Kant's Indirect Proof of Trascendental Idealism
by Paul Padovano

"Thus it is also false that the world (the sum total of all appearances) is a whole existing in itself. From which it follows that appearances in general are nothing outside our representations, which is just what we meant by their transcendental ideality," (Critique of Pure Reason B 535).¹

Kant claims that this result, the transcendental ideality of space and time, is a consequence of and solution to the contradictions present in the antinomies of pure reason. It will be the object of this paper to explore what the transcendental ideality of space and time means, how Kant is driven to adopt this position as a result of purported contradictions, and how convincing Kant's solution to this problem is. In the investigation of these three topics I will be focusing only on the First Antinomy and whether or not the contradictions present in this antinomy give us reason to declare the transcendental ideality of space and time. The first part of this paper will be concerned only with coming to an understanding of what the transcendental ideality of space and time amounts to. The second part will consist of my explication of the First Antinomy itself and its contradictions. In this section I will try to present in as strong a manner as possible Kant's claim that we become entangled in an unavoidable contradiction when we assume the transcendental reality of space and time. Where I think Kant falls short of this aim I will offer my own arguments. In the third section I will first explain how the transcendental ideality of space and time is a solution to the First Antinomy contradictions. Finally, in the second half of the third section I will show that Kant's solution ultimately fails because he is unable to prove that the contradictions that support the transcendental idealist solution are genuine.

I.

The first point that needs clarification is exactly what Kant means when he says that the world as the sum total of appearances does not exist in itself (the transcendental ideality of space and time). As we will see, appearances will be synonymous with the spatial-temporal world that we experience. To say that appearances don't exist in themselves is to say that the spatial-temporal world we experience is not a correct representation of the world independently of our consciousness of it. As Kant says in the Transcendental Aesthetic,
"the things that we intuit are not in themselves what we intuit them to be, nor are their relations so constituted in themselves as they appear to us; and that if we remove our own subject...space and time themselves would disappear," (B 59).
Here Kant makes the earlier stated association between objects as they appear to us (objects as we intuit them) and space and time.

It is also helpful to understand transcendental idealism by contrasting it with the antithetical position that Kant calls transcendental realism, which involves the claim that the spatial-temporal world as we experience it is a true representation of the world as it is constituted outside the consciousness of a subject. While it remains to be seen why Kant thinks that space and time are transcendentally ideal, we are at least now in a position to understand what the transcendental ideality of space and time amounts to.

Before delving into the second section in which the First Antinomy and its contradictions will be treated, two introductory remarks are necessary concerning Kant's concept of the world (equivalent henceforth to the phrases: the sum total of appearances, the empirical world, and the spatial-temporal world). The first clarification concerns what is included in the concept of the world. The world includes much more than the spatial and temporal boundaries of the earth, the solar system or the galaxy. Rather, it includes the entire world of appearances, or the entire empirical world.

The second clarification concerns what is excluded by the concept of the world. Just as the world-whole contains the entire empirical world, it excludes any posited world beyond the empirical. The world-whole includes "nothing other than the exposition of appearances, hence it does not concern the understanding's pure concept of a whole of things in general," (B 443). Thus, the world-whole includes nothing beyond the world that we experience, which for Kant is either in space and time or simply in time (B 37). This therefore excludes what Kant says might be contained in the concept "of things in general" which would include the possibility of a world not constituted by space or time. Such a world is not logically impossible but is something of which we can have no conception.

II.

Having explicated transcendental idealism and Kant's concept of the world we can now proceed to the First Antinomy contradiction. The basic structure of Kant's argument is as follows. First, he lays out the two opposing theses concerning the magnitude of the world in space and time. The thesis position is that the world has a beginning in time and a boundary in space (the world is finite in time and space). The antithesis position is that the world has no beginning in time and no boundary in space (the world is infinite or unbounded in space and time). After demonstrating this purported contradiction, Kant proceeds to explain that the existence of this contradiction demonstrates that the world cannot possibly exist as a whole because it can be neither a finite nor an infinite whole. If the world cannot exist as a spatial-temporal whole, the world is not described correctly as being spatial-temporal independently of the human subject. Transcendental idealism is therefore proven indirectly.

