Mind (journal)/Volume 33/Number 129/Space and Time: An Essay in the Foundations of Physics (I.)
MIND
A QUARTERLY REVIEW
OF
PSYCHOLOGY AND PHILOSOPHY
I.—SPACE AND TIME: AN ESSAY IN THE FOUNDATIONS OF PHYSICS (I).
By Jaroslav Císař.
Prefatory Note.[note 1]
Mr. Císař’s thesis for his doctorate at the University of Prague is written in the Czech language. He submitted to me the first draft of a free translation to obtain my opinion as to whether the line of thought, at which he had arrived independently, was sufficiently distinct from that which he had subsequently discovered in my Concept of Nature, to warrant its publication in English. I am decidedly of opinion that this is the case: in many respects our views are divergent, and where they agree Mr. Císař has emphasised considerations different from those on which I have founded my own arguments. But, apart from the minor consideration of its relation to my own work, the essay contains novel ideas which ought to be taken account of.
1st August, 1923.
- ↑ Note by the author: The present essay is a somewhat condensed version of a work which I recently presented as thesis for my doctorate of Natural Science at the University of Prague. When I put on paper the substance of it two years ago, and came across Professor Whitehead’s admirable book, I was in serious doubt whether I was not merely retracing a path which the author of the Enquiry and Concept of Nature had traversed in a much better way; but Professor Whitehead himself dispelled my doubts and encouraged me to pursue my line of thought. I cannot sufficiently acknowledge my deep gratitude and heavy indebtedness to him for the kind interest he has taken in my work, for the inspiration which I derived from my discussions with him, and for the encouragement he has so generously given me; without which this essay would have never been finished.
Jaroslav Císař.
Introduction.
1. The basis of the physical explanation of phenomena consists in a description of the phenomena in terms of relations and of the entities subordinate to these relations. The aim of physics is to furnish the simplest explanation possible, i.e., a description employing the minimum number of entities and relations; its ideal is thus to achieve an explanation which would describe all phenomena by means of a single entity and a single relation.
Every explanation must begin with a certain number of undefined conceptions, of which it is assumed that they are generally intelligible, and that they convey the same meaning to everyone. The concept of relation itself is the first of these undefined conceptions, and so is the concept of the related entity—relatum—which is thus presupposed at the same time as the subject of the relation; both concepts are of a purely logical character.
2. Physics, as ordinarily expounded, begins by postulating two fundamental relations, space and time, which it leaves undefined, assuming their general and unique intelligibility, and to which it endeavours to reduce all other relations. At the same time with them it assumes the existence of an entity which is the subject of spatio-temporal relationships, i.e., matter, which, after space and time have been assumed, can be defined by its fundamental property of impenetrability and the inability of the same particle of matter to be in two places at the same time.
The relations of space and time, which are thus made the basis of physics, are the result of a long and complicated process of abstraction, the beginnings of which are to be found somewhere in the dawn of animal intelligence; the current conception of them, however, is very far removed from that ideal simplicity and unique intelligibility which is assumed of them by physics, and which at a first glance seems to be their characteristic; this is attested by the fact that there is no consensus of opinion even as regards their assignment into the category of relations. All who have concerned themselves with the study of the foundations of physics, are aware what irreconcilable divergencies of opinion there exist concerning these two concepts, and in what a logical circle move the majority of the attempts at their definition. The definition of space and time as “pure extension” forms an example of such a hidden circle: the definition is devoid of meaning, if by the word “extension” we do not imply “space-ness and time-ness” which implication, however, renders our definition valueless.
3. It is obvious that such a lack of uniformity in its fundamental concepts cannot be very beneficial to a science; to what confusion it can lead is known to every reader of the popular and semi-popular expositions to which Einstein’s theory gave rise. Space and time begin to warp, to stretch, to shrink, and to pass through all possible deformations, without any knowledge on the part of the writer or reader what a deformation of space and time means or can mean.[1]
The present study is an attempt to arrive at a logically satisfactory definition of space and time which would also satisfy the requirements of physics. To accept space and time as undefined fundamental concepts of physics is not possible, owing to the previously mentioned lack of agreement as to their meaning; it is therefore necessary to derive them from some more fundamental and undefined concepts, as to the meaning of which there exists no such divergency of opinions.
