Editing Arthur Schopenhauer (section)

=== Mathematics ===
Schopenhauer's [[Mathematical realism|realist]] views on mathematics are evident in his criticism of 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 [[concept]]s to prove what he held was directly evident from [[intuition|intuitive perception]].

{{blockquote|text=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.<ref name="ReferenceB">''[[The World as Will and Representation]]'', vol. 2, ch. 13</ref>}}

Throughout his writings,<ref>"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&nbsp;..." – ''The World as Will and Representation.'', vol. 2, ch. 7, p. 88 (trans. Payne)</ref> Schopenhauer criticized the logical derivation of philosophies and mathematics from mere concepts, instead of from intuitive perceptions.

{{blockquote|text=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 a posteriori|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&nbsp;...}}

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.<ref>This comment by Schopenhauer was called "an acute observation" by [[T. L. Heath|Sir Thomas L. Heath]]. In his translation of [[Euclid's Elements|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 [[Hermann von Helmholtz|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".</ref>

{{blockquote|text=It surprises me that the eighth axiom,<ref>What Schopenhauer calls the eighth axiom is Euclid's Common Notion 4.</ref> "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 (logic)|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.<ref name="ReferenceB"/>}}

This follows [[Immanuel Kant|Kant]]'s reasoning.<ref>"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)</ref>