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 - twitter.com/JonAlanSchmidt On Fri, Oct 20, 2017 at 1:36 PM, Jerry LR Chandler < jerry_lr_chand...@icloud.com> wrote: > Gary: > > On Oct 20, 2017, at 12:48 PM, Jeffrey Brian Downard < > 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 >
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