is passed we know how toidentify a certain stream of events which maintain permanence ofcharacter, namely the character of being the situations of Cleopatra'sNeedle. Day by day and hour by hour we can find a certain chunk in thetransitory life of nature and of that chunk we say, 'There isCleopatra's Needle. ' If we define the Needle in a sufficiently abstractmanner we can say that it never changes. But a physicist who looks onthat part of the life of nature as a dance of electrons, will tell youthat daily it has lost some molecules and gained others, and even theplain man can see that it gets dirtier and is occasionally washed. Thusthe question of change in the Needle is a mere matter of definition. Themore abstract your definition, the more permanent the Needle. Butwhether your Needle change or be permanent, all you mean by stating thatit is situated on the Charing Cross Embankment, is that amid thestructure of events you know of a certain continuous limited stream ofevents, such that any chunk of that stream, during any hour, or any day, or any second, has the character of being the situation of Cleopatra'sNeedle. 

Finally, we come to the third statement, 'There are dark lines in theSolar Spectrum. ' This is a law of nature. But what does that mean? Itmeans merely this. If any event has the character of being an exhibitionof the solar spectrum under certain assigned circumstances, it will alsohave the character of exhibiting dark lines in that spectrum. 

This long discussion brings us to the final conclusion that the concretefacts of nature are events exhibiting a certain structure in theirmutual relations and certain characters of their own. The aim of scienceis to express the relations between their characters in terms of themutual structural relations between the events thus characterised. Themutual structural relations between events are both spatial andtemporal. If you think of them as merely spatial you are omitting thetemporal element, and if you think of them as merely temporal you areomitting the spatial element. Thus when you think of space alone, or oftime alone, you are dealing in abstractions, namely, you are leaving outan essential element in the life of nature as known to you in theexperience of your senses. Furthermore there are different ways ofmaking these abstractions which we think of as space and as time; andunder some circumstances we adopt one way and under other circumstanceswe adopt another way. Thus there is no paradox in holding that what wemean by space under one set of circumstances is not what we mean byspace under another set of circumstances. And equally what we mean bytime under one set of circumstances is not what we mean by time underanother set of circumstances. By saying that space and time areabstractions, I do not mean that they do not express for us real factsabout nature. What I mean is that there are no spatial facts or temporalfacts apart from physical nature, namely that space and time are merelyways of expressing certain truths about the relations between events. Also that under different circumstances there are different sets oftruths about the universe which are naturally presented to us asstatements about space. In such a case what a being under the one set ofcircumstances means by space will be different from that meant by abeing under the other set of circumstances. Accordingly when we arecomparing two observations made under different circumstances we have toask 'Do the two observers mean the same thing by space and the samething by time?' The modern theory of relativity has arisen becausecertain perplexities as to the concordance of certain delicateobservations such as the motion of the earth through the ether, theperihelion of mercury, and the positions of the stars in theneighbourhood of the sun, have been solved by reference to this purelyrelative significance of space and time. 

I want now to recall your attention to Cleopatra's Needle, which I havenot yet done with. As you are walking along the Embankment you suddenlylook up and say, 'Hullo, there's the Needle. ' In other words, yourecognise it. You cannot recognise an event; because when it is gone, itis gone. You may observe another event of analogous character, but theactual chunk of the life of nature is inseparable from its uniqueoccurrence. But a character of an event can be recognised. We all knowthat if we go to the Embankment near Charing Cross we shall observe anevent having the character which we recognise as Cleopatra's Needle. Things which we thus recognise I call objects. An object is situated inthose events or in that stream of events of which it expresses thecharacter. There are many sorts of objects. For example, the colourgreen is an object according to the above definition. It is the purposeof science to trace the laws which govern the appearance of objects inthe various events in which they are found to be situated. For thispurpose we can mainly concentrate on two types of objects, which I willcall material physical objects and scientific objects. A materialphysical object is an ordinary bit of matter, Cleopatra's Needle forexample. This is a much more complicated type of object than a merecolour, such as the colour of the Needle. I call these simple objects, such as colours or sounds, sense-objects. An artist will train himselfto attend more particularly to sense-objects where the ordinary personattends normally to material objects. Thus if you were walking with anartist, when you said 'There's Cleopatra's Needle, ' perhaps hesimultaneously exclaimed 'There's a nice bit of colour. ' Yet you wereboth expressing your recognition of different component characters ofthe same event. But in science we have found out that when we know allabout the adventures amid events of material physical objects and ofscientific objects we have most of the relevant information which willenable us to predict the conditions under which we shall perceivesense-objects in specific situations. For example, when we know thatthere is a blazing fire (_i. E. _ material and scientific objectsundergoing various exciting adventures amid events) and opposite to it amirror (which is another material object) and the positions of a man'sface and eyes gazing into the mirror, we know that he can perceive theredness of the flame situated in an event behind the mirror--thus, to alarge extent, the appearance of sense-objects is conditioned by theadventures of material objects. The analysis of these adventures makesus aware of another character of events, namely their characters asfields of activity which determine the subsequent events to which theywill pass on the objects situated in them. We express these fields ofactivity in terms of gravitational, electromagnetic, or chemical forcesand attractions. But the exact expression of the nature of these fieldsof activity forces us intellectually to acknowledge a less obvious typeof objects as situated in events. I mean molecules and electrons. Theseobjects are not recognised in isolation. We cannot well miss Cleopatra'sNeedle, if we are in its neighbourhood; but no one has seen a singlemolecule or a single electron, yet the characters of events are onlyexplicable to us by expressing them in terms of these scientificobjects. Undoubtedly molecules and electrons are abstractions. But thenso is Cleopatra's Needle. The concrete facts are the eventsthemselves--I have already explained to you that to be an abstractiondoes not mean that an entity is nothing. It merely means that itsexistence is only one factor of a more concrete element of nature. So anelectron is abstract because you cannot wipe out the whole structure ofevents and yet retain the electron in existence. In the same way thegrin on the cat is abstract; and the molecule is really in the event inthe same sense as the grin is really on the cat's face. Now the moreultimate sciences such as Chemistry or Physics cannot express theirultimate laws in terms of such vague objects as the sun, the earth, Cleopatra's Needle, or a human body. Such objects more properly belongto Astronomy, to Geology, to Engineering, to Archaeology, or to Biology. Chemistry and Physics only deal with them as exhibiting statisticalcomplexes of the effects of their more intimate laws. In a certainsense, they only enter into Physics and Chemistry as technologicalapplications. The reason is that they are too vague. Where doesCleopatra's Needle begin and where does it end? Is the soot part of it?Is it a different object when it sheds a molecule or when its surfaceenters into chemical combination with the acid of a London fog? Thedefiniteness and permanence of the Needle is nothing to the possiblepermanent definiteness of a molecule as conceived by science, and thepermanent definiteness of a molecule in its turn yields to that of anelectron. Thus science in its most ultimate formulation of law seeksobjects with the most permanent definite simplicity of character andexpresses its final laws in terms of them. 

