Time

Time

Time has frequently struck philosophers as mysterious. Some have even felt that it was incapable of rational discursive treatment and that it was able to be grasped only by intuition.

This defeatist attitude probably arises because time always seems to be mysteriously slipping away from us; no sooner do we grasp a bit of it in our consciousness than it has slipped away into the past. This entry will argue, however, that this notion of time as something that continually passes is based on a confusion.

St. Augustine’s Puzzles

The apparent mysteriousness of time can make puzzles about time seem more baffling than they are, even though similar ones arise in the case of nontemporal concepts. St. Augustine, in his Confessions, asks, “What is time?” When no one asks him, he knows; when someone asks him, however, he does not know.

St. Augustine’s Puzzles

He knows how to use the word “time” and cognate temporal words, such as “before,” “after,” “past,” and “future,” but he can give no clear account of this use. Trouble arises particularly from the form in which he puts his question: “What is time?” This looks like a request for a definition, and yet no definition is forthcoming.

However, most interesting concepts cannot be elucidated by explicit definitions. Thus, to explain the meaning of the word “length,” we cannot give an explicit definition, but we can do things that explain how to tell that one thing is longer than another and how to measure length.

In the same way, it is possible to give an account of the use of the word “time” even though it is not possible to do so by giving an explicit definition. In short, this puzzle of St. Augustine’s is not of a sort that arises peculiarly in the case of time. Beyond pointing this out, therefore, it is not appropriate here to go further into the matter.

temporal measurement

Augustine was also puzzled by how we could measure time. He seems to have been impressed by the lack of analogy between spatial and temporal measurement. For example, one can put a ruler alongside a tabletop, and the ruler and the tabletop are all there at once.

However, if one were to measure a temporal process, it would be done by comparing it with some other process, such as the movement of the hand of a watch. At any moment of the comparison, part of the process to be measured has passed away, and part of it is yet to be. It is not possible to get the thing to be measured in front of a person all at once, as one could with the tabletop.

Moreover, if two temporal processes are compared—say, a twenty-mile walk last week with a twenty-mile walk today—they are compared with two different movements of a watch hand, whereas two different tabletops are compared with the same ruler. Augustine is led to see a puzzle here because he demands, in effect, that non-analogous things should be talked about as though they were analogous.

non-analogous

In any case, the two things are not, in fact, as non-analogous as they appear to be at first sight. If we pass to a tenseless idiom in which material things are thought of as four-dimensional space-time solids, the difference becomes less apparent. For in the case of the tables we compare two different spatial cross sections of the four-dimensional object that is the ruler with spatial cross sections of the two tables.

Augustine seems to have been influenced by the thought that the present is real, although the past and future are not (the past has ceased to exist, and the future has not yet come to be); consequently, the measurement of time is puzzling in a way in which the measurement of space need not be (where the whole spatial object can be present now).

This thought that the present is real in a way in which past and future are not real—is part of the confusion of the flow or passage of time. This is not to say that presentism has not recently been intelligently defended, however implausibly, as by John Bigelow (1996). Apodeictic proof has rarely been possible in metaphysics, and we fall back eventually on trading plausibilities.

D. C. Williams

One of the central objections to presentism is the difficulty it has in analyzing cross-temporal statements such as “Smith will have come before you have finished breakfast.” Perhaps the most important objection relates to the explanatory value of four-dimensional space-time in relativity theory to be discussed below.

The Myth of Passage

We commonly think of time as a stream that flows or as a sea over which we advance. The two metaphors come to much the same thing, forming part of a whole way of thinking about time that D. C. Williams has called “the myth of passage”. If time flows past us or if we advance through time, this would be a motion with respect to a hypertime.

For motion in space is motion with respect to time, and motion of time or in time could hardly be a motion in time with respect to time. Ascription of a metric to time is not necessary for the argument, but supposing that time can be measured in seconds, the difficulty comes out clearly.

The Myth of Passage

If motion in space is feet per second, at what speed is the flow of time? Seconds per what? Moreover, if passage is of the essence of time, it is presumably the essence of hypertime, too, which would lead one to postulate a hyper-hypertime and so on ad infinitum.

