Schopenhauer's criticism of the proofs of the parallel postulate
historyArthur Schopenhauer criticized mathematicians' attempts to prove Euclid's Parallel Postulate because they try to prove from indirect concepts that which is directly evident from perception.
, Throughout his writings,"I wanted in this way to stress and demonstrate the great difference, indeed opposition, between knowledge of perception and abstract or reflected knowledge. Hitherto this difference has received too little attention, and its establishment is a fundamental feature of my philosophy…." Ibid., chap. 7. Schopenhauer criticized the logical derivation of philosophies and mathematics from mere concepts, instead of from intuitive perceptions.
,
In voicing his criticism, Schopenhauer naturally assumed that Euclid was using plane geometry which operates with lines drawn on a flat surface. Neither Schopenhauer nor Euclid was referring to Hyperbolic geometry, with its concave surface, or Elliptic geometry's convex surface. Although Schopenhauer could see no justification for trying to prove Euclid's Parallel Postulate, he did see a reason for examining another of Euclid's axioms.This comment by Schopenhauer was called "an acute observation" by Sir Thomas L. Heath. In his translation of The Elements, vol. 1, Book I, "Note on Common Notion 4," Heath made this judgment and also noted that Schopenhauer's remark "was a criticism in advance of Helmholtz' theory." Helmholtz had "maintained that geometry requires us to assume the actual existence of rigid bodies and their free mobility in space …" and is therefore "dependent on mechanics."
- Schopenhauer, Arthur, The World as Will and Representation, vol. 2, Dover, New York, 1966, ISBN 0-486-21762-0
- Heath, Sir Thomas L., The Thirteen Books of Euclid's Elements, vol. 1, Dover, New York, 1956, ISBN 0-486-60088-2