Beautiful Proofs¶
A collection of the most beautiful proofs I found:
A classic: Prove that an angle inscribed in a semicircle is a right angle.
Proof of the Pythagorean theorem. There are many ways to prove this. Probably the simplest is a geometric proof, other cool ones in my opinion are from Einstein, and Leonardo da Vinci.
Proof of the formula to find the roots of a parabola (geometric proof + algebra proof). Btw, a formula for finding roots of 5+ degrees is impossible! (Abel-Ruffini Theorem), so we are luck with parabolas. Also, imaginary numbers were invented to solve cubic equations.
Proof that there are more real numbers than integer numbers (Cantor’s diagonal argument)
Proof of the formula to calculate the Area of a sphere. Can be done with both Archimede and Cavalieri’s Principle
Other cool things:
The most common foundation of mathematics is based on the ZFC, which is a set of axioms to formalize set theory. The “C” in ZFC stands for the controversial axiom of choice. ZFC is formalized using first-order logic, adding the in symbol and defining nine axioms. An alternative formalization is Type Theory where Lambda calculus is based on, which is used by proof assistants and many programming languages.
Russell’s paradox: what about the set of all sets that do not contain themselves? This led to the formalization of zet theory by Zermelo, “the Z” in “ZFC”.
what about “this theorem is unprovable” (Gothel)? Math is incomplete! There are true statements that cannot be proven from axioms. Gothel later demonstrated that math is inconsistent too.
Due to Turing, math is also undecidable!