# Mind (journal)/Volume 33/Number 130/Space and Time: An Essay in the Foundations of Physics (II.)

Second part of the article. For the first part see 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 (II).
By Jaroslav Císař.
The Concepts of Time and Space.
27. If we concentrate our attention on the whole content of our mind, we find that this content exhibits one universal characteristic of which the mind is aware before it begins to analyse it into parts, the characteristic, namely, of activity, becoming, perpetual and all-pervading change. The content of our mind tells us in the first instance and before all else that something is going on; and whether we premise that it is going on within us or outside us—that this activity is the activity of the mind itself or that a part of this activity is the reflexion of an external fact—activity, becoming, is an incontestable fact which the mind observes in its content.
If we divide the content of the mind into the perceptual and non-perceptual, and, contenting ourselves with the “social” criterion of the difference between the two, say that the former is received by the mind from the external world, from nature, and that the latter is produced out of the mind itself, then nature appears to us in our perceptions as a process, as a swarm or succession of events, each of which is marked by this characteristic of its being.
Time.—28. If we reflect upon the content of the mind as a whole, and try to find some formative relation which is valid of this whole, we always come upon this dual character of the mind: the perceptual part of the content we can order under certain perceptual attributes, the non-perceptual part under non-perceptual attributes, but, as a general rule, we cannot order parts of the one class under an attribute of the other. ​If, however, we consider what is common to the ordering of both classes, and postulate the existence of an ordering relation for the whole content of the mind, perceptual and non-perceptual, we shall not have great difficulty in identifying this formative relation with ordering in time.
Time thus appears to us as an abstraction from that growth, change, activity, which is the first characteristic of which the mind becomes aware, when it looks into its own content; it is an abstraction from the ordering of the whole content of the mind. We may therefore define it as a formative principle, by the aid of which the whole content of the mind can be divided into slices common to its perceptual and its non-perceptual part, and which enables us to arrange the whole content of the mind in a one-dimensional continuum of these slices in such a way, that, of any two of them, we may say that one is after the other, and of any three of them, that one is between the other two.
29. This view conforms to the fact that the mind is able to say of any two parts A and B of its content whether they are (in the mind) contemporaneous or not, and, in the latter case, whether A is after B, or B after A. This fact remains a fact, whatever metaphysical explanation we give of it; it will depend upon our metaphysical position whether we ascribe to the mind the capacity to distinguish the temporal order of two different parts of its content, or to import this order between the parts; for us the important thing is that in any case this order ultimately exists, and from our standpoint we can regard the fact of the temporal ordering of the mind’s different parts and the division of the mind’s content into elements which are so ordered, as a basic datum which admits of no modification. The type of order under which the whole content of the mind is arranged in a time-series we call inner time.
30. The elements of the continuum which arises with the temporal ordering of the mind’s content we will call instants: the instant of this definition is not the instant of time of current phraseology, but the whole content of the mind of which we can say that it is simultaneous with a given element of experience. An instant in this sense will have a perceptual or a non-perceptual content; since for our purpose only the former is important, it will be as well to separate the two and give a special name to the perceptual content of an instant, ​which we will call a moment (analogously to the use of the term in photography). A moment will thus be the sum of elements of experience which are simultaneous with any element of experience, or, in other words, the sum of the elements of experience in a given instant. Perceptual Experience, then, ordered in accordance with the attributes of simultaneity and non-simultaneity, can be regarded as a one-dimensional continuum of moments, and physical time as an ordering principle, by which the moments in this continuum are ordered, a type of order, by which elements of experience are ordered in a one-dimensional continuum of moments.
30.1. It may be asked whether the conception of time or that of order is the more fundamental: I imagine that logically the latter, being the broader conception, is the more fundamental, but psychologically the former; psychologically, I think, the conception of order was derived from direct experience of (inner) time, which, as we have already indicated, is an inner, indisputable, and inevitable reality. There, I believe, we have also the reason for the psychological predominance of physical time over physical space, which we will define later: whereas logically both are merely variant types of order, psychologically the former is fixed far deeper in the constitution of the mind than the latter. Logically the relation of temporality is subordinate to the relation of order, as is evident from the fact that the fundamental relation “between,” while it is applicable to temporality, is applicable also to many other aspects of reality. Time is logically only a special instance of order where the relation “before-after” is expressed by the terms “earlier-later”.
