I can attest by spot-checking for small N that even most mathematicians have not been exposed to this idea. It’s the standard concept in mathematical logic, but for some odd reason, the knowledge seems constrained to the study of “mathematical logic” as a separate field, which not all mathematicians are interested in (many just want to do Diophantine analysis or whatever).
I’m surprised by this claim. Most mathematicians have at least some understanding of mathematical logic. What you may be encountering are people who simply haven’t had to think about these issues in a while. But your use of Diophantine analysis, a subbranch of number theory, as your other example is a bit strange because number theory and algebra have become quite adept in the last few years at using model theory to show the existence of proofs even when one can’t point to the proof in question. The classical example is the Ax-Grothendieck theorem, Terry Tao discusses this and some related issues here. Similarly , Mochizuki’s attempted proof of the ABC conjecture (as I very roughly understand it) requires delicate work with models.
I’m not going to say they haven’t been exposed to it, but I think quite few mathematicians have ever developed a basic appreciation and working understanding of the distinction between syntactic and semantic proofs.
Model theory is, very rarely, successfully applied to solve a well-known problem outside logic, but you would have to sample many random mathematicians before you could find one that could tell you exactly how, even if you restricted to only asking mathematical logicians.
I’d like to add that in the overwhelming majority of academic research in mathematical logic, the syntax-semantics distinction is not at all important, and syntax is suppressed as much as possible as an inconvenient thing to deal with. This is true even in model theory. Now, it is often needed to discuss formulas and theories, but a syntactical proof need not ever be considered. First-order logic is dominant, and the completeness theorem (together with soundness) shows that syntactic implication is equivalent to semantic implication.
If I had to summarize what modern research in mathematical logic is like, I’d say that it’s about increasingly elaborate notions of complexity (of problems or theorems or something else), and proving that certain things have certain degrees of complexity, or that the degrees of complexity themselves are laid out in a certain way.
There are however a healthy number of logicians in computer science academia who care a lot more about syntax, including proofs. These could be called mathematical logicians, but the two cultures are quite different.
I’m surprised by this claim. Most mathematicians have at least some understanding of mathematical logic. What you may be encountering are people who simply haven’t had to think about these issues in a while. But your use of Diophantine analysis, a subbranch of number theory, as your other example is a bit strange because number theory and algebra have become quite adept in the last few years at using model theory to show the existence of proofs even when one can’t point to the proof in question. The classical example is the Ax-Grothendieck theorem, Terry Tao discusses this and some related issues here. Similarly , Mochizuki’s attempted proof of the ABC conjecture (as I very roughly understand it) requires delicate work with models.
I’m not going to say they haven’t been exposed to it, but I think quite few mathematicians have ever developed a basic appreciation and working understanding of the distinction between syntactic and semantic proofs.
Model theory is, very rarely, successfully applied to solve a well-known problem outside logic, but you would have to sample many random mathematicians before you could find one that could tell you exactly how, even if you restricted to only asking mathematical logicians.
I’d like to add that in the overwhelming majority of academic research in mathematical logic, the syntax-semantics distinction is not at all important, and syntax is suppressed as much as possible as an inconvenient thing to deal with. This is true even in model theory. Now, it is often needed to discuss formulas and theories, but a syntactical proof need not ever be considered. First-order logic is dominant, and the completeness theorem (together with soundness) shows that syntactic implication is equivalent to semantic implication.
If I had to summarize what modern research in mathematical logic is like, I’d say that it’s about increasingly elaborate notions of complexity (of problems or theorems or something else), and proving that certain things have certain degrees of complexity, or that the degrees of complexity themselves are laid out in a certain way.
There are however a healthy number of logicians in computer science academia who care a lot more about syntax, including proofs. These could be called mathematical logicians, but the two cultures are quite different.
(I am a math PhD student specializing in logic.)
How about you survey mathematicians?