ESP the Similarities Between Space and Time

Mind Association

The Similarities Between Space and TimeAuthor(s): G. SchlesingerSource: Mind, New Series, Vol. 84, No. 334 (Apr., 1975), pp. 161-176Published by: Oxford University Press on behalf of the Mind AssociationStable URL: http://www.jstor.org/stable/2253384 .

Accessed: 07/02/2014 14:05

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

Oxford University Press and Mind Association are collaborating with JSTOR to digitize, preserve and extendaccess to Mind.

http://www.jstor.org

This content downloaded from 150.164.180.195 on Fri, 7 Feb 2014 14:05:01 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=oup

http://www.jstor.org/action/showPublisher?publisherCode=mind

http://www.jstor.org/stable/2253384?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


The Similarities Between Space and Time

G. SCHLESINGER

I

A number of philosophers have maintained that space and time are similar to a large degree. Goodman for example claims that

' . the analogy between space and time is indeed close. Duration is comparable to extent. A thing may vary in color in its different spatial or in its different temporal parts. A thing may occupy different places at one time or the same place at different times or may vary concomitantly in place and time. The relation between the period of time occupied by a thing during its entire existence and the rest of time is as fixed as the relation between the region the thing covers during its entire existence and the rest of space."

But at the same time he clearly admits that space and time are not similar in every respect.

R. Taylor expresses his belief in the similarity of space and time in terms of the similarity between temporal and spatial propositions. In his well known article 'Spatial and Temporal Analogies and the Concept of Identity' he claims that

'. . . many propositions involving temporal concepts which seem obviously and necessarily true are just as necessarily but not obviously true when formulated in terms of spatial relations.' 2

He then goes on to examine a number of temporal propositions which are true and whose spatial counterparts do not seem to be true thus apparently representing cases in which space and time do not resemble one another. Taylor is however entirely success- ful in showing in each case that the lack of parallelism is illusory, for what we thought to be the spatial proposition to correspond to a given true temporal proposition is not really the one, while the actual spatial counterpart is in fact true.

I Problems of Space & Time (ed. J. J. C. Smart), pp. 367-368. 2 Ibid. P. 381i.

6 I6I



i62 G. SCHLESINGER:

There are, however, two very important questions which have been left unanswered. The first question is which true temporal propositions do have true spatial analogues and which do not; or in what kind of properties space and time are similar to one another and in what kind of properties they are not. Taylor claims only that 'many propositions involving temporal concepts which are true do in spite of appearances have corresponding true propositions involving spatial concepts', but not all. But it is of greatest interest for us to know what characterizes those true temporal propositions to which true spatial propositions do correspond and those to which such propositions do not correspond; we should be able to answer the question by what criterion are propositions involving temporal concepts divided into the class the members of which remain true when formulated in terms of spatial relations and the class the members of which cease to be true when so formulated. If a demarcation principle between the two types of propositions exists and we are able to come up with a characteriza- tion of those temporal statements which do have symmetrical counterparts then it is indeed of interest to have a close look at any true temporal statement which has the characteristics of a statement which ought to have a true spatial counterpart but does not seem to have one, and enough motivation would be provided for making efforts to show that in a correct analysis such a counterpart does exist. But if there is no systematic way in which temporal propositions may be divided into two kinds and it just so happens that some temporal propositions remain true when reformulated in spatial terms and others do not then I can see little point in making elaborate efforts to show that some true temporal propositions in spite of appearances to the contrary have true spatial counterparts.

The second question to answer is why all those true temporal propositions which do have true spatial counterparts, do have them. If all such propositions can be recognized by their possessing certain characteristics it is of great interest to know why the possession of such characteristics is sufficient in ensuring that that true spatial counterparts exist. Whatever the reason will turn out to be it is to be expected that our knowledge of it should increase our understanding of the nature of space and time. The need to answer the first question would of course be entirely obviated if we embraced the Doctrine of the Complete Similarity of Space and Time according to which every necessarily true



THE SIMILARITIES BETWEEN SPACE AND TIME I63

temporal proposition remains true when reformulated in spatial terms. Those who are convinced by Taylor's article may wish to subscribe to this doctrine since he seems to do away with all the apparent counter-examples to the doctrine.