31. I think, however, that I ought to emphasise again that our definition is in no way intended to solve the question as to the metaphysical character of time; our aim is to find a definition which, without any metaphysical preconception, will tell us what time is in the perceptions with which we have to do in physics and what properties we can ascribe to it in consequence. As far as our definition goes up to the present, we can already reject as devoid of meaning any view which would affirm any deformation or discontinuity of time. Time is not a continuum or aggregate of which we could possibly predicate these properties; time is a mode of ordering a given totality of our perceptions, and it is this totality which may be discontinuous, or may have certain metrical properties. Time of itself cannot have metrical properties; and when we speak of “measuring time,” it is merely a somewhat misleading name for the attempt to find in the continuum, in which we arrange our time percepts, a structure which will permit of a ​unique determination of the “interval” between two moments. The problem arising out of this is closely connected with the problem of space and time, and its solution is very important for physics. This solution i is, however, not necessary for the mere definition of space and time, and we will postpone it for a separate study, in which we propose to take up the problem of measuring continuous aggregates, with especial reference to time and space.
Space.—32. Moment is the name we have given to the sum of elements of experience which, for the observer who is the subject of this Experience, are simultaneous with a given element, or, in other words, with such an ideal part of a given Experience as will constitute the perceptual content of the mind in an inner instant. We chose the name moment in order better to distinguish between an instant of non-perceptual and an instant of perceptual experience. We speak of the parts of a given Experience, which come within the same moment, as being co-momentary. Between two co-momentary events there exists no relation of temporal sequence; a moment is thus nature as it would appear to a given mind in an instant, a section of nature in which there is no temporality. Such a section is, indeed, an abstraction which the mind can explore only in idea; reflexion upon it occupies a finite section of the inner time series, whereas the moment itself occupies no such finite section.
33. In the concept of moment the mind has not yet reached a reality which is not further reducible: a moment, too, can be found to be divisible into parts, and between its separate parts relations of order can be found to subsist. Proceeding along the path indicated in the foregoing section, we will postulate in every moment a relation which is applicable as between all parts of the moment: then this will be a formative relation, and the mode of ordering the elements of experience in a given moment we call the Form of this moment. Brief reflexion leads us to the conclusion that this relation, which exists between the co-momentary parts of experience, can be identified with the relation which we ordinarily call spatial. Momentary space can thus be defined as the sum of those ordering relations of a given moment which can be predicated as existing between any and all parts of this moment.
What this definition in effect says is that all events and all objects, with which physics has to do, occur in the space (that is, in spatial relations to the other parts) of some moment; taken together with the definition of time, it tells us that, through the mediation of this moment, they likewise ​occur in time. It is, then, nothing but an affirmation of the fact that all percepts are in space and in time; no new truth certainly, but one strangely overlooked in most attempts to define space and time, which omit to notice the fact that space (or spatial relations) exists only in perception (or in objects which this perception represents, if our philosophy requires such reservations), and in ideas which are ideas of perceptions, and that “physical time” likewise exists only in perceptions. The essence of our definition lies in the fact that it defines space and time as relations between percepts; and it permits of the distinction between time and space by making time also a relation between the non-perceptual parts of the mind.
34. We know empirically that elements of experience forming a given moment can be ordered by the spatial relation in one-, two-, or at most three-dimensional continuous aggregates; every element of experience contained in a given moment can be uniquely determined by the correlation of three co-ordinates or by correlation with a three-dimensional continuum of real numbers; a moment of experience is thus a three-dimensional continuum of elements of experience.
A one-dimensional continuum of co-momentary elements, or instantaneous co-momentary “points,” we will call an instantaneous line, a two-dimensional an instantaneous surface, a three-dimensional an instantaneous body.
