Jon, List,
The questions lead, I think, to a natural progression. 1) We seek a mathematically adequate conception of continuity for the sake of developing a sufficiently rich conception of continuity for inquiry in logic and semiotics. 2) In turn, we seek a logically adequate conception of continuity for the sake of developing a sufficiently rich conception of continuity for inquiry in metaphysics. 3) Finally, we seek a philosophical conception of continuity that is adequate for logic and metaphysics for the sake of developing a sufficiently rich conception of continuity for inquiries in the special sciences. As such, figuring out what conception of continuity is adequate for pure mathematics and, in turn in applied mathematics is essential. For the sake of applying mathematics to positive questions in the sciences, a proper understanding of different kinds of mathematical models and forms of measurement are important for our inquiries in philosophy and the special sciences. After all, each of these cenoscopic or idioscopic sciences can be made rigorous only insofar as we are competent to employ the right sorts of mathematical models and forms of measurement in those areas of inquiry. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________ From: Jon Alan Schmidt <jonalanschm...@gmail.com> Sent: Saturday, August 3, 2019 5:09 PM To: peirce-l@list.iupui.edu Subject: Re: [PEIRCE-L] Is Synechism Necessary? (was Lecture by Terrence Deacon) Jeff, List: Thanks for your comments. I agree that the contemporary field of mathematics still seems to be mostly wedded to the set-theoretical approach, although I wonder if category theory offers an alternative more conducive to Peirce's late "topological" conception of continuity; Fernando Zalamea, for one, seems to think so. I tried to learn more about it a while back, but in all honesty, I never managed to attain a firm grasp of it. As is presumably evident from my history of posts, personally I am more interested in your second and third questions. From that standpoint, the possibility that set-theoretical notions might be sufficient for mathematically modeling continuity strikes me as largely beside the point. 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 Sat, Aug 3, 2019 at 6:37 PM Jeffrey Brian Downard <jeffrey.down...@nau.edu<mailto:jeffrey.down...@nau.edu>> wrote: Jon, List, In the opening remarks of the last lecture in RLT, Peirce frames three questions. Let me restate them in my own words. 1. What conception of continuity is needed for mathematics? 2. What conception of continuity is needed for a philosophical theory of critical logic and the larger theory of semiotics? 3. What conception of continuity is needed for metaphysics? We could add, what conception of continuity is needed for the special sciences, including the physical sciences as well as the human sciences? Focusing on the first question, I don't think the ongoing disputes about what conception of continuity is needed, for instance, to ground the hypotheses that lie at the basis of the calculus are "much ado about nothing". In the late 19th and early 20th century, many mathematicians and philosophers tried to defend a general approach that grounds the calculus on set-theoretical notions concerning the limit. Peirce, on the other hand, argued that the conception of the infinitesimal is more fundamental and less problematic--logically speaking. Those questions have continued to call out for more inquiry in the latter part of the 20th century and the first part of the 21st. I was in the audience when Matthew Moore delivered that presentation in Bogota. He seemed to be suggesting that set-theoretical notions involving certain types of infinity will suffice for all of mathematics. Given the fact that much inquiry in topology from the time of Poincaré to the present has moved in the direction of treating continuous paths and surfaces as point-sets, there are strong arguments that can be marshaled in support of this general approach to analyzing the mathematical conception of continuity as being sufficient for topology as well as for the calculus. I don't buy those arguments, but they are probably the majority opinion in most mathematics departments in the U.S. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354
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