Schopenhauer's criticism of the proofs of the Parallel Postulate  

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Kunstformen der Natur (1904) by Ernst Haeckel
Kunstformen der Natur (1904) by Ernst Haeckel

Schopenhauer's realist views on mathematics are evident in his criticism of the contemporaneous attempts to prove the parallel postulate in Euclidean geometry. Writing shortly before the discovery of hyperbolic geometry demonstrated the logical independence of the axiom—and long before the general theory of relativity revealed that it does not necessarily express a property of physical space—Schopenhauer criticized mathematicians for trying to use indirect concepts to prove what he held was directly evident from intuitive perception.

The Euclidean method of demonstration has brought forth from its own womb its most striking parody and caricature in the famous controversy over the theory of parallels, and in the attempts, repeated every year, to prove the eleventh axiom (also known as the fifth postulate). The axiom asserts, and that indeed through the indirect criterion of a third intersecting line, that two lines inclined to each other (for this is the precise meaning of "less than two right angles"), if produced far enough, must meet. Now this truth is supposed to be too complicated to pass as self-evident, and therefore needs a proof; but no such proof can be produced, just because there is nothing more immediate.

Throughout his writings, Schopenhauer criticized the logical derivation of philosophies and mathematics from mere concepts, instead of from intuitive perceptions.

In fact, it seems to me that the logical method is in this way reduced to an absurdity. But it is precisely through the controversies over this, together with the futile attempts to demonstrate the directly certain as merely indirectly certain, that the independence and clearness of intuitive evidence appear in contrast with the uselessness and difficulty of logical proof, a contrast as instructive as it is amusing. The direct certainty will not be admitted here, just because it is no merely logical certainty following from the concept, and thus resting solely on the relation of predicate to subject, according to the principle of contradiction. But that eleventh axiom regarding parallel lines is a synthetic proposition a priori, and as such has the guarantee of pure, not empirical, perception; this perception is just as immediate and certain as is the principle of contradiction itself, from which all proofs originally derive their certainty. At bottom this holds good of every geometrical theorem ...

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.

It surprises me that the eighth axiom, "Figures that coincide with one another are equal to one another", is not rather attacked. For "coinciding with one another" is either a mere tautology, or something quite empirical, belonging not to pure intuition or perception, but to external sensuous experience. Thus it presupposes mobility of the figures, but matter alone is movable in space. Consequently, this reference to coincidence with one another forsakes pure space, the sole element of geometry, in order to pass over to the material and empirical.

This follows Kant's reasoning.

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