Immanuel Kant famously argued that the equation 7+5=12 is an example of a synthetic a priori judgement. Indeed, he maintained that “arithmetical propositions are always synthetic.” ¹ But is this the case? By critically engaging Kant’s argumentation, I will uncover two prima facie problems with the idea that arithmetical propositions are synthetic. By reengaging these objections from Kant’s perspective, one will be overcome, but the other will seem to fortify. These problems are, first, that on account of their identity relation to the addenda, sums cannot be both indeterminate pre-synthesis and also determinate post-synthesis: they must be one or the other. Secondly, numbers connected by the conjunction operator entail a specific conjoined number. Wrestling with this last objection will produce a formidable objection rooted in set theory, namely, that addition logically entails union.

            The plan for this post is as follows. In the first section, we will outline and review Kant’s reasons for thinking that arithmetical propositions like 7+5=12 are synthetic. Here it will be important to keep in mind the indeterminacy of sums that he claims takes place. In the second section, we will critically engage Kant’s ideas and pose the two problems described above. Here we will also consider how Kant might have responded. The first objection will fall to the charge of ambiguity inasmuch as there are propositional and predicative logical structures in propositions, but the second will become a serious problem for Kant. In the concluding section, we will wrap up by summarizing our findings that Kant faces a steep, though perhaps not insurmountable climb. Indeed, while the dialectical maturity of an argument is just as important as its soundness, it does appear that Kant’s idea is in trouble.

1. Kant and Mathematical Knowledge

Recall that Kant distinguishes analytic judgements, which “say nothing in the predicate that wasn’t already thought—though less clearly—in the concept of the subject,” from synthetic judgements which do say more. ² At the heart of determining whether a judgement is analytic or synthetic by these criteria is the ability to tell when something counts as a subject or a predicate, and what all is included within a subject so identified. In the case at hand—7+5=12—we might think at first pass that Kant is regarding either ‘7’ or ‘5’ as the subject, with the other number thereby becoming the predicate. In that event, Kant would be taking arithmetical operations like addition to in some way obtain their result by predicating one number of another, thereby expanding our knowledge. Thus framed, it is evident that the concept of ‘5’ qua ‘5’ does not include ‘7’, which implies that predicating ‘7’ of ‘5’ creates something new that was not already in ‘5’.

However true this may be for Kant, his actual phraseology is not that the equation ‘7+5=12’ is synthetic because it involves predicating one number of the other, but rather because the sum of 7 and 5 is undetermined until the two are relevantly combined together. That is to say, crucially, the addenda are not for Kant already sufficiently combined in virtue of being connected by an arithmetical operator like addition. He will grant that the addition operator will combine 7 and 5 into a singular number, but he warns that this number is undetermined until it is counted. ³ This specific, counted number is our synthesis of principles of reason and sensible or pure intuitions. He will say that “however we might turn and twist our concept of the sum of 38976 and 45204 we could never find 84180 in it through mere analysis, without the help of intuition.” ⁴ We have to synthesize ‘7’ and ‘5’ in order to obtain ‘12’. In other words, ‘12’ is not in them individually or together for us to discover by analysis.

1. Two Identity Problems

There is a sense for Kant in which we genuinely learn new applications of our universal rules of syntheses, and so a corresponding and even robust idea of ‘discovery’. But here let us take discovery in the plainer sense of coming upon something already there, though hitherto unknown, and ask whether the results of arithmetical procedures are created or discovered? Is it the case, in other words, that ‘7’ and ‘5’ do not make ‘12’ until we add them? Or, rather, that they do make ‘12’ but because we add them? It will be helpful in discerning this matter to get clear on whether the equation has a propositional structure, and if so, whether it ascribes a property to the subject, or identifies one object with another.

In favor of there being a propositional structure underlying equations is their use of the equality or identity relation ‘=’. One part of the equation is being thereby identified with the other: it is to say of one that it is the other. This may bear for one a striking resemblance to the basic structure ‘S is P’ of propositions, (‘P(s)’ in first-order). Suppose then that this is at least one way in which equations are logically structured. What then is the predication involved here—an ascription of a property to a subject, or an identification as being the same? Obviously, the latter. But, then, two severe, though perhaps not insurmountable problems arise for Kant.

