Jon and Jeff, This is an important thread that clarifies many of the debates.
JAS
The new subject line is the title of the paper by Matthew E. Moore...
Thanks for those references. The concluding paragraph at the end of MEM's 2012 paper summarizes the issues: MEM
We wind up with an attitude that should have had a lot going for it in advance of any detailed examination of the vicissitudes of the Peircean continuum; for it is the attitude we should bring to the study of any great figure from the past. Peirce has left us, not any kind of final word, but a work in progress, one eminently worth carrying on, in the spirit of the one who started it. Which is to say that we must as resolutely critical, and as ruthless in paring away what does not work, as Peirce was at his best.
Peirce himself said that his oeuvre was a work in progress that he himself could not complete. He also insisted that every statement in science is fallible, and he would certainly include his own. As Moore said, Peirce left us "a work in progress, one eminently worth carrying on, in [his] spirit." Any attempt to "harmonize" his work would turn that living, growing spirit into a dead ghost. JBD
figuring out what conception of continuity is adequate for pure mathematics and, in turn in applied mathematics is essential.
No. There are infinitely many theories of pure mathematics. Gödel's undecidability theorem makes it doubtful that a single foundation is possible, desirable, or useful. In the early 20th c, Whitehead & Russell adopted set theory as the foundation for mathematics, but Leśniewski developed mereology as an alternative. For geometry, Whitehead and Tarski independently developed versions of mereology as the foundation. But after Gödel proved his undecidability theorem, the goal of a single universal foundation for all of mathematics became doubtful. As for applied mathematicians (which include scientists and engineers), they find the debates about foundations irrelevant for solving any practical problems. Newton, Leibniz, Einstein, Bohr, and Dirac are physicists who invented novel mathematical techniques long before the mathematicians discovered proofs to justify the computations. JBD
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.
Peirce developed outstanding theories of logic and semeiotic with just a topological version of continuity. There is no evidence that anything more would be significantly better. I agree with MEM. JBD
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.
I agree. But with the qualification that "a proper understanding of different kinds of mathematical models" does not privilege any one as proper and make all the others improper. As Peirce said, all mathematics is based on diagrammatic reasoning. Even by Cantor's standard, a countable infinity is Aleph 0, the infinity of real numbers is Aleph 1, and the infinity of possible diagrams is Aleph 2. You don't need a supermultitudinous infinity to realize that Aleph 2 is unimaginably huge. To claim that any finitely specified theory could serve as a foundation for all those diagrams is almost certainly false. JBD
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.
Actually, Abraham Robinson (whom MEM mentioned) showed that the theory of infinitesimals (as Newton, Leibniz, and Peirce used it) is mathematically sound. But Robinson's proof does not require a supermultitude. Aleph 1 (the infinity of real numbers) is sufficient. John
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