Hi urpflanze group:

OK, last time I listed a couple of points that Kant would need to address in order to defend his assertion that statements of mathematics--not just geometry, but even elementary arithmetical statements such as 7+5=12--are actually synthetic judgments.

The first point was that the intuition that one must resort to in a synthetic a priori judgment is more than a quirk of our own individual minds. Otherwise, it would seem that all kinds of intuition could be invoked in order to ground such a judgment. One wonders why the geometrical axioms are all agreed upon, when the mathematicians are employing possibly quite distinct intuitions in order to justify them. Frege's points against psychologism in the *Grundlagen* [G. Frege, *The Frege Reader*, ed. M. Beaney, Oxford: Blackwell, 1997, pp. 84-129.], and his development of first order predicate logic for the foundation of arithmetic beat back this first salient that Kant needs to establish. If there's interest, I'll lay out Frege's arguments later in the thread.

The second salient upon which Kant's synthetic a priori offensive needs to prevail is the idea that from concepts alone only analytic knowledge can be derived.

There are a number of pertinent remarks by Coffa. See his article: Alberto Coffa, “The emergence of logicism,” in Frege’s Philosophy of Mathematics, ed. William Demopoulos (Cambridge, MA: Harvard University Press, 1995), pp. 29-40.

[Let me see if the italics formatting works in the above paragraph; I've noticed that some apparently correctly formatted paragraphs, with pieces italicized, come out completely italicized in the yahoo groups message inventory. Something isn't right.]

Coffa writes: "Bolzano was the first to recognize the fallacy behind the principle of synthetic judgments. The crucuial step in Kant's inference for the need to appeal to intuition in synthetic judgments was the premise that from concepts alone only analytic knowledge can be derived" [Coffa, p. 34].

Coffa continues: "Astonishingly, there isn't a single argument in the *Critique* for this claim; all Kant says about it is that 'it is evident' (A47, B64)" [Coffa, p. 34]. I'm looking at my Kemp Smith translation of the Critique [Kant, Critique of Pure Reason, N. Kemp Smith, trans., New York: St. Martin's Press, 1965]. I see Kant saying on pp. 85-86 "...it is evident that from mere concepts only analytic knowledge, not synthetic knowledge, is to be obtained." He gives as examples that two straight lines cannot enclose a space and that with three straight lines a figure is possible. Can these propositions follow from the concepts alone? No way, Kant argues. "All your labour is vain; and you find that you are constrained to have recourse to intuition, as is always done in geometry. You therefore give yourself an object in intuition." [Kant, *Critique*, Kemp Smith, trans., p 86]

I like Coffa's next thrust: "What is evident, instead, is that Kant had confused *true in virtue of concepts* with *true in virtue of definitions*, or, in his own language, he had errroneously identified judgments whose predicate is not contained in their subject-concept with judgments that extend our knowledge (*Erweiterungsurteile*) [RLA translation: augmentative judgment].

It's interesting how central the mathematician Bernard Bolzano (1781-1848) is to all this, as Coffa indicates: "Against this, Bolzano was the first to make a point that even Frege would miss: that Kant's analytic judgments, far from exhausting the grounding power of the conceptual resources of our language, mobilize only a very modest fraction of them, the logical concepts" [Coffa, p. 34].

Here it seems that Kant only put his finger upon the problem; in fact, he was far from understanding how limited his own concept of analyticity really was.

Lengthy post, but I have to stop for a breath. Thanks, --Ron