Kant and Mathematics

This is an excerpt from a paper in which I argue that in the first Critique Kant does not give good reasons for thinking that the truths of mathematics are synthetic a priori.

Mathematical judgments, Kant states, are an example of synthetic a priori judgments (A 10/B 14). Judgments such as “5 + 7 = 12” and “a straight line is the shortest distance between two points” are a priori, since they are both necessary and universal. However, according to Kant they are also synthetic. He states that the concept “twelve” cannot be arrived at solely by examining the concepts of “five,” “seven,” and “addition,” and the concept “shortest distance between two points” cannot be found in the concept “straight line” (B 16). In both cases, Kant argues, we must appeal to an intuition in order to the truth of the judgment. When we add five and seven, we must have recourse to calculating aids such as our fingers, or the visualization of discrete points, etc. (B 15). And in the case of the straight line or other geometrical truths, we must have recourse to intuitions in order to arrive at the correct answer.

But what exactly does Kant mean by appealing to an intuition in this context? As he explains at A 19/B 33, by “intuition” he is specifically referring to the means by which a cognition relates to an object when the mind is presented with that object. In the case of mathematical truths, it is these kinds of intuitions (e.g. the processes of counting or of constructing a geometric figure, cf. Dicker (2004) 27) that allow us to see the truth or falsehood of our judgments, rather than an examination of the concepts involved (Guyer (2006) 61). So an intuition is something over and above the particular concepts involved in a judgment.

In the case of the straight line being the shortest distance between two points, Kant states that the concept of straight “contains nothing of quantity, but only a quality” (B 16), and concludes that this judgment is synthetic. However, a closer examination of the concept “straight line” will reveal the judgment to be analytic. Euclid defines a line as “breadthless length” (Elements I def. 1), and a straight line as “a line which lies evenly with the points on itself (Elements I def. 4); in other words, a straight line is a magnitude in one dimension “which represents the extension equal with (the distances separating) the points on it” (Heath (1956) 166). In addition, Kant’s contemporary Legendre followed Archimedes in simply defined a straight line as the shortest distance between two points (Heath (1956) 169). So the judgment that a straight line is the shortest distance between two points turns out to be analytic after all.

In the case of “5 + 7 = 12,” Kant states that the concept “twelve” cannot be arrived at simply by examining the concepts “five,” “seven,” and “addition,” but requires a process of counting. But it does not seem to me that the process of counting “goes beyond” (B 15) the concepts of the particular numbers and the concept of addition, but that it makes explicit what is already contained in the concepts involved. The concepts “five” and “seven” surely include the concept “magnitude,” and the combination of two particular magnitudes must yield another particular magnitude; in this case, the magnitude that we happen to denote by the concept “twelve.” It seems that e.g. when we represent the concept “five” to ourselves, we unavoidably do so by imagining a certain number of distinct objects (even if those objects are components of another object, as if for instance we represent “five” by visualizing a pie with five slices). I suggest that what Kant calls an “intuition” in this and in similar cases of counting is nothing more than the act of examining the concept of a particular number. That we require this kind of representation in order to perform calculations should be understood as an indication of the genesis of our concepts of numbers (or perhaps an indication of the limits of our particular cognitive capabilities), rather than a proof that mathematical knowledge must be arrived at with the aid of intuitions that go beyond the concepts involved.