Building Blocks for the Critique of Pure Reason: “Mathematical Judgements are all synthetic”.

This is one of the more difficult sections of the Introduction, and I can’t pretend to fully understand what’s at stake in Kant’s claim.

The fact that mathematical judgements are a priori should be obvious from previous posts. As Kant says:

It must first be remarked that properly mathematical propositions are always a priori judgements and are never empirical, because they carry necessity with them, which cannot be derived from experience.

The question, though, is whether such judgements are analytic or synthetic. It is worth quoting Kant’s argument in full:

To be sure, one might initially think that the proposition “7 + 5 + 12″ is a merely analytic proposition that follows from the concept of a sum of seven and five in accordance with the principle of cotradiction. Yet if one considers it more closely, one finds that the concept of the sum of 7 and 5 contains nothing more than the unification of both numbers in a single one, though it is not at all thought what this single number is which comprehends the two of them. The concept of twelve is by no means already thought merely by my thinking of that unification of seven and five, and no matter how long I analyze my concept of such a possible sum I will still not find twelve in it. One must go beyond these concepts, seeking assistance in the intuition that corresponds to one of the two, one’s five fingers, say, or (as in Segner’s arithmetic) five points, and one after another add the units of the five given in the intuition to the concept of seven. For I take first the number 7, and, as I take the fingers of my hand as an intuition with the concept of 5, to that image of mine I now add the units that I have previously taken together in order to constitute the number 5 one after another to the number 7, and thus see the number 12 arise. That 7 should be added to 5 I have, to be sure, thought in the concept of a sum = 7 + 5, but not that this sum is equal to the number 12. The arithmetical proposition is therefore always synthetic; one becomes all the more distinctly aware of that if one takes somewhat larger numbers, for it is then clear that, twist and turn our concepts as we will, without getting help from intuition we could never find the sum by means of the mere analysis of our concepts.

Rather than discuss the legitimacy of Kant’s claims, I want to think about precisely what he is claiming here.

Let’s start with the following claim:

One must go beyond these concepts, seeking assistance in the intuition that corresponds to one of the two, one’s five fingers, say, or (as in Segner’s arithmetic) five points, and one after another add the units of the five given in the intuition to the concept of seven.

Kant cannot mean that we literally have to look at our fingers to do arithmetic, otherwise judgement would not be a priori; rather, he is drawing our attention to the fact that that a process is involved in mathematical judgement.   Furthermore, this process is a synthesis of intuitions/combination of concepts, rather than a process of analysis and refinement of concepts.

Obviously with lower numbers this process is negligible – it doesn’t require much thought to work out that 7 + 5 = 12; however when we move to larger numbers, the process becomes more apparent: I cannot work out 1578 + 22567 without some thought. But is this thought a process of synthesis rather than analysis, as Kant claims? The statement ’1578 + 22567′ seems to contain all the information we need to reach a correct conclusion, suggesting that the judgement is analytic.  But the question is how we reach this correct conclusion.  Implicitly, Kant is claiming that this is not through an analysis of the concepts involved, but through a synthesis and amplification of them. To demonstrate this, try analysing whatever concepts you have of ’1578′, ‘+’, and ’22567′, individually. You will not reach the figure 24145 through this process alone, because it is not contained in any of the concepts by itself. In contrast, try the same procedure on whatever concepts you have of ‘bachelors’, ‘are’, and ‘unmarried’. Analysis of the first concept – ‘bachelors’ – should be sufficient to reach the correct conclusion that bachelors are indeed unmarried, because it is an integral part of the concept.

However this is not just synthesis in the sense of bringing together the statement into a sensible whole with normative force (i.e. seeing that ’1578′,  ‘+’,  and ’22567′ are combined in such a way that we ought to see that the answer is 24145).  As Kant says, it is not a question of what we should think, but what we actually think.  This is not a point about our inaptitude for mathematics; rather, the normativity of proposition does not mean that we follow it through to its conclusion by analysis alone.  Kant is suggesting that the reason we are capable of reaching a correct conclusion is because we have a priori structures available to us in pure intuition which provide us with the basis – the justification – for our reaching the conclusion that ’1578 + 22567 = 24145′.

Although each individual numerical concept doesn’t contain within it the concept of every other number,  a proper mathematical understanding does require us to have a conceptual (?) background sufficient for the manipulation of these numbers by various mathematical functions. We don’t pick up mathematical concepts one at a time, understanding the concept ’1′, then the concept ’2′, then ’3′, etc.  If this was how our understanding worked, basic arithmetic would be incredibly difficult.  Instead, our understanding of, say, the natural numbers and the mathematical functions that we apply to them, is based on something like a structured complex of concepts which must be available to us in toto if we have even a rudimentary understanding of arithmetic.  Mathematical judgement – according to Kant – requires a process of synthesis within this conceptual complex, even though the whole structure is available to us a priori.

The key question is whether this conceptual background is equivalent to, or at least analagous with, the transcendental structures which Kant will later claim shape any possible intuition for creatures like us, and which are available to us through reflection on the pure form of our intuition, i.e. a priori, without contemplating the content of any particular intuition – basing this judgement on a particular fact in the world.  Taking mathematical judgement as a paradigmatic example of synthetic a priori judgement, we can see that other cases of synthetic judgements should work in the same way.  They require a structured complex of concepts available to us a priori, such that our judgements take place through a process of synthesis within this a priori structure.  I will return to this in subsequent posts.

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