35. Since the temporal ordering of a given experience is independent of the spatial ordering of events in different moments of this experience, and vice versa, and since we have found that the temporal ordering of experience is one-dimensional and the spatial ordering three-dimensional, it is evident that the continuum of elements of experience which forms the totality of our perceptions is four-dimensional.
Elements of experience can thus be ordered in from one up to four-dimensional aggregates, of which three, mentioned in the foregoing paragraph, will be instantaneous, being composed of instantaneous co-momentary elements, and four enduring, or persistent, forming a one-dimensional continuum of which the elements are instantaneous points, instantaneous surfaces and instantaneous bodies; we call them progressive figures of the first, second, third and fourth order. Specially important for the construction of a geometry of Experience is the first figure of the second group, which is a progressive continuous aggregate of elements of experience (or instantaneous points), that is, the progressive figure of the first order, which we call a route.
The spatial relations of which, on the thesis here advanced, ​an intelligible account can be given, are spatial relations of co-momentary parts of Experience; the type of instantaneous space is once and for all given by the ordering of the moment to which it belongs, and which determines its differentiation from, or analogy with, the spaces of other moments of the same Experience.
36. From our definition of a spatial relation it is evident that spatial relations can be predicated as existing only between parts of the same moment. In spite of that, however, we are accustomed to speak of spatial relations between two events which are not co-momentary, thereby implying that the content and ordering of two distinct (not identical) moments are comparable. Such a comparison is in actuality rendered possible by the fact that moments of a given Experience constitute a psychologically continuous series; in a given psychological instant (specious present) there is present to the mind a whole series of very “near” moments, not at all clearly distinguishable from one another, which not only makes it possible to compare the content and ordering of moments in close proximity, but enables us to create a certain kind of space common to more moments than one and to speak of change of position (movement) as a reality given by immediate perception.
37. It would, however, be very difficult for the percipient mind to keep its bearings in an uninterrupted stream of events and to compare more distant moments (moments, that is, which are outside the mind’s field of vision in a given “psychological instant”) through the mediation of all the moments which fill the interval between them in its Experience.
Temporally, different parts of the mind vary, figuratively speaking, in the clearness with which they are apprehended by the mind: perceptions “present,” with those immediately adjacent in past and future (for in a given psychological instant the mind actually anticipates the perceptions of the immediate future) are the clearest, and this clearness is subject to a progressive diminution which is gradual with the past and very rapid with the future. Out of the past and the future there emerge only those swarms of perceptions which remain “the same” in a finite section of the time series of Experience, which, in other words, endure. The mind, we may say, has a special preference for permanence, for what in the ever changing stream of Experience remains the same; and following this preference, or actuated by this necessity, it looks for those properties and relations in Experience which endure.
​Impelled by this desire for permanence the mind has formed the concept of permanent space, which may be defined as the sum of the enduring spatial relations in a given Experience or finite section thereof (understanding by a finite section of Experience the whole intervening Experience between two “distant” moments). Permanent space thus appears as the space of an imaginary moment which the mind has allowed to become, as it were, petrified in thought, by preserving from given successive moments only those relations which remain unchanged in these moments. This space, or the imaginary moment which it represents, the mind carries over from moment to moment, compares it with momentary spaces, and in the spatial ordering of these moments notes its deviations with respect to permanent space.
When, therefore, we speak of two non-contemporaneous objects or events in space, it is simply an assertion of the fact, that we compare the position of these two events in their respective moments by means of corresponding positions in this petrified enduring moment, which we have unwittingly identified with each of the two moments.
38. In order that we may be able to distinguish between elements of real moments of experience and elements of this imaginary “enduring” moment, we call the latter elements timeless points. These will be imaginary elements of experience which will retain eternally the same spatial relations towards one another; in every momentary space which “underlies” them, one element of experience will correspond to (or coincide with) each point of permanent space, so that each timeless point will be a route of “instantaneous points”.
