Hey group:

From 7+5=12 is synthetic, Kant works his way up to making the case that proper metaphysical statements are synthetic, and they are one and all synthetic. Kant distinguishes between *properly* metaphysical judgments and metaphysical judgments. Many plain metaphysical judgments are analytic. If the concepts involved are metaphysical, but the judgments using them proceed from mere analysis, then the judgment is analytic, but still part of metaphysics. On the other hand, a judgment such as “All that is substance in things persists” is a properly metaphysical proposition, and it is synthetic. Metaphysics properly involves synthetic a priori propositions. These comprise its aim.

Now, this is where Frege comes in. Frege was novel, deep, and out of step with his contemporaries: Mathematics stands upon a foundation of logic—actually a bold claim! For consider the logic of Frege's contemporaries.

Logic in the late 19th century was fairly weak (Aristotelian syllogistic on the one hand [J. Lukasiewicz, *Aristotle’s Syllogistic: From the Statndpoint of Modern Formal Logic*, Oxford: Clarendon, 1951], or propositional logic, going back to the Megarian school and the Stoics, on the other [I.M. Bochenski, *Ancient Formal Logic*, Amsterdam: North-Holland, 1951, pp. 77ff]), and valid mathematical reasoning often seemed to surpass one or the other or, indeed, both of these alternative formalisms. Mathematics was informal; the proofs of the day are almost unreadable by modern (i.e. post-Frege, early 20th century) mathematicians. Steps without explicit justification were the rule rather than the exception, and the discipline seemed to rely upon a practicioner’s intuition.

A Wikipedia page: http://en.wikipedia.org/wiki/History_of_logic#Stoic_logic

The conclusion would thus be that it is the structure, capacity, and capability of the human mind—our *psychological* characteristics—that deliver mathematical proof. Out of this methodological inertness of mathematics is where Kant draws his most powerful arguments! (I would think of Plato, always turning to math for an example of how, generation after generation, mathematicians all seem to never overturn the conceptualizations, axioms, and proofs of their predecessors...there must be something standing outside of the math whizzes which their symbols really denote.)

All right, because of the state of logic and the sway of Kant, Frege was overlooked, ignored, and misunderstood. But, we do know mathematical facts. If we look at Kant's argument for why 7+5=12 is synthetic, there is a step in which the contemplator resorts to her own intuition to see that there is more in twelve and than the unification of two smaller numbers. Now maybe someone has the intuition the God exists. Maybe someone has the intuition that private property is theft. Maybe someone has the intuition that homosexuality is sinful. And so forth.

Here, one might also recall Wittgenstein's arguments about continuing an arithmetic series [Wittgenstein, Philosophical Investigations, (§§185-202)]. For what if someone insists that her intuition tells her that the series does not continue as we think it does? What if she insists that her intuition (do we know her intuition better than she does?) reveals that 5+7=12 is analytic, because the notion of compounding these two smaller numbers does contain the concept of twelve, but that 5+8=13 is synthetic, because there is something novel in thirteen that five and eight together just don't capture. Yeah, you know what she means.

Kant must argue two things:

(1) That this intuition is more than a quirk of our own individual minds;
(2) That from concepts alone only analytic knowledge can be derived.

So, here comes Frege, against all the intellectual wind of his time, that these intuitions are not quirks of our own mind, which (1) was a key point of Kant’s philosophy. Also, Frege put his finger on (2), asserting that analytic knowledge went beyond what could be extracted from concepts. In any case Frege was rejecting *psychologism*—the theory that the meaning of our concepts could be at all explicated in terms of the mental states of mathematicians (or ordinary folk, for that matter).

OK, the posts are still pretty distended. I have to feed the dogs. Thanks, --Ron