0
$\begingroup$

The title is the first sentence of Hermann Weyl's 1930 essay, "Levels of Infinity." He focuses on

"the distinction between actuality and potentiality, between Being and Possibility."

He opines

"the impossibility of grasping the continuum as a fixed being,"

and concludes his essay with the claim that

"we can only represent the completed infinite in the symbol."

My question is:

Q. To what extent is the claim that "Mathematics is the science of the infinite" accepted or justifiable by mathematicians today? Is his essay a relic of the period in which he penned it, or does it express a generally accepted viewpoint today?

I ask in light of the modern "rise of combinatorics"1 and of discrete mathematics.2

1Richard Guy: The rise of combinatorics (YouTube).

2 Ralston, Anthony. "The decline of calculus—The rise of discrete mathematics." In Mathematics Tomorrow, pp. 213-220. Springer, New York, NY, 1981. (Springer link.)

Improve this question
$\endgroup$
3
  • 8
    $\begingroup$ Voted to close... Primarily opinion-based. If we do this one, then we could ask about "Mathematics: The Science of Patterns" (Keith Devlin), and then "What is Mathematics?" (Courant & Robbins), and then "What is Mathematics Really?" (Reuben Hersch). So ... this is not a topic for a question here. Perhaps requesting references about "what is mathematics" may be OK. $\endgroup$
    – Gerald Edgar
    Commented Dec 3, 2018 at 0:02
  • 1
    $\begingroup$ So the underlying question in the light of other MO Qs you've asked is perhaps to what extent can continuum methods be legitimately supplanted by discrete methods--a reversal of the historical evolution of the calculus? $\endgroup$
    – Tom Copeland
    Commented Dec 3, 2018 at 0:58
  • 2
    $\begingroup$ I thought I read somewhere a traditional formula: the science of number and space (which I think has resonances going back to ancient Greece: the quaternity "arithmetic, geometry, music, astronomy" seems very suggestive). Then there's Thurston, who seems more modern: “the theory of formal patterns” (see his essay "Proof and Progress..."). But permit me to remind that MO is not well-adapted to discussion-y questions, interesting though this one could be for a discussion. [IMHO Weyl's description sounds much too limited.] $\endgroup$
    – Todd Trimble
    Commented Dec 3, 2018 at 1:50

1 Answer 1

Reset to default
4
$\begingroup$

I think there is a fairly straightforward answer to this question coming from reverse mathematics. According to Wikipedia, "finitistic reductionism" is represented by the system WKL${}_0$, and reverse math people know a lot about what mathematics can be done in WKL${}_0$ and what mathematics requires substantially stronger systems. Whether WKL${}_0$ corresponds to finitism exactly may be debatable, but in any case we know a lot about what can and can't be done around this level.

To answer the question directly, I think it's fair to say that most ordinary mathematics can be done in WKL${}_0$, and nearly all of it can be done in ACA${}_0$, which is predicative if not finitistic (although full predicativism goes substantially further than this).

Improve this answer
$\endgroup$
4
  • $\begingroup$ By "predictive", did you mean "predicative"? $\endgroup$
    – Andrés E. Caicedo
    Commented Dec 3, 2018 at 2:46
  • $\begingroup$ WKL asserts existence of infinite sets... $\endgroup$
    – Bjørn Kjos-Hanssen
    Commented Dec 3, 2018 at 3:57
  • $\begingroup$ Surely $RCA_0$ is the realm of finitism? $\endgroup$
    – David Roberts ♦
    Commented Dec 3, 2018 at 9:12
  • $\begingroup$ I think the idea is that WKL${}_0$ is supposed to be finitistically "reducible" because it is conservative over PRA for $\Pi^0_1$ sentences. In other words, we can trust it for the sentences we care about and just play a game with other sentences. Yes, I agree that RCA${}_0$ is broadly considered to be finitistically acceptable. $\endgroup$
    – Nik Weaver
    Commented Dec 3, 2018 at 13:08

Not the answer you're looking for? Browse other questions tagged .