the two gradients of biconditional theorems

Common notions are at ground state. Therefore, the ‘necessity’ branch of a discovered equivalence (biconditional) is always more difficult than the ‘sufficiency’ branch, because showing necessity means going from the ground state to the excited state. For example, Euclid gave the sufficient condition for an even number to be perfect, but it wasn’t until two millennia later that Euler showed the condition to be necessary. Another example: obtaining the integral representation of the logarithm is more work that merely showing that the integral representation works. Another example: the Pythagorean Theorem is much more difficult than its converse. Another example: showing that a closed and bounded set of numbers is compact is more work than showing that a compact set of numbers is closed and bounded. Another well-known example is Wilson’s Theorem. For n to be a prime, it is easy to show that it is sufficient that it divide (n – 1)! + 1, but showing that this condition is necessary is vastly more difficult. Quick takeaway: The converse of a theorem is not necessarily proved in a way similar to the proof of the theorem.




[Proof Theory]