Ben, Phyllis, Thank you both for your answers. I appreciate your insights.
Ben, I will check out the Gilman article you mentioned. I didn't know about it, but it sounds like it could be helpful. I believe that Peirce's answer to the paradox lies in his notion of theorematic deduction. However, I'm also having trouble understanding what he means by that. I'm hoping that the Gilman article will shed some light on this. Furthermore, I think it would be helpful to put his answer in perspective, taking into account the history of the problem and the subsequent development of logic. Best regards, Matias El sáb, 19 de ago de 2023, 09:24, Ben Udell <baud...@gmail.com> escribió: > > > > > > > > * I just found B.I. Gilman's article at Google Books. The whole article > was accessible to me here in the USA. > https://books.google.com/books?id=dPhl9SLIU54C&pg=PA38&lpg=PA38 > <https://books.google.com/books?id=dPhl9SLIU54C&pg=PA38&lpg=PA38> I'll try > to see (not immediately!) what to think of it. Best, Ben On 8/19/2023 7:22 > AM, Ben Udell wrote: Matias, Phyllis, One does often start with guessing, > retroduction, etc., in trying to solve a mathematical problem, be the > problem trivial or deep. However this guesswork or the like is usually not > formalized in publications. Occasionally a mathematician publishes a > mathematical conjecture, and some have been pretty important. One of > Peirce's students Benjamin Ives Gilman whom Peirce got published in Studies > in Logic (1883) > https://archive.org/details/studiesinlogic00gilmgoog/page/n15/mode/2up?ref=ol&view=theater > <https://archive.org/details/studiesinlogic00gilmgoog/page/n15/mode/2up?ref=ol&view=theater> > did not make a career in logic but did author a published (1923) article > "The Paradox of the Syllogism Solved by Spatial Construction" Mind, New > Series, Vol. 32, No. 125 (Jan., 1923), pp. 38-49 (12 pages) Published By: > Oxford University Press https://www.jstor.org/stable/2249497 > <https://www.jstor.org/stable/2249497> and I've meant to get hold of it and > read it because the general question interests me. Peirce thought highly of > Gilman; and Gilman in that article may reflect, explicitly or implicitly, > Peirce's views on novelty in deduction. Gilman claimed to have solved the > problem! It certainly is a problem. Who would bother with explicit, > deliberately weighed deduction if it did not produce conclusions with > aspects at least mildly surprising or with at least a jot of depth, > nontriviality? It's an instance of a broader paradox. Induction actually > (as opposed to seemingly like some deduction) adds claims; in Peirce's > later view it should conclude with verisimilitude a.k.a likelihood > http://www.commens.org/dictionary/term/verisimilitude > <http://www.commens.org/dictionary/term/verisimilitude> - which, as far as > I can tell, is to say that it ought to seem UNsurprising despite going > beyond the premises, or as Peirce put it, resemble the facts already in the > premises. Similar remarks can be made about abductive inference. I tend > to think that all reasoning depends for its value in part on > characteristics that resist being exactly quantified or exactly defined and > which are in some sort of tension, some sort of counterbalance, with the > inference mode's distinctive or definitive entailment-related structure. > I've noticed that the question of "seeing" or "not seeing" deductive > implications is sometimes discussed as the question of logical omniscience > and the lack thereof, for example by Sergei Artemov and Roman Kuznets in > "Logical omniscience as infeasibility", Annals of Pure and Applied Logic, > Volume 165, Issue 1, January 2014, Pages 6-25 > https://www.sciencedirect.com/science/article/pii/S0168007213001024 > <https://www.sciencedirect.com/science/article/pii/S0168007213001024> . > Best, Ben On 8/18/2023 9:08 PM, Phyllis Chiasson wrote: * > > * Wouldn't this be true for all of nature versus the all of discovery? > Discovery is human and therefore retroductive (as are "newspapers and great > fortunes"). Nature is. On Fri, Aug 18, 2023, 4:14 PM Matias > <matias....@gmail.com> <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 * > > _ _ _ _ _ _ _ _ _ _ > ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to > peirce-L@list.iupui.edu . > ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to > l...@list.iupui.edu with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the > message and nothing in the body. More at > https://list.iupui.edu/sympa/help/user-signoff.html . > ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and > co-managed by him and Ben Udell. >
_ _ _ _ _ _ _ _ _ _ ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu . ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to l...@list.iupui.edu with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the body. More at https://list.iupui.edu/sympa/help/user-signoff.html . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.