38.1. A one-dimensional aggregate of timeless points of a given permanent space (a timeless line) will intersect each one of the moments belonging to it in an instantaneous line, a two-dimensional aggregate (a timeless surface) in an instantaneous surface, and a three-dimensional aggregate (a timeless body) in an instantaneous body. In the Experience belonging to this space a point will form a progressive figure of the first order, a timeless line a progressive figure of the second order, a timeless surface a progressive figure of the third order, and a timeless body a figure of the fourth order.
Movement.—39. It is only through the idea of permanent space that the ideas of “rest and motion in space” and the idea of “matter” acquire any meaning. In a moment everything is at rest; to speak of change of position in a moment is meaningless, since the position of a given event ​in a given moment is the sum of its relations to the remaining parts of this moment and is given once for all. Matter also cannot be defined in momentary space. Only when we compare the ordering of two different moments by means of permanent space can we see that a group of elements of experience, which are distinguishable by the same characters, coincides in both moments with a certain group of points of the permanent space of these moments, while another group of elements, which coincided in the first moment with a group of timeless points A, coincides in the second moment with another group of timeless points B. That we express by saying that one particle, on which we fastened our attention, remained at rest and a second moved from position A to position B in the enduring space which we are using for our comparison. Motion is thus change of position in permanent space, and rest is absence of motion.
Matter.—40. A definition of motion is incomplete without a definition of the subject of motion, which is matter (in the broadest sense of the term); we will, therefore, briefly indicate here how a definition of matter can be arrived at.
Of a group of attributes, which can be identified in all the elements of any progressive continuous figure of elements of experience, we say that it persists or endures, and call it a permanent group of attributes.
As we have already indicated above the mind is disposed, or compelled, to seek or form such enduring, unchanging characteristics of experience. If we call the progressive one-dimensional figure of elements of experience, which carries an enduring group of attributes, a perceptual particle—we can define a material particle as a perceptual particle composed of impenetrable elements of experience, where impenetrable elements of experience signify those elements of experience which cannot form a common ingredient of two different material particles.
A Few Conclusions.
41. In the definition of space and time we have achieved the aim which we set up at the beginning of this essay, and ​at the same time we have laid broad foundations for the construction of a “natural geometry,” or “geometry of experience,” as we can now call physics. For its complete construction it would be necessary to supplement our inquiry by determining the properties of the relation of “interval”—the proper metrical relation—a task which, as we have already stated, will not be attempted in this essay. At this point it may, however, be useful to pause and enquire as to the relation of our definition to the views prevailing in contemporary physics, and as to the manner in which it solves the differences which arise out of the divergency of these views. I am purposely using the word views, and not definitions, because, as we noted at the beginning, we should vainly search for a satisfactory (useful) definition of space and time in text-books of physics; only with the greatest difficulty shall we find it in some text-books of logic and mathematics. Text-books of physics usually assume that definitions of these fundamental concepts belong to, and are given by, philosophy; philosophy, however, has given several answers, the majority of which, besides being of no use for physics, to the latter’s great misfortune also contradict one another: and physics only inherits the controversies arising out of these philosophical—or better, metaphysical—divergencies, as we can easily see in the controversy raging around the Einstein theory of relativity, or the Planck theory of quanta. The basis of these controversies, and the motive of the passionate opposition which they encounter in certain circles, does not lie in the disagreement of these theories with physical facts, but in their disagreement with the metaphysical preconceptions of their opponents; it is only a repetition of history, for from the same motives there arose the opposition to the Newton theory of gravitation, to the principle of conservation of energy, and to other principles now generally recognised. Besides the fact that the controversy is to a considerable extent a controversy about the aim and nature of physical theory in general—a point which we can only mention here in passing—its source in my opinion is a noetical, and not a physical, disagreement of the disputing parties as to the nature of space and time, a disagreement rendered considerably more acute by some of the philosophical expositors of the new theories.