A quick explication of the logical structure of each thesis is in order before proceeding to the content of each position. As Kant says numerous times, the proponents of both positions are successful only when they go on the attack. In a particularly colorful explanation of this Kant says,
"These...assertions thus open up a dialectical battlefield, where each party will keep the upper hand as long as it is allowed to attack, and will certainly defeat that which is compelled to conduct itself merely defensively," (B 450).
Translating this analogy into logical terms, we have two opposing positions, each of which proceeds by indirect proof. Neither position aims to establish its own position directly, but rather attempt to prove faults in the opponent's thesis.

Both positions in the First Antinomy begin with a disjunction. Either the world is finite in space and time (F) or the world is infinite in space and time (I). A proponent of the thesis position (F) proves his point by setting up the disjunction: either (F) is true or (I) is true. He then proves his position by a reductio ad absurdum. He assumes the opposite (I) of what he wants to prove and in so doing draws out a contradiction in his opponent's position. The finitist would therefore assume that the world is infinite in space and time and demonstrate that this position leads to a contradiction. Using the law of the excluded middle and the fact that his opponent's position is contradictory, he concludes that his position must be true. This is the same tactic that the proponent of the antithesis position (I) uses. Thus we have the following logical structure:

The Finitist Position:
The world is either finite or infinite in space and time: I or F
The world cannot be infinite in either space or time: -I
Therefore the world is finite in space and time: F

The Infinitist Position: The world is either finite or infinite in space and time: I or F
The world cannot be finite in either space or time: -F
Therefore the world is infinite in space and time I

Having laid out the logical structure of the arguments used by both sides in the First Antinomy we can proceed to the details and content.

First Antinomy
Thesis Position: "The world has a beginning in time, and in space it is also enclosed in boundaries," (B 454).
Thesis Proof of the bounded-ness (finitude) of the world in time:
1. Assume the world has no beginning in time
2. If this is the case then up to any given moment in time an eternity has passed (an infinite series of events has elapsed)
3. But the infinity of a series consists precisely in the fact that it can never come to completion through the successive synthesis of a given unit
4. Therefore an infinite series of world-events cannot have passed
5. Thus the world must be bounded in time (have a beginning in time) (B 454)

As noted earlier, Kant attempts to prove each position by assuming the opponent's position and drawing out a contradiction. Here he assumes that the world has no beginning in time. The heart of Kant's argument is that this leads to a conceptual contradiction that cannot be maintained. If the world has no beginning in time then this requires that the world contain an infinite series of events in time. In other words, it requires a completed infinity. But this contradicts the very concept of infinity. Therefore the world in time cannot possibly be unbounded or infinite.

The rest of this section will be devoted to explaining this conceptual contradiction. I will begin by defining the concept of infinity. Then I will talk about two different conceptions of infinity which are important to distinguish in order to see Kant's argument properly. Finally I will show how the world-whole in time must fall under one of these conceptions of infinity which as a result generates a contradiction. Thus we can conclude from this contradiction that the world cannot be unbounded (infinite) in time.

Kant defines the concept of infinity saying, "The true concept of infinity is that the successive synthesis of unity in the traversal of a quantum can never be completed," and that this "quantum thereby contains a multiplicity that is greater than any number," (B 460). The concept of infinity is essentially that of a quantity or series, all of whose members can never be successively enumerated. The set of natural numbers is an infinite set since for any given arbitrarily high natural number, there are always larger natural numbers. The traversal of the members of the set of natural numbers has no end; there is no highest natural number. Here we have to be careful, as Kant notes, to exclude what he terms defective concepts of infinity (B 458). Because the concept of infinity is that of a never-ending series, it is not the concept of a determinate number or a maximum number. In fact, it is not a number at all, but is rather simply a relation to an arbitrarily assumed unit whose succession never comes to an end (B 458). Thus to think of infinity as an extremely high number would be false since that would be to make infinity into a determinate number.