In this paper I propose to show that the doctrine is false-: there are necessarily true temporal propositions which do not have true spatial counterparts. I shall also show that there is an easy and obvious way to characterize those true temporal propositions which have symmetrical counterparts. It will also become perfectly evident why all those true temporal propositions which have symmetrical counterparts must have them.

II

Before being able to make any progress we must know how to determine whether a given true temporal statement has a true spatial counterpart and in order to do this we must make sure that we can always produce the correct statement into which a temporal statement must be turned to yield its spatial counterpart. The general principle is that in order to produce the spatial counterpart of a temporal statement we much change every temporal term into the corresponding spatial term and every spatial term into the corresponding temporal term. It may not be always so obvious how to do this since we may often overlook terms which do not seem to be referring to temporal or spatial predicates yet do have such predicates hidden in them. The following example illustrates how an error of translation can easily be committed through a failure to recognize a term as being temporal or spatial and being in the need of inversion to produce a complete translation of a temporal statement into its spatial counterpart:

'A further disanalogy between "here" and "now" which shows that there is no spatial analogy to temporal becoming is due to there being no spatial analog to waiting. If I wait for n-time units my use of "now" denotes a different time regardless whether I move or not. But the spatial analog of this is nonsense since there is no sense to "waiting for or through space" '.1

The temporal statement under review in this case is 'If I wait n-time units my use of "now" denotes a different time, regardless

R. Gale, '"Here" and "Now"', Monist (July I969), p. 409.



164 G. SCHLESINGER:

whether I move or not' and the presumed spatial counterpart is 'If I wait n-space units my use of "here" denotes a different place regardless whether I move or not'. The latter statement not only fails to be true but is even devoid of meaning. The failure of the true temporal statement under review to translate into a true spatial statement shows in Gale's opihion that space and time are radically different.

But of course before the question of the existence of a true analogue can be settled we must make sure that we are doing the translation correctly and this cannot be the case unless we realize that the terms 'wait' and 'move' have temporal and spatial terms hidden in them which have to be made explicit and be properly inverted. The term 'wait n-time units' in the first statement stands for 'occupy temporal positions t1 through tn' and 'move' stands for 'shift my spatial position'. Accordingly the first statement, with all its temporal and spatial terms laid bare runs 'If I occupy temporal positions t1 through t, then my uses of "now" at t1 and at tn denote different times regardless whether I shift my spatial position, that is regardless whether I make the two utterances at the same place or not'. The exact spatial counterpart of this statement is 'If I occupy spatial positions Pi through Pn then my uses of "here" at Pi and Pn denote different places regardless whether I shift my temporal position, that is regardless whether I make the two utterances at the same time or not'. This second statement is of course as true as the first one. There is therefore no disanalogy between space and time with respect to the relations referred to in the two statements.

It should also be noted that everyone will acknowledge that there is a certain disanalogy between space and time but it is of a kind which cannot serve as a counter-example to the thesis that space and time are radically alike. The temporal statement 'If we know that an object occupies two non-adjacent points in time, t, and t2 and is continuously extended temporarily then we can name other points in time which it also occupies' is true. How- ever its spatial counterpart 'If we know that an object occupies two non-adjacent points in space, P1 and P2 and is continually extended spatially then we can name other points in space which it also occupies' is false. This lack of analogy is obviously brought about by the fact that time is one-dimensional and space is multi- dimensional. Consequently if an object occupies t1 and t2 it occupies all the temporal positions which exist between t1 and t2.




But if an object occupies Pi and P2 it does not necessarily occupy all the points between them only all the points on one of the infinitely many lines which connect Pi and P2* In order to show that space and time were really radically different we would have to show that a true temporal statement cannot be trans- formed into a true spatial statement even in one-dimensional space.

Let me cite one other example where it may seem that space and time are disanalogous and where it is not at once clear what mistake of translation had been committed:

'If the birth and death of A and B coincide in time then every time occupied by A is also occupied by B and vice versa'.

