The classical theorists resemble Euclidean geometers in a non-Euclidean world who, discovering that in experience straight lines apparently parallel often meet, rebuke the lines for not keeping straight as the only remedy for the unfortunate collisions which are occurring. Yet, in truth, there is no remedy except to throw over the axiom of parallels and to work out a non-Euclidean geometry.
Mathematics is a presuppositionless science. To found it I do not need God, as does Kronecker, or the assumption of a special faculty of our understanding attuned to the principle of mathematical induction, as does Poincaré, or the primal intuition of Brouwer, or, finally, as do Russell and Whitehead, axioms of infinity, reducibility, or completeness, which in fact are actual, contentual assumptions that cannot be compensated for by consistency proofs.
My Design in this Book is not to explain the Properties of Light by Hypotheses, but to propose and prove them by Reason and Experiments: In order to which, I shall premise the following Definitions and Axioms.
My fundamental axiom of speculative philosophy is that materialism and spiritualism are opposite poles of the same absurdity-the absurdity of imagining that we know anything about either spirit or matter.
To be sure an European woman would blush to her fingers' ends at the very idea of appearing publicly stark naked; but education and prejudice are everything, since it is an axiom, that where there is no feeling of self-reproach, there can assuredly be no shame.
As a leader... I have always endeavored to listen to what each and every person in a discussion had to say before venturing my own opinion. Oftentimes, my own opinion will simply represent a consensus of what I heard in the discussion. I always remember the axiom; a leader is like a shepherd. He stays behind the flock, letting the most nimble go out ahead, whereupon the others follow, not realizing that all along they are being directed from behind.
What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else. Roughly speaking, people know that it deals with numbers, figures, with relations, operations, and that its formal procedures involving axioms, proofs, lemmas, theorems have not changed since the time of Archimedes.
The opinions that the price of commodities depends solely on the proportion of supply and demand, or demand to supply, has become almost an axiom in political economy, and has been the source of much error in that science.