The idea of time as passing is connected with the idea of events changing from future to past.We think of events as approaching us from the future, whereupon they are momentarily caught in the spotlight of the present and then recede into the past. Yet in normal contexts it does not make sense to talk of events changing or staying the same.

Roughly speaking, events are happenings to continuants—that is, to things that change or stay the same. Thus, we can speak of a table, a star, or a political constitution as changing or staying the same. But can we intelligibly talk of a change itself as changing or not changing?

differential calculus

It is true that in the differential calculus we talk of rates of change changing, but a rate of change is not the same thing as a change. Again, we can talk of continuants as coming into existence or ceasing to exist, but we cannot similarly talk of a “coming-into-existence” itself as coming into existence or ceasing to exist.

It is nevertheless true that there is a special class of predicates, such as “being past,” “being present,” “being future,” together with some epistemological predicates such as “being probable” or “being foreseen,” with respect to which we can talk of events as changing.

Significantly enough, these predicates do not apply to continuants. We do not, for example, naturally talk of a table or a star as “becoming past” but of its “ceasing to exist.”

the passage of time

There is something odd about the putative properties of pastness, presentness, futurity, and the like, whereby events are supposed to change.One might conjecture that the illusion of the passage of time arises from confusing the flow of information through our short-term memories with a flow of time itself.

TOKEN-REFLEXIVE EXPRESSIONS. Leaving aside the epistemological predicates, we may suspect that the oddness arises because the words “past,” “present,” and “future,” together with “now” and with tenses, are token-reflexive, or indexical, expressions.

That is, these words refer to their own utterance. If italics are allowed to indicate tenselessness in a verb, then if one says, “Caesar crosses the Rubicon,” the speaker does not indicate whether the crossing is something before, simultaneous with, or after the assertion.

temporal position

Tenseless verbs occur in mathematics where temporal position relative to a person’s utterance is not even in question. Thus, we can say, “2 + 2 is equal to 4” not because we wish to be noncommittal about the temporal position of 2 + 2 as being 4 but because it has no temporal position at all.

The token-reflexiveness (or more generally the indexicality) of the word “past” can be seen, for example, if a person who said that a certain event E is past could equally well have said, “E is earlier than this utterance.”

Similarly, instead of saying, “E is present,” he could say, “E is simultaneous with this utterance,” and instead of “E is future,” he could say, “E is later than this utterance.” The phrase “E was future” is more complicated.

spoken truly

It means that if someone had said, “E is future” or “E is later than this utterance,” at some appropriate time earlier than the present utterance (the utterance which we now refer to as “this utterance”), he would have spoken truly.

Thus, if we say that in 1939 the battle of Britain was in the future, we are putting ourselves into the shoes of ourselves as we were in 1939, when, given a certain amount of prescience, we might have said truly, “The battle of Britain is later than this utterance.”

Apart from this imaginative projection, we are saying no more than that the battle of Britain is later than 1939. Another way of dealing with this problem, one that is preferred by Michael Tooley (1997) would be to interpret the token reflexive expressions as referring not to utterances but to times of utterance.

genuine properties

It follows that there is a confusion in talking of events as changing in respect of pastness, presentness, and futurity. These are not genuine properties, which can be seen if the token-reflexiveness is made explicit. “E was future, is present, and will become past” goes over into “E is later than some utterance earlier than this utterance, is simultaneous with this utterance, and is earlier than some utterance later than this utterance.”

Here the reference is to three different utterances. However, if we allow simultaneity, being later, and being earlier as relations to times as well as events we could render the tensed sentence above by saying, “E is later than some time earlier than this utterance, is simultaneous with this utterance, and is earlier than some time later than this utterance.”

Also, the troubling sentence “Once there were no utterances” could go over to “There are times earlier than this utterance when there were no utterances.” A failure to recognize the direct or indirect indexicality of words such as “past,” “present,” and “future” can lead us to think wrongly of the change from future to past as a genuine change, such as the change in position of a boat that floats down a river.

the illusion of time

Nevertheless, there is probably a deeper source of the illusion of time flow. This is that our stock of memories is constantly increasing, and memories are of earlier, not of later, events. It is difficult to state this matter properly because we forget things as well as acquire new memories.