This statement is true. The spatial counterpart seems to be:

'If the birth and death of A and B coincide in space then every place occupied by A is also occupied by B and vice versa'.

This statement is false and its falsity is not a consequence of the multi-dimensionality of space for it is easily seen that the statement would be false in one-dimensional space too.

The mistake can however be uncovered once we realize that the spatial counterpart of 'to be born at tl' is not 'to be born at Pi'. The phrase 'to be born at t1' means 'to have t1 as the most extreme point in time occupied by an individual' (for if an individual is born at t1, then it follows that he does not extend temporally in one direction beyond t,) but 'to be born at Pi' does not mean 'to have Pi as the most extreme point in space occupied by an individual'. Bearing this in mind we realize that the correct counterpart of our temporal statement would run along these lines: 'If Pi is the most extreme point in one direction beyond which there is no point which is occupied by either A or B at any time and P2 is the most such extreme point in the other direction then (in one-dimensional space) every place occupied by A at one time or another is also occupied by B at some time and vice versa'.

III

In addition to the examples Taylor has brought in his article quoted above to show that seeming violations of the thesis of the similarity between space and time can on a closer analysis be shown not to constitute such violations and in addition to our own



i66 G. SCHLESINGER:

two examples there are also some instructive examples to be found in an article by James W. Garson, 'Here and Now', in the Monist (i969) pages 469-477. In view of all this one may be inclined to think that on a careful enough analysis every true proposition involving temporal concepts remains true if properly formulated in terms of spatial relations (as long as we are not exploiting the multi-dimensionality of space). This, however, cannot be maintained.

An effective counter-example to the Doctrine of the Complete Similarity Between Space and Time is presented by the statement 'Temporal positions are ordered, time has a direction' which is true while its spatial counterpart 'Spatial positions are ordered, space has a direction' is not true. Given three points in time t1, t2 and t3 there are many physical manifestations through which it is possible to tell in which order these points occur; given three points in space P., P2 and p3 similar physical signs do not exist to reveal the order of these points in space. For example given that at t1 a certain match is in perfect condition while at t2 it is in flames and at t3 it consists of burnt out ashes then we know that the order of these points is t,-t2-t3. Consider however the spatial counterpart of this situation which is having a match occupying points Pi, P2 and p3: at Pi the match is unscathed, at P2 it is in flames while at p3 it is reduced to ashes; it is impossible to know the spatial order of these points.

In attempting to reply to this objection one may perhaps want to say that ultimately all those phenomena which provide time with a direction are connected with the principle about the increase of entropy. Now it is a fact of physics that states with increasing amounts of entropy are associated with increasingly more advanced positions in time while different states of entropy may occur in space in any order. This fact is to be taken as indicative of a given feature of entropy rather than of the nature of time and space. In other words it is not the case that time as such is different from space but rather that entropy is differently related to time than to space. In the particular instance of the match for example it might be said that the asymmetry we have found is not indicative of any intrinsic differences between space and time itself but rather of the fact that the burning of the match is a process which is asymmetrically related to time and space. The different temporal parts of the burning match are strictly ordered among themselves while the different spatial parts of the




burning match may occur in any order; the half burnt stage of the match must be at a point in time which is between the time of the unscathed stage and the time of the fully burnt stage while the half burnt spatial part of the match may or may not be at a place which lies between the place a,t which the match is unscathed and the place at which the match is fully burnt. Thus we may expect to find innumerably many events, processes and entities the properties of which are different in relation to time than in relation to space but these examples only reveal a difference in the laws that govern the behaviour of these particulars with respect to time and with respect to space and not an intrinsic difference between time and space themselves.

