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Are modern proponents of formalism associated with an ontoglogical opinion regarding numbers?

If they view mathematics as the process of manipulating string according to agreed upon rules, there is no initial need for semantics. In a way you don't need the notion of the number three to do some work, which involved the expression "xxx". And you can make comparison rules like

'consider "xxx,xxxx", which holds "xxx" and "xxxx" and then remove one symbol "x" on the left and the right hand side iteratively and choose left or right according to which has no symbols left first'.

Some points why I have a problem seeing an answer: On the wikipedia page, for example, it says

"According to formalism, the truths expressed in logic and mathematics are not about numbers, sets, or triangles or any other contensive subject matter — in fact, they aren't "about" anything at all. They are syntactic forms whose shapes and locations have no meaning unless they are given an interpretation (or semantics)."

I'm not sure if this statement should be interpreted in such a way that the mere possibility of, for example, considering the number of symbols in a string and the possibility of abstraction of comparing two string lengths, should imply that there is some existence and truth to 3 and 4 and then 3<4 respecively.

The observation that, in the last line, here in this question I can only tell you what I'm talking about by introducing new symbols and strings "3", "4" and "3<4" obscures or amplifies the problem: If one interacts with other people, when can one really talk about the abstract concept, if I need to denote them by strings, or words, sounds or other signals anyway. The longer one talks about it, the more the any possitive onthological assumptions seems unnecessary. And I could kind of translate the problem to my own thought as well. After all, am I comunicating here, or do I write text on a computer, which only, maybe, gets read by other people later. That language always has to get expressed (or already is in a state of an expression) is a case for formalism being enough to talk about.

But if it's the case that formalists don't feel the need to talk about existence of other things than their strings (i.e. if there is "nothing real" to any number etc.), is for example ultrafinitism considered to be a radical position. They are talked about as having a very strong restricting point of view, but if there are other mathematicans/logicans/philosophers around, which discard the concept altogether, that is people whos position is totally vertical to the believers position, then ultrafinitism is not particlularly radical, one one step down in the hierarchy ideas that deny certain things.

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It would appear that Formalism could be viewed as a species of Mathematical Fictionalism, and would require no ontological commitments whatsoever. Both linked SEP articles are full of further details.

  • But the fictionalists, as described in the second article, believe in the truth of mathematical statements. It says "Indeed, platonists and fictionalists don't disagree about any semantic theses". Whereas formalists don't have to (or are able to) make such judgements. At best, they can give meaning to the word "true" via the formal notion of proof, but then there are unprovable statements, which are nevertheless considered true by platonists. – Nikolaj-K Aug 24 '12 at 14:51
  • @NickKidman It is strange to hear that fictionalists believe in the truth of mathematical statements. The most famous proponent (originator?) of Mathematical Fictionalism--- Hartry Field in Science Without Numbers ---advances a mathematical error theory. On his view, all mathematical sentences are (strictly speaking) false. This isn't an issue, however, because they are "correct", which is all we need for science. What correctness amounts to for field is that the mathematical theories are conservative extensions of true theories uncommitted to mathematical objects. – Dennis Jan 8 '13 at 0:07
  • Actually, looking at the SEP article, it confirms my suspicion: "As it was defined above, fictionalism is the view that (a) our mathematical sentences and theories do purport to be about abstract mathematical objects, as platonism suggests, but (b) there are no such things as abstract objects, and so (c) our mathematical theories are not true." – Dennis Jan 8 '13 at 2:17
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A strict formalist view usually can not say when the formal rules for a system are inconsistent. Assuming the principle of explosion, inconsistency leads to the system breaking down, with all well formed formulae derivable. If, on the other hand, one believes in the existence of say, the natural numbers, then one can say that the Peano system will not lead to a contradiction. So the formalist view of the various ontological debates would be that there are different positions on which formal systems do not break down - PA, ZFC, large cardinal axioms, etc.

But since the formalist does not have any way of knowing which position holds, my guess is that ultrafinitism will NOT be seen as radical. The surprising position of ultrafinitism is that even the Peano axioms could lead to a contradiction, since they allow arbitrarily large numbers beyond any normal experience. This will be radical only if one is commited semantically to the natural numbers.

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