With a very old man there may well be a net diminishing of his stock of memories, and yet he does not feel as if time were running the other way. This suggestion is therefore tentative and incompletely worked out.

Possibly we confuse a flow of information through our short-term memories with a flow of time itself. The subordinate question of why our memories are of the past, not of the future, is an extremely interesting question in its own right and will be answered in a later section.

tenseless verbs

TENSES. Not only words such as “past” and “future” but also tenses can be replaced by the use of tenseless verbs together with the phrase “this utterance.” Thus, instead of saying, “Caesar crossed the Rubicon,” we could have said, “Caesar crosses the Rubicon earlier than this utterance.” For the present and future tenses we use “simultaneous with this utterance” and “later than this utterance.” Of course, this is not a strict translation.

If one person says, “Caesar crosses the Rubicon earlier than this utterance,” that person refers to his utterance,whereas if another person says, “Caesar crossed the Rubicon,” she is implicitly referring to her utterance.

Nevertheless, a tensed language is translatable into a tenseless language in the sense that the purposes subserved by the one, in which utterances covertly refer to themselves, can be subserved by the other in which utterances explicitly refer to themselves.

spoken language

A second qualification must be made. In the case of spoken language the token or “utterance” can be taken to be the actual sounds. In a written language the “token,” the configuration of ink marks, is something that persists through time. By “this utterance” we must therefore, in the case of written language, understand the coming-into-existence of the token or perhaps the act of writing it.

It has sometimes been objected that this account will not stand because “this utterance” means “the utterance which is now,” which reintroduces the notion of tense. There does not seem to be any reason, however, why we should accept this charge of circularity.

We have as good a right to say that “now” means “simultaneous with this utterance” as our opponent has to say that “this utterance” means “the utterance which is now.” The notion of an utterance directly referring to itself does not seem to be a difficult one.

indexical expressions

Tenses and their cognates may be seen to be indexical expressions. The truth conditions of sentences containing them cannot be given by translation into a nonindexical language. Nevertheless they can be given in a nonindexical metalanguage.

The idea derives from Donald Davidson and is advantageous because there is a recursively specifiable infinity of sentences in a language but not of utterances or inscriptions. Equally with the token reflexive account it removes the mystery that one might feel about tenses and cognate expressions.

Tensers, such as Quentin Smith (1993), argue that the words “past,” “present” and “future” refer to intrinsic properties of events, though Smith defines “past” and “future” in terms of “present.”

spoken by person

This makes him in a sense a presentist, though only a mild one as he does not deny the reality of the past and future. Davidson’s suggestion for the semantics of tenses is to say that (say) “I will come” is true as (potentially) spoken by person P at time t if and only if P comes later than t.

As Heather Dyke, in her doughty defense of the token-reflexive approach, has remarked, without the “potentially” (of which critics of modal logic may be suspicious) the Davidsonian schema comes out trivially true in cases where (say) “I will come” is not uttered by P at t. Perhaps one might reply that trivial truth is still truth and so harmless, or one might treat the Davidsonian schema as an idealization.

Dyke has urged that one should abandon aspirations of the old token reflexive theory for a translation of tensed sentences into tenseless ones but argue that a tensed sentence states the same fact about the world as can be stated by a tenseless one. Thus she wants a semantics based on tokens of sentences, not sentences, and so abandons recursiveness.

D. H. Mellor

A similar appeal to the notion of “fact” is made by D. H. Mellor in his influential Real Time II (1998), where he says that ontology can be separated from considerations of semantics. Of course this metaphysical notion of “fact” has been thought problematic, as by Davidson himself.

Nevertheless, the difference between the token reflexive account and the metalinguistic one is not of great ontological significance. Dyke contests arguments by Quentin Smith (1993), who has been an immensely prolific defender of the tensed notion of time.

