Schopenhauer's criticism of the proofs of the parallel postulate
- Schopenhauer's criticism of the proofs of the parallel postulate
Arthur 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.
Notes
Bibliography
* Arthur Schopenhauer, "The World as Will and Representation", Volume 2, Dover, ISBN 0-486-21762-0
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