Having defined the notion of infinity we can now distinguish between two ways of talking about things which we call infinite. Although Kant does not appear to make this distinction explicitly in the proof for the finitude of the world's time-series, I believe that it helps us to see his argument more clearly. I will call the first of these two different conceptions of infinity an empirically real infinity and the second an ideal infinity.

An empirically real infinity is the notion of there being a whole given in experience that contains an infinite number of units. We can take the example of this sheet of paper. Here we have a determinate whole, more specifically a whole that takes up about 8 by 11 inches of space. Now if I were to say that this whole contained an actual infinite number of parts in it (we will leave for later a discussion of the possibility of this) then this would be something that contained an empirically real infinity. It would be a determinate empirical object given as a whole that actually did contain an infinite number of parts.

The second conception of infinity is that of the ideal infinity. The idea behind this notion of infinity would not be that of an empirically given whole with an infinite number of parts, but rather the notion of an indeterminate continuation of a series. Infinite sets would fall under this conception of an ideal infinity. When I talk about the set of natural numbers I say that it contains an infinite number of members within it. In other words we have what Kant would call a set that could never come to completion through a successive synthesis. We call this set an infinite set despite the fact that the whole set is never empirically given. Rather, the set of natural numbers is what I am calling an ideal infinity. I have a rule that allows me to continue this series to an indeterminate extent.

Thus we have seen that there are two ways of talking about infinite things. The first is the idea of a whole given in experience that contains an infinite number of parts or has an infinite magnitude. The second is the idea of an unending indefinite extension that is never given as a whole. Now we do not require that some series, such as mathematical series, be completed. But, as we'll see, with the time series that extends from the past to the present it is necessary that it be completed. In other words, we require that it be given as a determinate whole (an empirically real infinity) and not just contain the idea of an indeterminate extension (an ideal infinity). This is what is going to draw us into a contradiction because it requires a completed infinity.

Now we can return to our discussion of the time series to see why it must be regarded as completed. We know that this moment in time is empirically real. In order for it to exist all preceding time must have elapsed (be completed). If all preceding time had not elapsed, then this moment would never have come into existence. This must be the case since we're assuming the transcendental reality of time; we're assuming that our experience of a moment of time as being able to exist only as a result of completed past time is also true of the world independently of our experience of it. But if the world has no beginning we would have an infinite time sequence, which therefore couldn't come to completion. But this completion is exactly what is required in order for this moment in time to exist. The contradiction arises because the time series must be completed in order for this moment to exist and because an infinite time-series (an unbounded time series) cannot come to completion.

Thus, we will now quickly summarize the results of this argument for the finitude of the world in time. When we assume that the world has no bound in time this requires that an infinite series of states of things has elapsed. The reason why this may not at first appear to be impossible is that we talk about and manipulate mathematically infinite sets or what I have called ideal infinities. However, we do not require that these ideal infinities ever be given as completed wholes. Nevertheless, if the world is unbounded in time it must contain an empirically real infinity and not an ideal infinity. This is how we generate the contradiction because we cannot have an elapsed infinite time series (an empirically given infinity). Therefore the world must be finite in time.

Having argued for why the world cannot contain an infinite time-series (why the world has a beginning) Kant goes on to argue for the bounded-ness of space. Again I will lay Kant's argument out and then elaborate on it:

Thesis Position: The world is finite (bounded) in space
1) Assume that the world is infinite or unbounded in space
2) This means that the world must be an infinite given whole of simultaneously given things
3)Any object not given to us in experience as a bounded whole can be thought only by the completion of the synthesis of its parts.
4)The spatial world as a whole is not given to us in experience
5)Therefore the world in space as a whole can be cognized by us only by the completion of the synthesis of its parts
6)This would require a completed infinite time series, which according to the earlier argument is contradictory
7)Therefore an infinite aggregate of spatial things cannot be regarded as a given whole
8)Therefore the world is not infinite in its extension but is rather bounded (B 454, 456)

