Thanks, Jon! OK, the passages speak for themselves.
Of course, the qualified phrase, ‘… of the nature of…’ leaves the meaning vague. My feeling is that CSP’s remarks are now out of date in the sense that many forms of mathematical reasoning are used in different structural forms - sets, groups, rings, vector spaces etc. with different modes of reasoning, even about addition and multiplication. Just the consequences of further inquiry… Cheers Jerry > On Oct 20, 2017, at 2:15 PM, Jon Alan Schmidt <jonalanschm...@gmail.com> > wrote: > > Jerry C., List: > > Here is the first passage that comes to my mind, probably because it was the > key text for my articles on "The Logic of Ingenuity." > > CSP: Of late decades philosophical mathematicians have come to a pretty just > understanding of the nature of their own pursuit. I do not know that anybody > struck the true note before Benjamin Peirce, who, in 1870, declared > mathematics to be "the science which draws necessary conclusions," adding > that it must be defined "subjectively" and not "objectively." A view > substantially in accord with his, though needlessly complicated, is given in > the article "Mathematics," in the ninth edition of the Encyclopaedia > Britannica. The author, Professor George Chrystal, holds that the essence of > mathematics lies in its making pure hypotheses, and in the character of the > hypotheses which it makes. What the mathematicians mean by a "hypothesis" is > a proposition imagined to be strictly true of an ideal state of things. In > this sense, it is only about hypotheses that necessary reasoning has any > application; for, in regard to the real world, we have no right to presume > that any given intelligible proposition is true in absolute strictness. On > the other hand, probable reasoning deals with the ordinary course of > experience; now, nothing like a course of experience exists for ideal > hypotheses. Hence to say that mathematics busies itself in drawing necessary > conclusions, and to say that it busies itself with hypotheses, are two > statements which the logician perceives come to the same thing ... Now the > mathematician does not conceive it to be any part of his duty to verify the > facts stated. He accepts them absolutely without question. He does not in the > least care whether they are correct or not ... Thus, the mathematician does > two very different things: namely, he first frames a pure hypothesis stripped > of all features which do not concern the drawing of consequences from it, and > this he does without inquiring or caring whether it agrees with the actual > facts or not; and, secondly, he proceeds to draw necessary consequences from > that hypothesis. (CP 3.558-559, 1898; italics in original, bold added) > > I suspect that if Peirce had written this paragraph a few years later, when > he was being more careful about distinguishing existence and reality, he > would have substituted something like "existing world" or "actual world" for > "real world." Here is another relevant passage. > > CSP: Now all necessary reasoning, whether it be good or bad, is of the > nature of mathematical reasoning ... all necessary reasoning, be it the > merest verbiage of the theologians, so far as there is any semblance of > necessity in it, is mathematical reasoning. (CP 5.147-148, EP 2:206, 1903) > > Peirce essentially defined the mathematical realm as encompassing all > circumstances in which necessary reasoning can be done. > > Regards, > > Jon Alan Schmidt - Olathe, Kansas, USA > Professional Engineer, Amateur Philosopher, Lutheran Layman > www.LinkedIn.com/in/JonAlanSchmidt > <http://www.linkedin.com/in/JonAlanSchmidt> - twitter.com/JonAlanSchmidt > <http://twitter.com/JonAlanSchmidt> > On Fri, Oct 20, 2017 at 1:36 PM, Jerry LR Chandler > <jerry_lr_chand...@icloud.com <mailto:jerry_lr_chand...@icloud.com>> wrote: > Gary: >> On Oct 20, 2017, at 12:48 PM, Jeffrey Brian Downard <jeffrey.down...@nau.edu >> <mailto:jeffrey.down...@nau.edu>> wrote: >> >> Gary F., Mike, List, >> >> Should we expand the claim about mathematical objects? Gary F says: "That >> includes mathematical and other imaginary objects, which may be intelligible >> without being perceptible by the senses. Indeed it is only in the >> mathematical realm that necessary reasoning can be done, because the objects >> of pure mathematics have no being except what they are defined to have." > > I concur with Jeffrey’s definition, which,I think, is widely accepted. > > In addition, I am curious about your Peircian grounding of the assertion: >> it is only in the mathematical realm that necessary reasoning can be done, > > Do you have specific passages in mind? > > Cheers > > Jerry > > ----------------------------- > 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 the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at > http://www.cspeirce.com/peirce-l/peirce-l.htm . > > > >
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