# The interplay between semantics and syntax in the study of Mathematics

In Mathematics, if one were to ask a question such as what a "set" is, then the formal answer would be around the lines "an object which obeys a list of axioms". The obeying of axioms is captured by certain logical formula involving the set is valid.

Also in the context of mathematics, the question like "where does the set exist in the physical world" is non sense. A set existing, to my understanding, simply means a logical formula involving the definition of that set is well formed.

I conclude that the most formal answers for questions regarding sets is simply questions of syntax.

This also leads me to the following: the semantic aspect of mathematics is only important when figuring out what is the right definition. Once the theory is made, there is no aspects of semantics in the theory itself. That is a mathematical definition says no more than the logical formula.

# Question in more words

Is my inference about the nature of relation between semantics and syntax valid? What are considered to be the most important prespectives on this matter by mainstream ph£losophers ?

• Wouldn't model theory collapse into proof theory if this were so? OTOH, perhaps one could construe model theory as higher-order proof theory, or even that a theory can sustain multiple syntaxes. Nov 14, 2022 at 20:11
• "Similarly now that most of the Mathematics community is past the platonicism phase". Is that really true?
– cody
Nov 14, 2022 at 20:29
• OK I take that back @cody google.com/url?sa=t&source=web&rct=j&url=https://… Nov 14, 2022 at 20:33
• What do you mean by "formal answer"? If you mean "rigorously exact", as people often mean, then that's not the formal answer. What that is is the answer within a formal system. However, sets existed before formal systems and are not defined by them. Nov 14, 2022 at 22:28
• I take it for granted that axioms are used in context of a foundational formal system. Also, while I do agree there is a primitive conception of set which exists outside formal system, I think we have developed way past that Nov 14, 2022 at 22:38