# Relation of Mathematical Propositions to Natural Language

Treating Natural Language as a language game, what role does it play in our understanding of mathematics? Does natural language provide meaning to mathematics?

Does a proof of a conjecture, say FLT, which is essentially an arranged (legal) collection of mathematical symbols, by itself, without our interpretation, means a proof of FLT? If it doesn't, does there exist a possibility that WE, within the same calculus, can provide it some other justified meaning, one which is very different from FLT?

That is: does meaning of mathematical propositions, of whatever sort it may be, reside solely with in the mathematical propositions: how the proposition is structured, and relations between symbols and their relative positions? (Loosely, think Wittgenstein's Picture Theory). OR only a natural language can provide any meaning to it? (If this is the case, aren't mathematical statements tautologies?)

The issue I am concerned with is if description of mathematical results in natural language is a valid exercise, and if it is of any concrete importance to a

1. Mathematician,
2. And, in principle, to God (an infinitely intelligent being who can compute infinite operations concurrently and instantly).

Will be helpful if answers to this question also include remarks on the above two points on mathematician and God.

• Language (as a system of signs), natural or otherwise, can not provide meaning to anything, but mathematical propositions can be translated into natural language (very lengthily). It is not mere description, formalisms are set up in natural language. As a language game, mathematics itself provides partial meaning to its propositions, although it is enhanced by its interconnections with other games, including natural language ones (since mathematical propositions guide applications). Beyond that it is unclear what you are asking. May 14 '19 at 10:52
• @Conifold What I want to ask is: 1. If a hypothetical being exists who can perform infinite computations concurrently and instantly, i.e. effectively knows all true propositions of mathematics, of what value can natural language be to him? 2. If proof of FLT is a proof of FLT, because we interpret it to be, or it is a proof by virtue of its structure -> since proof was used to deduce the truth value, (previous steps and axioms were used) the idea of FLT's Truth must reside within the proof's *structure*(since truth was discovered, not created). I hope it makes my point clearer.
– Ajax
May 14 '19 at 12:58
• The issue is "on the table" of philosophers since a long time... You can see Proof-Theoretic Semantics for "an alternative to truth-condition semantics. It is based on the fundamental assumption that the central notion in terms of which meanings are assigned to certain expressions of our language, in particular to logical constants, is that of proof rather than truth." May 14 '19 at 13:21
• There is a difference between formal proof of a syntactic formula, which is just a chain connecting it to axioms according to mechanical rules, and proof as understood semantically. As Giaquinto put it, "without an interpretation of the language of the formal system the end-formula of the derivation says nothing; and so nothing is proved", see his piece in Mancosu volume, p.26. On this conception, not even God can whip up semantics out of syntax alone. May 15 '19 at 4:44
• I would say empty symbols, rather than tautologies. However, syntax does bestow at least some minimal semantics upon formulas, by establishing a web of derivations that connect them to the axioms, and each other, see inferential semantics. Rapaport defends an even more radical view that all semantics, natural language included, is reducible to syntax, by mapping syntax of a new domain unto the syntax of an already familiar one, see his Understanding Understanding. May 15 '19 at 9:34