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In this entry in the stanford encyclopedia of philosophy, it is stated that the theory of nominalist formalism deals with the metatheory problem of formalism as follows:

Commendably, Goodman and Quine do not shy away from the metatheory problem, the difficulty that syntax and metamathematics itself seems as ontologically rich and committed to abstract objects as arithmetic. On the contrary, they face it squarely and attempt to make do entirely with an ontology of concrete objects, finitely many such objects in fact. (However they do assume fairly powerful mereological principles, essentially universal composition: they assume that any fusion of objects, however scattered or diffuse, is also an object in good standing.)

With much ingenuity they try to develop a syntax which “will treat mathematical expressions as concrete objects” (ibid.)—as actual strings of physical marks—and give concrete surrogates for notions such as ‘formula’, ‘axiom’ and ‘proof’ as platonistically defined. However they do not address the issue of the application of mathematics, construed in this concrete, formalistic fashion.

To me, this passage seems to suggest that this version of formalism takes actual physical representation to be the objects of mathematical study. This passage in particular appears to confirm this hypothesis:

Secondly, what can Goodman and Quine say about a sentence such as

2^2^2^2^2^2 + 1 is prime ?

(That is—with ‘2^n’ representing ‘2 to the power n’—‘[2^(2^(2^(2^(2^2))))]+1 is prime’; cf. Tennant, 1997 p. 152.) They cannot deny the sentence exists, for there is the token before our very eyes. But there are strong grounds for thinking that no concrete proof or disproof will exist, for the only methods available may use up more time, space and material than any human could have at her disposal, perhaps than actually exists.

So, is it indeed true that nominalist formalism proposes that the formal element of mathematics exist as actual, physical objects? If so, how exactly does this solve the metatheory problem?

  • If mathematics is only based on "actual, physical objects" ,,, it is physics. – Mauro ALLEGRANZA Apr 18 at 9:59
  • This version takes actual physical marks to be the objects of metamathematical study, according to the quote. How much time or room a proof takes is unclear, however, since one can use abbreviations, just as you use for the sentence itself, or metaproofs that there is a proof. It is also irrelevant since they are not advocating finitism. – Conifold Apr 18 at 16:28
  • Thank you for your response. However, I still cannot fathom how saying physical marks are the object of study can be justified - How would, in this case, one distinguish concept from representations? – Aryaman Gupta Apr 18 at 19:18
  • I do not think this is quite what they are saying. Nominalism is usually interpreted fictionally: it is not that physical marks are the object of study in mathematics, but rather that mathematics has no object of study. The talk of sets and numbers is euphemistic, sentences "about" numbers are not about anything at all, the whole enterprise is a piece of fiction (language game, as Wittgenstein put it) for playing a formal game with physical tokens, like chess. After Quine analytic nominalism quickly evolved into fictionalism. – Conifold Apr 18 at 22:58
  • If so, then what exactly is the role of physical marks in Nominalist formalism (as presented above)? – Aryaman Gupta Apr 19 at 6:37

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