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 - twitter.com/JonAlanSchmidt On Sat, Aug 3, 2019 at 6:37 PM Jeffrey Brian Downard < [email protected]> 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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