Is mathematics a syntax?

Question

I have read that syntax is symbol and semantics is meaning those symbols convey.

Is mathematics a syntax?
Where is semantics in mathematics? What gives meaning to mathematics, to those symbols I write?

Probably any meaning Mathematics can ever have is the behaviour of its syntax only? But if syntax has a characteristic behaviour, is it a pure syntax after all? So it is a meaning oriented syntax which defines itself.

Or Mathematics is just an ideal syntax! If so, where is the semantics?

I can't figure out the interface of syntax and semantics pertaining to mathematics. Which part is syntax and which part is semantics? And what about logic?

Arithmetic is the "theory of numbers". Formulas used in theorems are made according to the syntax. Formulas "speak of" numbers and their properties, which are the semantics of the theory. — Mauro ALLEGRANZA, Nov 22 '18 at 13:25
@MauroALLEGRANZA What constitutes the syntax? Is it a pure syntax? — Ajax, Nov 22 '18 at 13:40
The syntax of e.g. arithmetic is that of first-order arithmetic. In practice, we use many abbreviations. — Mauro ALLEGRANZA, Nov 22 '18 at 13:46
@MauroALLEGRANZA Are you saying that Peano Axioms constitute the syntax for some arithmetic? — Ajax, Nov 22 '18 at 13:50
Example : Ethan Bloch, The real numbers and real analysis, Springer (2011), page 3. — Mauro ALLEGRANZA, Nov 22 '18 at 14:13
Example : Terence Tao, Analysis I, Springer (3rd ed 2016), page 15. — Mauro ALLEGRANZA, Nov 22 '18 at 14:15
The syntax are the rules for writing formulas. A theory is made of : language, specific mathematical axioms, and logical rules of inference to prove theorems from axioms. — Mauro ALLEGRANZA, Nov 22 '18 at 14:16
You can see the post Formal System and Formal Logical System and Does a formal system having inference rules imply that it is a logic system ? — Mauro ALLEGRANZA, Nov 22 '18 at 15:29
As well as the post Are axioms assumed to be true in a formal system ? — Mauro ALLEGRANZA, Nov 22 '18 at 15:29

score 1 · Answer 1 · answered Nov 27 '18 at 10:57

I'm not a logician, a mathematician nor a philosopher but maybe I can try to (quite subjectively) answer to the question. It depends on what you call Mathematics and Syntax.

The separation between syntax and semantics isn't the only way to understand formal systems, if it works well for programming languages, it doesn't explain everything for Logic and Mathematics.

Is mathematics a syntax?

There's no objective answer but I would say that mathematics is first made of social conventions to convince other people with arguments we call proofs. To convey our thoughts we use written language thus syntax. Hovewer, syntax isn't necessary but sufficient to do mathematics. Between two humans the communication is done through language but we can also communicate with ourselves through our mind. Is any form of syntax is still involved? I don't know. To sum up, mathematics can be done through syntax but may exist without it.

The language of mathematics can indeed still be turned into a global syntax accepted by everyone but in the real world it is the interaction between local communities of mathematicians who make the existence of mathematics.

Where is semantics in mathematics? What gives meaning to mathematics, to those symbols I write?

I think it's quite clear that there's no semantics. Logic is different because we have a pre-conception of what logic is and what is right, wrong, true, false. The semantic is the meaning/format we want to give. But is there any meaning we want to give to mathematics? Unlike logic, mathematics is free of ideals (such as truth).

I think your intuition is right. Following Wittgenstein's philosophy, the meaning of mathematics may be only be determined by its use ("meaning is use" paradigm). Note that there're many communities in mathematics using symbols following their own conventions.

And what about logic?

I think that question is too broad. You may be interested in the Curry-Howard correspondence; a formal correspondance between computer programs and logical systems. Logical systems may be seen as emerging from coherent computational behaviours and interactions (no divergence). In mathematics and our everyday life, logical arguments are also what makes our discussion converge to a conclusion. Logical systems formally corresponds to type systems preventing infinite reductions. Semantical explanations work but doesn't explain why logic is as it is. There're few works in that direction (Ludics, Geometry of Interaction, Transcendental Syntax, ...) mainly coming from Jean-Yves Girard.

Is mathematics a syntax?

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