42. In rough outline, the current conceptions of space and time, in so far as they are at all important for physics, may be placed in two classes, parallel with whose lines of division runs also the line distinguishing the followers of the new theories from their opponents. A good, though not an ​exhaustive (as will be seen in our own case) criterion of a concept of space and time belonging to the first or the second class, is the answer to the question whether physical space (and time) can or cannot be deformed, i.e., whether or not it is possible to find physical space in which different axioms hold for different places. The answer given by the first class—the Kantian idealists—is in the negative; space and time, they say, are “forms of our intuition,” independent of the external world, and pre-existing to all perception, and they can be only such as are given us; it is not possible for our intuition of the properties of space to be dependent upon our physical theories. The answer of the second class is in the positive: space is a physical fact, the properties of which can be ascertained (measured) by physical means: that is the attitude adopted by the empiricists, sensationalists and the whole of the Mach and Einstein school. Our own reply is that the question has no meaning, because space, according to our definition, can have as few physical or metrical properties as, for instance, the alphabetical order of the vowel-sounds; nevertheless, our point of view does not result in the rejection of the Einstein theory which, in my opinion, can be reconstructed on the basis of our conception of space and time in a manner more satisfactory from the noetic standpoint than the theory is at present. Within the space of the present essay we can only briefly justify our point of view.
43. The fundamental shortcoming of the idealistic point of view, and the cause of its failure lies in its complete irrelevance to the whole question: the decision as to the validity of the laws (postulates) of, e.g., Euclidean or non-Euclidean geometry for physical experience does not depend upon the answer to the question whether this geometry is given us by intuition or by the external fact, but upon the answer to the question what this geometry—irrespective of the way it is given—is: the problem of the structure (metrical properties) of the space in which we place our physical experience remains unaffected whether this space be inside or outside the mind. It is for this reason that Kant’s point of view as to the form of physical space cannot differ from that of Riemann or Einstein.
44. The second standpoint—the classical formulation of which we find in Riemann’s “Habilitazionsschrift”—starts ​from the assumption that space—an aggregate of points—is a physical entity, an actual object of perception, the properties of which can be examined by physical means, and which can be measured in a way similar to that in which we measure matter or energy, i.e., as a quantity. My objection to this viewpoint is two-fold: In the first place it seems to me that space as conceived by this theory—in fact every conception of space as a quantity consisting of points, and for that reason also the current definition of space of pure geometry—degenerates into a logical circle and is therefore worthless as a definition—through failing to realise that nothing can be a relation and a relatum at the same time. As long as we do not define point independently of space, space defined as an aggregate of points becomes a relation between things (points) which in their turn are only relations between its different parts, i.e., between themselves; pure geometry and many philosophers are clearly conscious of this circle.
Secondly, even if we assume that we can arrive at a definition of point (as a space relatum—that which is spatially related) independently of space, and that we can remove this logical circle, Einstein’s space is, in so far as it remains a continuum of points, open to further objections:
(a) Either this continuum of points will be a thoroughly homogeneous one, and then it cannot be applied for the explanation of gravitation in Einstein’s way; or,
(b) It will be heterogeneous and then it becomes impossible to distinguish space from its “contents,” i.e., from matter and perceptions; space then becomes a superfluous higher structure of Experience, because the only property by which, if defined as a continuum of points, it can be distinguished from matter, is its homogeneity. The space-time of Einstein is a kind of medium, which by its heterogeneity acquires the properties of matter, becoming superfluous if we wish to describe the phenomena in terms of matter, and useless, if we wish to describe them in terms of something else.
45. Assuming that we have defined “point” independently of space (i.e., so that the definition of space is not presupposed in that of point) we have two possibilities open to us: either space is an aggregate of such points, or a relation between them. The followers of the sensationalist theory of space maintain that it is necessary to postulate points as the constituents of empty space: as those parts of space in which there is no matter. In dealing with this objection great caution is necessary: in substance it is a question whether empty space can be subject to physical, sense experience. Here, I think, the sensationalist theory of space ​shares the error of Kant: empty space cannot be imagined, much less examined physically. But it is necessary to distinguish here clearly between the space (point-space) which in accordance with our nomenclature can be called momentary, (therefore an aggregate of points in a given moment), and the space, which in accordance with the conventional terminology we have called permanent space (therefore the aggregate of permanent points, i.e., routes, consisting of points of the various momentary spaces). As far as the latter is concerned, the objection of its “undistinguishableness” from matter and therefore superfluousness comes into play in its full weight; if this space has different properties in its various parts, it will be possible to ascertain these properties only when something enters these parts (a particle of matter, energy, like a ray of light)—but how is it possible to distinguish a space which has an influence on what enters it from a material medium? If we reply that such a distinction is not necessary, and still adhere to the point-aggregate conception of space, we shall get another space in which this medium is placed, and in a similar way a whole hierarchy of spaces in an infinite regress.