4. The majority of definitions which have undertaken this task are very unsatisfactory; some, because they are logically defective, like the one mentioned above, others, because they yield nothing to our knowledge of physics, and hence are useless for its purpose. The answers given to the problem of defining space and time by the philosophers are mostly inadequate because, instead of a definition applicable to physics, they attempt to ascertain the metaphysical status of these two concepts; so, e.g., Kant’s famous attempt.
The present study is based upon the point of view that all ontological or metaphysical speculations are entirely and completely irrelevant to physics; physics produces its laws irrespective of whether the entities described by them are realities existing outside the mind, or merely its products. The truth of a given law of gravitation remains unaffected, whether we presuppose that the bodies subject to it are real things, existing even when we do not exist, or that they are an illusion of our mind.
5. Physical science is conditioned by perception; in perception it has its roots, percepts form its material, which it arranges, analyses and “explains” by reducing them to the smallest attainable number of elementary (i.e., further irreducible) percepts and relations.
The conception of percept is here to be understood in a somewhat broader meaning than that of mere sense-data simplified and schematised by a process of apprehending; percept in our use of the term must comprise also hypothetical entities, such as atoms, electrons, ether waves, etc., as long as by these names we understand genuine objects of possible sense-experience, unapprehended directly because of the imperfection of our senses, but inferable from sense-experience as a whole, and not physical fictions, serving as useful pseudo-perceptual symbols of certain mathematical expressions of natural processes. We must, of course, not forget that the idea of perception, even as currently understood, is very complex, and represents the result of a complicated activity of simplification, generalisation, intention, imagination, etc., to which it is subjected before it emerges from the rough and formless sense-data, in which it “enters” consciousness; these details, however, belong to the domain of psychology and logic, and although important also for the present study, they are not essential to it, and would lead too far, if we were to occupy ourselves with them.
6. Perception presupposes the existence of a percipient, of a mind; the necessity of distinguishing the perceptual contents of the mind from its unperceptual contents then leads to the postulate of a greater number of percipients—of minds—and thus to the pre-supposition of the so-called “external world”: the denial of this external world leads to a denial also of plurality of minds—solipsism, which philosophy endeavours to avoid for reasons mainly ethical, and which physics cannot accept if it does not wish to deprive itself of the most useful criterion of perceptuality.
Speculations concerning the ontology of the external world, its mode of existence, or the manner in which it can act upon the mind, are metaphysical speculations, entirely irrelevant to physics. Some of them are of vital importance to ethics, none of them are of vital importance to physics; the latter, as we remarked above, measures and arranges its material—percepts—irrespective of whether they are the result of the action of some real existence “in itself,” or the mere product of a mind which supposes that it apprehends.
These speculations, I conjecture, do not condition even the possibility of philosophy of the physical science, the aim of which is to determine the fundamental concepts of physics and the field of its jurisdiction; from ontological speculations physics does not need to take over any but those conclusions upon which all metaphysical systems are agreed, with the exception of absolute scepticism, which cannot logically be refuted, but which must be rejected on account of its absolute sterility. Roughly speaking, these assumptions are two in number: (1) there exist a plurality of percipients—minds capable of knowing; (2) there exists something which is not perceiving mind—the object of knowledge.
7. To these assumptions every philosophy of science which desires to have any social value—to be valid for more than a single mind—must add a postulate, which in the existence of what is not mind (the object of knowledge) demands the existence of something which as a characteristic of the object of knowledge is common to all minds, whether as a characteristic directly perceived, or inferred. By the external world, or briefly by the World, we understand this characteristic of the object of knowledge, which (at least potentially) is the common property of all minds, and on the identity of which they all agree.
7.1. The agreement of the characteristics of the object of cognition, postulated by this definition of the external world, implies the independence of the external world from the individual percipient, and accordingly the invariance of the external world with respect to a given type of mind (if by this type of mind we mean one subject to similar laws of thought), but not the independence of the external world from mind in general. Variously constituted minds will, in all probability, construct various “external worlds.” A world independent of mind in general would be a world common to all possible types of mind; but as minds constructed diversely from ours are not only unknown to us, but even impossible for us to imagine in respect of all possibilities of their type, such an absolute external world is not only unknowable, but a concept devoid of content.
