2. The consensus view: Kant embraces NAITS

In the secondary literature, Henry Allison’s (Reference Allison1983) Kant’s Transcendental Idealism has become the locus classicus for the consensus view that Kant rejects actually infinite tota synthetica as conceptually incoherent. Allison begins from the observation that Kant makes an important distinction between wholes that result from a combination or unification of pre-given parts (tota synthetica) and wholes that are given prior to their parts (tota analytica).Footnote ⁴ Space and time are paradigmatic examples of tota analytica, for in space and time (as Kant understands them) the ‘parts are possible only in the whole, and not the whole through the parts’ (A483/B466). In contrast, the material world (as it is treated in the antinomies) is a totum syntheticum, for it is a whole formed from a combination of prior parts.Footnote ⁵ As Kant writes, the idea of the material world is the idea of ‘[t]he absolute completeness of the composition [Zusammensetzung] of the given whole of all appearances’ (A415/B443).Footnote ⁶ Because the material world requires a composition, it is a totum syntheticum rather than a totum analyticum. And for proponents of the current consensus, this is a crucial part of Kant’s explanation of why space and time can be actually infinite, while the material world cannot (per the first antinomy’s thesis argument). As the argument goes, tota synthetica can exist only as a product of the complete composition or combination of their parts, and actually infinitely many parts can never be completely combined, on pain of contradiction. In contrast, space and time are given as wholes prior to their parts, and so they do not depend on composition; hence, their actual infinity is unproblematic.Footnote ⁷

As just noted, one thing motivating proponents of the current consensus is their desire to explain why Kant thinks space and time can be actually infinite, while the material world cannot. However, for those who read Kant as a proponent of NAITS, a further crucial consideration is that NAITS would seem to relieve Kant of an otherwise damning criticism, namely, that the thesis argument of the first antinomy presupposes transcendental idealism. Consider Allison’s discussion. According to Allison, the thesis argument of the first antinomy seems to make a questionable claim when it says that ‘the infinity of a series consists precisely in the fact that it can never be completed through a successive synthesis’ (A426/B454). For unless one had already assumed that synthesis is a mental activity performed by finite human minds, it is not clear why one would take an infinite synthesis to be impossible (Allison Reference Allison1983: 42-3). And unless one had already accepted transcendental idealism, it is not clear why one would grant that what can be synthesised by finite minds determines answers to questions about how the spatiotemporal world can be.Footnote ⁸ Since the antinomies are supposed to show that the commitments of transcendental realism lead to contradictions, this is a problem; their arguments ought not to infer from facts about what we cannot represent to conclusions about what cannot exist (or so proponents of the current consensus argue).Footnote ⁹

In response to these worries, Allison argues that the claims concerning ‘synthesis’ in the thesis argument should be read as expressing conceptual truths about the notion of infinity. That is, when Kant says (in the voice of a transcendental realist) that an infinite successive synthesis cannot be completed, he means to be saying that the very thought of a complete infinite combination is conceptually impossible or logically incoherent. And if it is correct that the very concept of infinity excludes the concept of completeness in this way, then actually infinite tota synthetica can be ruled out without presupposing transcendental idealism. After all, transcendental realists and transcendental idealists alike hold that contradictory states of affairs cannot obtain in the world, so if an actually infinite totum syntheticum is a contradiction in terms, then transcendental realists cannot complain that the thesis argument begs the question. As Allison summarises his solution, in the thesis argument ‘the critique of the infinitistic position turns on a conceptual claim and has nothing to do with the presumed psychological impossibility of grasping or comprehending the infinite’ (Reference Allison1983: 43).

Finally, note that Allison also takes Kant’s putative endorsement of NAITS to be supported by his claim in the remark on the thesis argument that ‘the true (transcendental) concept of infinity is that the successive synthesis of unity in the traversal of a quantum can never be completed’ (A432/B460). According to Allison, Kant intends to establish with this remark that the notion of ‘inexhaustibility’ is built into the very concept of the infinite. Since it is also built into the concept of a totum syntheticum that pre-given parts have been brought together completely by a synthesis or combination, it follows according to Allison that the notion of an infinite totum syntheticum is a contradiction in terms. As Allison argues, ‘[s]ince [the material universe] is conceived as a totum syntheticum (it could hardly be regarded as a totum analyticum), the thought of the complete enumeration or “synthesis” of its parts, which is built into this concept, contradicts the thought of inexhaustibility, which is similarly built into the concept of the infinite’ (Reference Allison1983: 43). To summarise then, for proponents of the current consensus, Kant thinks the idea of an actually infinite material universe embodies a contradiction because it is the contradictory idea of an actually infinite totum syntheticum; whatever is a totum syntheticum is by definition the product of the complete combination of its parts, while whatever has infinitely many parts by definition cannot be completely combined (or so the argument goes).Footnote ¹⁰