46. In the case of a momentary space the conception of empty space loses its sense altogether: into an empty momentary space, i.e., into a momentary space (point-aggregate) in which there is nothing, nothing will ever come, and it is therefore superfluous to maintain that the space is there, and that it has such and such (unascertainable) properties. The postulate of the existence of points of empty space has meaning in the pre-Einsteinian space of classical mechanics, where every eternal point is a possible recipient of a particle of matter; the moment, however, we amalgamate space and time into one continuum, as it is amalgamated in Einstein’s theory, empty space becomes a superfluous multiplication of entities, a useless fiction, satisfying perhaps the requirements of æesthetics (not even transcendental) but not of logics and physics. Empty space, as empty time, is an empty word, with no meaning save that of a possibility of ordering relations: if we say that between the particle A and the particle B there is an empty space, it means that there exists the possibility of placing a particle X between them; evidently it has a meaning only in the case of permanent space.
47. Convincing ourselves that the concept of space as an aggregate of points is untenable, we are reduced to the solution, developed in the preceding paragraphs: space and time is a relation between things, i.e., between parts of the contents ​of our mind; the difference between them being, that while time is an abstraction of the whole—perceptual and non-perceptual—content of the mind, space is an abstraction from the first only, from the events, from Experience. The definition of this relation is given by its being the only relation, common to all and any two parts of this content. Space and time as such are not subjects of physical experience, they are pure concepts, abstractions from experience; therefore they have no physical properties, properties which it would be possible to examine and determine by physical means. Properties which can be determined and measured by physical means are not properties of space and time, but of that which is “in” space and time (which means, according to our definition, that which is arranged in spatio-temporal order)—i.e., of Experience. It is, therefore, meaningless to speak about the “measurement” of space and time, still less about their metrical properties: what we are doing when we “measure” space (or time) is (assuming for the moment that we know what we mean by the word measurement)—measuring Experience, having regard only to its order in respect of space (or alternatively time), and leaving aside its order of time (or alternatively space).
47.1. At this point it is necessary to remark that order itself is not the only relation which it would be possible to call spatial or temporal, i.e., which would be common to all parts of Experience; the second relation of this kind—as long as we have not analysed Experience into its elements—is extension. Extension is so often considered to be the property defining space and time that it is necessary briefly to justify our choice of order as the space—and time—defining relation. The reasons by which we were led in our choice are roughly these: in the first place, extension is a characteristic of Experience as a whole, but loses all meaning when we think of Experience as analysed into (unextended) elements, and here the relation of order is at a great advantage. Secondly, extension as a pure space and time characteristic appears in the Geometry of Experience when we introduce the relation of interval between two elements, and it can be considered to be only another facet of the same relation of which one facet is order: this being understood, we can finally say that we were led by the same reasoning which compels us to consider analysis situs to be a more fundamental branch of geometry than metric geometry. That, of course, may be a matter of opinion: I do not doubt that it is possible to proceed in the contrary diirecton and arrive at results equally fundamental.
​47.2. The theory of space and time which we have thus reached is, of course, not complete; for completeness there is lacking a discussion of this second spatial and temporal (formative) relation of extension, or better, (to distinguish from the use of the term extension in the sense defined at the beginning of this essay) space and time magnitude, depending on the concept of distance and forming the basis of measurement and metric discussion upon which we have touched above in passing, but which is not to form a part of the present essay; we hope to devote to it a new study to which the present will form, as it were, an introduction. The method which must be followed in this development we shall indicate in the remaining few paragraphs.