Again when we seek definitely to express the relations of events whicharise from their spatio-temporal structure, we approximate to simplicityby progressively diminishing the extent (both temporal and spatial) ofthe events considered. For example, the event which is the life of thechunk of nature which is the Needle during one minute has to the life ofnature within a passing barge during the same minute a very complexspatio-temporal relation. But suppose we progressively diminish the timeconsidered to a second, to a hundredth of a second, to a thousandth of asecond, and so on. As we pass along such a series we approximate to anideal simplicity of structural relations of the pairs of eventssuccessively considered, which ideal we call the spatial relations ofthe Needle to the barge at some instant. Even these relations are toocomplicated for us, and we consider smaller and smaller bits of theNeedle and of the barge. Thus we finally reach the ideal of an event sorestricted in its extension as to be without extension in space orextension in time. Such an event is a mere spatial point-flash ofinstantaneous duration. I call such an ideal event an 'event-particle. 'You must not think of the world as ultimately built up ofevent-particles. That is to put the cart before the horse. The world weknow is a continuous stream of occurrence which we can discriminate intofinite events forming by their overlappings and containings of eachother and separations a spatio-temporal structure. We can express theproperties of this structure in terms of the ideal limits to routes ofapproximation, which I have termed event-particles. Accordinglyevent-particles are abstractions in their relations to the more concreteevents. But then by this time you will have comprehended that you cannotanalyse concrete nature without abstracting. Also I repeat, theabstractions of science are entities which are truly in nature, thoughthey have no meaning in isolation from nature. 

The character of the spatio-temporal structure of events can be fullyexpressed in terms of relations between these more abstractevent-particles. The advantage of dealing with event-particles is thatthough they are abstract and complex in respect to the finite eventswhich we directly observe, they are simpler than finite events inrespect to their mutual relations. Accordingly they express for us thedemands of an ideal accuracy, and of an ideal simplicity in theexposition of relations. These event-particles are the ultimate elementsof the four-dimensional space-time manifold which the theory ofrelativity presupposes. You will have observed that each event-particleis as much an instant of time as it is a point of space. I have calledit an instantaneous point-flash. Thus in the structure of thisspace-time manifold space is not finally discriminated from time, andthe possibility remains open for diverse modes of discriminationaccording to the diverse circumstances of observers. It is thispossibility which makes the fundamental distinction between the new wayof conceiving the universe and the old way. The secret of understandingrelativity is to understand this. It is of no use rushing in withpicturesque paradoxes, such as 'Space caught bending, ' if you have notmastered this fundamental conception which underlies the whole theory. When I say that it underlies the whole theory, I mean that in my opinionit ought to underlie it, though I may confess some doubts as to how farall expositions of the theory have really understood its implicationsand its premises. 

Our measurements when they are expressed in terms of an ideal accuracyare measurements which express properties of the space-time manifold. Now there are measurements of different sorts. You can measure lengths, or angles, or areas, or volumes, or times. There are also other sorts ofmeasures such as measurements of intensity of illumination, but I willdisregard these for the moment and will confine attention to thosemeasurements which particularly interest us as being measurements ofspace or of time. It is easy to see that four such measurements of theproper characters are necessary to determine the position of anevent-particle in the space-time manifold in its relation to the rest ofthe manifold. For example, in a rectangular field you start from onecorner at a given time, you measure a definite distance along one side, you then strike out into the field at right angles, and then measure adefinite distance parallel to the other pair of sides, you then risevertically a definite height and take the time. At the point and at thetime which you thus reach there is occurring a definite instantaneouspoint-flash of nature. In other words, your four measurements havedetermined a definite event-particle belonging to the four-dimensionspace-time manifold. These measurements have appeared to be very simpleto the land-surveyor and raise in his mind no philosophic difficulties. But suppose there are beings on Mars sufficiently advanced inscientific invention to be able to watch in detail the operations ofthis survey on earth. Suppose that they construe the operations of theEnglish land-surveyors in reference to the space natural to a being onMars, namely a Martio-centric space in which that planet is fixed. Theearth is moving relatively to Mars and is rotating. To the beings onMars the operations, construed in this fashion, effect measurements ofthe greatest complication. Furthermore, according to the relativisticdoctrine, the operation of time-measurement on earth will not correspondquite exactly to any time-measurement on Mars. 

I have discussed this example in order to make you realise that inthinking of the possibilities of measurement in the space-time manifold, we must not confine ourselves merely to those minor variations whichmight seem natural to human beings on the earth. Let us make thereforethe general statement that four measurements, respectively ofindependent types (such as measurements of lengths in three directionsand a time), can be found such that a definite event-particle isdetermined by them in its relations to other parts of the manifold. 

If (p₁, p₂, p₃, p₄) be a set of measurements of this system, then theevent-particle which is thus determined will be said to have p₁, p₂, p₃, p₄ as its co-ordinates in this system of measurement. Suppose that wename it the p-system of measurement. Then in the same p-system byproperly varying (p₁, p₂, p₃, p₄) every event-particle that has been, orwill be, or instantaneously is now, can be indicated. Furthermore, according to any system of measurement that is natural to us, three ofthe co-ordinates will be measurements of space and one will be ameasurement of time. Let us always take the last co-ordinate torepresent the time-measurement. Then we should naturally say that (p₁, p₂, p₃) determined a point in space and that the event-particle happenedat that point at the time p₄. But we must not make the mistake ofthinking that there is a space in addition to the space-time manifold. That manifold is all that there is for the determination of the meaningof space and time. We have got to determine the meaning of a space-pointin terms of the event-particles of the four-dimensional manifold. Thereis only one way to do this. Note that if we vary the time and take timeswith the same three space co-ordinates, then the event-particles, thusindicated, are all at the same point. But seeing that there is nothingelse except the event-particles, this can only mean that the point (p₁, p₂, p₃) of the space in the p-system is merely the collection ofevent-particles (p₁, p₂, p₃, [p₄]), where p₄ is varied and (p₁, p₂, p₃)is kept fixed. It is rather disconcerting to find that a point in spaceis not a simple entity; but it is a conclusion which follows immediatelyfrom the relative theory of space. 

Furthermore the inhabitant of Mars determines event-particles by anothersystem of measurements. Call his system the q-system. According to him(q₁, q₂, q₃, q₄) determines an event-particle, and (q₁, q₂, q₃)determines a point and q₄ a time. But the collection of event-particleswhich he thinks of as a point is entirely different from any suchcollection which the man on earth thinks of as a point. Thus the q-spacefor the man on Mars is quite different from the p-space for theland-surveyor on earth. 