7.2. It is clear then that neither the name of “external world,” nor its definition, conceals any supposition as to the mode of existence of the object of cognition; even if I presuppose (and this is a matter of indifference for physics) that only percepts can be an object of immediate knowledge, that therefore the mind can know only its own states, it is a matter of indifference whether I conjecture that these states are the mind’s own creations, or that they are caused by some external entity, existing outside the mind and independent of it. Physics, as we have already said above, is concerned only with percepts, and not with the manner in which they enter the mind, or arise within the mind; and the view to which we give the preference in this matter is a matter of personal predilection; as far as physics is concerned, our choice will remain without consequences. If it better satisfies our metaphysical preconceptions and habits, or preferences, we shall not commit any error if by the external world we understand some common source of percepts, the action of which on cognately constituted minds gives rise to analogous states within these minds; that the se states are analogous is inferred by each of the given minds from contact with others. Such a view of the external world as a source and cause of percepts, I think, actually predominates with the majority of physicists, perhaps implicitly with the majority of them; from the point of view of physics it cannot be erroneous, while from the point of view of natural philosophy it means a certain facilitation of the processes of thought by fixing a symbol for a comparatively concrete concept. It must, however, be clearly remembered that from the point of view of physics we are making no new assumption, but that we only replace the postulate of the correspondence between the contents and relations of two minds (which must exist in both minds, if it is to be a correspondence) by the postulate that a certain part of these minds is common to both; whether “within them” or “beyond them” is a question of metaphysics, with no bearing upon physics at all.
The Concepts of Extension and Order.
8. From what we have said so far, it is clear that it is necessary to consider the space and time of physics as real to the same extent as that to which we consider as real the perceptual content of our minds, or the “external world” of physics, as defined in previous paragraphs. This means that we are not in the least concerned with their transcendental reality. Physics knows only perceptions, and physical space and time must therefore be something within these perceptions: if we did not exist, space and time could, but, as far as physics is concerned, need not, exist.
According to the theory held in the present essay, space and time are an abstraction from the perceptual content of the mind, an abstraction resting upon the fundamental logical concept of order. The concept of order presupposes the existence of entities among which it can exist as a relation; and that again assumes the existence of another relation, which is usually called the relation of externality, and for which—to avoid the too spatial implication of the word externality—in what follows we shall employ the term ‘exclusiveness’. The totality of entities which lend themselves to a given type of order, form a collection, or an aggregate; and if the order of a given aggregate is to be unambiguous, the members of this aggregate must be uniquely determined, and must be mutually exclusive: they must be what we shall later denote by the term elements of the aggregate. Externality and ‘elementarity’ are only different consequences of the same relation, which we denote by the term extension, and which thus together with the relation of order, will constitute the basis of our definition of space and time.
To formulate this definition, to find what in our perceptions constitutes space and time, we shall have to undertake an analysis of the perceptual datum; and for this purpose we must collect and define a number of concepts which will serve us as the tools and foundations of our analysis. As we wish to eliminate the danger of a possible logical circle in our definition, we must be on the alert to prevent any specially spatio-temporal concepts from creeping surreptitiously into our definition of space and time through the material on which this definition is to be based; consequently great stress will be laid upon the purely formal, non-perceptual character of our basic materials; hence also the purely logical, formal character of this section.
Extension.—9. The first fundamental and undefinable, although generally and unambiguously intelligible relation, which we shall place at the basis of our investigation, is the relation of whole to part, which we shall briefly denote as the relation of inclusiveness. This relation is purely formal: although it holds good as a special case in the world of our perceptions, it can be also predicated of non-perceptual, purely conceptual entities: of numbers, speech, sensations, etc.; it therefore does not of itself contain anything spatial or temporal.
Extension we shall define as the possibility of inclusiveness: a given entity will be said to possess extension, to be extended, if taken by itself it admits of the relation of inclusiveness, i.e., if it is divisible into parts. The relation of extension is implied in the relation of inclusiveness, and is therefore as purely formal a concept as the latter; but although, being void of all spatial and temporal content, it possesses the characteristics necessary for the purposes of our definition, it is not adequate to define space and time. To say, for instance, that space is pure extension, would be equal to maintaining that any extended pure concept, e.g., human speech, has spatial relations; the fact that all extended entities admit of spatial representation is not, as we can readily see, a proof of the spatiality of extension: the spatiality which appears in such representation, is not an attribute of extension, but of the symbols employed.
10. The term entity we have so far employed to denote any existens which is not contrary to the laws of thought, but without any further limitation; it can be of a physical as well as only an abstract, purely conceptual nature. In what follows we shall be concerned mainly with entities which admit of division into parts; and the term entity we shall henceforward use to denote any existens, whether physical or purely conceptual, possessing extension. This means that, of classes of pure concepts, only those will be termed entities, which it is possible to define as collections, and not solely by intension.
