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Philosophy

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    Mathematical formal systems have non-logical symbols and axioms on top of the underlying system of inference ("logic"). Those introduce operations in particular, but they are not "valid within logical system", one can introduce whatever one wants and then use the logical system as an inference machine. This inference machine can itself be made into a mathematical object and studied, and that is what mathematical logic does, but studying it mathematically is distinct from using it as logic. Commented Aug 31, 2020 at 22:03
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    You should be told upfront that almost all logical systems fall under MATHEMATICAL LOGIC which is math. Philosophy had the origin of logical systems. The first one was called Aristotelian logic which did not use mathematics nor any symbols as Mathematical logic today uses. Philosophy teaches logic differently. So it is likely confusing to beginners because most people REFUSE to use the correct term for what it is: it is MATHEMATICAL LOGIC. The subject name is NOT LOGIC. There are other logic systems which may differ in rules. ARISTOTELIAN LOGIC still works but used for a different purpose. Commented Sep 1, 2020 at 3:04
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    Formal logic was “symbolic” from the start: see Aristotle but a significant symbolization was achieved only after the development of modern algebra, starting with Boole. Commented Sep 1, 2020 at 8:35
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    Modern mathematical logic is “mathematical” in two ways: because it is a highly mathematical discipline and because it is used to formalize mathematical theories and study their properties as mathematical objects. Commented Sep 1, 2020 at 8:37
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    @wolf-revo-cats I admittedly have an expansive view of mathematics that does not match the colloquial use, but then general public still often thinks of math as all about geometric shapes and number crunching. So I would say that Hamiltonian dynamics or the formal part of Chomsky's grammar are mathematical parts of physics and linguistics, respectively. I think this is more justifiable philosophically than the haphazard colloquial notion derived from traditional family resemblance. But to the extent that Llullian (or physicist's or linguist's) art is art it is beyond math. Commented Sep 1, 2020 at 19:38