In the Theaetetus 195e-196-b Socrates confronts Theaetetus with an apparent paradox: surely it’s impossible, he has Theaetetus admit, for anyone to think that 11 is 12. But, he notes, it happens all the time that someone tries to add 5+7 and comes up with 11 — and isn’t “5+7” the same as (=) 12? How can this be? This apparent paradox forces them to abandon the empiricist “Wax Block” model of mind and turn to the (much stranger) “Aviary,” since now it turns out that false judgment is produced not only by our mismatching past and present sense data, but can occur even in comparisons of purely mental objects. But the conviction that this situation — i.e., of forming a false judgment about an arithmetic problem — is paradoxical depends, so far as I can tell, entirely on the idea that statements of arithmetic are analytic truths — if “5+7=12” is analytic, then judging that “5+7=11” would be like claiming, “This bachelor is married.” But we’re constantly making mistakes in arithmetic (especially with larger numbers, as Theaetetus notes, 196b), whereas a competent English speaker would scarcely ever think (de dicto) that bachelors are married. So what’s the difference?
The first person, so far as I know, to explain the principle that distinguishes these two kinds of error was Immanuel Kant, and, fascinatingly, he seems to have explained himself in explicit conversation with Plato. Early in the Critique of Pure Reason, he writes, “Mathematical judgments are one and all synthetic a priori” (A 10/B 14, p. 55). In analytic judgments, of course, the predicate is “contained” in the subject; not so in synthetic (A 7/ B 11, 51). Analytic judgments are “elucidatory,” synthetic, “expansive” (Ibid.). Kant illustrates his claim about mathematical judgments with the same proposition that Socrates used: “7 + 5 = 12” is a synthetic proposition, he insists (B 15, p. 56). Why? “If we look closely, we find that the concept of the sum of 7 and 5 contains nothing more than the union of the two members into one; but in [thinking] that union we are not thinking in any way at all what that single number is that unites the two…We become aware of this all the more distinctly if we take larger numbers” (B 16, 57).