11. If we exclude the case where an entity A can be its own part (“part” therefore meaning “proper part”) there exist, as is shown in every text-book of logic, four possibilities of extensional relation of entity A to entity B:
(a) a part of A is B (B is a part of A); or
(b) vice versa; or
(c) a part of A is also a part of B, in which case the reverse is also true; or
(d) there exists no part of A which is also a part of B, and vice versa.
The first case is only the reverse of the second, but the relation is the same, and we have already applied to it the name inclusiveness; the relation described by the third case we shall call intersection, and that indicated by the fourth we shall denote as exclusiveness. Inclusiveness, exclusiveness, and intersection, are three different aspects of the purely formal relation of extension, and are therefore equally free of all spatial and temporal content.
12. Every entity is defined by a number of attributes, by the totality of which it is distinguished from all entities not identical with it. These attributes can be divided into two classes: those attributes which are common to all parts of a given entity, and by virtue of which (with respect to which) this entity has extension; and attributes, which are not common to all parts of a given entity and which, in so far as the extension of the entity is concerned, can be neglected. The totality of the attributes of the first class in a given entity we shall call its extensional characteristic. It follows from our definition that in every entity it will be possible to find either parts which still are entities (have extension) with respect to the extensional characteristic of the original entity, and also parts which are without extension with respect to this characteristic, or only the latter; these latter we shall call elementary parts, or briefly elements of the given entity. Thus, for instance, the entity “the family of Mr. X,” consists of entities: “the parents X,” “the children X,” etc.; and of the elements: Mr. AX., Mrs. MX., their son FX, their daughter BX, etc.; the entity “colour” will have parts which in their turn are entities, for instance the colour of sodium light, but also parts which have no extension with respect to the general characteristic which makes colour an extended entity—therefore elementary parts—e.g., the colour Na1.
It is therefore possible to consider every entity as an aggregate of entities, which are its parts, or an aggregate of elements, which, although its parts, are no longer entities with respect to its extensional characteristic.
12.1. A given entity can have a limited or an unlimited number of elements, and will thus be either a finite or an infinite collection. In the first case it is easy to find all of its elements by means of the relation of inclusiveness (applied repeatedly as a relation of part of a part until a part is reached which does not admit of a further application of this relation); in the case of infinite aggregates the task is a more complicated one, and we shall later—in a concrete perceptual case—attempt to find a different method.
Co-intersection, dissection and complementary parts.— 13. In what follows we shall find it useful to have at our disposal a few concepts based on the relation of extension. In the first place we shall use the term co-intersection of n mutually intersecting entities to denote the entity which is the aggregate of all parts which all the n entities in question have in common. Again, if in a given entity R we can find two non-intersecting entities A and B such that no part of R can be found which is not either a part of A or a part of B, or wholly composed of two parts, one of which is a part of A and one a part of B, we say that we have dissected R into A and B; A and B are called complementary parts of R, and each is called the other’s complement in R (written: B is co-A, and A is co-B).
Order.—14. The second relation which we shall place in the foundations of our logical analysis, is the relation of order. The concept of order is based upon the properties of a relation expressed by the words “before and after” or “between”. Between these two relations there is a close logical connection: If A is before (or after) B, and B is before (or after) C, then B is between A and C. From this, and the transitiveness of the relation “before and after,” and from the other properties of the relation “between,” as they are usually given in works on the foundations of mathematics, the whole theory of order can be developed. The exact logical formulation of the properties of these order-relations offers considerable difficulties with which we shall not concern ourselves here, referring the reader to standard works on the matter; in these works we can also satisfy ourselves that the order-relations can be defined without any reference to space and time, and are intrinsically independent of these concepts; the concept of order is there fore purely formal and its use in the definition of space and time cannot involve us in a vicious circle.
14.1. We say that a given aggregate of any elements (or mutually exclusive parts) is ordered, if of any two of its elements we can say that one is before the other, or, what is the same thing, that one is after the other; or if of any three elements of the aggregate we can say that one is between the other two. A given aggregate of elements is said to be properly ordered, when the case “X is between A and B” excludes the cases “B is between X and A” and “A is between X and B”'; that is, if we say that A is before B, we cannot also say that B is before A.
