Axioms of mathematics and incompleteness..
|Painting by Escher.|
The so-called axioms of mathematics are the few thought determinations which mathematics needs for its point of departure. Mathematics is the science of magnitudes; its point of departure is the concept of magnitude. It defines this lamely and then adds the other elementary determinations of magnitude, not contained in the definition, from outside as axioms, so that they appear as unproved, and naturally also as mathematically unprovable. The analysis of magnitude would yield all these axiom determinations as necessary determinations of magnitude. Spencer is right in as much as what thus appears to us to be the self-evidence of these axioms is inherited. They are provable dialectically, in so far as they are not pure tautologies.
Dialectics of nature