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I've done so far a few courses in logic and formal verification, and I've always wondered: mathematical logic, at least as Hilbert envisioned, tries to model mathematics. Formally define what a "true" statement is, or why proving something (at least in a sound system) makes it true.

But, every logic course uses mathematics in its essence. The use of sets, functions and series' (as with inductive groups, for example) - is already mathematics, in a field that tries to model exactly that.

Isn't that circular logic?

sadcat_1
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  • Interesting question:) – Napoleon Bonaparte Feb 20 '24 at 21:35
  • I don't see a contradiction - but I am not an expert for mathematical logic. To "model" need not be circular here, and of course it depends what is meant by this - "to model". In the naive sense, prime numbers model number theory, although number theory is used to define prime numbers. But this is certainly a bad comparison. – Dietrich Burde Feb 20 '24 at 21:41
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    Have you seen this and this? If not they're worth having a look at. If they don't answer your doubts, then it will help if you explain what they leave to be desired. I think it's basically a misconception that the whole point of logic is to "build up" maths from the ground up - it's a big field with many different things going on, though that is one part of it. To really grasp the answer takes some "mathematical-philosophical maturity" that you should trust will come with time. – Izaak van Dongen Feb 20 '24 at 21:56
  • It's all just symbol shunting. Any meaning we ascribe to it is metaphorical. – CyclotomicField Feb 20 '24 at 22:46
  • That's why it's called metamathematics. It is the mathematics of mathematics. The formal language is the language of which we speak and the metalanguage is the language in which we speak. We have a formal language of set theory but we can speak about it in English. – John Douma Feb 20 '24 at 23:57
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    How can linguistics logic try to model human language, when human language is used to write linguistics? – Mauro ALLEGRANZA Feb 21 '24 at 06:54
  • Yes, some may describe that as 'circular'. Depending on your goal, that may be an unwanted feature. In some sense one may consider this problem to be solvable. Perhaps you would be interested in philosophical investigations of some sort. In my experience, most people are just not very epistemologically minded, when for instance they play around with symbols of some sort (though of course, mistakenly, they may often find that they are). As far as I understand, part of what Hilbert and others had in mind when inventing proof theory is something that can be called a $\textit{reduction}$. – Ettore Feb 23 '24 at 20:24

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It is a common misconception many logic students pass by.

Let us start with an example: Every microprocessor we use has its own design, and we can implement it in different ways: Either physically with silicon wafers, virtually with emulators, or even over Minecraft. Nobody says the design of a microprocessor is circular or contradictory. However, people become confused if a "microprocessor" becomes a "formal theory."

Formal theories, like first-order logic, set theories, or type theories can be viewed as sets of axioms and formal rules. We can implement them as computer programs (like Coq or Lean, which implement a type theory) or over another theory (as a formal theory, an interpretation, or a model.) Mathematical logic handles formal theories as objects implemented over a metatheory, and we can ask what can hold for 'implemented' formal theories.

This line of understanding collides with a common belief that mathematical logic is a field trying to build up mathematics from the ground. It was true that mathematical logic in the early 20th century focused on building a theory ensuring the consistency of mathematics. However, it changed around the advent of Hilbert's program and the incompleteness theorem.

Hilbert formulated a program for defining a complete foundation for mathematics, and he assumed that the 'perfect' foundation should satisfy some desirable properties (like soundness and completeness.) It might be disputable that "we proved a completeness of $T$ from a metatheory" ensures "$T$ is complete indeed in the reality" (The word "reality" is also quite ambiguous, although it should be clear in the Platonist view.) However, establishing the completeness of $T$ over some metatheory is good evidence of the preference of $T$ over other theories without any justification.

However, Gödel proved that we cannot formulate a complete theory (more precisely, sound, complete, and recursively axiomatizable extension of $\mathsf{PA}$.) Hilbert's program turned out not to work, but Hilbert's view on analyzing a formal theory over mathematics turned out to be useful.

By that tendency, modern logic textbooks teach students about the properties of formal theories over a metatheory, and they do not try to build a formal theory encompassing the entire mathematics.

Hanul Jeon
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