Dear list members, I am trying to contextualize Peirce's reference to the long-standing conflict between the notion of mathematical reasoning and the novelty of mathematical discoveries. I would appreciate any information that traces the history of this problem.
Here are two citations in which Peirce mentions such a conflict: "It has long been a puzzle how it could be that, on the one hand, mathematics is purely deductive in its nature, and draws its conclusions apodictically, while on the other hand, it presents as rich and apparently unending a series of surprising discoveries as any observational science. Various have been the attempts to solve the paradox by breaking down one or other of these assertions, but without success." (Peirce, 1885, On the Algebra of Logic, p. 182) "It was because those logicians who were mathematicians saw that the notion that mathematical reasoning was as rudimentary as that was quite at war with its producing such a world of novel theorems from a few relatively simple premisses, as for example it does in the theory of numbers, that they were led,--first Boole and DeMorgan, afterwards others of us, -to new studies of deductive logic, with the aid of algebras and graphs." (NEM 4:1) I know that I am asking a basic question, but thank you for your time. Best regards, Matías A. Saracho
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