3. The thinkability of actually infinite tota synthetica in the Transcendental Dialectic

It is clear that Kant conceives of space and time as actually infinite tota analytica, and it is clear that he intends to deny that the spatiotemporal world exists as an actually infinite totum syntheticum. Nonetheless, in this section, I argue that Kant does not rule out the possibility of all actually infinite tota synthetica on purely conceptual grounds, and so he does not embrace NAITS. On the contrary, according to Kant, human reason has a non-contradictory idea of an actually infinite totum syntheticum in its idea of an infinite whole series of conditions, and this idea plays an important explanatory role in generating the antinomies. Thus, as I argue, a proper appreciation of how the antinomies arise in fact requires acknowledging that Kant thinks actually infinite tota synthetica are conceptually possible (even if an actually infinite spatiotemporal world is not).

To see that Kant embraces the logical possibility of actually infinite tota synthetica, consider the following passage from the first section of the antinomies, where Kant says that we can think (denken) of the unconditioned (das Unbedingte) in two ways when the conditioned and its conditions form a series.Footnote ¹¹ Having argued that pure reason pursues objects answering to the idea of the unconditioned in the antinomies (such that each of the thesis and antithesis positions asserts the existence of something unconditioned), Kant writes:

In an important footnote, Kant expands on why even an infinite series of conditions can be unconditioned as follows:

Together, these passages allow us to draw two important conclusions. First, they establish that Kant does not believe the idea of an infinite whole series of conditions is a conceptual or logical impossibility. As Kant writes, we can think of an infinite series of conditions as being ‘absolutely unconditioned’ qua ‘absolute whole’. That is, we can think of a series of conditions that is both infinite and whole. It is infinite in the sense that every member is conditioned by a further condition. It is whole in the sense that ‘outside it there are no more conditions regarding which it could be conditioned’, i.e., it contains all the conditions of the relevant kind.Footnote ¹⁴ Since whatever we can think is not contradictory in Kant’s account (see Bxxvifn and Bxxviii), it follows that the idea of an infinite whole series of conditions is not a contradictory idea in his view.

But given this, we can draw a second important conclusion: an actually infinite totum syntheticum must not be a contradictory notion in Kant’s view. We can draw this conclusion because a series of conditions satisfies Kant’s conception of a totum syntheticum. Why is this? Recall that a totum syntheticum is a whole resulting from the combination or composition of its parts, where the parts precede the whole. As Kant puts it in a reflection, a totum syntheticum is ‘that whose composition, as to its possibility, is grounded on its parts, which can also be thought without any composition’ (Refl 3789, 17: 293.9-11). But the idea of a whole series of conditions is precisely an idea of this kind of object. A series of conditions is a collection of pre-given items (conditions) brought together or composed (via conditioning relations) to form a whole (the series). Thus, if an infinite whole series of conditions is a conceptual possibility, as Kant claims it is, then so too is an infinite totum syntheticum. Hence, Kant does not think that the very notion of an infinite totum syntheticum is a contradiction in terms. We can think (even if we cannot cognise) an actually infinite totum syntheticum.

Notice further that these conclusions show that two notions of totality are in fact applicable to the idea of an infinite and yet unconditioned whole series of conditions. First, the series of conditions qualifies as a whole or totality in the sense that it contains all of the conditions of the relevant type. In virtue of this, Kant argues, it can be thought of as unconditioned (recall Kant’s explanation at A417-8/B445fn that ‘outside it there are no more conditions regarding which it could be conditioned’). But second, the series also qualifies as a totality in the sense that it is a plurality of things brought together to form a unity. This feature of the series is what qualifies it as a totum syntheticum , or synthetic whole. That is, it is a series made up of constituent conditions, which metaphysically precede the whole series that they form, and they form a series in virtue of the conditioning relations that order them and unite them together. Moreover, in virtue of being united in this way, the category of totality can also be thought of as applying to the series. According to Kant, the category of totality is ‘nothing other than plurality considered as unity [die Vielheit als Einheit betrachtet]’ (B111). And when we think of an unconditioned infinite whole series of conditions, we are thinking of something to which this notion applies (at least in its unschematised form). We are thinking of a plurality of items (the conditions) as united (via conditioning relations) to form a unity (the whole series). So, in Kant’s view, an actually infinite series of conditions is thinkable as a totality in two senses. It is thinkable as containing all of the conditions of the relevant type (i.e., it is thinkable as the complete collection of conditions). And it is thinkable as structured into a unified whole (since it is thinkable as formed from conditions that are unified via the conditioning relations that structure the series).Footnote ¹⁵

4. Is the idea of an infinite whole series of conditions really a logically coherent idea of an infinite totum syntheticum?