So far in speaking of space we have been talking of the timeless spaceof physical science, namely, of our concept of eternal space in whichthe world adventures. But the space which we see as we look about isinstantaneous space. Thus if our natural perceptions are adjustable tothe p-system of measurements we see instantaneously all theevent-particles at some definite time p₄, and observe a successionof such spaces as time moves on. The timeless space is achieved bystringing together all these instantaneous spaces. The points of aninstantaneous space are event-particles, and the points of an eternalspace are strings of event-particles occurring in succession. But theman on Mars will never perceive the same instantaneous spaces as the manon the earth. This system of instantaneous spaces will cut across theearth-man's system. For the earth-man there is one instantaneous spacewhich is the instantaneous present, there are the past spaces and thefuture spaces. But the present space of the man on Mars cuts across thepresent space of the man on the earth. So that of the event-particleswhich the earth-man thinks of as happening now in the present, the manon Mars thinks that some are already past and are ancient history, thatothers are in the future, and others are in the immediate present. Thisbreak-down in the neat conception of a past, a present, and a future isa serious paradox. I call two event-particles which on some or othersystem of measurement are in the same instantaneous space 'co-present'event-particles. Then it is possible that A and B may be co-present, and that A and C may be co-present, but that B and C may not beco-present. For example, at some inconceivable distance from us thereare events co-present with us now and also co-present with the birth ofQueen Victoria. If A and B are co-present there will be some systemsin which A precedes B and some in which B precedes A. Also therecan be no velocity quick enough to carry a material particle from A toB or from B to A. These different measure-systems with theirdivergences of time-reckoning are puzzling, and to some extent affrontour common sense. It is not the usual way in which we think of theUniverse. We think of one necessary time-system and one necessary space. According to the new theory, there are an indefinite number ofdiscordant time-series and an indefinite number of distinct spaces. Anycorrelated pair, a time-system and a space-system, will do in which tofit our description of the Universe. We find that under given conditionsour measurements are necessarily made in some one pair which togetherform our natural measure-system. The difficulty as to discordanttime-systems is partly solved by distinguishing between what I call thecreative advance of nature, which is not properly serial at all, and anyone time series. We habitually muddle together this creative advance, which we experience and know as the perpetual transition of nature intonovelty, with the single-time series which we naturally employ formeasurement. The various time-series each measure some aspect of thecreative advance, and the whole bundle of them express all theproperties of this advance which are measurable. The reason why we havenot previously noted this difference of time-series is the very smalldifference of properties between any two such series. Any observablephenomena due to this cause depend on the square of the ratio of anyvelocity entering into the observation to the velocity of light. Nowlight takes about fifty minutes to get round the earth's orbit; and theearth takes rather more than 17, 531 half-hours to do the same. Hence allthe effects due to this motion are of the order of the ratio of one tothe square of 10, 000. Accordingly an earth-man and a sun-man have onlyneglected effects whose quantitative magnitudes all contain the factor1/10⁸. Evidently such effects can only be noted by means of the mostrefined observations. They have been observed however. Suppose wecompare two observations on the velocity of light made with the sameapparatus as we turn it through a right angle. The velocity of the earthrelatively to the sun is in one direction, the velocity of lightrelatively to the ether should be the same in all directions. Hence ifspace when we take the ether as at rest means the same thing as spacewhen we take the earth as at rest, we ought to find that the velocity oflight relatively to the earth varies according to the direction fromwhich it comes. 

These observations on earth constitute the basic principle of the famousexperiments designed to detect the motion of the earth through theether. You all know that, quite unexpectedly, they gave a null result. This is completely explained by the fact that, the space-system and thetime-system which we are using are in certain minute ways different fromthe space and the time relatively to the sun or relatively to any otherbody with respect to which it is moving. 

All this discussion as to the nature of time and space has lifted aboveour horizon a great difficulty which affects the formulation of all theultimate laws of physics--for example, the laws of the electromagneticfield, and the law of gravitation. Let us take the law of gravitationas an example. Its formulation is as follows: Two material bodiesattract each other with a force proportional to the product of theirmasses and inversely proportional to the square of their distances. In this statement the bodies are supposed to be small enough to betreated as material particles in relation to their distances; and weneed not bother further about that minor point. The difficulty to whichI want to draw your attention is this: In the formulation of the law onedefinite time and one definite space are presupposed. The two masses areassumed to be in simultaneous positions. 

But what is simultaneous in one time-system may not be simultaneous inanother time-system. So according to our new views the law is in thisrespect not formulated so as to have any exact meaning. Furthermore ananalogous difficulty arises over the question of distance. The distancebetween two instantaneous positions, _i. E. _ between two event-particles, is different in different space-systems. What space is to be chosen?Thus again the law lacks precise formulation, if relativity is accepted. Our problem is to seek a fresh interpretation of the law of gravity inwhich these difficulties are evaded. In the first place we must avoidthe abstractions of space and time in the formulation of our fundamentalideas and must recur to the ultimate facts of nature, namely to events. Also in order to find the ideal simplicity of expressions of therelations between events, we restrict ourselves to event-particles. Thusthe life of a material particle is its adventure amid a track ofevent-particles strung out as a continuous series or path in thefour-dimensional space-time manifold. These event-particles are thevarious situations of the material particle. We usually express thisfact by adopting our natural space-time system and by talking of thepath in space of the material particle as it exists at successiveinstants of time. 

We have to ask ourselves what are the laws of nature which lead thematerial particle to adopt just this path among event-particles and noother. Think of the path as a whole. What characteristic has that pathgot which would not be shared by any other slightly varied path? We areasking for more than a law of gravity. We want laws of motion and ageneral idea of the way to formulate the effects of physical forces. 

In order to answer our question we put the idea of the attracting massesin the background and concentrate attention on the field of activity ofthe events in the neighbourhood of the path. In so doing we are actingin conformity with the whole trend of scientific thought during the lasthundred years, which has more and more concentrated attention on thefield of force as the immediate agent in directing motion, to theexclusion of the consideration of the immediate mutual influence betweentwo distant bodies. We have got to find the way of expressing the fieldof activity of events in the neighbourhood of some definiteevent-particle E of the four-dimensional manifold. I bring in afundamental physical idea which I call the 'impetus' to express thisphysical field. The event-particle E is related to any neighbouringevent-particle P by an element of impetus. The assemblage of all theelements of impetus relating E to the assemblage of event-particles inthe neighbourhood of E expresses the character of the field ofactivity in the neighbourhood of E. Where I differ from Einstein isthat he conceives this quantity which I call the impetus as merelyexpressing the characters of the space and time to be adopted and thusends by talking of the gravitational field expressing a curvature in thespace-time manifold. I cannot attach any clear conception to hisinterpretation of space and time. My formulae differ slightly from his, though they agree in those instances where his results have beenverified. I need hardly say that in this particular of the formulationof the law of gravitation I have drawn on the general method ofprocedure which constitutes his great discovery. 