DURATION. The philosophical notion of duration seems to be heavily infected with the myth of passage. Thus John Locke in his Essay concerning Human Understanding (1690) says that “duration is fleeting extension”. In the early nineteenth century, Henri Bergson made the notion of duration (durée) central in his philosophy.

physical time

According to him, physical time is something spatialized and intellectualized, whereas the real thing, with which we are acquainted in intuition (inner experience), is duration. Unlike physical time, which is always measured by comparing discrete spatial positions—for example, of clock hands—duration is the experienced change itself, the directly intuited nonspatial stream of consciousness in which past, present, and future flow into one another.

Bergson’s meaning is unclear, partly because he thinks that duration is something to be intuitively—not intellectually—grasped. Duration is closely connected in his thought with memory, for in memory, Bergson says, the past survives in the present.

Here he would seem to be open to the objection, urged against him by Bertrand Russell in his History of Western Philosophy (1945), that he confuses the memory of the past event with the past event itself, or the thought with that which is thought about.

simply about

Even though the Bergsonian notion of duration may be rejected because of its subjectivism and because of its close connection with the notion of time flow or passage, there is nevertheless a clear use of the word “duration” in science and ordinary life. Thus, in talking about the duration of a war, we talk simply about the temporal distance between its beginning and its end.

MCTAGGART ON TIME’S UNREALITY. The considerations thus far adduced may well be illustrated by considering how they bear on John McTaggart Ellis McTaggart’s well-known argument for the unreality of time, which was put forward in an article in Mind (1908) and in his posthumous Nature of Existence (1927). For McTaggart, events are capable of being ordered in two ways.

First, they can be ordered in respect to past, present, and future. He calls this ordering of events “the A series.” Second, events can be ordered in respect to the relations “earlier than” and “later than.”He calls this “the B series.” McTaggart then argues that the B series does not by itself give all that is essential to time and that the A series is contradictory.

Battle of Waterloo

Neither leg of his argument can stand criticism. His reason for saying that the B series misses the essence of time is that time involves change and yet it always is, was, and will be the case that the Battle of Hastings, say, is earlier than the Battle of Waterloo. It has already been shown, however, that it is not just false but also absurd to talk of events’ changing.

The Battle of Hastings is not sempiternally earlier than the Battle of Waterloo; it simply is (tenselessly) earlier than it. The notion of change is perfectly capable of being expressed in the language of the B series by saying that events in the B series differ from one another in various ways.

Similarly, the proposition that a thing changes can be expressed in the language of the B series by the statement that one spatial cross section of it is different from an earlier one, and the proposition that it does not change can be expressed by saying that earlier and later cross sections are similar to one another.

events change

To express the notion of change, we are therefore not forced to say that events change. Nor, therefore, are we forced into referring to the A series, into saying that events change (in the only way in which we can plausibly say this) in respect to pastness, presentness, and futurity.

Nevertheless, if we do retreat to the language of the A series, we can perfectly well do so without contradiction. Just as McTaggart erred by using tensed verbs when talking of the B series, he in effect made the correlative error of forgetting tenses (or equivalent devices) when talking of the A series.

For the contradiction that he claimed to find in the A series is that because any event is in turn future, present, and past, we must ascribe these three incompatible characteristics to it; but an event cannot be future, present, or past simpliciter but only with reference to a particular time—for example, one at which it was future, is present, and will be past.

tenseless language

If we restore the tenses, the trouble with the A series disappears. Unsuccessful though McTaggart’s argument is, it provides an excellent case study with which to elucidate the relations between tensed and tenseless language.

Space-time

The theory of relativity illustrates the advantages of replacing the separate notions of space and time by a unified notion of space-time. In particular, Minkowski showed that the Lorentz transformations of special relativity correspond to a rotation of axes in space-time.

He showed how natural the kinematics of special relativity can seem, as opposed to Newtonian kinematics, in which, in effect, we should rotate the time axis without correspondingly rotating the space axes.

Space-time

Since the theory of relativity it has become a commonplace to regard the world as a four-dimensional space-time manifold. Nevertheless, even in the days of Newtonian dynamics, there was nothing to prevent taking this view of the world, even though it would not have been as neat as it is in relativity theory.

