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.
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2 comments:
Nice, Dan
My concern (and you knew i'd have one!) is that you misunderstand the role of synthesis in the operative notion of synthetic a priori.
Concerning any judgment of a synthetic a priori form, there's always going to be a WAY of explicating the concepts involved so as to make the judgment appear analytic. If you're a Kantian, synthetic a priori forms of judgment make empirical consciousness possible, so there's some sense in which analyzing the relevant concepts will seem to reveal the content of the judgment itself. Take "Every event has a cause". Since we're unable to fully wrap our heads around the notion of an uncaused event, once the situation is fully described, it can begin to look as though the judgments "X has a cause", "X is an effect of some Y" are contained in the judgment "X is an event". But what we actually see here, and what I want to suggest on behalf of your discussion of mathematics, is that we are actually just observing the ways in which the concepts themselves reflect our cognitive structure as synthetic, conceptual, spontaneous intellects.
Keep in mind, from the Transcendental Deduction, that the original a priori synthesis is the combinatorial nature of cognition, which prevents our apprehension of the world from being that of a blooming buzzing confusion. It's because we're able to combine, synthesize disparate intuitions into discrete objects of cognition that we're able to have experience at all. And this sort of synthetic cognition is exemplified par excellence in mathematical reasoning.
Now, that said, a few concessions.
First, the worries you point to are very common, and i've never seen them expressed so clearly as i see them here. I suspect your paper will do very well.
Second, as I read over my own analysis, i note a couple of things. First, it's hard to justify what I've written here with the simple, cursory characterization of synthetic and analytic judgments that Kant himself gives. What's more, if my analysis is to be taken as right (which it's probably not) the question arises as to whether there's even such things as analytic judgments. And if that's a consequence of my view, it's probably a reductio. Nonetheless, I do think that making sense of the synthetic judgment, which will be necessary to adjudicate the claim that arithmetic judgments are synthetic judgments, will require an understanding, more sophisticated than my own, of the role Kant gives to synthesis in his picture of cognition.
I think you should send this to Lissa, down under. I can send her the link if you like.
Raleigh
You write:
"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."
This seems to me to be the right thing to say about Kant's view of mathematical concepts. It also seems to me that this description is nothing like Kantian analysis. What you call "the act of examining" the concept of number by "visualizing" a particular object that falls under it *just is* what Kant means to say when he says that we must "construct" our mathematical concepts in intuition. This act of visualization is certainly not meant to be included in Kant's conception of analysis.
I think it is wrong to suppose that Kant’s theory of mathematical cognition must be thought of as either wholly intuitive or else wholly conceptual. Kant clearly believes that we have mathematical concepts: I can have the concept TRIANGLE, by which I represent a class of qualitatively similar spatial regions. But Kant’s point is that the concatenation of THREE-SIDED, CLOSED, PLANAR and FIGURE does not alone tell me that the interior angles of triangles sum to 180 degrees. In order to know that – i.e., in order to know what “I already think” through my concept – I must, as you say, “visualize” and perform operations on concrete particulars that fall under TRIANGLE. I think it would be a mistake to think that this counts as analysis. For nothing like this process is, by Kant’s lights, required to know analytic judgments (like “all bachelors are unmarried).
Kant’s appeal to intuition is not designed to show that “has angles that sums to 180 degrees” does not in some sense necessarily “belong to” the concept TRIANGLE. So I think you’re right to notice that a particular kind of examination of a mathematical concept will “reveal” the properties the objects in its extension necessarily have. But this sense of “belong to” is not conceptual containment. Or, if it is, then I think you need to do more than assert that a process of visualization is tantamount to Kantian analysis.
I think you're inclined to call mathematical judgments analytic because you recognize that they must be necessarily true - indeed, I think that's the insight your examples rely on. But that alone won't show that they're analytic, especially in Kant's context, since one of his aims is to show that there are synthetic necessities. I think Kant's theory of concept construction is designed to show that the process of visualization you describe is required and that such visualizations reveal the necessity of the judgments thereby produced. The visualization proves syntheticity, and the impossibility of any other visualization proves necessity.
So I don't think you've undermined Kant's account by showing that mathematical cognition requires a process of visualization. To the contrary, I think you've (perhaps not entirely intentionally) offered a kind of *explanation* of the way in which intuition plays a role in mathematical judgment.
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