Einstein showed how to express the characters of the assemblage ofelements of impetus of the field surrounding an event-particle E interms of ten quantities which I will call J_{11}, J_{12}(=J_{21}), J_{22}, J_{23}(=J_{32}), etc. It will be noted thatthere are four spatio-temporal measurements relating E to itsneighbour P, and that there are ten pairs of such measurements if weare allowed to take any one measurement twice over to make one suchpair. The ten J's depend merely on the position of E in thefour-dimensional manifold, and the element of impetus between E andP can be expressed in terms of the ten J's and the ten pairs of thefour spatio-temporal measurements relating E and P. The numericalvalues of the J's will depend on the system of measurement adopted, but are so adjusted to each particular system that the same value isobtained for the element of impetus between E and P, whatever be thesystem of measurement adopted. This fact is expressed by saying that theten J's form a 'tensor. ' It is not going too far to say that theannouncement that physicists would have in future to study the theory oftensors created a veritable panic among them when the verification ofEinstein's predictions was first announced. 

The ten J's at any event-particle E can be expressed in terms of twofunctions which I call the potential and the 'associate-potential' atE. The potential is practically what is meant by the ordinarygravitation potential, when we express ourselves in terms of theEuclidean space in reference to which the attracting mass is at rest. The associate-potential is defined by the modification of substitutingthe direct distance for the inverse distance in the definition of thepotential, and its calculation can easily be made to depend on that ofthe old-fashioned potential. Thus the calculation of the J's--thecoefficients of impetus, as I will call them--does not involve anythingvery revolutionary in the mathematical knowledge of physicists. We nowreturn to the path of the attracted particle. We add up all the elementsof impetus in the whole path, and obtain thereby what I call the'integral impetus. ' The characteristic of the actual path as comparedwith neighbouring alternative paths is that in the actual paths theintegral impetus would neither gain nor lose, if the particle wobbledout of it into a small extremely near alternative path. Mathematicianswould express this by saying, that the integral impetus is stationaryfor an infinitesimal displacement. In this statement of the law ofmotion I have neglected the existence of other forces. But that wouldlead me too far afield. 

The electromagnetic theory has to be modified to allow for the presenceof a gravitational field. Thus Einstein's investigations lead to thefirst discovery of any relation between gravity and other physicalphenomena. In the form in which I have put this modification, we deduceEinstein's fundamental principle, as to the motion of light along itsrays, as a first approximation which is absolutely true for infinitelyshort waves. Einstein's principle, thus partially verified, stated in mylanguage is that a ray of light always follows a path such that theintegral impetus along it is zero. This involves that every element ofimpetus along it is zero. 

In conclusion, I must apologise. In the first place I have considerablytoned down the various exciting peculiarities of the original theory andhave reduced it to a greater conformity with the older physics. I do notallow that physical phenomena are due to oddities of space. Also I haveadded to the dullness of the lecture by my respect for the audience. Youwould have enjoyed a more popular lecture with illustrations ofdelightful paradoxes. But I know also that you are serious students whoare here because you really want to know how the new theories may affectyour scientific researches. 

CHAPTER IX

THE ULTIMATE PHYSICAL CONCEPTS

The second chapter of this book lays down the first principle to beguarded in framing our physical concept. We must avoid viciousbifurcation. Nature is nothing else than the deliverance ofsense-awareness. We have no principles whatever to tell us what couldstimulate mind towards sense-awareness. Our sole task is to exhibit inone system the characters and inter-relations of all that is observed. Our attitude towards nature is purely 'behaviouristic, ' so far asconcerns the formulation of physical concepts. 

Our knowledge of nature is an experience of activity (or passage). Thethings previously observed are active entities, the 'events. ' They arechunks in the life of nature. These events have to each other relationswhich in our knowledge differentiate themselves into space-relations andtime-relations. But this differentiation between space and time, thoughinherent in nature, is comparatively superficial; and space and time areeach partial expressions of one fundamental relation between eventswhich is neither spatial nor temporal. This relation I call 'extension. 'The relation of 'extending over' is the relation of 'including, ' eitherin a spatial or in a temporal sense, or in both. But the mere'inclusion' is more fundamental than either alternative and does notrequire any spatio-temporal differentiation. In respect to extension twoevents are mutually related so that either (i) one includes the other, or (ii) one overlaps the other without complete inclusion, or (iii)they are entirely separate. But great care is required in thedefinition of spatial and temporal elements from this basis in order toavoid tacit limitations really depending on undefined relations andproperties. 

Such fallacies can be avoided by taking account of two elements in ourexperience, namely, (i) our observational 'present, ' and (ii) our'percipient event. '

Our observational 'present' is what I call a 'duration. ' It is the wholeof nature apprehended in our immediate observation. It has therefore thenature of an event, but possesses a peculiar completeness which marksout such durations as a special type of events inherent in nature. Aduration is not instantaneous. It is all that there is of nature withcertain temporal limitations. In contradistinction to other events aduration will be called infinite and the other events are finite[10]. Inour knowledge of a duration we distinguish (i) certain included eventswhich are particularly discriminated as to their peculiarindividualities, and (ii) the remaining included events which are onlyknown as necessarily in being by reason of their relations to thediscriminated events and to the whole duration. The duration as a wholeis signified[11] by that quality of relatedness (in respect toextension) possessed by the part which is immediately under observation;namely, by the fact that there is essentially a beyond to whatever isobserved. I mean by this that every event is known as being related toother events which it does not include. This fact, that every event isknown as possessing the quality of exclusion, shows that exclusion is aspositive a relation as inclusion. There are of course no merelynegative relations in nature, and exclusion is not the mere negative ofinclusion, though the two relations are contraries. Both relations areconcerned solely with events, and exclusion is capable of logicaldefinition in terms of inclusion. 

[10] Cf. Note on 'significance, ' pp.  197, 198. 

[11] Cf. Ch.  III, pp.  51 et seq. 

Perhaps the most obvious exhibition of significance is to be found inour knowledge of the geometrical character of events inside an opaquematerial object. For example we know that an opaque sphere has a centre. This knowledge has nothing to do with the material; the sphere may be asolid uniform billiard ball or a hollow lawn-tennis ball. Such knowledgeis essentially the product of significance, since the general characterof the external discriminated events has informed us that there areevents within the sphere and has also informed us of their geometricalstructure. 

Some criticisms on 'The Principles of Natural Knowledge' show thatdifficulty has been found in apprehending durations as realstratifications of nature. I think that this hesitation arises from theunconscious influence of the vicious principle of bifurcation, so deeplyembedded in modern philosophical thought. We observe nature as extendedin an immediate present which is simultaneous but not instantaneous, andtherefore the whole which is immediately discerned or signified as aninter-related system forms a stratification of nature which is aphysical fact. This conclusion immediately follows unless we admitbifurcation in the form of the principle of psychic additions, hererejected. 