Boundary.—15. With the aid of the concept of dissection it is possible to define the concept of boundary, which is of the greatest importance in the theory of ordered aggregates; the concept is one of the most difficult to define in a purely formal way, and, as in the case of order, for its exact definition we shall refer the reader to standard works on the foundations of mathematics. We shall here content ourselves with an incomplete definition which we think sufficient for our purpose, and define the boundary of the entity R in the entity X (of which R is a part), as that characteristic of R, which R has in common with co-R, and by virtue of which R is also distinct from co-R.
The definition is sufficient to enable us to see that the boundaries of two entities can intersect (that two entities can have a part of their boundary in common). Of two non-intersecting entities, which have a part of their boundary in common, we shall say that they are joined; two non-intersecting entities which are not joined we shall call separate. Further, we shall say that an entity R is enclosed in the entity X, if every part of R is a part of X, and if for every part of R (say r) there can be found two parts of co-R, (say a and b) such that r is between a and b. It can be readily seen that if R is an entity enclosed in X, the whole of its boundary will be a part of the boundary of co-R, the rest of the boundary of which will be defined when the boundary of X (being considered a part of a larger entity) is defined; but in practice, as a rule, the boundary of X (which may have no boundary) does not concern us, and all we need is the boundary of some enclosed part of it.
Connectedness and Continuity.—16. This takes us directly to the very important concept of continuity, which the concept of boundary enables us to define. We shall say that an entity is connected, if any and every one of its possible dissections gives us two joined entities; then by the continuity of a given entity we understand the fact that every one of its possible divisions (division meaning repeated dissection) gives us an aggregate of entities such that no matter which member or collection of members obtained by the division be taken (save the collection which is the complete entity), such member or collection will always be joined to at least one other member of the aggregate. From this it follows that given any two separate, connected parts of a continuous aggregate X, say A and B, it is always possible to find another connected part of X, say C, which will form a connected entity with A and B (is either joined to both or intersects both). A continuous aggregate of elements is called a continuum.
Ordinal Characteristic.—17. In an earlier paragraph we found that a given entity possessed extension by virtue of a certain class of attributes, which we denoted by the term extensional characteristic; similarly in the case of ordered aggregates we must assume the existence of something with respect to which the aggregate is ordered—by virtue of which the elements of the aggregate are arranged in one particular way and in no other. Without making any assumption as to the nature of this something, which we may, analogously to the extensional case, call the ordinal characteristic (or the ordering relation), we can readily see that this characteristic must in a certain way be the same for all elements of the aggregate, and in a certain way different for every one of the elements: the sameness of the characteristic for a given aggregate will depend upon the nature of this aggregate; the difference for the various elements will be either a result of convention, of a rule which we arbitrarily set up for the case in question, or—as is the case when the aggregate is given us already ordered—a result of comparison of the various elements, and consequent determination of the characteristic in which they all differ. Usually the comparison consists in measuring the amount of a certain quality possessed by the various elements; and we say that the various elements differ as to the “degree” or “intensity” of the ordinal characteristic, and are ordered according to the variation of that intensity.
17.1. It is very difficult to state more than this about the ordinal characteristic without a loss of generality; the definition of the characteristic in each special case will, as we said before, depend upon the nature of the aggregate in question, and the determination of the manner of variation of this characteristic will be a matter of practical physics, of practical possibilities and necessities, which will give rise to a certain method of ‘measurement’. One suggestion which offers itself at this point arises from the fact of the sameness of the ordinal characteristic for all elements of the aggregate: in this respect the ordinal characteristic invites comparison with the extensional one, and it is suggested, that if the two are not identical in all cases, it should be possible for them to be the same at least in some cases. An investigation of the relation of the ordinal to the extensional characteristic offers an interesting field of research, into which it is, however, not our intention to enter in the present essay.
17.2. Neither shall we make it a part of our present task to study the equally interesting and more important problem of measurement, which is concerned with quantitative determinations of the variation of the ordinal characteristic; it will here suffice if we say that the possibility of measurement of a given continuous aggregate is dependent upon the existence of a certain relation between every two elements of the continuum, which relation it is possible to express numerically, and to compare as to equality or inequality between various pairs of elements; this relation we denote by the term “interval,” and spatial “distance” is a particular case of it.
Dimensionality.—18. The ordinal characteristic of a given aggregate may be simple or complex, as its variation between two arbitrary elements of the aggregate can be determined by a single comparison or by a number of comparisons mutually independent; thus, two colours may be compared as to their wave lengths and as to their intensity (energy), and no amount of measurement of the one will give us any idea of the magnitude of the other. The number of such mutually independent comparisons necessary to determine the relation between two arbitrary elements in the continuum is called the dimensionality of the aggregate: thus, where one determination is sufficient, the aggregate is said to be one-dimensional, where two are necessary, two-dimensional, three, three-dimensional, etc.