If we pass to the four-dimensional way of looking at things, it is important not to be confused about certain conceptual matters. Confusion will arise if the tenseless way of talking, appropriate to the four-dimensional picture, is mixed with our ordinary way of talking of things as enduring substances, “the permanent in change.”

In ordinary language the word “space” itself is used as the name of a continuant.We can say, for example, that a part of space has become, or has continued to be, occupied. Space-time, however, is a “space” in a tenseless sense of this word, and because time is already in the representation, it is wrong to talk of space-time as itself changing.

light signal

Thus, in some expositions of relativity it is said that a certain “world line” is a track along which a material body moves or a light signal is propagated. The body or light signal, however, cannot correctly be said to move through space-time.

What should be said is that the body or the light signal lies (tenselessly) along the world line. To talk of anything’s moving through space-time is to bring time into the story twice over and in an illegitimate manner.

When we are talking about motion in terms of the spacetime picture, we must do so in terms of the relative orientations of world lines. Thus, to say that two particles move with a uniform nonzero relative velocity is expressed by saying that they lie (tenselessly) along straight world lines that are at an angle to one another.

moving backward

Similarly, the recent conception of the positron as an electron moving backward in time is misleading because nothing can move, forward or backward, in time.What is meant is that the world lines of a positron and electron, which are produced together or which annihilate one another, can be regarded as a single bent world line, and this may indeed be a fruitful way of looking at the matter.

In popular expositions of relativity we also read of such things as “consciousness crawling up the world line of one’s body.” This is once more the confusion of the myth of passage and, hence, of the illegitimate notion of movement through space-time.

It is instructive to consider how H. G. Wells’s time machine could be represented in the space-time picture. A moment’s thought should suffice to indicate that it cannot be represented at all. For if a line is drawn extending into the past, this will simply be the representation of a particle that has existed for a long time.

time machine

It is not surprising that we cannot represent a time machine because the notion of such a machine is an incoherent one. How fast would such a machine flash over a given ten-second stretch? In ten seconds or minus ten seconds? Or what? No sensible answer can be given, for the question is itself absurd.

The notion also involves the contradiction, pointed out by D. C. Williams in his article “The Myth of Passage” (1951) that if a person gets into a time machine at noon today, then at 3 a.m., say, that person shall be both at 3 p.m. today and at, say, a million years ago. There is nevertheless a more consistent notion of time travel though misleadingly so called. A person as a space-time entity might lie along a bent-back world line.

It might curve back and then would go back to your great grandmother’s time and then a bit forward while you saw your great grandmother. Paradox lurks because if the great grandmother had been shot you would not have existed.

David Lewis

David Lewis has proposed a banana skin solution. Since you could not have shot your great grandmother some accident, such as your slipping on a banana skin or your pistol jamming, must have prevented you from harming her. One would wish, however, for a solution of the paradox by reference to the laws of nature.

Though D. H. Mellor ably defends the four-dimensional ontology in his Real Time II, he nevertheless says something that may puzzle four-dimensionalists—for example, that a person from birth to death, or a stone over a long period of time, is said to have a certain property at time t, but not that a mere time slice or temporal stage of the person or stone has the property.

The puzzle is perhaps resolved if we note that Mellor thinks of the thing S as reidentifiable or a sortal as discussed by Peter Strawson. This is understandable because a child could hardly—and an adult could not easily—reidentify the mereological fusion of a bird, a bishop, and Mount Everest. Even so, the four-dimensionalist need not discern a difference between “S is A at t” and “S at t is A.”

F. P. Ramsey

The time slice may be referred to by reference to the salient four-dimensional object of which it is a slice. Mellor rightly stresses the importance for agency and practical matters of notions of reidentifiable sortals and for the determination of the strengths of beliefs and desires by a method originally due to F. P. Ramsey.

Absolute and Relational Theories

Isaac Newton held to an absolute theory of space and time, whereas his contemporary Gottfried Wilhelm Leibniz argued that space and time are merely sets of relations between things that are in space and time.