In the thesis proof for the bounded-ness of the world in space, Kant seems to have taken a different path than the one taken in proving the bounded-ness of the world in time. The difference is that Kant's argument for the necessary bounded-ness of the world in time rests purely on a conceptual contradiction, while his argument for the finitude of the world in space rests on a reference to a subject's cognizing capacities. Thus I take his argument for time to be without reference to subjectivity unlike his argument for space. In fact, such a reference to subjectivity is out of place here since it is precisely the assumption that the spatial-temporal world exists outside the consciousness of a subject that is supposed to create the contradiction in the first place. The point is not to prove that we cannot cognize a world infinite in space, but rather that a world infinite in spatial extension cannot exist. Thus, in order for Kant to show that there really is a contradiction with the assumption that the spatial world exists as a whole outside of the subject's experience, he must continue without reference to a subject. In this section I will first discuss my interpretation of Kant's treatment of space and then offer my own argument that I believe Kant should have made in order to successfully prove that the world cannot have infinite spatial extension.

In his proof Kant states that if we assume the world has no boundary in space then there must be an infinite given whole of simultaneously existing things. After this Kant immediately turns to the cognizing capacities of the subject. In the third step in the proof as constructed above Kant says that any empirical object not given to us in intuition as a bounded whole can be thought only by the completion of the synthesis of its parts. Kant is indicating that there are two ways for a subject to cognize the magnitude of an empirical whole; either it is given immediately in intuition as a whole or we construct or synthesize its parts to produce a whole (B 456). When Kant says that something is given in experience as a whole he says that this whole is constituted by "boundaries, which would of themselves constitute this totality in intuition" (B 460). This page that I am looking at right now is a whole given in experience. Therefore I don't construct its totality by synthesizing its parts, but rather the whole is given to me immediately in experience.

If an empirical object is not given to us as a whole in intuition, Kant says that we can think the whole only through the completion of the successive synthesis of its parts. Kant doesn't explain what he means by "thinking" such empirically whole objects. But he seems to have the following in mind. "Thinking" doesn't have anything to do with representing the object in imagination since the objects in question have magnitudes too large to be intuited. "Thinking" here has to do with the way in which we account for or justify our use of our concepts of empirical wholes. I am justified by calling empirical objects "whole" when I intuit them as wholes. But with wholes that are not experienced as such Kant says, "we must give an account of our concept" and that we can establish the possibility of these wholes only "through the successive synthesis of its parts" (B 460). Therefore Kant seems to be indicating that we can justify calling something an empirical whole only if we have determined it to be such a whole. For objects that are not intuited as wholes this can be done only by actually determining its total spatial extension. This requires the completion of the measurement of its parts (the completion of the successive synthesis of its parts).

Now the world as a whole is never given to us in experience. Therefore, in order for us to cognize it, or "think" it as a whole, we must not only synthesize its parts, but most also complete this synthesis. If the world is unbounded or infinite in space, it will take an infinite amount of time for us to bring this synthesis to completion. But we have seen that infinity is precisely the concept of a synthesis or series that is never completed. Therefore, an unbounded world can never be cognized as a whole by a subject. Thus, we can cognize the world as a whole only if it is finite in space. From this Kant concludes that the world must be finite in space.

Nonetheless we are operating under the assumption that the spatial-temporal world exists outside of the consciousness of a subject and as such there should not be any reference to a subject's cognitive abilities. Kant's argument as recreated above seems to be simply that we can never measure an infinite space and thus can never cognize it as a whole. At most this can prove that we can never adequately account for our idea of the world-whole (complete the measurement of its magnitude) if it is infinite. It does not immediately prove that the world independent of us cannot be infinite in space. I believe Kant could and should have argued for the impossibility of the infinitude of the world in space without reference to the subject. Such an argument would run in the following way:

1)If the spatial world exists it has a size or magnitude
2)We are assuming the transcendental reality of space, and are therefore assuming space does indeed exist outside the consciousness of a subject
3)Therefore the spatial world has a size (magnitude)
4) Every magnitude or size is determinate. If this were not the case then we would have an indeterminate or incomplete magnitude, which is a contradictory concept.
5)Therefore the world in space must have some determinate magnitude.
6)The concept of infinity is that of a series that never comes to completion, i.e. it is not a determinate value.
7)Thus any infinite magnitude would be equivalent to an indeterminate size, which is contradictory.
8) Therefore all determinate magnitudes must be finite.
9) Therefore the world in space cannot have an infinite magnitude.
10)Thus the world in space must have a finite magnitude.