There are several ways in which a proponent of the current consensus might resist the conclusion that our idea of an infinite whole series of conditions is a logically coherent idea of an actually infinite totum syntheticum. In this section, I consider two especially salient lines of resistance. According to the first line of resistance, an actually infinite series of conditions can be conceived, but because it is infinite, it cannot really be conceived as a totum syntheticum. According to the second line of resistance, we have an idea of an actually infinite totum syntheticum, but this idea is ultimately shown to be incoherent; that is, it is a contradictory idea after all.

First consider the suggestion that an actually infinite series of conditions cannot really be represented as a totum syntheticum because it is infinite. This is in fact how Allison understands the thesis argument of the first antinomy in his classic defence of the consensus view. According to Allison, the first antinomy’s thesis argument begins from the assumption that the series of past world-states is infinite. But as Allison continues:

Here, Allison suggests that Kant’s own view is that the series of past world-states is infinite, and for this reason it cannot be a totum syntheticum. Notice that there are in fact two ways of understanding this claim. First, one might take it to mean that the series cannot in fact have the structure of a totum syntheticum because it is infinite; it must rather be a totum analyticum .Footnote ¹⁶ This clearly is not how Allison intends for us to read Kant, for in his view, the world as treated in the antinomies can ‘hardly be regarded as a totum analyticum’ (Reference Allison1983: 43).Footnote ¹⁷ I agree with Allison on this point, for Kant explicitly describes the ideas of the unconditioned treated in the antinomies as ‘world-concepts (Weltbegriffe)’ (A408/B434), and throughout his career he describes the concept of a world as the concept of a composite, which (since composition builds from parts to wholes) must be a totum syntheticum.Footnote ¹⁸

A second alternative more in line with Allison’s own intentions goes as follows. Because the series is infinite, it cannot be a true totum. Or as Allison himself puts the point, ‘since, as infinite, the series has only one end, it cannot constitute a totality’ (Reference Allison1983: 44). Note also that a further consideration one might take to support this line of resistance is Kant’s commitment to the idea that the concept of the infinite as such is not problematic; only its application to the material world is contradictory. As one might reason, this should make us think that considering a series of conditions as infinite is unproblematic in exactly the same sense in which considering the infinite as such is unproblematic: we have a perfectly coherent concept of both. And given this, one might be tempted to conclude, Kant’s view must be that problems arise only when we attempt to combine the concept of an infinite series of conditions with the concept of a totum or totality. That is, as one might argue, the concept of an infinite series of conditions has the same status as the concept of the infinite (it is perfectly coherent), but the concept of an infinite series of conditions considered as a totum (or totality) has a different status – it is an incoherent concept.Footnote ¹⁹

However, this argument fails to appreciate the significance of the A417-8/B445-6 passage. First, as explained above, an infinite series of conditions does qualify as a totality in the sense of being a plurality of items brought together to form a unified whole; its status as a totum syntheticum indicates that it is a totality in this sense. But second, we have also seen that Kant thinks an infinite series of conditions coherently can be thought as the complete collection containing all the conditions of the relevant type for a given conditioned thing. Consider again Kant’s claim that we can think of an infinite whole series of conditions as unconditioned. As Kant puts it, we can think of an infinite whole [ganze] series of conditions as one ‘outside [of which] there are no more conditions regarding which it could be conditioned’, and our ability to represent an infinite series in this way explains why we can represent it as unconditioned (A417-8/B445fn). But thinking of an infinite whole series of conditions in this way – i.e., thinking of it as containing all of something’s conditions, such that outside of the series no conditions are missing – is precisely to think of an infinite series of conditions as a complete collection or totality of conditions in the relevant sense.Footnote ²⁰ So, although one can consider an infinite series of conditions without considering it as a totality in the relevant sense, Kant is explicit at A417-8/B445-6 that our idea of the unconditioned in its infinite manifestation is an idea of an infinite series of conditions considered as a complete totality.Footnote ²¹

