There is no royal road to science and only those who do not dread of its fatiguing climb have a chance of gaining its luminuos summits.
-Karl Marx

Jan 31, 2014

Axioms of mathematics and incompleteness..

Painting by Escher.[1]
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.

Engels

Dialectics of nature

(A reflection on mathematics by engels which later was explored (independent of Engels) by Godel in his Incompleteness theorem.)

[1] Hofstadter calls this Escher work a “pictorial parable for Godel's Incompleteness Theorem.