– what does it mean for a theory to be “proven” mathematically?

Question

– what does it mean for a theory to be “proven” mathematically?

In: Mathematics

Answer 1

You prove theorems with mathematics, not theories.

A proof can be done mathematically a variety of ways. We used to prove the pythagorean theorem by a rearrangement proof that was taught in 6th grade geometry. We lost the will to properly teach geometry in American schools (if you are American) because everyone became obsessed with getting kids into calculus (where understanding geometry is crucial) so we replaced geometry with algebra.

Here is an example of a rearrangement proof:
[https://www.youtube.com/watch?v=mijd9BWVF40](https://www.youtube.com/watch?v=mijd9BWVF40)

Here is an example of a proof by similar triangle:
[https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-pythagorean-proofs/v/pythagorean-theorem-proof-using-similarity](https://www.khanacademy.org/math/geometry/hs-geo-trig/hs-geo-pythagorean-proofs/v/pythagorean-theorem-proof-using-similarity)

Basically, you take well known axioms that are long established, and you prove your conjecture by applying these axioms in creative ways. Once you have proved it using axioms, it moves from conjecture to theorem. You don’t have to prove axioms in your proof. Like, with using the similarity proof, you have to demonstrate that you have properly created similar triangles, but you don’t have to prove the axioms of similar triangles, you can simply wield them.

Using one of those proofs, among many available for this theorem, proves the theorem mathematically. We end a proof by saying Q.E.D – quod erat demonstratum, or roughly “that which was to be demonstrated.”