If we remodel Kant's thesis argument for space like this, then we have an argument more like the thesis argument for the finitude of time. In other words, it would rest purely on a conceptual contradiction and would make no reference to whether or not a subject could cognize such a thing.

Two objections can be raised to the proof I have offered for the impossibility of the infinitude of the world in space and Kant's proof for the impossibility of the infinitude of the world in time. The first is that both proofs really only demonstrate an epistemological limitation and not a metaphysical impossibility. In other words, these proofs only demonstrate that we cannot determine the world to be infinite in space or time. However this is not the case. Neither Kant's argument in reference to time nor my argument in reference to space is concerned with whether or not we can know space or time to be infinite. Regardless of our knowledge of the world in space and time, it cannot possibly have an infinite spatial or temporal magnitude because in the case of time this would require a completed infinity and in the case of space it would require an indeterminate size or magnitude. Both of these are contradictory concepts and thus cannot occur regardless of our knowledge of them.

A second objection that might be raised would be that while these proofs demonstrate a conceptual or logical contradiction, this does not therefore entail a metaphysical impossibility. Such an objection would arise from viewing logical contradictions as limitations of our cognizing capacities, but not limitations of the world independent of our knowledge of it. This would be tantamount to saying that although a round square is a logically contradictory concept (and not therefore cognizable by us), it is still metaphysically possible. I do not have a further justification for the fact that logical contradictions imply metaphysical impossibilities other than to re-state my position that a logically contradictory thing cannot exist. As long as we are unwilling to grant the possibility of the existence of contradictory things like round-squares, lengths with no extension, existent non-existences, and one-dimensional spheres, we must also be unwilling to grant the possibility of completed infinities and indeterminate sizes or magnitudes.

This brings to an end our discussion of the thesis position in the First Antinomy. The finitude of the world in space and time were both proven by assuming the infinitude of the world in space and time and drawing out contradictions. We can now proceed to the antithesis position. While I will ultimately disagree with Kant on the antithesis position, my aim in the next few pages will simply be to present Kant's argument in the most favorable light. Only later will I return to his argument and demonstrate its faults.

First Antinomy
Antithesis: "The world has no beginning and no bounds in space, but is infinite with regard to both time and space," (B 455).
Antithesis Proof of the infinitude or unbounded-ness of the world in time
1) Suppose that the world has a beginning in time
2) The concept of a beginning requires a preceding time in which some thing was not existing
3)Therefore a beginning of the world requires a preceding time in which the world did not exist
4)This is an empty time
5)But nothing can arise in an empty time since prior to anything being in time there is no criterion of the existence or non-existence of time
6)Therefore the world cannot have a beginning in time but must rather extend back into the past infinitely (B 455)

The antithesis position argument for time rests on the requirement of an empty time if the world is to be finite in time. Thus, my explication will center on what exactly an empty time is and why such a thing is problematic for Kant. The arguments and details that I offer to support Kant's claim are not actually offered by Kant, but I think they underlie Kant's claims.

Before going into why an empty time is a problematic concept, it is necessary to examine Kant's conception of a beginning. According to Kant, the concept of a beginning requires a preceding time in which some existence was not (B 455). If I say that it began to rain at 12 p.m. then this requires that there was a time prior to 12 p.m. when it was not raining. Otherwise, there would be no grounds for saying that the rain began at 12 p.m. Rather it would simply be continuing at 12 p.m. This is also the case with the world. If I am to talk about a beginning of the world, then there must necessarily be a period of time in which the world did not exist. But here we require a preceding empty time. In other words, it doesn't demand as in the case of the rain that it was simply not raining during a previous time interval, but rather it demands that there is absolutely nothing in the time interval preceding the beginning of the world.