48. It may appear that the conclusions reached in the above paragraphs are of little value, since they are valid for the time and space of a single mind, whereas different minds may arrive at a different ordering of the elements of their Experience, and thus render impossible any conclusions as to the space and time of the external world, which may be entirely different from the space and time of individual Experiences. That is, I think, an error arising from the mistaken judgment that the external world is given as an unordered aggregate of elements of Experience which individual minds first reduce to order, thus creating the space and time of their Experience. In reality the process is the reverse; the external world is given to minds as a whole, a coherent and connected structure of events, in which individual minds first seek out elements of Experience and their ordering; and whatever the metaphysical character of the external world—whether it is the work of some transcendent being acting upon a mind of the same structure, or whether it is some common creation of the human subconsciousness— the “Form” of the world is already there when it is presented to us in perception. Since, then, we have defined the external world as that part of the perceptual content of the mind which is common to all minds, and since the uniform ordering of its elements in every experience is the fundamental condition upon which part of Experience can be held in common by different minds (if the elements of Experience only, but not their ordering, were common to different minds, it would be impossible to decide whether anything was common to them at all), we can affirm as an axiom the principle, that the Form of Experience will be common to all minds and therefore a property of the external world independent of the individual mind; from this we can affirm a priori that two observers will agree in their “situational” description of phenomena (that is, a description which is ​content with determining the order and position of given elements in the sense of an analysis situs). But that is also all that can be said a priori; whether two observers will coincide in their judgment as to the simultaneity of two elements of Experience must for the time being remain an open question. We must not forget, too, that we have hitherto paid no attention to the metrical properties of a physical continuum and we cannot a priori preclude the possibility of difference in this direction without offending against the methodological principle of economy in hypotheses, and unduly circumscribing possibilities which are in no way at variance with the laws of our thought.
49. But even here we must not go too much to extremes; the metrical part of our study, which we have deferred to a later work, and which is of the greatest importance for physics, lies in investigating the relation of “interval” (distance) between two elements of a given continuum; this relation in a multidimensional continuum is a certain mathematical function of its components, which we will call “distances,” and which enable us uniquely to denote individual elements by co-ordinates already having more than a descriptive function. It would, indeed, be meaningless to affirm that we are free to premise that the interval between two elements will be different for two observers: an interval is an interval, that is, something in the given continuum, and though different observers may call it by different names, their speaking of it in different ways does not alter it. We may state this fact in different words, or, if we wish, lay it down as a postulate, saying that the total interval between any two elements of Experience is independent of, or invariant with respect to, the observer. It is otherwise with the partial “distances”: all that is required in their case is that a certain function of them, including them all, should produce the same total interval. Here the main problem will be whether there is in a single Experience something (a structure) which directly determines these partial “distances,” or whether it is left to the observer to determine them (in our case spatially and temporally) as well as he can. In reality it appears that this freedom is left to the observer, or, properly speaking, imposed upon him; and upon this freedom depend those differences in estimating the spatial and temporal lengths which appear so paradoxical in the theory of Relativity. Even this freedom, however, I maintain, is not absolute, being limited by the principle, which I call the principle of maximum uniformity, and which requires space-time distances to be so determined as to permit, in relation to one another, of the most uniform possible description of Experience.
​The Law of Motion.—50. As we have already gone thus far, we will venture upon a further digression which will enable us to formulate the most important law of physics, the law of motion of a free body. By the attribute free we denote a body or material particle which does not (or in so far as it does not) meet any other material particle in its course. Given any two non-contemporaneous elements of experience forming this particle, the law of motion has to determine the remaining elements of Experience forming it. In current terminology, given any two points in the course of a material point, the law of motion has to determine the remaining points of this course. What will this course be? Can it be determined a priori? I believe it can, if we exclude the possible intervention of some unknown power (e.g., so-called force). It is found by the following process of reasoning: There is an infinite number of possible paths by which two elements of Experience may be joined, and the course which a particle of matter chooses must itself afford the ground for its choice. There must be something in this course which distinguishes it from all other possible courses, otherwise there would be no a priori reason why it did not choose one of the remaining courses. What property can we find in this course to distinguish it from all other possible ones? A clue is afforded us in this direction by the fact of the multiplicity of observers; a course is a course, i.e., a something, some condition of the “external world,” which has no reference to who observes it, and must be the same for all. Its affinity with the interval between two elements of experience is evident; and we shall find that what distinguishes one material particle from others (one path connecting two points from other possible paths) is the interval between its various elements, and that the path which a free material point will take, is determined by the interval between any two of its elements.