The foregoing argument may seem quite reasonable. After all everyone would have to agree that an asymmetry in the way a given particular is related to space and time need not always be interpreted as symptomatic of a disanalogy between space and time. Consider for example a complex piece of machinery like a television set which occupies a certain amount of space and whose life extends over a period of many years. Suppose we chopped off several years from its existence. Such an act would not radically interfere with the status of the remaining temporal part of the set. All we would have done is to have produced a T.V. set with a shorter life span, but the remaining temporal part of the set would just as much constitute a proper T.V. set as the original, temporally untruncated, set. Suppose on the other hand that we reduced the amount of space occupied by the set by removing for example its entire front part. This act would substantially interfere with the status of the remaining spatial part. The part of the set with which we would now be left would not simply be a T.V. set which occupied less volume but a non-functioning wreck. Thus it is quite obvious that the different spatial parts of a T.V. set hang together and are mutually dependent on one another in a way in which its different temporal parts are not. But at the same time it seems clear that most people would not interpret this as a manifestation of a fundamental difference in the nature of space and time itself. The difference is merely a consequence of the peculiar nature of a complex mechanical system like a T.V. set: temporally it consists of a stretched out series of virtually identical parts each such part constituting a full fledged, short lived, T.V. set. Spatially however it does not constitute an assembly of identical parts; its various parts occupying different



I68 G. SCHLESINGER:

regions of space play different roles in contributing to the assembly as a whole functioning as a T.V. set.

Nevertheless it is obvious that it is impossible to defend the Doctrine of the Complete Similarity of Space and Time by claiming that the fact that there is order in time is not a feature of time itself but rather of all those particulars which are ordered in time. What is basically wrong with such a defence that if it were permissible then the Doctrine would become vacuously true. Any counter-example produced to the Doctrine could at once be mechanically explained away as indicative not of any real difference between space and time as such but rather of one between the relationships the physical particular, which manifests the difference in question, has to space and time. Consider for instance our example concerning the birth and death of A and B which coincided in space and time yet exhibited differences with respect to the former and the latter. It was entirely superfluous to subject the notions of 'birth' and 'death' to analysis and show that these can be unpacked to reveal asymmetrical references to time and space. The problem could have easily been dismissed by simply saying that all that has been discovered is that the physical events of birth and death are asymmetrically related to space and time but not that intrinsically there is any difference between space and time. In other words if such a defence were to be accepted then the fact that it can automatically be advanced in every instance would empty the Doctrine of all substance. While it may not be clear whether the Doctrine is true or false it certainly is intended to convey something of considerable substance.

Returning for a moment to the example about the T.V. set: the only reason why in this case the asymmetry we found should not be assigned to space and time itself is because there happen to exist physical counterparts to particulars such as T.V. sets which are exactly conversely asymmetrical with respect to space and time. The existence of two kinds of particulars-particulars which when truncated temporally have their remaining parts un- affected but which when truncated spatially have their remaining parts radically changed and another kind of particulars for which the loss of a temporal part is crucial but not the loss of a spatial part-is what determines essentially that we have not found any fundamental disanalogy here between space and time. A counterpart in the relevant sense to a particular like a T.V. set would be a symphony heard throughout an extended region of space and




time. Suppose we halved the space throughout which the symphony was heard: this would leave us with the same symphony heard but in fewer places. It would however be quite different if we chopped off a considerable temporal chunk from the symphony, say, the first and the last movement: what we would be left with would not merely be a shorter symphony but no symphony at all. Were it not a fact that we could produce such a physical counterpart to a mechanical system and were it the case that all particulars had essentially the features of T.V. sets with respect to temporal and spatial truncation we might well be inclined to interpret this as indicative of a basic difference in the nature of space and time.

Seeing how the previous attempt to defend the Doctrine fails one may be tempted perhaps to seek to defend it by claiming that instances like burning matches do not constitute counter-examples since terms like 'half burnt' and 'completely burnt' contain covert references to time. Once we recognize this we are in a position to formulate the correct spatial counterpart for the temporal statement expressing the existence of order among the different temporal stages of burning, which equally expresses the existence of order among the different spatial stages of the burning and is true. The temporal statement asserting that there is order among tl, t2 and t3 amounts after all to 'If t1 marks the beginning of the burning process of a match, t2 the middle and t3 the end of that process then the order among these instances is t,-t2-t3'. But clearly the spatial counterpart of this statement is 'If Pi marks the place which is at one end of a burning match, P2 the place which is in the middle and p3 which is at the other end of the match then the order among these points is P1-P2-P3' which is also true.