Note that this conclusion also is not impugned by the fact that Kant rejects the idea of an infinite number. According to Kant, ‘the concept of a number [Zahl] (which belongs to the category of allness) is not always possible wherever the concepts of multitude and of unity are (e.g., in the representation of the infinite)’ (B111). With this in mind, one might be tempted to argue that Kant must rule out infinite numbers because he thinks that the concept of an infinite totality is absurd. As one might reason, Kant thinks numbers are finite because he thinks numbers are always reachable by counting, and this seems to suggest the availability of an argument that goes as follows. Just as an infinite number is impossible because one cannot completely count through all the finite numbers to reach an infinite number, so too an infinite totality in general is impossible because one can never completely enumerate infinitely many things.Footnote ²² In fact, one might even argue that Kant just means to say at B111 that the category of totality does not apply to the infinite at all (even though the categories of unity and plurality do).Footnote ²³

However, Kant’s rejection of infinite numbers does not entail that all magnitudes are finite or that every totum syntheticum has a magnitude that is representable by a number. His claim at B111 also does not entail that infinite multiplicities cannot be represented as totalities at all. For, first, the claim that the concept of number belongs to the category of totality does not entail that the concept of number is applicable whenever the concept of totality is. The concept of number might be more determinate than the concept of totality and hence might be more restricted in its application (e.g., in virtue of being a schematised concept, as Kant suggests at A142-3/B182). Second, Kant in fact has a longstanding commitment to the view that some magnitudes (namely, infinite ones) cannot be represented by a number, and he does not hold that only tota analytica can have such magnitudes. Instead, even in the early part of his career, Kant is at pains to stress that tota synthetica should not be defined so as to entail their finitude. Consider Kant’s claim in the Inaugural Dissertation that it is a ‘prejudice’ of our ‘cognition of quantity’ that we mistakenly believe ‘every actual multiplicity can be given numerically, and thus every magnitude is finite’ (2: 435). In saying this, Kant suggests that magnitudes in general are not limited to those that can be represented with number.Footnote ²⁴ Now add to this Kant’s early criticisms of Baumgarten’s finitistic conception of a world. According to Kant, Baumgarten wrongly builds finitude into his very definition of a world, for ‘[i]t is not necessary that the finitude of the world, which is yet to be proven, is brought into the definition’ (MH, 28: 39). In saying this, Kant suggests that the very concept of a world does not include the mark of finitude; hence, neither does the very concept of a totum syntheticum. Thus, Kant in fact embraces the logical possibility of actually infinite tota synthetica, and he takes this to be compatible with his claim that the idea of an infinite number is ‘absurd’.Footnote ²⁵

Let us turn now to a second strategy for resisting the conclusion that our idea of an infinite whole series of conditions is a coherent idea of an actually infinite totum syntheticum. Namely, one might argue that the upshot of the antinomies is that our idea of an actually infinite whole series of conditions is not a logically coherent idea after all. One might develop this line of resistance as follows. First, one might note, Kant goes out of his way to stress in the A417-8/B445 footnote discussed above that ‘[t]he absolute whole of the series of conditions for a given conditioned’ is ‘a problematic concept, whose possibility has to be investigated’. But as one might then argue, the upshot of the antinomies is precisely that our ideas of the unconditioned turn out to be ideas of impossible objects. For example, we find out in the first antinomy that a series of conditions with an infinite magnitude is in fact impossible, for we establish the possibility of a magnitude by successively synthesising its elements, and one of the lessons of the first antinomy is that ‘the infinity of a series consists precisely in the fact that it can never be completed through a successive synthesis’ (A426/B454). Thus, one might reason, although the idea of an infinite series of conditions initially looks like the idea of a possible object, we find out through the antinomy’s investigation that it is actually impossible. Hence, the idea of an actually infinite totum syntheticum is an incoherent idea after all.

This objection can be answered by invoking Kant’s important distinction between real and logical possibility. According to Kant, we establish the real possibility of an object (i.e., that it could really exist in space and time) by showing that it is compatible with the forms of experience (i.e., the forms of intuition and the categories).Footnote ²⁶ As Kant puts it at A596/B624fn, an object is really possible if ‘the objective reality of the synthesis through which [its] concept is generated … rests on principles of possible experience.’ Or, as he puts it earlier in the Critique, an object is really possible if its concept is one for which ‘an example from experience’ can be thought (A290-1/B347). Arguably, this means that to prove the real possibility of an actually infinite series of conditions, we must show that it could actually exist in space and time, and this would require completing an infinite successive synthesis of all of its parts. But as we have seen, Kant holds this to be impossible (recall A432/B460).

