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People who study mathematical logic make arguments about logic itself. So it seems that people take for granted an "intuitive logic" (otherwise, how would they form arguments?). So the observation is this: there is an accepted way of reasoning all throughout math, which I'm calling "intuitive logic", and we apply that same reasoning towards logic itself.

My concern is this:

Can "intuitive logic" be studied? Any form of reasoning about "intuitive logic" would have to use "intuitive logic". This seems paradoxical because reasoning/argumentation is used as a method of justification and it doesn't really make sense to provide justification for something that has been taken for granted.

I think my question is basically equivalent to

How to justify the use of logic?

  • "People who study natural languages make statements about language itself. Is this paradoxical ?" NO: it is the only way to reason about any topic whatsoever. – Mauro ALLEGRANZA Dec 2 '17 at 15:35
  • You don't need natural language to form a statement about natural language. "Statements about natural language" can be written in a formal, non-natural language. So this is not really analogous to what I'm saying. Correct me if I misunderstood? – isthisreallife Dec 2 '17 at 15:41
  • So are you saying that you cannot learn a foreign language, like e.g. Italian using a Grammar textbook written in e.g. English, because it is not "formalized" ? Interesting... How did people learned how to read and write before the "discovery" of math logic ? – Mauro ALLEGRANZA Dec 2 '17 at 15:44
  • Regarding the question's title:YES, we can. We do it usually. – Mauro ALLEGRANZA Dec 2 '17 at 15:45
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    and I've never seen mathematicians be particular about how they reason. The discipline of "mathematical logic" is reasoned about like any other mathematical discipline. The type of reasoning that is used throughout math is what I'm calling "intuitive logic". – isthisreallife Dec 2 '17 at 20:03
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We have to assume some finitary arithmetic, otherwise there's no way to even begin making sense of symbols, formulas, etc.

According to Hilbert, there is a privileged part of mathematics, contentual elementary number theory, which relies only on a “purely intuitive basis of concrete signs.” Whereas the operating with abstract concepts was considered “inadequate and uncertain,” there is a realm of

extra-logical discrete objects, which exist intuitively as immediate experience before all thought. If logical inference is to be certain, then these objects must be capable of being completely surveyed in all their parts, and their presentation, their difference, their succession (like the objects themselves) must exist for us immediately, intuitively, as something which cannot be reduced to something else.

These objects were, for Hilbert, signs. The domain of contentual number theory consists in the finitary numerals, i.e., sequences of strokes. These have no meaning, i.e., they do not stand for abstract objects, but they can be operated on (e.g., concatenated) and compared. Knowledge of their properties and relations is intuitive and unmediated by logical inference. Contentual number theory developed this way is secure, according to Hilbert: no contradictions can arise simply because there is no logical structure in the propositions of contentual number theory.

The intuitive-contentual operations with signs forms the basis of Hilbert's metamathematics. Just as contentual number theory operates with sequences of strokes, so metamathematics operates with sequences of symbols (formulas, proofs). Formulas and proofs can be syntactically manipulated, and the properties and relationships of formulas and proofs are similarly based in a logic-free intuitive capacity which guarantees certainty of knowledge about formulas and proofs arrived at by such syntactic operations. [...]

The question of what Hilbert thought the epistemological status of the objects of finitism was is equally difficult. In order to carry out the task of providing a secure foundation for infinitistic mathematics, access to finitary objects must be immediate and certain. Hilbert's philosophical background was broadly Kantian, as was Bernays's, who was closely affiliated with the neo-Kantian school of philosophy around Leonard Nelson in Göttingen. Hilbert's characterization of finitism often refers to Kantian intuition, and the objects of finitism as objects given intuitively. [...] Both Bernays and Hilbert justify finitary knowledge in broadly Kantian terms (without however going so far as to provide a transcendental deduction), characterizing finitary reasoning as the kind of reasoning that underlies all mathematical, and indeed, scientific, thinking, and without which such thought would be impossible.

https://plato.stanford.edu/entries/hilbert-program/

  • We could take the manipulation of formulas and symbols as primitives rather than the arithmetic of natural numbers as primitive. That's even how we're taught arithmetic. – Hurkyl Apr 18 at 23:05
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Mathematical logic isn't logic, but a formalisation thereof; one view is to think of it as mathematics as inspired by logic, in the same way that mathematics has been inspired by physics but is not synonymous with it - indeed they often have difficult understanding each other despite the laymans view that they are virtually synonymous.

There is such a thing as naive set theory, and mathematicians that aren't set theorists use this all the time when they reason about sets. Likewise, one can say that there is a naive or intuitive logic that mathematicians use when they reason about mathematics.

Logic remain a field of study in philosophy, and it isn't neccessarily mathematical; linguistics and semantics for example come into it as well as others.

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