Our 'percipient event' is that event included in our observationalpresent which we distinguish as being in some peculiar way ourstandpoint for perception. It is roughly speaking that event which isour bodily life within the present duration. The theory of perceptionas evolved by medical psychology is based on significance. The distantsituation of a perceived object is merely known to us as signified byour bodily state, _i. E. _ by our percipient event. In fact perceptionrequires sense-awareness of the significations of our percipient eventtogether with sense-awareness of a peculiar relation (situation) betweencertain objects and the events thus signified. Our percipient event issaved by being the whole of nature by this fact of its significations. This is the meaning of calling the percipient event our standpoint forperception. The course of a ray of light is only derivatively connectedwith perception. What we do perceive are objects as related to eventssignified by the bodily states excited by the ray. These signifiedevents (as is the case of images seen behind a mirror) may have verylittle to do with the actual course of the ray. In the course ofevolution those animals have survived whose sense-awareness isconcentrated on those significations of their bodily states which are onthe average important for their welfare. The whole world of events issignified, but there are some which exact the death penalty forinattention. 

The percipient event is always here and now in the associated presentduration. It has, what may be called, an absolute position in thatduration. Thus one definite duration is associated with a definitepercipient event, and we are thus aware of a peculiar relation whichfinite events can bear to durations. I call this relation 'cogredience. 'The notion of rest is derivative from that of cogredience, and thenotion of motion is derivative from that of inclusion within a durationwithout cogredience with it. In fact motion is a relation (of varyingcharacter) between an observed event and an observed duration, andcogredience is the most simple character or subspecies of motion. To sumup, a duration and a percipient event are essentially involved in thegeneral character of each observation of nature, and the percipientevent is cogredient with the duration. 

Our knowledge of the peculiar characters of different events dependsupon our power of comparison. I call the exercise of this factor in ourknowledge 'recognition, ' and the requisite sense-awareness of thecomparable characters I call 'sense-recognition. ' Recognition andabstraction essentially involve each other. Each of them exhibits anentity for knowledge which is less than the concrete fact, but is a realfactor in that fact. The most concrete fact capable of separatediscrimination is the event. We cannot abstract without recognition, andwe cannot recognise without abstraction. Perception involvesapprehension of the event and recognition of the factors of itscharacter. 

The things recognised are what I call 'objects. ' In this general senseof the term the relation of extension is itself an object. In practicehowever I restrict the term to those objects which can in some sense orother be said to have a situation in an event; namely, in the phrase'There it is again' I restrict the 'there' to be the indication of aspecial event which is the situation of the object. Even so, there aredifferent types of objects, and statements which are true of objects ofone type are not in general true of objects of other types. The objectswith which we are here concerned in the formulation of physical laws arematerial objects, such as bits of matter, molecules and electrons. Anobject of one of these types has relations to events other than thosebelonging to the stream of its situations. The fact of its situationswithin this stream has impressed on all other events certainmodifications of their characters. In truth the object in itscompleteness may be conceived as a specific set of correlatedmodifications of the characters of all events, with the property thatthese modifications attain to a certain focal property for those eventswhich belong to the stream of its situations. The total assemblage ofthe modifications of the characters of events due to the existence of anobject in a stream of situations is what I call the 'physical field' dueto the object. But the object cannot really be separated from its field. The object is in fact nothing else than the systematically adjusted setof modifications of the field. The conventional limitation of the objectto the focal stream of events in which it is said to be 'situated' isconvenient for some purposes, but it obscures the ultimate fact ofnature. From this point of view the antithesis between action at adistance and action by transmission is meaningless. The doctrine of thisparagraph is nothing else than another way of expressing theunresolvable multiple relation of an object to events. 

A complete time-system is formed by any one family of paralleldurations. Two durations are parallel if either (i) one includes theother, or (ii) they overlap so as to include a third duration common toboth, or (iii) are entirely separate. The excluded case is that of twodurations overlapping so as to include in common an aggregate of finiteevents but including in common no other complete duration. Therecognition of the fact of an indefinite number of families of paralleldurations is what differentiates the concept of nature here put forwardfrom the older orthodox concept of the essentially unique time-systems. Its divergence from Einstein's concept of nature will be brieflyindicated later. 

The instantaneous spaces of a given time-system are the ideal(non-existent) durations of zero temporal thickness indicated by routesof approximation along series formed by durations of the associatedfamily. Each such instantaneous space represents the ideal of nature atan instant and is also a moment of time. Each time-system thus possessesan aggregate of moments belonging to it alone. Each event-particle liesin one and only one moment of a given time-system. An event-particle hasthree characters[12]: (i) its extrinsic character which is its characteras a definite route of convergence among events, (ii) its intrinsiccharacter which is the peculiar quality of nature in its neighbourhood, namely, the character of the physical field in the neighbourhood, and(iii) its position. 

[12] Cf. Pp.  82 et seq. 

The position of an event-particle arises from the aggregate of moments(no two of the same family) in which it lies. We fix our attention onone of these moments which is approximated to by the short duration ofour immediate experience, and we express position as the position inthis moment. But the event-particle receives its position in moment Min virtue of the whole aggregate of other moments M{'}, M{''}, etc. , in which it also lies. The differentiation of M into a geometry ofevent-particles (instantaneous points) expresses the differentiation ofM by its intersections with moments of alien time-systems. In this wayplanes and straight lines and event-particles themselves find theirbeing. Also the parallelism of planes and straight lines arises from theparallelism of the moments of one and the same time-system intersectingM. Similarly the order of parallel planes and of event-particles onstraight lines arises from the time-order of these intersecting moments. The explanation is not given here[13]. It is sufficient now merely tomention the sources from which the whole of geometry receives itsphysical explanation. 

[13] Cf. _Principles of Natural Knowledge_, and previous chapters of thepresent work. 

The correlation of the various momentary spaces of one time-system isachieved by the relation of cogredience. Evidently motion in aninstantaneous space is unmeaning. Motion expresses a comparison betweenposition in one instantaneous space with positions in otherinstantaneous spaces of the same time-system. Cogredience yields thesimplest outcome of such comparison, namely, rest. 

Motion and rest are immediately observed facts. They are relative in thesense that they depend on the time-system which is fundamental for theobservation. A string of event-particles whose successive occupationmeans rest in the given time-system forms a timeless point in thetimeless space of that time-system. In this way each time-systempossesses its own permanent timeless space peculiar to it alone, andeach such space is composed of timeless points which belong to thattime-system and to no other. The paradoxes of relativity arise fromneglecting the fact that different assumptions as to rest involve theexpression of the facts of physical science in terms of radicallydifferent spaces and times, in which points and moments have differentmeanings. 