However, notice that even if an actually infinite totum syntheticum can be shown to be really impossible in this way, it does not follow that Kant thinks we lack a proof of its logical possibility. On the contrary, Kant holds that an object is logically possible if its concept can be thought, which is to say if we have a concept of it that does not ‘cancel[] itself out’ (A292/B348-9). Our idea of an infinite whole series of conditions is not self-cancelling in this way, for as the A417-8/B445-6 passage proves, we in fact can think of an actually infinite totum syntheticum in an idea of reason.

This said, let me stress here that Kant’s embrace of the logical coherence of actually infinite tota synthetica does not entail that he thinks an actually infinite spatiotemporal world is even logically possible. For although a spatiotemporal world is a totum syntheticum, there are (at least logically) possible tota synthetica that are not spatiotemporal worlds. For example, we think of a totum syntheticum that is not a spatiotemporal world when we think of a totum syntheticum composed of things in themselves (which is at least logically possible). Reason also thinks a totum syntheticum that is not yet a spatiotemporal world in its idea of an actually infinite unconditioned series of conditions (discussed at A417-8/B445-6); this is the idea of a totum syntheticum of conditions that is not yet determined as to any specific spatiotemporal properties. And we can see Kant’s case for the unthinkability of an actually infinite spatiotemporal world in the thesis argument of the first antinomy. As we have seen, Kant argues that ‘[t]he true (transcendental) concept of infinity is that the successive synthesis of units [Einheit] in the traversal [Durchmessung] of a quantum can never be completed’ (A432/B460, my underlining). This is plausibly interpreted as the claim that it is logically or absolutely impossible for a temporally successive process to be actually infinite. After all, Kant held even prior to the first Critique that even an infinite intellect could not complete an infinite successive synthesis; and as he argued at that time, what no intellect whatsoever can represent is absolutely impossible in the sense of being ‘unthinkable’.Footnote ²⁷

So, the argument against the conceptual possibility of an actually infinite spatiotemporal world can be understood as follows. According to Kant, a spatiotemporal world is both a spatially and a temporally extended object, and its temporal magnitude depends on the successive combination of the world-states that determine the length of its history. As Kant puts it, the temporal magnitude of the spatiotemporal world is a result of its previous states having ‘elapsed’ (abgelaufen) or ‘passed away’ (verflossen) (A426/B454). But as the first antinomy’s thesis argument shows, an infinite successive combination can never be completed, and for this reason ‘an infinitely elapsed world series [unendliche verflossene Weltreihe] is impossible’ (A426/B454). That is, because of the way in which a temporally infinite world would require an infinite and yet completed temporal succession of states, it is a conceptual impossibility.Footnote ²⁸ This argument can go through without ruling out the possibility of actually infinite tota synthetica that do not require a temporally successive combination of their parts.Footnote ²⁹

Finally, notice that this interpretation does not commit Kant to saying that we cannot represent any kind of infinite temporal series at all. Rather, on this reading, his arguments leave open the possibility that we can represent an infinite series of items ordered in temporal relations such as the relations of ‘before than’ and ‘later than’, since such a representation need not involve a successive process. They also allow for our ability to represent the infinity of time as a whole in our original intuition of it (A32/B48). Kant’s definition of the infinite at A432/B460 prohibits only the possibility of representing an infinite multiplicity via a successive mode of representation and the related phenomenon of representing a complete infinite succession as such.Footnote ³⁰

5. NAITS’s putative advantage for interpreting the first antinomy

Thus far, I have argued that Kant is not a proponent of NAITS, the view that all actually infinite tota synthetica are conceptually impossible or logically incoherent. As I have argued, Kant speaks unambiguously of our possession of an idea of an actually infinite whole series of conditions (which is our idea of the unconditioned in its infinite manifestation). And since this idea is an idea of a totum syntheticum, his view is not that the very notion of an actually infinite totum syntheticum is an unthinkable contradiction in terms. I have also noted in the discussion above that these conclusions are compatible with the claim that an actually infinite temporally successive process is incoherent. In light of this, it is possible for Kant to argue that a temporally infinite spatiotemporal world is absolutely impossible, even if it is not the case that all actually infinite tota synthetica can be ruled out on purely conceptual grounds.