But it is by no means necessary to render the statement expressing the existence of an order among tl, t2 and t3 the way we have just done. The terms 'unscathed', 'half burnt' and 'completely burnt' may be replaced by terms which are unquestionably free entirely of all temporal reference. The terms we may substi- tute would be the full chemical descriptions of the state of a match in perfect condition sl, in half burnt condition S2, and its state when it has completely turned into ashes S3. Accordingly the true temporal statement which represents a counter-example to the doctrine runs: 'If at t1 an object is in state sl, at t2 is in state s2 and at t3 in state s3 then the order of this times is t1-t2-t3.'



I70 G. SCHLESINGER:

The obvious spatial counterpart of this statement is 'If at Pi an object is in state sl, at P2 in state S2 and at p3 in state S3 then the order of these places if P1-P2-P3' which of course may not be true.

Lastly let me consider a defence which would be based on the claim that the fact that the spatial counterpart of the true temporal statement under review may not be true does not violate the Doctrine of the Similarity of Space and Time because that Doctrine refers only to similarities in properties which are necessarily possessed by time but not in properties time possesses contingently only. The law about the behaviour of entropy is a physical and not a logical law and could conceivably be false. In the particular case of the match it is quite conceivable that the process of burning be reversed and turned into a process of 'unburning' in which a heap of ashes is enveloped in flames and gradually emerges into a perfectly unscathed match.

At first this line of defence might appear quite reasonable but the more one thinks about it the less plausible it becomes. Ad- mittedly it is not easy to say exactly how far we can go on depriving time of properties it has by virtue of the fact that laws of physics happen to be what they are and it would still continue to retain its conceptually essential features of temporality. It seems clear, however, that there must be a limit to this process somewhere and if we went on depriving time of too many properties associated with it we would end up with something we would no longer recognize as time. Admittedly if the law concerning the increase in the entropy of closed systems had a few exceptions this would not yet drastically transform the nature of time, however, we could not dispense with all those phenomena which endow time with a direction and be left with what would still recognizably be time.

What I am trying to say is that it is undoubtedly true that the fact that this or that particular process is ordered in time and hence its different temporal parts are indicative of the way the moments which these parts occupy are themselves ordered, is a contingent matter only. However the fact that in general there are phenomena which prQvide time with an order and direction is not merely a contingent matter, for were it otherwise our familiar concept of time would break down.

It seems quite clear that the notion of 'direction' and even more so the notion of 'order' is conceptually vital for the notion of



THE SIMILARITIES BETWEEN SPACE AND TIME 171

time. The relations 'before' and 'after' have generally been acknowledged as being the most fundamental temporal relations, which means that time deprived of these relations would cease to be time. But of course if we deprived time of order we would deprive it of much more than just the relations 'before' and 'after'. In an orderless time it is not merely that the burning process of the match might be reversed but it could also happen that its half burnt state occurred first, its unscathed state next, followed at once by its completely consumed state. The resulting universe would not merely resemble what one sees when a film of a sequence of events is run in reverse but rather what one would see if the pictures of the film, had been completely mixed up. A universe like this would be too strange for us to handle with our existing conceptual machinery.

We may also make brief reference to memory. Memory is one of the major phenomena which endow time with direction and order. If tl, t2 and t3 are past times then these are ordered in our memory firstly in accordance with the varying vividity with which they are etched in our minds and more importantly these are ordered by the fact that we may remember now that at t3 we remembered events occurring at t2 and t1 while at t2 we remembered only events of t1 and not t3 and that at t1 both t2 and t3 were thought of as lying in the future. Imagine if you can what would result if we interfered with memory supposing not only that we remembered all events with equal intensity without distinguishing them in accordance with their relative distance from the present but also that past and future events were etched equally vividly in our memories. This would be sufficient to eliminate the relations of before and after, destroy the notions of past and present, and do in addition something which would play much havoc with our conceptual framework: we would lose the appearance of the present continually shifting its position, we would be deprived of the flow of time. Surely, time thus robbed of all its characteristic features would cease to be time.