The source of order has already been indicated and that of congruence isnow found. It depends on motion. From cogredience, perpendicularityarises; and from perpendicularity in conjunction with the reciprocalsymmetry between the relations of any two time-systems congruence bothin time and space is completely defined (cf. _loc. Cit. _). 

The resulting formulae are those for the electromagnetic theory ofrelativity, or, as it is now termed, the restricted theory. But there isthis vital difference: the critical velocity c which occurs in theseformulae has now no connexion whatever with light or with any other factof the physical field (in distinction from the extensional structure ofevents). It simply marks the fact that our congruence determinationembraces both times and spaces in one universal system, and therefore iftwo arbitrary units are chosen, one for all spaces and one for alltimes, their ratio will be a velocity which is a fundamental property ofnature expressing the fact that times and spaces are really comparable. 

The physical properties of nature are expressed in terms of materialobjects (electrons, etc. ). The physical character of an event arisesfrom the fact that it belongs to the field of the whole complex of suchobjects. From another point of view we can say that these objects arenothing else than our way of expressing the mutual correlation of thephysical characters of events. 

The spatio-temporal measurableness of nature arises from (i) therelation of extension between events, and (ii) the stratified characterof nature arising from each of the alternative time-systems, and (iii)rest and motion, as exhibited in the relations of finite events totime-systems. None of these sources of measurement depend on thephysical characters of finite events as exhibited by the situatedobjects. They are completely signified for events whose physicalcharacters are unknown. Thus the spatio-temporal measurements areindependent of the objectival physical characters. Furthermore thecharacter of our knowledge of a whole duration, which is essentiallyderived from the significance of the part within the immediate field ofdiscrimination, constructs it for us as a uniform whole independent, sofar as its extension is concerned, of the unobserved characters ofremote events. Namely, there is a definite whole of nature, simultaneously now present, whatever may be the character of its remoteevents. This consideration reinforces the previous conclusion. Thisconclusion leads to the assertion of the essential uniformity of themomentary spaces of the various time-systems, and thence to theuniformity of the timeless spaces of which there is one to eachtime-system. 

The analysis of the general character of observed nature set forth aboveaffords explanations of various fundamental observational facts: (α) Itexplains the differentiation of the one quality of extension into timeand space. (β) It gives a meaning to the observed facts of geometricaland temporal position, of geometrical and temporal order, and ofgeometrical straightness and planeness. (γ) It selects one definitesystem of congruence embracing both space and time, and thus explainsthe concordance as to measurement which is in practice attained. (δ) Itexplains (consistently with the theory of relativity) the observedphenomena of rotation, _e. G. _ Foucault's pendulum, the equatorial bulgeof the earth, the fixed senses of rotation of cyclones and anticyclones, and the gyro-compass. It does this by its admission of definitestratifications of nature which are disclosed by the very character ofour knowledge of it. (ε) Its explanations of motion are morefundamental than those expressed in (δ); for it explains what is meantby motion itself. The observed motion of an extended object is therelation of its various situations to the stratification of natureexpressed by the time-system fundamental to the observation. This motionexpresses a real relation of the object to the rest of nature. Thequantitative expression of this relation will vary according to thetime-system selected for its expression. 

This theory accords no peculiar character to light beyond that accordedto other physical phenomena such as sound. There is no ground for such adifferentiation. Some objects we know by sight only, and other objectswe know by sound only, and other objects we observe neither by light norby sound but by touch or smell or otherwise. The velocity of lightvaries according to its medium and so does that of sound. Light moves incurved paths under certain conditions and so does sound. Both light andsound are waves of disturbance in the physical characters of events; and(as has been stated above, p.  188) the actual course of the light is ofno more importance for perception than is the actual course of thesound. To base the whole philosophy of nature upon light is a baselessassumption. The Michelson-Morley and analogous experiments show thatwithin the limits of our inexactitude of observation the velocity oflight is an approximation to the critical velocity 'c' which expressesthe relation between our space and time units. It is provable that theassumption as to light by which these experiments and the influence ofthe gravitational field on the light-rays are explained is deducible _asan approximation_ from the equations of the electromagnetic field. Thiscompletely disposes of any necessity for differentiating light fromother physical phenomena as possessing any peculiar fundamentalcharacter. 

It is to be observed that the measurement of extended nature by means ofextended objects is meaningless apart from some observed fact ofsimultaneity inherent in nature and not merely a play of thought. Otherwise there is no meaning to the concept of one presentation of yourextended measuring rod AB. Why not AB′ where B′ is the end Bfive minutes later? Measurement presupposes for its possibility natureas a simultaneity, and an observed object present then and present now. In other words, measurement of extended nature requires some inherentcharacter in nature affording a rule of presentation of events. Furthermore congruence cannot be defined by the permanence of themeasuring rod. The permanence is itself meaningless apart from someimmediate judgment of self-congruence. Otherwise how is an elasticstring differentiated from a rigid measuring rod? Each remains the sameself-identical object. Why is one a possible measuring rod and the othernot so? The meaning of congruence lies beyond the self-identity of theobject. In other words measurement presupposes the measurable, and thetheory of the measurable is the theory of congruence. 

Furthermore the admission of stratifications of nature bears on theformulation of the laws of nature. It has been laid down that these lawsare to be expressed in differential equations which, as expressed in anygeneral system of measurement, should bear no reference to any otherparticular measure-system. This requirement is purely arbitrary. For ameasure-system measures something inherent in nature; otherwise it hasno connexion with nature at all. And that something which is measuredby a particular measure-system may have a special relation to thephenomenon whose law is being formulated. For example the gravitationalfield due to a material object at rest in a certain time-system may beexpected to exhibit in its formulation particular reference to spatialand temporal quantities of that time-system. The field can of course beexpressed in any measure-systems, but the particular reference willremain as the simple physical explanation. 

NOTE: ON THE GREEK CONCEPT OF A POINT

The preceding pages had been passed for press before I had the pleasureof seeing Sir T.  L. Heath's _Euclid in Greek_[14]. In the originalEuclid's first definition is

 σημειον εστιν, ου μερος ουθεν. 

I have quoted it on p.  86 in the expanded form taught to me inchildhood, 'without parts and without magnitude. ' I should haveconsulted Heath's English edition--a classic from the moment of itsissue--before committing myself to a statement about Euclid. This ishowever a trivial correction not affecting sense and not worth a note. Iwish here to draw attention to Heath's own note to this definition inhis _Euclid in Greek_. He summarises Greek thought on the nature of apoint, from the Pythagoreans, through Plato and Aristotle, to Euclid. Myanalysis of the requisite character of a point on pp.  89 and 90 is incomplete agreement with the outcome of the Greek discussion. 

[14] Camb. Univ. Press, 1920. 