Co-ordinates.—19. An important consideration which offers itself to us in this respect is the fact that the elements of one n-dimensional continuum can be brought into a unique one-to-one correspondence with the elements of any other n-dimensional continuum; in particular, a one-dimensional continuum can be so correlated with the series of real numbers, an n-dimensional continuum with an n-dimensional series of numbers. Such correlation, the assignment to each element of the given continuum of an element of a corresponding number-series (containing n real numbers in an n-dimensional series) we call the introduction of co-ordinates, and the numbers thus assigned the co-ordinates of the respective elements. It follows from the properties of the number-continuum that there is an infinite number of possible correlations of a given continuum to its number-series: it is the task of the theory of measurement to indicate a method by which, once a certain finite number of elements is assigned its co-ordinates, the co-ordinates of any other element can be determined uniquely. But even before this method is known, co-ordinates serve a certain purpose, namely as mere names of the elements to which they are assigned: they are purely descriptive.
With this preliminary equipment we can embark upon our main task of analysing our perceptions and extracting from them that something which we so unconcernedly call space and time.
Analysis of Perception.
20. The thirst of the mind for knowledge of something outside itself must be regarded as a disposition inherent in the nature of mind. The way in which mind comes into contact with the object of its knowledge is a mystery which neither physics nor philosophy will probably ever solve. If we accept the postulate of the existence of the external world as something which is common to all minds, but independent of each individual one, then perceptions are the result of contact between the mind and this external world, and the bodily senses, themselves parts of the external world, enlarged both in the scope and the precision of their powers by physical instruments, are the media through which the contact is effected.
In perception the external world is presented to the mind as a phenomenon, a something of which the first quality distinguishable by the mind is variety, manifoldness, or, in other words, divisibility into parts. Consciousness of this manifoldness results from the recognition that in perception, as it is found in the mind, there is more than the mind is immediately aware of; and it arises, we may suppose, from the limitation of the mind, from its inability to comprehend totality, to seize the whole of being as such, in its entirety. Thus, in consequence of our definition of extension (par. 9), we may say that nature, the external world, possesses for the mind, extension; whether we say that this extension is discovered in nature, or imposed upon nature by the mind, will depend upon our philosophic views or preconceptions.
From the effort of the mind to apprehend something outside itself we must infer that the awareness of something which is not mind, the awareness, that is, that the mind itself is not the whole of reality, but only a part of it, constitutes a fundamental characteristic of its being; it is in this internal evidence that we have to look for proof that it is not commensurate with the universe, which fact will then account for its inability to comprehend the totality of being as a single whole (if we are justified in supposing that, in order to comprehend the whole universe at once, the mind would have to be commensurate with it), and for the necessity of dividing this totality into fractions which it can comprehend.
21. Since we discover extension in the whole of our perceptual data, it follows that of parts of the world as such we can predicate everything which we can predicate of an entity having extension, i.e., inclusiveness, exclusiveness, and intersection of its various parts, and that we can postulate in them the existence of elementary parts, i.e., parts which have no extension as regards any attribute by virtue of which they are parts of our perceptual data. A satisfactory definition of such an elementary part will be attempted in a later paragraph.
The relations which are deducible from the fundamental relation of extension do not, however, exhaust the logical relations which the mind discovers in (or imports into) the totality of its perceptions. Of the parts of our perceptual data which can be called separate, it is hardly possible, on the basis of the relation of extension, to say more than that they have no common part; but for that very reason we cannot usefully apply this relation to the elements of our experience, to relations of which we endeavour to reduce phenomena. The problem is made easier by the fact that the mind discovers in (or imports into) its perceptual data a second fundamental relation, namely that of order, i.e., we can arrange individual parts or components of our perceptual data in series according to many different characteristics of given perceptions.
22. For reasons of convenience, we will give a name to the aggregate of perceptions of a given mind calling it the Experience[2] belonging to this mind, or, in short, the Experience of a given mind. Any part of its perceptual data which the mind is able to distinguish in its Experience, we will call an event. When we thus distinguish an individual part, or event, in a given experience, we have, in the first instance, to determine its boundary, as we defined that term in the foregoing chapter; so that we can define an event as a delimited part of the perceptual data of a given mind, and Experience as the aggregate of events which the mind distinguishes, or delimits, inits perceptual data. An elementary part of a given perceptual datum we call an element of experience.