Newton misleadingly and unnecessarily expressed his absolute theory of time in terms of the myth of passage, as when he confusingly said, “Absolute, true and mathematical time, of itself and from its own nature, flows equably without relation to anything external” (Principia, in the Scholium to the Definitions of Mathematical Principles of Natural Philosophy).

Absolute and Relational

The special theory of relativity has made it impossible to consider time as something absolute; rather, it stands neutrally between absolute and relational theories of space-time. The question as between absolute and relational theories of space-time becomes especially interesting when we pass to the general theory of relativity.

According to this theory, the structure of space-time is dependent on the distribution of the matter in the universe. In most forms of the theory there is nevertheless a residual space-time structure that cannot be thus accounted for.

A curvature is usually attributed to spacetime even in the complete absence of matter, and the inertia of a body, according to this theory, depends in part on this cosmological contribution to the local metrical field and hence not solely on the total mass of the universe, as a purely relational theory would require.

Graham Nerlich

Research on this question is still going on, and until it has been decided, Mach’s principle (as Einstein called it), according to which the spatiotemporal structure of the universe depends entirely on the distribution of its matter, will remain controversial. But even if Mach’s principle were upheld, it might still be possible to interpret matter, in a metaphysical way, as regions of special curvature of space-time.

Graham Nerlich (1994) has given a striking and simple argument against those who, like Leibniz, defend relational theories by asking how one could tell whether everything had not doubled in size. He pointed out that this depends on the assumption that space is Euclidean. Relational theorists usually make the relevant relation that of cause and effect.

If this is defined by the use of counterfactual propositions one may object that the murkiness or contextual nature of these contrasts with the absolute theory’s reliance on the limpid clarity of geometry. Here I use “absolute” to contrast with ‘relational’ not as contrasted with “relativistic.”

Michael Tooley

An objection to a causal theory of time is that there could be uncaused events and that there are uncountably more space-time points than there are events. Michael Tooley separately assumes an ontology and topology of instants of time, but uses a causal theory to define temporal direction.

Time and the Continuum

An absolute theory of space-time, as envisaged above, need not imply that there is anything absolute about dis- tance (space-time interval). Because of the continuity of space-time, any space-time interval contains as many space-time points as any other (that is, a high infinity of them); space and time do not possess an intrinsic metric, and there must always be an element of convention in definitions of congruence in geometry and chronology, as Adolf Grünbaum has pointed out.

This means that the same cosmological facts can be expressed by means of a variety of space-time geometries, provided that they have the same topological structure. (Topology is that part of geometry which treats only of those properties of a figure which remain the same however that figure is transformed into a new one, with the sole restriction that a point transforms into one and only one point and neighboring points transform into neighboring ones. Thus, the surface of a sphere and that of a cube have the same topology, but that of a sphere and that of an infinite plane do not.)

Time and the Continuum

ZENO AND CANTOR. The continuity of space and time can be properly understood only in terms of the modern mathematical theory of infinity and dimensionality. Given the concepts available to him, Zeno rightly rejected the view that an extended line or time interval could be composed of unextended points or instants.

In modern terms it may be said that not even a denumerable infinity of points can make up a nonzero interval. Cantor has shown, however, that there are higher types of infinity than that which belongs to denumerable sets, such as the set of all natural numbers.

Cantor showed that the set of real numbers on a line, or segment of a line, is of a higher type of infinity than is the set of natural numbers. Perhaps the right cardinality of “dimensionless points” can add up to a nonzero length. This answer is on the right track. Nevertheless, the cardinality of a set of points does not by itself determine dimensionality.

sets of points

For example, Cantor showed that there is a one-to-one mapping between the points of a plane and the points of a line. However, a mathematical theory of dimension has been developed that accords with our intuitions in assigning 0, 1, 2, 3, and so on, dimensions respectively to points, lines, planes, volumes, and so on, and which also assigns dimensions to other sorts of sets of points.

For example, the set of all rational points on a line has dimension 0. So does the set of all irrational points. In these cases an infinity of “unextended points” does indeed form a set of dimension 0. Because these two sets of points together make up the set of points on a line, it follows that two sets of dimension 0 can be united to form a set of dimension 1.