Having seen that beginnings require preceding time intervals in which "some existence was not," we need to examine the conceptual requirements of a time interval. All time intervals must be characterized by duration. When I say 5 minutes passed since I had dinner there is a duration involved. When I say that the world is preceded by an empty time then this empty time must also be a duration of time. There is also a conceptual requirement of any duration, namely that it must be composed of moments of time. Furthermore, in order for there to be moments of time, these moments must be distinguished from each other (otherwise we would not have moments of time and thus no duration).

Now in order to have distinguished moments of time there must be a criterion of distinguishability. This criterion of distinguishability is not provided by the moments of time in themselves, but is rather only provided by distinguishable content that fills time. In other words it is what happens in the world at different moments of time that makes these moments of time distinguishable. So it is not that moments of time themselves are different, but rather that they are different by virtue of the content of these moments (the world occurrences).

One might object to this requirement by saying, "Of course moments of time are distinguishable without content that fills these moments. One moment is distinguished from another because it came before the other moment." But this misses the point. It is not possible to judge that one moment of time preceded another unless the moments of time had content. We only know, for example, that the moment of time in which sand filled one end of an hour glass preceded the moment of time in which the sand filled the other end of the hour glass because of a change in the content that filled these moments. The moments of time in themselves are not marked out as either preceding or following one another.

Having seen that a beginning requires a preceding time interval in which a thing was not, that a time interval requires duration, and that duration requires moments of time distinguished from one another, we can now see why Kant believes empty time is a problematic concept. If there is absolutely no content, no events, no occurrences in time, then there is no criterion by which there can be distinguished moments of time. Hence there is no criterion by which to judge a duration of empty time as having passed and therefore no criterion for the existence of an empty time. Therefore, we cannot require that the world be preceded by an empty time. Since the world cannot conceivably be preceded by an empty time, the world cannot be bounded or have a beginning in time. Kant concludes that if the world can have no beginning in time then it must extend back without bound in time and therefore must be infinite.

Kant's antithesis position on space is argued for very similarly, only here it turns on the conception of an empty space. His argument is as follows:

Antithesis Proof of the unbounded-ness or infinitude of the world in space
1) Suppose the world is bounded in space (finite).
2) This requires that the world exist in an empty space.
3)But the world is the idea of a whole outside of which there is no object of intuition to which the world is related.
4)Therefore a bounded world requires that the world be in relation to no object. In other words it requires a relation without there being anything to which it is related.
5)Such a relation is incoherent and so the world cannot be bounded by an empty space.
6)Therefore the world cannot be bounded and must be infinite in extension. (B 455, 457)

Kant's argument here hinges on the notion of an empty space. According to Kant, a world bounded in space requires that it be bounded by empty space. But because Kant believes this notion to be "nothing," (B 457), he concludes that the world cannot be bounded in space.

Before we get into why empty space is a problematic concept for Kant, we need to first look at the concept of a bound that he is working with. Any bounded or shaped object not only contains space within its bounds, but also requires further space outside of its bounds. If I tell someone to imagine a square they might draw the object in their imagination and in doing so have space contained within the square and space lying outside of the square. It is a requirement that any shaped figure have space outside of it. The same occurs when we talk about the entire spatial world having a bound, namely that this boundary requires that we posit further space outside of this bound in order to give a shape to the world.

Having seen that any shaped or bounded object must contain space outside of it, we can go on to investigate how empty space becomes problematic. First, like any bounded object, there must be space outside the boundary of the world in space. Second, because the world-whole is a whole outside of which there is no object of intuition, what is outside of this boundary cannot be more space. Otherwise this would just be more of what was contained in the world, namely intuitable space. So we require that space be related to something outside of its bounds and also that this relationship be characterized as "a relation of the world to no object" (Kant's emphasis, B 457). Thus when we assume the world to have a boundary we run into a contradiction - namely that it must have further space outside of it and yet cannot have further space outside of it. Therefore, the notion of a bounded or shaped world-whole is contradictory and so the world cannot have a boundary. Kant concludes from this that the world in space must be unbounded or infinite.

We have now reached the end of the discussion of the First Antinomy contradiction. The contradiction lies precisely in the fact that the concept of a world-whole cannot "yield a determinate infinite, nor yet something determinately finite (something absolutely bounded)," (B 546). Kant believes that the solution to this contradiction, namely that the world cannot exist as a whole in space or time, lies in the recognition of the transcendental ideality of space and time.

