A calculus of the absurd

Chapter 5 Proof

We must not believe those, who today, with philosophical bearing and deliberative tone, prophesy the fall of culture and accept the ignorabimus [that we cannot know whether something is true or false]. For us there is no ignorabimus, and in my opinion none whatever in natural science. In opposition to the foolish ignorabimus our slogan shall be “Wir müssen wissen - wir werden wissen” [we must know — we will know]

— David Hilbert, radio address in 1930

The human mind is incapable of formulating (or mechanizing) all its mathematical intuitions, i.e., if it has succeeded in formulating some of them, this very fact yields new intuitive knowledge, e.g., the consistency of this formalism. This fact may be called the “incompletability” of mathematics. On the other hand, on the basis of what has been proved so far, it remains possible that there may exist (and even be empirically discoverable) a theorem proving machine which in fact is equivalent to mathematical intuition, but cannot be proved to be so, nor even be proved to yield only correct theorems of finitary number theory

Kurt Goedel, remarks on his incompleteness theorems

5.1 Proving identities

To prove an identity, pick one of the sides of the identity, and then apply the rules of algebra to work to the other side. Once you’ve picked a side, you have to stick with it; you can’t use both sides (because you’d be accepting what we want to prove as being true in order to prove that it’s true which would then not be a proof).