The first problem is that since arithmetical equations use the identity relation, it cannot be that their relata refer to different things. Indeed, Leibniz’s Law implies that it cannot even be that something is true of one relatum that is not true of the other because identity has it that the relevant terms are indiscernible, or interchangeable salva veritate, or (where applicable) possessed of the same truth conditions. However, even without this application of the identity of indiscernibles, it simply cannot be that ‘7’ and ‘5’ are both equal to an undetermined sum prior to synthesis, and also that their sum is equal to ‘12’ afterwards. Kant said, “One might think that the proposition 7 + 5 = 12 is analytic, and that it follows according to the law of contradiction from the concept of the sum of 7 and 5.” ⁵ Quite so.

The second problem is that if there truly are ‘7’ and ‘5’, then this just is for there to be ‘12’, since their conjoined amount seems entailed by their in fact being a conjunction. Of course, if these numbers are not conjunctively present, that is one thing. But, in that case, neither could they be getting added together either. No matter the numbers Kant has in mind—even 38976 and 45204—if they do in fact form a conjunction, then it seems there is a specific number they conjoin into; their conjunction, however unknown that is to us beforehand.

In reply to the first objection, Kant might agree that this is all well and good, but that the relevant predication structure is given away by the addition sign, not by the equal sign. That is, it may be that at the level of identity, the first part of the continued example is identical to ‘12’, and cannot precede or differ from it. But it is rather ‘5’ or ‘7’ that is being predicated of the other here, not either or both of ‘12’. In that case, perhaps the structure of the equation could form like so: ‘P(s) ↔ r’, as in 5(7) ↔ 12, or, for logicians disinclined to see identity as predicative, perhaps as (s & p) ↔ r, as in (5 & 7) ↔ 12.

In reply to the second objection, Kant could simply respond that while addition corresponds to union in set theory, conjunction corresponds to intersection. The objection thus seems to rest on an ambiguity. The mere presence of conjunction does not entail addition or union, though the fact of addition would entail union. This final point is perhaps the most difficult for Kant to resolve. The union of ‘7’ and ‘5’ is not an undetermined amount, it is of the strictest necessity, ‘12’. Perhaps, though, Kant would not be interested in disputing this fact, but rather explaining it. Counterfactually, if we did not count, the union would either be empty or not there at all. It is necessarily ‘12’, then, but because it is so counted. However, there is no logically prior moment for a union to be empty: insofar as there is a union, it just is the totality of its subsets. And if there is no union at all, neither therefore is there any addition.

1. Conclusion

In conclusion, it seems that Kant’s characterization of arithmetical propositions as synthetic faces a very steep climb indeed. One of the most natural ways of reading equations propositionally flows from their use of the ‘=’ sign. But this identity relation sharply eschews any notions of sums referring to different things in being both undetermined and determinate depending on the time, world, person, or whatever. However, supposing the logical structure of equations is deeper or more complex than propositional and requires something like predicative cartography, still, addition entails union. Perhaps Kant can account for this with increasingly strained or ad hoc stipulations such as counterfactually grounded syntheses, or some such. But the motivations for doing so quickly become suspect, and there seems no way around this rock-bottom result: if there is no specific union, there is no addition; and if there is addition, there is specific union. Tertium non datur?

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1. Immanuel Kant, Prolegomena to Any Future Metaphysics, trans. Jonathan Bennett (2017), par. 2. Project Gutenberg Ebook, https://www.earlymoderntexts.com/assets/pdfs/kant1783.pdf. p. 9. ↩︎
2. Immanuel Kant, Prolegomena to Any Future Metaphysics, par. 2, p. 7. ↩︎
3. Immanuel Kant, Prolegomena to Any Future Metaphysics, par. 2, p. 9. ↩︎
4. Immanuel Kant, Prolegomena to Any Future Metaphysics, par. 2, p. 8. ↩︎
5. Immanuel Kant, Prolegomena to Any Future Metaphysics, par. 2, p. 8. ↩︎