Having recourse to the formula used above, we can define an element of experience as a part of Experience which has no extension as regards any attribute, by virtue of which it is a part of Experience; this definition is, however, unsatisfactory in so far as these attributes are not clearly defined, and we must therefore replace it by a better one.
23. If we define a spherical event[3] as a connected event, the co-intersection of which with any other spherical event which it intersects is itself a spherical event, we can define an element of experience as the co-intersection of a class of spherical events which have a common co-intersection, and which comprise every spherical event having a common co-intersection with all the members of this class.
23.1. We arrive at this definition in the following way:—Ideally we can regard every event as part of another event, and vice versa we can regard every event as composed of other events which are its parts. The ideal delimitation of an event depends on nothing in the events themselves, on no particular characteristic of Experience, but may be arbitrarily determined by the cognitive mind. From this it is evident that a given event or a given section of Experience can be divided ideally an infinite number of times, and that every section of Experience will contain an infinite number of (possible) spherical events, which will have an infinite number of mutual co-intersections; and these, by our definition of spherical event, will themselves be spherical events. If we take a co-intersection of any number of mutually intersecting spherical events, we can always find a spherical event, the co-intersection of which with these original events will be a part of the original co-intersection; and by increasing the number of such spherical events we can obtain as small a co-intersection as we please of a finite aggregate of spherical events. If we include in this aggregate every spherical event which has a common co-intersection with all the members of the aggregate, we tend to an ultimate element which is the co-intersection of the infinite class, an element which will not be further divisible into parts and which will not intersect any element other than itself; in other words, an element of experience.[4]
23.2. That an element of experience has not parts can be proved as follows: Let us suppose that an element of experience R has parts A and B; then it is possible to find a spherical event X which does not contain the whole of R, but only one of its parts. Since a co-intersection of n spherical events is, by definition, contained in each one of these events, the event X is not a member of the class of events, the co-intersection of which is the element of experience R; but since an element is a co-intersection common to all spherical events which mutually intersect, such an event X cannot exist; therefore the element R cannot have parts.
24. As the number of events comprising a given element of experience is infinite, an infinite number of determinations would be required to identify a given element. Now that is impossible in perception; in perception, therefore, we substitute in place of an element of experience a very small event representing the perceptual limit of our sense faculties, and comprising the given element together with a number of elements which cannot be distinguished from it in perception; to determine such a perceptual element of experience a finite number of relations between this element and the other elements is sufficient. The sum of the relations of a given element to the remaining elements we call its position in a given Experience, and the sum of those relations which suffice for its approximate determination, in the way indicated above, its approximate position.
24.1. It may be, indeed, and it most probably is so, that, even apart from the imperfection of our senses, the mind is more limited in perception than in thought, and that there is inherent in Experience itself a certain characteristic, which uniquely divides Experience, or a given section of it, into a finite number of delimited events, which the mind can further divide in thought but not in perception. Since such a characteristic must needs be independent of the mind we cannot predict it a priori, and its existence or non-existence can only be proved empirically; this characteristic—without making for the present any assumption as to its existence or non-existence—we call the atomicity of Experience.
From the manner in which we arrived at elements of experience it is evident that these elements will form a continuum; this continuum will, however, only be conceptual and thus does not preclude the possible perceptual atomicity of Experience. Even if it were ascertained by exact perceptual analysis that Experience is atomic, it will nevertheless suit us better, from the point of view of facilitating mathematical analysis, to regard it as continuous, with the understanding, of course, that results based on this hypothesis are only approximately valid, like statistical averages within aggregates of great numbers of elements, the individual significance of which in relation to the whole is infinitesimally small.
25. The parts of perceptual and non-perceptual content of a given mind present a number of attributes which can be arranged in series and enable us to order these parts; such ordinal characteristics are, for example, pleasurableness, colour, sound, intensity, and soon. We facilitate the ordering of the parts of the mind’s content according to this or that characteristic by correlating the various “degrees” of the attribute with the series (or part of a series) of natural numbers, which is nothing but an ultimate abstraction from an ordered aggregate, of which the elements have become disembodied into mere concepts of position in a series, devoid of all perceptual content.