These brief considerations show that the statement 'Time has direction and time has order' is not merely true but necessarily true for if we deprived time of all those properties which lent it direction and order we would destroy the concept of time alto- gether.

Let me cite another counter-example to the Doctrine of the Similarity of Time and Space, an example which shows that time



172 C. SCHILESINGER:

is in some ways ontologically mnore fundamental and indispensable than space. 'There are particulars which have temporal positions and extension but no spatial position and extension' is true. But the spatial counterpart of this statement "rhere are particulars which have spatial positions and extension but no "temporal position and extension' is false. There are various kinds of particulars which illustrate the truth of the first statement, to one kind of particulars belong mental particulars such as a sense of well- being or distress. It is meaningful to speak of a sense of well-being starting at a given time and having such and such a duration and similarly with distress. It is meaningless, however, to speak about the spatial edges or the area occupied by a sense of well being or distress. It may legitimately be claimed that we are dealing here with a necessarily true temporal statement which has no spatial counterpart. Admittedly there are possible universes in which mental phenomena do not exist; still the existence of mental phenomena is necessarily possible. On the other hand it does not merely happen to be the case that there are no particulars with spatial positions and extension and which lack temporal position and extension; it seems inconceivable that there be such.

I shall also make brief reference but not go into the details of another counter-example of a somewhat controversial nature. It concerns the unity of time. Everything that is in time is temporally related to everything else in time. But there could be disparate spaces and spatial entities of one space could be entirely un- related spatially to such entities in another space. Disparate times are however inconceivable.'

IV I have not referred to all the counter-examples there are to the

Doctrine of the Complete Similarity Between Space and Time nor have I spent very much time in elaborating upon the counter- examples I have referred to. But there is no need really for lengthy discussions, for on a proper understanding of matters it becomes fully evident-witbout the help of any examples-that, with the exception of certain respects, time and space should differ from one another in indefinitely many ways. Before setting out on the clarification of this point let me just hint in a very

Cf. A. Quinton, 'Spaces & Times', Philosophy (I962), pp. 130-147. Some have disputed his claim.



THE SIMILARITIES BETWEEN SPACE AND TIME 173

crude fashion why one should expect the Doctrine to fail: after all time is time and space is space, so why should not the two be different from one another?

In order to arrive at a correct understanding of why there must be a certain class of properties with respect to which space and time cannot fail to be fully analogous let us consider a typical aspect of space and time in which the two completely resemble one another. The statement 'Nothing can occupy two places at the same time' is commonly regarded as necessarily true. Is its spatial counterpart true, i.e. is 'Nothing can occupy two times at the same place' true too? Before we can answer this question we must realize that the first statement is true only if understood in a certain manner but not otherwise. A physical object which is spatially extended can have and will have its different parts occupying different places at the same time. So the first statement is true only if it refers either to things which occupy infinitely small amounts of space or if we refer to all the points occupied by its different parts as one place. Under parallel restrictions the second statement is true too. An infinitesimally short lived entity cannot occupy two times at the same place; also if we decide to call the whole period during which an entity is at any single loca- tion one time then even a temporally extended particular cannot occupy two times at the same place.

All this is obviously true and furthermore is true not only about space and time but about every pair of continua containing some occupant. Suppose there is an occupant which has an extended existence in two continua X and Y and any point x in X occupied by the occupant is related to some point y in Y occupied by the same occupant by the relation represented as y f(x). Suppose that at some point y1 the continuant occupies many points in X then it is correct to say that the occupant 'is at two places in X at the same point in Y'. On the other hand if y - f(x) is such a function that for any particular value of x there is a unique value of y and for every value of y there is a unique value of x than the occupant 'cannot be at two places in one continuum while at one place in the other continuum'. Finally, even if corresponding to a particular point occupied in Y many points in X were occupied and corresponding to a particular point occupied in X many points in Y were occupied, we might decide to call the whole set of points in X which correspond to a single point in Y the same place in X and vice versa, and then



174 G. SCHLESINGER:

once more it would be true that 'it is impossible for an occupant to be in two places in one continuum while at one point in the other continuum'.