In this section, I consider whether abandoning the current consensus in this way – i.e., arguing that Kant does not embrace NAITS – is unacceptable because it makes the first antinomy’s thesis argument question-begging against the transcendental realist. As I will argue, the reading I have offered does show that the spatial side of the thesis argument may not speak to transcendental realists in terms they would accept, but closer examination of the consensus view reveals that it is not obviously better off in this regard. Moreover, the reading I offer fares better than does the current consensus in a different respect, since it offers a better story of why Kant thinks we so easily become embroiled in antinomies in the first place.

Recall that proponents of the current consensus argue that if Kant embraces NAITS, then the first antinomy’s thesis argument does not illicitly slide from observations about what we can represent to conclusions about how the world can be. As the argument goes, if Kant’s talk of synthesis in the first antinomy refers to what finite minds like ours can accomplish, then the transcendental realist can reject them out of hand. But if Kant means to articulate purely conceptual claims with his talk of synthesis, then no such problems arise. According to proponents of the current consensus, the claim that an infinite successive synthesis is impossible can be rendered as the claim that the very concept of a complete infinite combination is incoherent. And since the concept of a totum syntheticum includes the idea of a complete combination of its elements, an actually infinite totum syntheticum can be rejected as a contradiction in terms.

The arguments of this article suggest that the temporal side of the thesis argument would be acceptable to transcendental realists, since, as I have argued, it rules out a temporally infinite world by arguing that a temporally successive infinite combination is impossible simpliciter (not merely impossible for finite minds to represent). However, the spatial side of the thesis argument does not seem to fare so well on the reading I have been recommending. For in the spatial part of the argument, Kant (speaking in the voice of a transcendental realist) seems to infer from the impossibility of an infinite successive synthesis to the impossibility of an infinite simultaneous combination of things. And as one might object, it is not clear why the spatial extent of the world should depend on a successive combination from the point of view of transcendental realism. Consider the text of the thesis argument itself. Kant writes that if the world is spatially infinite, then ‘in order to think the world that fills all space as a whole, the successive synthesis of the parts of an infinite world would have to be regarded as completed, i.e., in the enumeration [Durchzählung] of all coexisting things, an infinite time would have to be regarded as having elapsed, which is impossible’ (A428/B456). But as one might object, the transcendental realist can deny that the spatial extent of the world is determined by what can be enumerated or counted through (durchgezählt) in a temporally successive process. Indeed, as one might object, even if we can represent magnitudes only via temporally successive counting processes (which cannot be actually infinite), it is not clear why transcendental realists must accept that this determines the kinds of magnitudes that can exist.Footnote ³¹

Here, I want to bracket the question whether it might, in fact, be acceptable to some transcendental realists to argue that magnitudes that cannot be measured also cannot exist. As I see it, this depends on questions about the precise content of transcendental realism, which I cannot settle here. Instead, what I want to argue is that the consensus view is, in relevant respects, on a par with mine when it comes to making the thesis arguments depend on claims about what minds like ours can represent. Hence, the consensus view does not have the clear advantage it is often taken to have.

Recall that the consensus view holds that Kant articulates purely conceptual claims about the notion of infinity via the language of ‘synthesis’. Thus, as Allison has put it, the first antinomy’s thesis argument ‘has nothing to do with the presumed psychological impossibility of grasping or comprehending the infinite’ (Reference Allison1983: 43). However, as Allison himself also admits, construing the argument in this way ‘does not eliminate all reference to mind’ (Reference Allison1983: 44). The reason for this is that the interpretation depends on a theory of concepts according to which concepts are defined through intellectual procedures involving synthesis. As Allison puts it in his discussion of the concept of a totum syntheticum, ‘the concept of a totum syntheticum is here operationally defined in terms of the intellectual procedure through which it is conceived, much as geometrical figures were thought to be given ‘real’ or ‘genetic’ definitions through the articulation of the rules for their construction’ (Reference Allison1983: 43).