Strictly speaking, it is even inaccurate to talk of “unextended points.” It is sets of points that have dimension. A line is a set of points, and the points are not parts of the line but members of it.

modern theory

The modern theory of dimension shows that there is no inconsistency in supposing that an appropriate nondenumerable infinity of points makes up a set of greater dimensionality than any finite or denumerable set of points could.

The theory of the continuum implies that if we take away the lower end of a closed interval, what is left is an open interval, an interval without a first point. In fact, Zeno’s premises in his paradox of the dichotomy do not lead to paradox at all but are a consistent consequence of the theory of the continuum.

Motion is impossible, according to the paradox of the dichotomy, because before one can go from A to B, one must first get to the halfway mark C, but before one can get to C, one must get to the halfway mark D between A and C, and so on indefinitely. It is concluded that the motion can never even get started. A similar argument, applied to time intervals, might seem to show that a thing cannot even endure through time.

fatal objection

The fallacy in both cases comes from thinking of the continuum as a set of points or instants arranged in succession. For if a continuous interval had to consist of a first, second, third, and so on point or instant, then the dichotomy would provide a fatal objection.

However, points or instants do not occur in succession, because to any point or instant there is no next point or instant. Such considerations enable us to deal with Zeno’s paradox of Achilles and the tortoise, in which similar difficulties are supposed to arise at the latter end of an open interval.

KANT’S ANTINOMIES. A related paradox is Kant’s first antinomy, in his Critique of Pure Reason (1929 [1781]). As was shown by Edward Caird (1889) in his commentary on Kant’s Critique, the antinomies (or paradoxes which Kant had constructed about space, time, and causality) were as important as Hume’s skeptical philosophy in arousing Kant from his “dogmatic slumbers.”

Critique of Pure Reason

Kant’s first antinomy relates to both space and time; the concentration here is on Critique as it relates to time. There are two antithetical arguments. The first states that the world had a beginning in time, whereas the second, with equal plausibility, seems to show that the world had no beginning in time.

The first argument begins with the premise that if the world had no beginning in time, then up to a given moment an infinite series of successive events must have passed. But, says Kant, the infinity of a series consists in the fact that it can never be completed. Hence, it is impossible for an infinite series of events to have passed away.

It can be seen that Kant’s argument here rests partly on the myth of passage. Kant thinks of the world as having come to its present state through a series of past events, so that an infinite succession would therefore have had to be completed. Otherwise, he would have been just as puzzled about the possibility of an infinite future as about an infinite past, and this does not seem to have been the case.

last member

Just as the sequence 0, 1, 2 ... can never be completed in the sense that it has no last member, the sequence ——, –2, –1, 0 cannot be completed in the sense that it has no first member. This is not to say, of course, that an infinite set need have either a first or last member.

Thus, the set of temporal instants up to, but not including, a given instant, has neither a first nor last member. However, Kant is clearly thinking not of the set of instants but of a sequence of events, each taking up a finite time.

The set of instants does not form a sequence because there are no instants that are next to one another. Kant’s definition of infinity, besides being objectionably psychologistic, is clearly inapplicable to infinite sets of entities which do not form a sequence, such as the points on a line or a segment of a line.

antithetical argument

Concerning an infinite set of events which form a sequence, however, Kant is not justified in supposing that its having a last member is any more objectionable than its having a first member. There is a perfect symmetry between the two cases once we rid ourselves of the notion of passage—that is, of the one-way flow of time.

In Kant’s antithetical argument, he argues that the world cannot have had a beginning in time, so that, contrary to the thesis of the antinomy, there must have been an infinity of past events.

His reason is that if the world had begun at a certain time, all previous time would have been a blank and there would be no reason that the world should have begun at the time it did rather than at some other time.

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Previously, Leibniz had used the same argument to support a relational theory of time. If time is constituted solely by the relations between events, then it becomes meaningless to ask questions about the temporal position of the universe as a whole or about when it began.

In an absolute theory of time (or of space-time) Kant’s problem remains, but further discussion of it cannot be pursued here because it would involve a metaphysical discussion of causality and the principle of sufficient reason.

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