0
$\begingroup$

In this answer and the comments Joel David Hamkins talks about a conflict between Cantor-Hume principle and Euclid's principle. He writes:

This principle [Cantor-Hume] is often defended as a fundamental principle for any concept of "the number of".

Those intuitive strict inequalities are hugely discussed (hundred of papers) in the philosophical literature surrounding Hume's principle, with the main point being the tension between that principle and Euclid's principle, asserting that the whole is (strictly) greater than the (proper) parts.

Galileo had thrown up his hands in confusion at the conflict between Euclid's principle and Cantor-Hume, opting ultimately to keep Euclid. It seems numerosity is motivated perhaps by the same idea.

Schanuel's paper on geometric measure theory says the following:

Of these two advances, Euler's has been by far the more important; but we seem, most of us, to have spent more effort retraining our intuitions to incorporate Cantor's ideas than Euler's.

Indeed, for instance, extremely useful geometric value, Euler's characteristic (and Betti numbers) exists exactly because Euler rejected Cantor-Hume principle.

So, what are the enormous advantages of Hume princople that made it universally adopted despite it effectively contradicts foundations of modern geometric measure theory?

$\endgroup$
10
  • $\begingroup$ Geometric measure theory is built on set theory, so they do not contradict each other. Nor does Schanuel claim that, he only regrets that measure gets less attention than cardinality. His preceding sentences are:"Since the time of Euclid, there have been two great advances in our notion of cardinal number. From Cantor we learned to count infinite discrete sets, and from Euler we learned to count extended bodies." The twists and turns in the adoption of Hume's principle are described in Mancosu's Measuring the Size of Infinite Collections. $\endgroup$
    – Conifold
    Apr 24, 2023 at 11:17
  • $\begingroup$ @Conifold but the article does not discuss infinite discrete sets. So one can imply from the text that the author thinks that Euler's approach is better suited for continuum than Cantor's. $\endgroup$
    – Anixx
    Apr 24, 2023 at 11:32
  • $\begingroup$ One cannot imply anything about the not discussed other than that the author's focus is elsewhere. Nothing prevents one from using both cardinality and measure to study continuum, or just one of them, if that's enough for the problems at hand. There is no universal "better", only better for this or that. $\endgroup$
    – Conifold
    Apr 24, 2023 at 11:59
  • $\begingroup$ @Conifold well, mathematically speaking, yes. But even from your (extremely interesting) sourse: "Conflict occurs when both are used to capture the same notion of size. Since ONE–ONE has won against SUBSET on account of Cantor’s successful theory of infinite sets, Katz sets out to develop an approach that will vindicate SUBSET even for infinite collections and in particular for the collection of subsets of natural numbers." So, it is established that Cantor's approach has won. This can be seen from voting on Stackexchange sites as well. $\endgroup$
    – Anixx
    Apr 24, 2023 at 12:12
  • 1
    $\begingroup$ So what? Informal notions are vague and often split into several precise ones when formalized. Cardinality "won" in pure set theory because it does not require extra structure to define. But in geometry, probability, even analysis, it "lost", and is used far less frequently than measure. "Size" is still used informally to refer to both depending on context, when one talks about the size of a square they typically do not mean cardinality. And Mancosu is talking about "winning" over numerosity, not measure. $\endgroup$
    – Conifold
    Apr 24, 2023 at 12:24

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.