As should be clear, although this interpretation makes the first antinomy’s thesis position depend on conceptual claims, it does this only by shifting mentalistic constraints on possibility to the account of concept definition. That is, it makes the first antinomy’s thesis argument depend on a theory of concept definition according to which intellectual procedures involving successive synthesis define concepts such as <infinity> and <totum syntheticum>. But to make our intellectual procedures the determinants of the marks of concepts in this way is another way of making the first antinomy’s arguments depend on assumptions about what minds like ours can represent. Thus, upon closer inspection, the current consensus does not have the clear advantage it is often taken to have. According to the consensus view, mentalised constraints on possibility enter the thesis argument via its assumed account of concept definition, which limits coherent conceptual contents to what can be synthesised in our intellectual grasping procedures (procedures which involve successive synthesis). According to the view I have been recommending, the spatial part of the thesis argument infers directly from the claim that we cannot grasp a magnitude in successive synthesis to the conclusion that the magnitude is impossible. Either way, features of our representational capacities are being called upon to limit what magnitudes can exist in space and time.Footnote ³²

Finally, let me also point to one further advantage my view has over the current consensus, which is that it can help to explain why we so easily become embroiled in antinomies in the first place. Consider the following. If Kant embraces NAITS, then the very idea of an actually infinite totum syntheticum is contradictory. But this is presumably something we can see just by thinking about the ideas of the infinite and of tota synthetica, respectively. However, if this is the case, then it is not obvious why we would find it tempting to think that the spatiotemporal world can be actually infinite. After all, we ought to be able to see right away that an infinite totum syntheticum is contradictory – i.e., we ought to be able to see that an infinite totum syntheticum is contradictory even prior to thinking about any particular spatiotemporal manifestation that an infinite totum syntheticum might take (and indeed prior to the arguments of the thesis positions).

In contrast, if Kant rejects NAITS and holds that the generic notion of an actually infinite totum syntheticum is perfectly coherent, then we can readily understand why we are so tempted to think that the spatiotemporal world can be actually infinite. Namely, since the idea of an actually infinite totum syntheticum is not itself incoherent (and in fact is one of our core representations of the unconditioned), when not on our guard, we can easily slide from the coherence of this idea to the belief that an actually infinite spatiotemporal world is possible. But the spatiotemporal world has determinations that the generic idea of a totum syntheticum does not have, and once we consider the antinomial arguments (and what it would mean for the spatiotemporal world in particular to be infinite), we see that such as world is impossible. Indeed, as I have argued, the coherence of the idea of an actually infinite totum syntheticum is crucial to Kant’s account of how the antinomies get up and running in the first place, for each antinomy depends on the assumption that a series of conditions is either unconditioned and finite or unconditioned and infinite. That is, each antinomy depends on the assumption that an actually infinite totum syntheticum is at least logically possible.

6. Upshots for Kant’s views on the infinite

If the arguments above are correct, then Kant is not a proponent of NAITS, and he instead holds that actually infinite tota synthetica are conceptually possible (even if some spatiotemporal manifestations of actually infinite tota synthetica are not). In this section, I conclude by drawing out some upshots for our overall understanding of Kant’s views on the infinite.

The first upshot is that Kant does not believe the very concept of the infinite excludes the concept of completeness (contra Allison). This is an implication of rejecting NAITS. If the concept of the infinite can coherently be combined with the concept of a totum syntheticum, and if the concept of a totum syntheticum includes the mark of completeness (insofar as a totum syntheticum is the result of the complete combination or composition of its parts), then the mark of completeness cannot be excluded by the concept of infinity as such.

We might be surprised by this result insofar as Kant seems to associate infinity with incompleteness in his discussion of the ‘true (transcendental) concept of infinity’ (A432/B460). There, Kant says that the true (transcendental) concept of infinity is that ‘the successive synthesis of units in the traversal [Durchmessung] of a quantum can never be completed’ (A432/B460). But while this initially looks like a way of defining the infinite as incomplete, we should instead understand the passage as follows. According to Kant, an infinite magnitude is greater than any finite unit of measurement repeated any finite number of times, and so it is greater than any magnitude that one could measure through (durchmessen) (given that measurement occurs successively in Kant’s account).Footnote ³³ And crucially, this is not to say that the infinite as such is incomplete; it is just to say that the process of measuring an infinite magnitude can never be completed.

We should also be careful not to conclude that because the infinite does not by definition exclude the concept of completeness, it must therefore be complete by definition. For Kant does not appear to be committed to this view. According to Kant, an infinite magnitude can be thought of as complete (as is the case in our idea of an unconditioned infinite whole series of conditions), but this is not the same as saying that it must be or that the mere idea of an actually infinite magnitude is the idea of a complete magnitude. Consider again Kant’s discussion of the infinite in his remark on the first antinomy’s thesis argument. There, Kant says that by an ‘infinite whole’ (unendlichen Ganzen) one understands ‘not how great [ wie groß ] it is’, but rather one ‘thinks only its relation to an arbitrarily assumed unit, in respect of which it is greater than all number’ (A431-2/B459-60). Following Smyth (Reference Smyth2023), I think we should take seriously Kant’s suggestion here that infinite wholes are appraised as strictly greater than all finite ones, but this is not to assign to them a determinate (measurable) size of their own. And given this, we can leave open the question whether an infinite magnitude is complete or incomplete when we say that it is infinite. An infinite magnitude is by definition greater than any finite one, but unless more is specified, it might or might not be complete.

