> A philosophical issue of not being able to prove a negative, you really can't prove that there is not a way to solve a problem fast. You can definitely prove

> A philosophical issue of not being able to prove a negative, you really can't prove that there is not a way to solve a problem fast. You can definitely prove negatives, and we do so all the time. There's the undecidability of the halting problem, for example: there is no algorithm that can be expressed in a Turing-complete language that can determine if a particular program will halt or not. Another fun one is the unsolvability of the quintic. You know how there's a formula for solving a quadratic equation (https://en.wikipedia.org/wiki/Quadratic_formula https://en.wikipedia.org/wiki/Quadratic_formula)? Well, there's also one for order 3 polynomials (cubics) and order 4 polynomials (quartics). But order 5 (quntics)? There is no formula that can solve quintic equations using addition, subtraction, multiplication, division, exponents, and radicals (square roots, cube roots, etc) in the general case. The theorem actually goes even further by providing explicit examples: the equation x^5 + x^3 + 1 has a root, approximately equal to -0.83762, which cannot be expressed in terms of the operations I listed above. This is all the consequence of Galois theory.