*
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
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 - 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 .
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> 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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