We can clarify this point by considering Kant’s discussion of space and time as infinite wholes given prior to their parts. It is tempting to think that space and time must be given as complete infinite wholes in intuition, since, as Kant says, they are given as magnitudes in which the ‘parts are possible only in the whole [im Ganzen], and not the whole through the parts’ (A438/B466). As one might reason, if space and time are given as infinite wholes in this way, then they must be complete, for unless they were given in their entirety prior to their parts, they could not properly be said to be given as wholes at all.

While I am somewhat sympathetic with this line of reasoning, we should be sensitive to Kant’s effort to distinguish between space as it is represented in intuition and space as it is represented by reason in the idea of absolute space. According to Kant, we have (in addition to our pure intuition of space) an idea of absolute space, and this idea represents space as an unconditioned whole with reference to which all motion is determined (e.g., see MFNS, 4: 559). In the Critique of the Power of Judgement, Kant further elaborates on reason’s representation of infinite space as follows. According to Kant, ‘the voice of reason [die Stimme der Vernunft] … requires totality [Totalität] for all given magnitudes’ and it ‘does not exempt from this requirement even the infinite (space and past time), but rather makes it unavoidable for us to think of it (in the judgment of common reason) as given entirely (in its totality) [als ganz (seiner Totalität nach) gegeben zu denken]’ (5: 254). Kant continues several lines down: ‘what is most important is that even being able to think of it as a whole [als ein Ganzes auch nur denken zu können] indicates a faculty of the mind which surpasses every standard of sense. … even to be able to think the given infinite without contradiction requires a faculty in the human mind that is itself supersensible’ (5: 254). Here, Kant seems to suggest both (a) that reason surpasses what intuition alone represents when it thinks the infinite as a given complete totality and (b) that reason can think such a totality without contradiction. Putting aside other interpretative questions about the idea of absolute space, I take this to support the conclusion that representations of the infinite do not always include the kind of completeness that reason represents in its ideas of totality. The idea of absolute space (an idea of reason) represents a completeness that our pure intuition of space does not.Footnote ³⁴

Finally, notice that the conclusions of this article also allow us to refine our understanding of Kant’s place in seventeenth- and eighteenth-century debates about the infinite. Leibniz famously argued that an actually infinite multiplicity of things cannot constitute a true whole, and many historians of philosophy take this rejection of actually infinite totalities to be unchallenged until the time of Cantor (when actually infinite sets and transfinite cardinalities came to be widely accepted). According to Leibniz, actually infinite ‘multiplicities’ are possible, but in the case of the actually infinite, ‘this multiplicity is not a number or a single whole’ (Leibniz Reference Leibniz2000: lxii).Footnote ³⁵

However, if this article’s arguments are correct, then Kant’s thinking about the infinite is not Leibnizian in this way.Footnote ³⁶ For rather than embracing Leibniz’s argument for NAITS, Kant holds that we can very well conceive of an infinite multiplicity or collection of items as constituting a whole – as we have seen, Kant holds that we can think of an actually infinite series of conditions as an ‘absolute whole … outside of which there are no more conditions’ (A417-8/B445fn). And in thinking of such a series, we are thinking of a totality that is both complete (in the sense of containing all the conditions of the relevant type) and unified into a whole (since it is a series of conditions brought together and ordered via conditioning relations). None of this is to say that we can sensibly cognise such a whole, but it is to say that an actually infinite totality is logically or conceptually possible – and, as we have seen, the reason for this is that whatever we coherently can think cannot be ruled out on merely conceptual grounds in Kant’s view. Of course, this falls short of saying that Kant anticipates nineteenth-century developments in set theory, for Kant had nothing like Cantor’s conception of cardinality for infinite sets. But amid other profound differences, Kant shared with Cantor one important judgement. Like Cantor (and unlike Leibniz), Kant held that there is nothing incoherent in the idea of collecting together actually infinitely many items to form a whole. Indeed, reason (Vernunft) represents actually infinite wholes in its idea of an unconditioned infinite whole series of conditions.