On Jul 2, 2014, at 10:23 PM, Clark Goble <[email protected]> wrote: > > On Jul 2, 2014, at 10:06 PM, Clark Goble <[email protected]> wrote: > >> Putnam’s perhaps Peirce inspired semi-empirical methods in mathematics >> paper. I rather enjoyed that paper but some noted that it was a bit after >> the fact given the reality of how computers were already being used in >> mathematical proof. As I recall Putnam’s paper came out somewhat after the >> Four Color Proof that brought a lot of attention to the role of computers in >> proofs. > > Correction. I just checked my bookshelf and my memory was *way* off. Putnam > wrote on this way back in the 1970s. Far before the Four Color Theorem proof. > My apologies. That’ll teach me to go by memory.
One last note and then I’ll go silent since this just isn’t my expertise. (Apologies for the replies to my previous posts - just wanted to make corrections before someone took me to task) The 2011 Presidential Address by Cheryl Misak did a nice job on relating the indispensability arguments (of which Putnam’s for mathematic is but one example) to pragmatism. It’s in vol 47 no 3 of Transactions for those interested. It doesn’t get at Putnam’s argument but is a nice background for the more general philosophical approach. Peirce is a but more restrained than many other pragmatists about such arguments though. He sees it more as desperation or hope in contrast to James who was a bit more excessive. And of course Kantians had been making such transcendental arguments long before the pragmatists. With regards to Peirce on the sign and returning this all to the thesis about the unreasonable effectiveness of mathematics, I suspect a large part of this can be generalized to the question of when a sign is an index or an icon. Peirce touches somewhat on this relative to Euclid in his Neglected Argument. Deduction has two parts. For its first step must be by logical analysis to Explicate the hypothesis, i.e. to render it as perfectly distinct as possible. This process, like Retroduction, is Argument that is not Argumentation. But unlike Retroduction, it cannot go wrong from lack of experience, but so long as it proceeds rightly must reach a true conclusion. Explication is followed by Demonstration, or Deductive Argumentation. Its procedure is best learned from Book I. of Euclid's Elements, a masterpiece which in real insight is far superior to Aristotle's Analytics, and its numerous fallacies render it all the more instructive to a close student. It invariably requires something of the nature of a diagram; that is, an "Icon," or Sign that represents its Object in resembling it. It usually, too, needs "Indices," or Signs that represent their Objects by being actually connected with them. But it is mainly composed of "Symbols," or Signs that represent their Objects essentially because they will be so interpreted. Demonstration should be Corollarial when it can. An accurate definition of Corollarial Demonstration would require a long explanation; but it will suffice to say that it limits itself to considerations already introduced or else involved in the Explication of its conclusion; while Theorematic Demonstration resorts to a more complicated process of thought. While not directly related to the discussion at hand, one can I think see here how Peirce thinks about mathematics and its connection to icons and indices.
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