NOTE: ON SIGNIFICANCE AND INFINITE EVENTS

The theory of significance has been expanded and made more definite inthe present volume. It had already been introduced in the _Principles ofNatural Knowledge_ (cf. Subarticles 3. 3 to 3. 8 and 16. 1, 16. 2, 19. 4, andarticles 20, 21). In reading over the proofs of the present volume, Icome to the conclusion that in the light of this development mylimitation of infinite events to durations is untenable. This limitationis stated in article 33 of the _Principles_ and at the beginning ofChapter IV (p.  74) of this book. There is not only a significance of thediscerned events embracing the whole present duration, but there is asignificance of a cogredient event involving its extension through awhole time-system backwards and forwards. In other words the essential'beyond' in nature is a definite beyond in time as well as in space [cf. Pp.  53, 194]. This follows from my whole thesis as to the assimilationof time and space and their origin in extension. It also has the samebasis in the analysis of the character of our knowledge of nature. Itfollows from this admission that it is possible to define point-tracks[_i. E. _ the points of timeless spaces] as abstractive elements. This isa great improvement as restoring the balance between moments and points. I still hold however to the statement in subarticle 35. 4 of the_Principles_ that the intersection of a pair of non-parallel durationsdoes not present itself to us as one event. This correction does notaffect any of the subsequent reasoning in the two books. 

I may take this opportunity of pointing out that the 'stationary events'of article 57 of the _Principles_ are merely cogredient events got atfrom an abstract mathematical point of view. 

INDEX

_In the case of terms of frequent occurrence, only those occurrences areindexed which are of peculiar importance for the elucidation ofmeaning. _

 A [_or_ an], 11

 Abraham, 105

 Absolute position, 105, 106, 114, 188

 Abstraction, 33, 37, 168, 171, 173; extensive, 65, 79, 85

 Abstractive element, 84; set, 61, 79

 Action at a distance, 159, 190

 Action by transmission, 159, 190

 Active conditions, 158

 Activity, field of, 170, 181

 Adjunction, 101

 Aggregate, 23

 Alexander, Prof. , viii

 Alexandria, 71

 Alfred the Great, 137

 Anticipation, 69

 Anti-prime, 88

 Apparent nature, 31, 39

 Area, 99; momental, 103; vagrant, 103

 Aristotelian logic, 150

 Aristotle, 16, 17, 18, 24, 197

 Associate-potential, 183

 Atom, 17

 Attribute, 21, 26, 150

 Awareness, 3

 Axiom, 36, 121

 Axioms of congruence, 128 et seqq. 

 Bacon, Francis, 78

 Behaviouristic, 185

 Bergson, 54

 Berkeley, 28

 Between, 64

 Beyond, 186, 198

 Bifurcation, vi, 30, 185, 187

 Boundary, 100; moment, 63; particle, 100

 Broad, C.  D. , viii

 Calculation, formula of, 45, 158

 Cambridge, 97

 Causal nature, 31, 39

 Causation, 31, 146

 Centrifugal force, 138

 Change, uniformity of, 140

 Character, extrinsic, 82, 89, 90, 113, 191; intrinsic, 80, 82, 90, 113, 191

 Charge, 160

 Closure of nature, 4

 Coefficient of drag, 133

 Coefficients of impetus, 183

 Cogredience, 110, 188

 Coherence, 29

 Comparison, 124, 125, 143, 189

 Complex, 13

 Conceptual nature, 45; space, 96

 Concrete facts, 167, 171, 189

 Conditioning events, 152

 Conditions, active, 158

 Congruence, 65, 96, 118, 120, 127, 196

 Continuity, 157; Dedekindian, 102; of events, 76; of nature, 59, 76

 Convention, 121

 Convergence, 62, 79; law of, 82

 Conveyance, 154, 155

 Co-present, 177

 Covering, 83

 Creative advance, 178

 Critical velocity, 193, 195

 Curvature of space-time, 182

 Cyclone, 194

 Dedekindian continuity, 102

 Definite, 53, 194, 198

 Delusions, 31, 38

 Delusive perceptual object, 153

 Demarcation of events, 144

 Demonstrative phrase, 6

 Descriptive phrase, 6, 10

 Differential equations, 196

 Discrimination, 14, 50, 144

 Diversification of nature, 15

 Duddington, Mrs, 47

 Duration, 37, 53, 55, 186

 Durations, families of, 59, 73, 190

 Dynamical axes, 138

 Einstein, vii, 102, 131, 164, 165, 181, 182, 183, 184, 191

 Electromagnetic field, 179

 Electron, 30, 146, 158, 171

 Element, 17; abstractive, 84

 Elliptical phraseology, 7

 Empty space, 145

 Entity, 5, 13

 Equal in abstractive force, 83

 Error, 68

 Ether, 18, 78, 160; material, 78; of events, 78

 Euclid, 85, 94, 197

 Euler, 140

 Event, 15, 52, 75, 165; percipient, 107, 152, 186

 Event-particle, 86, 93, 94, 172, 191

 Events, conditioning, 152; continuity of, 76; demarcation of, 144; ether of, 78; infinite, 197, 198; limited, 74; passage of, 34; signified, 52; stationary, 198; stream of, 167; structure of, 52, 166

 Exclusion, 186

 Explanation, 97, 141

 Extended nature, 196

 Extension, 22, 58, 75, 185

 Extensive abstraction, 65, 79, 85

 Extrinsic character, 82, 89, 90, 113, 191; properties, 62

 Fact, 12, 13

 Factors, 12, 13, 15

 Facts, concrete, 167, 171

 Family of durations, 59, 63, 73; of moments, 63

 Faraday, 146

 Field, gravitational, 197; of activity, 170, 181; physical, 190

 Finite truths, 12

 Fitzgerald, 133

 Formula of calculation, 45, 158

 Foucault, 138, 194

 Four-dimensional manifold, 86

 Fresnel, 133

 Future, the, 72, 177

 Galileo, 139

 Geometrical order, 194

 Geometry, 36; metrical, 129

 Gravitation, 179 et seqq. 