A given attribute is “simple” (or one-dimensional) if all its degrees can be uniquely determined by their correlation with members of a single series of natural numbers, “complex” (or n-dimensional) if members of more than one (that is n) series of natural numbers are required for the unique determination of an element which possesses this attribute; a one-dimensional relation gives rise to a one-dimensional aggregate or continuum (according as it comprises a finite or an infinite and continuous number of degrees) and an n-dimensional relation similarly to an n-dimensional aggregate or continuum.
26. If, within the world of our perceptions, we try to order parts of our perceptual data under any ordinal characteristic arbitrarily chosen from among those which we immediately apprehend, we find that we can order (that is, correlate with one or more series of natural numbers) under this characteristic a determinate part of the totality of events into which the mind divides its Experience; but we can as a general rule expect that in a given Experience we shall always be able to find events which remain outside the ordering, that is, outside the system of co-ordinates which we have chosen. The requirement of simplicity and uniformity, upon which a satisfactory physical description of a phenomenon is based, would be disregarded to a considerable extent, if we arbitrarily took as the basis of our description any one of the characteristics immediately apprehended: instead of one, uniform description we should obtain a whole series of descriptions, each one of which would only describe the Experience partially; and we should never know how much of the Experience still remained to be described, since we could not be certain that we should not eventually succeed in discovering events distinguished by an attribute which could not be reduced to the one we had selected. In order to satisfy that requirement we must find an ordering relation—an ordinal characteristic—which could be predicated as existing between any and all parts of Experience, and which could therefore serve as an ordering principle for all events. The discovery of such a relation will be attempted in the following section; meanwhile we will postulate its existence in Experience and call it a formative relation. A formative relation is thus an ordering principle which can subsist between any and all parts of Experience whatever; the “Form” of a given aggregate of events will then be the sum of relations of which the existence can be predicated between all parts of this aggregate without exception.
26.1. The conception of the Form of an Experience is without meaning unless we determine the kind of events into which we divide this Experience, as a given Experience can be divided into events in an infinite number of ways; among these ways, however, there exists only one division which is uniquely determined for every Experience, and that is the division into elements of experience. We may therefore define the Form of an Experience as the sum of formative relations existing between the elements of this Experience.
26.2. In this connexion we must recur to the result we reached in our consideration of an ordering relation towards the end of the preceding section (17.1), and inquire whether the relation, which we have called formative, is a purely ordinal, or at the same time also an extensional characteristic. At this juncture I see no clear logical ground for my view—which may turn out to be the denial of mutual independence between order and extension—but it seems to me that the answer to our question must admit the latter alternative; although the element of experience, as defined in this chapter, is an abstraction and an intersection of colours, sounds, touches, and all other possible primitive sensations, I do not see how it could be attainable unless there were present in all these parts of Experience a characteristic of a special kind proper to these phenomena and enabling us to speak of the intersection of a given colour with a given sound. I think, therefore, we may safely define the Form of Experience as that attribute, in virtue of which we can divide Experience into elements (or distinguish them in Experience), and by reference to which we can also uniquely order all these elements.
- ↑ Space and time are even said to be discontinuous! Some authors (among them Poincaré) admit the possibility of “atoms of time” as a result of the Quantum theory (even a book written with such rare lucidity and common sense as Eddington’s Space, Time and Gravitation, is not flawless in this respect).
- ↑ I use the term Experience (with a capital) as the best substitute I can find in English for the Czech word “dění” which I used when I conceived the work in my own language. The word “dění” conveys much better my meaning, its English equivalent being “the something that is going on”. “Experience” shares with it the advantage of being very non-committal as to any particular view of its mode of existence.
- ↑ The term “spherical” here does not imply the geometrical properties of a curved surface: a cube, for instance, in so far as it is an event, is a spherical event.
- ↑ Prof. Whitehead, who very kindly discussed with me a considerable part of the present work, on the basis of this definition proposed an alternative definition of an element of experience which, while accepting mine, made room for his view that there can be classes of events which, though they stand to one another in the relation of inclusiveness and so of intersection have no co-intersection common to all; his proposal is as follows: “An element of experience is a class of spherical events such that(1) each of its finite sub-classes has a co-intersection;(2) every spherical event, which has a co-intersection with each finite sub-class of a given class, itself belongs to this class.”As will be seen, this definition agrees with mine except that Prof. Whitehead calls an element of experience a class of events which have a given co-intersection, whereas I call an element this co-intersection itself. Both definitions have their advantages and disadvantages; I prefer my original definition, because it agrees with our current view of a point, and because it has its justification in perception, in that it leads us direct to the perceptual limit of the compass which we usually understand by the conception of point.