How prevalent among philosophers and mathematicians, perhaps even among academics at large, is the formalist perspective on mathematics introduced I think by David Hilbert at the beginning of the 20th century?
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4As the quip goes, "mathematicians are platonists, non-mathematicians are formalists". But according to 2020 PhilPapers poll of (mostly) analytic philosophers, only 6% "accept or lean towards" formalism. Maybe some of it is colored by distaste for the folk "game formalism" (meaningless game of symbols), which is pretty close to incoherent in metatheory and leaves applications of mathematics inexplicable. There are sophisticated versions of formalism that are current, see SEP.– ConifoldCommented Nov 15 at 11:20
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@Conifold Applications inexplicable? Clarity is necessary for abstracting, abstracting necessary for ranging applications, and aren't the most fictional/game interpretations the least obstructed? I would think game formalism and fictionalism are closely related by being the most unobstructed in this sense; as up to the narrator philosophy.stackexchange.com/questions/118900/…– J KusinCommented Nov 15 at 20:02
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1@JKusin As SEP describes it, it is not about any abstracting. One picks arbitrary rules out of thin air and tracks what results if they are followed:"The game formalist sticks with the view that mathematical utterances have no meaning; or at any rate the terms occurring therein do not pick out objects and properties and the utterances cannot be used to state facts. Rather mathematics is a calculus in which ‘empty’ symbol strings are transformed according to fixed rules." Fictionalism is quite a different position.– ConifoldCommented Nov 15 at 21:24
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@conifold I thought "the terms occurring therein do not pick out objects and properties and the utterances cannot be used to state facts" is shared w/ hard-road fictionalism, "a) our mathematical sentences and theories do purport to be about abstract mathematical objects, as platonists suggest, but (b) there are no such things as abstract objects, and so (c) our mathematical theories are not true. Thus, the idea is that sentences such as “4 is even” are false, or untrue". shorturl.at/hj0D4 We all speak what the platonists purport (a).– J KusinCommented Nov 15 at 22:27
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1@JKusin In fictionalism, the rules are not picked out of thin air, the utterances do have meaning, albeit non-literal, and they have reasonable ways to interpret logical consequence. Unlike naive game formalists, who stick to concrete formal derivations and run into problems with strict finitism. SEP discusses these issues in the last section of the linked article.– ConifoldCommented Nov 16 at 1:50
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3 Answers
See Philip Davis and Reuben Hersh's quote [page 359 of the 1995 Edition of The Mathematical Experience (1981)]:
“The typical working mathematician is a Platonist on weekdays and a Formalist on Sundays.”
answered Nov 15 at 10:53
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2One might say: what naturalism is to the methodology of the natural sciences, formalism is to the methodology of mathematics. It is not necessarily the metaphysical side of the process, and there are ways to blur the boundaries between formal and abstract zones of concepts, so even a metaphysical formalist might not be so in an absolute, but only primary, sense (thinking that most mathematics is not concerned with abstract objects per se, while allowing for some abstracta here and there, etc.). Commented Nov 15 at 11:08
I imagine many mathematicians haven't given the matter much thought at all, but among those who have, the great divide is in the attitude toward an independent realm, or independent reality, of mathematical objects, and especially infinitary ones. Some mathematicians accept such a reality, and others reject it. For convenience, we will refer to the former as Platonists.
The non-Platonist camp includes two rather distinct groups: the Intuitionists, who following Brouwer reject logical tools such as the Law of Excluded Middle (LEM), and classical mathematicians who do not question the legitimacy of classical logic (including LEM). The latter group is often described as Formalists. This group includes Sir Michael Atiyah, who wrote in 2006:
"The idea that there is a pure world of mathematical objects (and perhaps other ideal objects) totally divorced from our experience, which somehow exists by itself is obviously inherent nonsense."
The Platonist camp includes such luminaries as Barry Mazur and Alain Connes. Mazur once mentioned, as evidence in favor of Platonism, that when he works with the mathematical entities involved in his research, they feel so real. Connes adopts a different tactic: he posits the existence of what he refers to as Primordial Mathematical Reality (PMR; I will spare you the French term), postulates certain properties with regard to such PMR, and derives certain consequences (notably, that Robinsonian infinitesimals have no right of residence in PMR; however, Connesian infinitesimals do).
How does one gauge the prevalence of Formalist or Platonist views among mathematicians (who are interested in the question to begin with)? One way would be to visit a site where one finds many mathematicians, such as MSE or MO, and judge what their philosophical position may be based on their reactions found there.
For example, the question https://math.stackexchange.com/questions/2801663/minimal-requirements-for-platonist-views-of-the-standard-model-of-set-theory-lea challenges the views of the Platonists, and the distribution of down-votes to up-votes (11 to 15) indicates that there may be approximately equal numbers of Platonists and Formalists.
Complementing @Mauros quote: IMO the working mathematician in pure mathematics does not care about Platonism or formalism, not at all and not even on Sundays :-)
If necessary and if he/she is a creative mind than they develop some new and interesting branch of mathematics and solve open problems in mathematics. And possibly, a posteriori they show that the new ideas require to change some existing axiom or to add a new axiom.