III.

In this section of the paper I will first discuss how transcendental idealism is a solution to the First Antinomy contradictions and then whether or not Kant was right in declaring space and time to be transcendentally ideal. Kant makes the point that if we become entangled in an irresolvable contradiction then it is possible that this contradiction arises because of a misdirected premise. When we say that the world-whole can neither be bounded (finite) nor unbounded (infinite) we have arrived at a contradiction. If neither element in the disjunction can be true than we are working with a false premise. If I say that some object either smells good or does not smell good, both propositions may be false, namely if the object has no smell at all (B 503). Kant also claims that the solution to the contradiction in the First Antinomy lies in a false premise, namely that there exists a spatial-temporal world-whole (B 504). Such a conclusion allows him to posit the transcendental ideality of space and time (space and time are not features of the world-in-itself but are rather elements contributed by the subject to experience).

Kant's outline for this argument is extremely brief. I will lay it out and then comment on it briefly.

Kant's argument for the transcendental ideality of space and time
1)If the world is a whole existing in itself, then it is either finite or infinite
2)The world can neither be finite nor infinite
3)Thus it is false that the world (sum total of spatial-temporal appearances) is a whole existing in itself
4)We can conclude that the spatial-temporal world is transcendentally ideal (B 534-5)

If the world independent of the consciousness of a subject is described correctly as being spatial and temporal then this world must be either finite or infinite in space and time. But as we have seen this spatio-temporal world is incapable of being either according to Kant. Therefore, space and time cannot be transcendentally real. Nevertheless, because space and time certainly exist in our consciousness, their existence must be described as being transcendentally ideal.

At this point I have gone through all of Kant's arguments and attempted to present them in as favorable a manner as possible. I will now consider whether or not Kant was right about the transcendental ideality of space and time. Kant's solution is only valid if there is in fact an absolutely unavoidable contradiction, namely if the world really cannot be a finite or an infinite whole. But this is where his argument fails. I believe that Kant's argument for time and my argument for space have proven the impossibility of the world's infinitude in space and time. Nonetheless, in the section on the antithesis position, Kant has not proven that the world cannot be finite in space and time. As we will see, his antithesis position arguments do not demonstrate metaphysical or logical impossibilities, but only epistemological limitations. Therefore, since space and time may still be finite, the contradiction collapses and with it the solution of the transcendental ideality of space and time.

We can now return to Kant's antithesis position arguments for space and time to see why his arguments are only epistemological. First we will deal with Kant's argument for the impossibility of a boundary in time. Kant argued that time could not be bounded since any boundary requires a preceding time, which, if there were no world during this time, would be an empty time. We saw why Kant took this concept to be nonsensical. All time requires durations of time, which in turn requires distinguishable moments of time, which in turn requires events in time.

There are two ways to respond to Kant's antithesis position depending on the view we take of time. The first suggests that time can pass without our being able to determine that it has passed. This would mean that an empty time, while not knowable, is nonetheless a metaphysical possibility. Kant has argued that an empty time must be composed of distinguished moments of time, which can be distinct only if there is content filling these moments of time. But this is only a necessary condition for our being able to determine that time has passed. While we certainly could not determine time to have passed if there was no content filling it, this does not mean that time cannot still be passing. Thus, Kant's argument seems to require that we be able to empirically verify a time interval for it to be possible. And since we cannot verify an empty time because we require content to fill moments of time, Kant thinks this proves that an empty time is impossible. Nonetheless, the question is not whether or not we can determine that an empty time has passed (an epistemological question), but rather whether an empty time can exist (a metaphysical question).

