Schopenhauer's criticism of the proofs of the parallel postulate

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Arthur Schopenhauer (22 February 1788 – 21 September 1860) criticized mathematicians' attempts to prove Euclid's Parallel Postulate because they try to prove from indirect concepts that which is directly evident from 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[1] (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,[2] 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 ….

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.[3]

It surprises me that the eighth axiom,[4] "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.[5]


  1. ^ What Schopenhauer calls the eleventh axiom is Euclid's Fifth Postulate.
  2. ^ "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.
  3. ^ 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."
  4. ^ What Schopenhauer calls the eighth axiom is Euclid's Common Notion 4.
  5. ^ ”Motion of an object in space does not belong in a pure science, and consequently not in geometry. For the fact that something is movable cannot be cognized a priori, but can be cognized only through experience.” (Kant, Critique of Pure Reason, B 155, Note)