 Gravitational field, 197

 Greek philosophy, 16; thought, 197

 Gyro-compass, 194

 Heath, Sir T.  L. , 197

 Here, 107

 Idealists, 70

 Immediacy, 52; of perception, 72

 Impetus, 181, 182; coefficients of, 183; integral, 183

 Inclusion, 186

 Individuality, 13

 Infinite events, 197, 198

 Inge, Dr, 48

 Ingredient, 14

 Ingression, 144, 145, 148, 152

 Inherence, 83

 Inside, 106

 Instant, 33, 35, 57

 Instantaneous plane, 91; present, 72; spaces, 86, 90, 177

 Instantaneousness, 56, 57

 Intersection, locus of, 90

 Intrinsic character, 80, 82, 90, 113, 191; properties, 62

 Ionian thinkers, 19

 Irrelevance, infinitude of, 12

 Irrevocableness, 35, 37

 It, 8

 Julius Caesar, 36

 Junction, 76, 101

 Kinetic energy, 105; symmetry, 129

 Knowledge, 28, 32

 Lagrange, 140

 Larmor, 131

 Law of convergence, 82

 Laws of motion, 137, 139; of nature, 196

 Leibnizian monadology, 150

 Level, 91, 92

 Light, 195; ray of, 188; velocity of, 131

 Limit, 57

 Limited events, 74

 Location, 160, 161

 Locke, 27

 Locus, 102; of intersection, 90

 London, 97

 Lorentz, H.  A. , 131, 133

 Lossky, 47

 Manifold, four-dimensional, 86; space-time, 173

 Material ether, 78; object, 169

 Materialism, 43, 70

 Matrix, 116

 Matter, 16, 17, 19, 20, 26

 Maxwell, 131, 133

 Measurableness, 196; of nature, 193

 Measurement, 96, 120, 174, 196; of time, 65, 140

 Measure-system, 196

 Memory, 68

 Metaphysics, 28, 32

 Metrical geometry, 129

 Michelson-Morley, 195

 Milton, 35

 Mind, 27, 28

 Minkowski, viii, 131

 Molecule, 32, 171

 Moment, 57, 60, 88

 Momental area, 103; route, 103

 Momentum, 105

 Motion, 105, 114, 117, 127, 188, 192

 Multiplicity, 22

 Natural philosophy, 29, 30

 Natural science, philosophy of, 46

 Nature, 3; apparent, 31, 39; causal, 31, 39; conceptual, 45; continuity of, 59, 76; discrimination of, 144; extended, 196; laws of, 196; passage of, 54; stratification of, 194, 196; system of, 146

 Newton, 27, 136, 139, 140

 Object, 77, 125, 143, 169, 189; delusive perceptual, 155; material, 169; perceptual, 153; physical, 155, 157; scientific, 158, 169; uniform, 162

 Occupation, 22, 34, 36, 100, 101

 Order, source of, 192; spatial, 95, 194; temporal, 64, 95, 194

 Organisation of thought, 79

 Outside, 63, 100

 Paradox, 192

 Parallel, 63, 127; durations, 190

 Parallelism, 95, 191

 Parallelogram, 127

 Paris, 87, 138

 Parliament, 120

 Part, 14, 15, 58

 Passage of events, 34; of nature, 54

 Past, the, 72, 177

 Perception, 3

 Perceptual objects, 149, 153

 Percipience, 28

 Percipient event, 107, 152, 186, 187

 Period of time, 51

 Permanence, 144

 Perpendicularity, 117, 127, 193

 Philosophy, 1; natural, 29, 30; of natural science, 46; of the sciences, 2

 Physical field, 190; object, 155, 156, 157

 Physics, speculative, 30

 Place, 51

 Plane, 191; instantaneous, 91

 Plato, 16, 17, 18, 24, 197

 Poincaré, 121, 122, 123

 Point, 35, 89, 91, 114, 173, 176

 Point-flash, 172, 173

 Point of space, 85

 Point, timeless, 192

 Point-track, 113, 198

 Pompey, 36

 Position, 89, 90, 92, 93, 99, 113, 191; absolute, 105, 106, 114, 188

 Potential, 183; associate-, 183

 Predicate, 18

 Predication, 18

 Present, the, 69, 72, 177; instantaneous, 72; observational, 186

 Primary qualities, 27

 Prime, 88

 Process, 53, 54; of nature, 54

 Psychic additions, 29, 187

 Punct, 92, 93, 94

 Pythagoreans, 197

 Quality, 27

 Quantum of time, 162

 Quantum theory, 162

 Ray of light, 188

 Reality, 30; of durations, 55, 187

 Recognition, 124, 143, 189

 Rect, 91, 92

 Recurrence, 35

 Relative motion, 117; velocity, 130

 Relativity, 169; restricted theory of, 193

 Rest, 105, 114, 188, 192

 Rotation, 138, 194

 Route, 99; momental, 103; straight, 103

 Russell, Bertrand, 11, 122, 123

 Schelling, 47

 Science, 2; metaphysical, 32

 Scientific objects, 149, 158, 169

 Secondary qualities, 27

 Self-congruence, 196

 Self-containedness of nature, 4

 Sense-awareness, 3, 67

 Sense-object, 149, 170

 Sense-perception, 3, 14

 Sense-recognition, 143, 189

 Series, temporal, 66, 70, 85, 178

 Set, abstractive, 61, 79

 Significance, 51, 186, 187, 188, 194, 197, 198

 Signified events, 52

 Simplicity, 163, 173

 Simultaneity, 53, 56, 196

 Situation, 15, 78, 147, 148, 152, 160, 189

 Solid, 99, 101, 102; vagrant, 101

 Sound, 195

 Space, 16, 17, 31, 33, 79; empty, 145; timeless, 86, 106, 114; uniformity of, 194

 Spaces, instantaneous, 86, 90

 Space-system, 179

 Space-time manifold, 173

 Spatial-order, 95

 Spatio-temporal structure, 173

 Speculative demonstration, 6

 Speculative physics, 30

 Standpoint for perception, 107, 188

 Station, 103, 104, 113

 Stationary events, 198

 Straight line, 91, 114, 191; route, 103

 Stratification of nature, 187, 194, 196

 Stream of events, 167

 Structure of events, 52, 166

 Structure, spatio-temporal, 173

 Subject, 18

 Substance, 16, 18, 19, 150

 Substratum, 16, 18, 21

 Symmetry, 118, 126; kinetic, 129

 System of nature, 146

 System, time-, 192

 Tarner, Edward, v, 1

 Temporal order, 64, 95, 194

 Temporal series, 66, 70, 85

 Tensor, 182

 Terminus, 4

 The, 11

 Theory, quantum, 162

 There, 110, 189

 This, 11

 Thought, 3, 14

 Timaeus, the, 17, 20, 24

 Time, 16, 17, 31, 33, 49, 79; measurement of, 140; quantum of, 162; transcendence of, 39

 Time-series, 178, also cf. Temporal series

 Time-system, see Time-series, also 91, 97, 104, 179, 192

 Timeless point, 192; space, 86, 106, 114, 177

 Totality, 89

 Transcendence of time, 39

 Transmission, 26, 28; action by, 159, 190

 Tubes of force, 146

 Unexhaustiveness, 50

 Uniform object, 162

 Uniformity of change, 140; of space, 194

 Vagrant area, 103; solid, 101

 Veblen and Young, 36

 Velocity, critical, 193, 195; of light, 131, 195; relative, 130

 Volume, 92, 101

 When, 107

 Where, 107

 Whole, 58

 Within, 63

 Young, Veblen and, 36