We can use our second objection if someone were to respond that it is absurd to talk of time passing without content filling it. In other words, time is a measure of change; if there is no change, as expressed through the content filling time, then there is no time at all. One might then claim that a beginning to the world requires a time preceding it, yet such a preceding time is impossible since there is no content filling time and thus no change. If we go on this definition of time as a measurement of change then we can agree with the second half of this position, namely that a time in which there is no content filling it is indeed not time at all. But we do not have to agree that a beginning to the world requires a preceding time. If there is no content filling time and thus no change it makes no sense to say that the beginning of the world was preceded by time. Thus, the beginning of the world coincided with the beginning of time and it makes no sense to ask what was prior to this beginning. Rather, we can simply say that change arose and with it the measurement of time. What makes us think there must be a time preceding the beginning of the world is simply that we cannot experience and therefore cannot imagine what such a beginning would be like. All beginnings of which we are conscious are always preceded by a time in which the thing that began did not exist. But this is simply a limitation of our experience and not a metaphysical impossibility.

The same two objections can be used in tandem with respect to Kant's argument for the impossibility of the bounded-ness of the world in space. Kant argued that this would require an empty space outside of the bounded world, which is ultimately an incoherent notion. If we run the first objection then we can say that just as time can pass without our being able to determine its having passed (an empty time), space can exist without empirical matter filling it (empty space). In this case the world would be bounded by empty space. We would not be able to imagine such a bound because we can only imagine a bound with further content filled space outside of it. However, this demonstrates only a limitation of our imaginative capacities, i.e. an epistemological limitation, and not a metaphysical impossibility.

Nonetheless one can object by claiming that talk of space without matter filling it is absurd. We can now run the second objection. Kant seems to think that all space must require further space outside of it. But this is only a requirement that must be met if we are to picture or imagine a shape or bounded object. There could very well be a finite space without further space outside of it. It wouldn't make sense to ask what is outside of it because there is nothing outside of it. We certainly wouldn't be able to picture the shape of the universe in our imagination, but this is merely a limitation of our imagination.

Another problem with Kant's proof for the impossibility of the finitude of the world in space is that he conflates unbounded-ness with infinitude. However, these are not equivalent concepts. We can show that these two concepts are not co-extensive by constructing a model of the universe that is represented by the surface of a torus. We begin by picturing the world as the surface of a sheet of paper. There are bounds on four sides of this paper, which Kant would not allow (since bounds require further space outside these bounds). We can proceed to role up the sheet of paper such that two of its ends touch to form a tube. Now the surface of this shape contains bounds on two sides instead of four. We can then wrap the tube around so that its ends meet, thus eliminating the two bounds that it had previously. We end up with a torus and if the world is conceptualized as the surface of a torus, then we have a space that is finite yet not bounded. It is not bounded because at any point in space there is further space beyond the point. Yet the world would not therefore contain an infinite number of things in it, or an infinite number of spatial units.

Therefore Kant has not proven that a world finite in space or time is a metaphysical impossibility. If we see time as not requiring changing content, then although we cannot determine that an empty time has passed it can nevertheless still pass without our ability to determine that it has. If we suppose that space can exist without matter filling it, then the empirical world could be bounded by empty space even though we could not imagine such a boundary. If we define time in terms of changing content, then a beginning to the world coincides with the measurement of time and it makes no sense to ask what preceded it. If we suppose that space without matter is an absurd concept, then the edges of the universe constitute the edge of space with no further space outside of it. A final objection to Kant's antithesis argument for space would be that he conflates un-boundedness with infinitude. And as we've seen we can have a spatial model that is unbounded yet not infinite (the surface of a torus).

We have now seen the nature and cogency of Kant's First Antinomy indirect proof of the transcendental ideality of space and time. Kant attempts to prove the transcendental ideality of space and time by showing that the empirical world can be neither a finite whole nor an infinite whole, which must be the case if the world is correctly described as being spatial-temporal outside the consciousness of a subject. While Kant's argument for time and my argument for space demonstrate the impossibility of the infinitude of the world in space and time, Kant does not prove the impossibility of the finitude of the world in space and time. The reason is that he demonstrates only epistemological limitations and not metaphysical impossibilities. Therefore, while Kant's entire project is not wasted (since the proof of the impossibility of a world infinite in spatial or temporal magnitude is not trivial by any means), his First Antinomy indirect proof of transcendental idealism ultimately fails since the First Antinomy contradictions are only apparent.