If we reverse the sequence of our reasoning and postulate the length of the course of a free body as the measure of the distance between two elements of Experience, we have introduced into the physical continuum a geometry, in which a straight line (a geodetic line) is defined by the course of a free material particle, that is, where the simplest law of motion to which we can attain (“a free material particle moves in a geodetic line”) is at the same time a determinant of the metrical properties of Experience, which will make the geometry of our continuum the simplest.
Physics and Geometry.—51. At this point it will not be inappropriate to touch upon the relation of physics to geometry—a question which has given rise to so many violent controversies in the past forty years. In the first place, it will be ​necessary to make clear to ourselves the definition of geometry as it will appear if we accept the standpoint of the above investigations and the definition of space and time arising from them: geometry will no longer be a “science of the properties of space” as it is usually defined, but a “science of the properties of ordered aggregates, or continua”. These continua will possess different (metrical) properties depending upon the way they are ordered and upon the way we define “interval” in them—upon the postulates we lay down in the foundations of our geometry.
One of such continua is our perceptual Experience: it is the investigation of its properties which is the task of physics. Between physics and geometry there is only this difference: that in geometry we can give orders up to a certain limit, i.e., as far as we are not offending against internal consistency, whereas in physics we must obey orders.
We say that geometry is a priori, deductive, physics a posteriori, inductive, founded on experimental knowledge; the truth seems to be that both sciences are to a certain extent inductive, based on experience, from which they generalise, abstract, and to a certain extent deductive, building on concepts and postulates obtained by generalisation. The difference between geometry and physics (I speak of course only of theoretical physics) is therefore not, as generally assumed, so great as regards their material: both deal with concepts, i.e., abstractions, generalisations from experience: the difference between them being, that geometry can do with its concepts whatever it pleases, as long as it remains consistent with itself, while physics must strive to arrive at conclusions which agree with physical experience, to make its concepts correspond to certain physical experience.
Finally, geometry once more approaches physics when, to facilitate its work, it borrows from physics its aids: the perceptual representation of its concepts; but here must be kept in mind that this representation is but an approximation to the ideal concepts of geometry, just as (if I may be allowed to present the fact of physical simplification of phenomena in this reverse order) actual empirical physical experience is but a rough approximation to the ideal Experience of physical theory.
* ↑ It is perhaps hardly necessary to emphasise that the term “content of the mind” is chosen purely for want of a better expression: it must not be taken to signify that the mind is a kind of receptacle, in which thoughts, perceptions, feelings, and the like, are contained.
* ↑ This definition implies that we postulate the possibility of comparing attributes which are not co-momentary and identifying their “likenesses,” in a sense purely qualitative and not quantitative. Even if there were no empirical grounds for this postulate, it would be perfectly legitimate; since without it thought itself, as well as science, would be impossible. Even if objects changed, which we imagine unaffected by time, it would not be the change which would matter so much as our collective judgment; a red patch of colour will be red as long as everyone calls it red, even though some higher being may see it as changing its colour with time.
* ↑ I do not know whether of Kant himself; my criticisms are aimed at those Kantians who hold that geometry is determined by the properties of space, or, better, that geometrical postulates determine or express these properties, and that they, together with the intuition of space, are given us a priori.
* ↑ With due apologies to those who define geometry as a purely formal branch of logic, having nothing to do with the so-called geometrical “intuition” of lines, points, etc., but with pure hypothetical judgments: “If we assume A, then B follows”. If we admit this definition of geometry, why differentiate geometry from other branches of pure mathematics? For our present purpose we shall prefer to think of geometry as the science of points, lines, angles, etc.