/lit/ - Literature

Is there a relationship between Kant's breakdown of propositions into analytic and synthetic judgments, the tug-of-war about whether Aristotelian intension needs to be a part of logic as much as "taxonomic" extension, Peirce's breakdown of deductive reasoning into corollarial and theorematic reasoning, and the debate in geometry over whether there are cases where constructions are necessary in proofs?

I was reading Euclid's Elements and thinking about what various philosophers had to say about Euclidean geometry, logic itself and the philosophy of mathematics (Aristotle, Kant, Peirce, Russell, etc.). I can't help but think there's something about the nature of logic and spatiotemporal intuition that links these all together. And I don't think the invention of non-Euclidean geometry solves, dismisses, or complicated the core issue.

The problem is that, every time I see the analytic-synthetic distinction argued, the distinction seems subjective in nature. Does the predicate of the proposition reveal something new about the subject, or is it "contained" in it? Of course, Kant views mathematics as a collection of synthetic propositions because of its connection to the pure intuitions of space and time.

However, the problem with Kant's formulation of the analytic-synthetic distinction, if you already know the proposition, then all propositions can be analytic. And even an analytic proposition is "synthetic" if it's the first time that you learned about, let's say, what the word "bachelor" means. This is where Peirce comes in:
>Generally, Peirce divided deduction in two: on the one hand, deduction is either necessary or probable (deductive reasoning about probabilities), and on the other hand, deduction is either corollarial or theorematic. Corollarial deduction is reasoning “where it is only necessary to imagine any case in which the premisses are true in order to perceive immediately that the conclusion holds in that case.” Theorematic deduction “is deduction in which it is necessary to experiment in the imagination upon the image of the premiss in order from the result of such experiment to make corollarial deductions to the truth of the conclusion” (MS L 75, 1902).

Here's the kicker: if Peirce's distinction between corollarial and theorematic reasoning holds true, then we have (what I think) is the real intention behind Kant's formulation of the analytic-synthetic statement: the explanation of what mathematics is and how it is linked to the bedrock of conscious structures. We wouldn't be able to reduce geometrical proofs to propositional axioms. We would HAVE to perform the constructions, even if we "know" what they entail. Through the involvement of the imagination, we have Kant's pure intuitions, in some form, sneak back into the picture. And you know what also can't be ignored if imagination, the realm of qualia, is a part of reasoning? Aristotle's notion of intension.