Matias, List: Although I cannot offer "any information that traces the history of this problem" as requested, I can suggest Peirce's own explanation of it.
CSP: Deductions are of two kinds, which I call *corollarial *and *theorematic*. The corollarial are those reasonings by which all corollaries and the majority of what are called theorems are deduced; the theorematic are those by which the major theorems are deduced. If you take the thesis of a corollary,--i.e. the proposition to be proved, and carefully analyze its meaning, by substituting for each term its definition, you will find that its truth follows, in a straightforward manner, from previous propositions similarly analyzed. But when it comes to proving a major theorem, you will very often find you have need of a *lemma*, which is a demonstrable proposition about something outside the subject of inquiry; and even if a lemma does not have to be demonstrated, it is necessary to introduce the definition of something which the thesis of the theorem does not contemplate. (CP 7.204, 1901) See also NEM 4:1-12 (1901), which begins with the second quotation below; CP 4.233 (1902), where Peirce proposes that "corollarial, or 'philosophical' reasoning is reasoning with words; while theorematic, or mathematical reasoning proper, is reasoning with specially constructed schemata"; and especially CP 4.612-616 (1908), where he discusses at length "the step of so introducing into a demonstration a new idea not explicitly or directly contained in the premisses of the reasoning or in the condition of the proposition which gets proved by the aid of this introduction," which he calls "a theoric step." As he writes in another contemporaneous manuscript ... CSP: Everybody knows that mathematics, which covers all necessary reasoning, is as far as possible from being purely mechanical work; that it calls for powers of generalization in comparison with which all others are puny, that it requires an imagination which would be poetical were it not so vividly detailed, and above all that it demands invention of the profoundest. There is, therefore, no room to doubt that there is *some *theoric reasoning, something unmechanical, in the business of mathematics. I hope that, before I cease to be useful in this world, I may be able to define better than I now can what the distinctive essence of theoric thought is. I can at present say this much with some confidence. It is the directing of the attention to a sort of object not explicitly referred to in the enunciation of the problem in hand. (NEM 3:622, 1908) Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Fri, Aug 18, 2023 at 6:14 PM Matias <matias....@gmail.com> wrote: > 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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