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- Jul 28, 2014Hi urpflanze group:I'm thinking that I need to restate one of the principles that Kant needs to defend. The second point, from message #310 was
(2) That from concepts alone only analytic knowledge can be derived.I think now that this is really Kant's definition of analytic, as the predicate being included in the concept of the subject. What he really needs to argue is that this is the *only* way to get analytic knowledge. So, I should restate this as follows:(2') Analytic knowledge can only be derived from concepts alone.Now, we can see that proof of certain ideas from arithmetic, as Frege argued, such as the associativity of addition and so forth, can be proven from certain other axioms using first-order predicate logic and a combination of notions such as zero, successor, and induction. Analytic judgments can come from a combination of logical and conceptual mechanisms, not just analysis of a particular concept. Bolzano, in the example of the intermediate value theorem, produces a proof from elementary concepts such as continuity and limit of a principle that Kant would consider synthetic and based on intuition. But, it need not be based on intuition, and the point is that the special Kantian claim that arithmetic, geometry, and all of mathematics is synthetic begins to collapse.Thanks & sorry for the misstep. --RonOn Thursday, July 24, 2014 9:45 PM, "wavelets@... [urpflanze]" <urpflanze@yahoogroups.com> wrote:
Hi Neil:
Thanks for following along.
I'm not sure how to address your ruminations vis-a-vis analytic judgments just yet. Post-Quine, the analytic judgments don't really exist, so that's one way to say that they are not important. I'm thinking that Wittgenstein would disagree on this point. (As do I, since I think that Quine does not have it quite right.) That is, some argument can be explicative in regard to the concepts without being ampliative, introducing something new, and the explication can come about through a combination of logical derivations and posited definitions. Kant interprets all of this through the lens of Aristotelian syllogistic, which is naturally given to interpret things by containment and exclusion.
So, it's not surprising that he has an artificially restricted view of analyticity. If we think about some mathematical examples, it is not intuitively evident that the order of every subgroup of a finite group divides the order of the group. I don't see this latter as being enclosed in the concept of a finite group. Yet it follows, of course, from the axioms of a group, the restriction that the group be finite, and the notion of cosets partitioning the set of elements of a group. This is a simple, straightforward example.
OK, on the Kantian other hand, would we say that Lagrange's Theorem is intuitive? No, I think, of course not. Without the proof and the supplementary notion of cosets, it is inattainable through intuition. Thus, it follows from logical derivation, and is contained, to follow Kant's lead a bit, not in the concept of a finite group, but in all of these: The concept of a group, being a finite group, there being a coset, and being derivable by first-order logical rules of derivation from the axioms supporting the above concepts. Lagrange's Theorem is analytic, in that it follows from the concept of a finite group through a complex series of derivations following first-order quantificational logic.
Thi! s brings us back to Wittgenstein's remark in the Philosophische Bemerkungen. Wittgenstein is going to endorse Bolzano and refute both Kant and Frege.
Thanks, --Ron - << Previous post in topic Next post in topic >>