Wednesday, March 20, 2013

Math is reducible to logic

A math encyclopedia article on Logicism says:
One of the trends in the foundations of mathematics whose object is to justify mathematics by reducing its initial concepts to those of logic. The idea of reducing mathematics to logic was expressed by G. Leibniz (end of the 17th century). ...

On the whole the attempt to reduce mathematics to logic was not successful. As K. Gödel showed [3], no formalized system of logic can be an adequate basis for mathematics (cf. Gödel incompleteness theorem).
That last sentence would be a surprise to K. Gödel -- he did not prove that.

Yes, Mathematics was reduced to Logic, and Gödel helped with crucial work. Math knowledge is established with proofs, and proof are formalized with logic. A suitable logic system is Zermelo–Fraenkel set theory. If a math encyclopedia can misrepresent what math is all about, no wonder the public misunderstands it.

Alternative axiomatizations of Mathematics include Von Neumann–Bernays–Gödel set theory and Tarski–Grothendieck set theory. NGB has the advantage of allowing classes of objects that are too big to be sets, and of using only finitely many axioms. TG allows certain inaccessible cardinals that are popular in algebraic geometry, and allows types that are useful in computer theorem checking.

While these systems have some differences, all of the big theorems of modern mathematics can be formalized and proved in any of them.

1 comment:

  1. If there are multiple valid reductions, indeed infinite methods of reduction, and these are inconsistent, then is it fair to say that math is in fact reducible?