But what has just been said completely reveals the scope of the resemblance between space and time and provides -a full ex- planation why such resemblance has to exist. Space and time are both continua and possess therefore all the properties continua in general possess; events, processes and objects, which are occupants of space and time, also have all the properties which are characteristic of occupants of continua in general. Any statement which is true about an occupant of a pair of continua X and Y (where X and Y have only general continuum properties without any extra properties which would turn them into a specific kind of continuum) must be translatable into a pair of true statements (and not just into a single true statement since X and Y are inter- changeable) specifically referring to space and time. This means that, if there is a true statement asserting that an occupant which has a specified relation R to any continuum X which it occupies must have a specified relation S to any second continuum Y which it also occupies, then it must be the case that it is true of some occupant of space-time that if it has relation R to time then it has relation S to space and also true of some occupant that if it has relation R to space then it has relation S to time.

This can also be put slightly differently thus: a temporal statement which is true, not by virtue of any special time-like property of time or by virtue of any space-like property of space, but exclusively by virtue of those characteristics of space and time which these have as an inevitable consequence of the fact that each one of them constitutes a continuum, must have a true spatial counterpart since time and space are continua to equal degrees.

Now while space and time possess all the properties continua possess in general it is also true that time possesses an indeter- minate number of extra properties which it does not share with any other continuum and by virtue of which time is specifically time and not just a continuum. The same is true about space. As long as we are dealing with continuum properties, we are dealing with what constitute a common denominator of space and time and hence we shall find space strictly resembling time. But as soon as we consider the time-like properties of time, i.e. properties which are not imposed upon time simply because it is a




continuum but are those by virtue of which time becomes the unique continuum it is, and as soon as we consider the space-like properties of space, there is no reason whatever why there should be, and consequently we should not expect at all to find, similarities between the two.

V

It is therefore quite a simple task when faced with a true temporal statement, to determine whether or not it will of necessity have a true spatial counterpart. First of all we have to try and reformulate the statement in pure continuum terms making a universal statement concerning any continuum-in case the temporal statement refers to time alone, or concerning any pair of continua-in case the temporal statement refers to space as well as time.

What are continuum terms? All occupants of continua in general occupy points, may stretch or extend over distances which can be divided into intervals; the extremities of two occupants may coincide in which case the occupants are congruent, or have only one of their extremities coincide in which case they are adjacent; they may also be inside one another or overlap or have no common points. Thus terms like 'point', 'stretch', 'extension', 'distance', 'interval', 'extremity', 'coincide', 'congruent', 'adjacent', 'overlap' are continuum terms. Typical temporal terms or spatial terms may be translated into these: the temporal term 'duration' is equivalent to the continuum term 'extension'; 'to occur at t' is equivalent to 'to be at point t'; 'are simultaneous' is equivalent to 'are at the same point' and the spatial term 'the place of O' is equivalent to 'the set of points occupied by O' and so on. -After we have translated the temporal statement into the

equivalent continuum statement we must examine whether the latter is necessarily true of all continua. If it is then the true temporal statement has of necessity a true spatial counterpart; otherwise it does not.

For example the temporal statement 'Nothing can occupy two places at the same time' is true if we call all the points occupied by all the parts of an object as the same place. We have seen in the previous section how this,statement is to be reformulated in pure continuum terms and that the resulting statement is of necessity true. It follows therefore necessarily that the spatial counterpart of our temporal statement is true, and this we indeed found was the case.



I76 G. SCHLESINGER: SPACE AND TIME

On the other hand, for example, the statement 'There is order in time' translates into something like 'The various points of occupants in all continua are ordered in a special way' which of course is not necessarily true. The spatial counterpart 'There is order in space' need not be, and indeed is not, true..-

UNIVERSITY OF NORTH CAROLINA AT CHAPEL HILL



Documents

ESP the Similarities Between Space and Time