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Hyperbolic geometry
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=== Philosophical consequences === The discovery of hyperbolic geometry had important [[philosophical]] consequences. Before its discovery many philosophers (such as [[Hobbes]] and [[Spinoza]]) viewed philosophical rigor in terms of the "geometrical method", referring to the method of reasoning used in [[Euclid's Elements|Euclid's ''Elements'']]. [[Kant]] in [[Critique of Pure Reason#Space and time|''Critique of Pure Reason'']] concluded that space (in [[Euclidean geometry]]) and time are not discovered by humans as objective features of the world, but are part of an unavoidable systematic framework for organizing our experiences.<ref>{{cite book |last=Lucas |first=John Randolph |author-link=John Lucas (philosopher) |title= Space, Time and Causality |year=1984 |page=149 |publisher=Clarendon Press |isbn=0-19-875057-9}}</ref> It is said that Gauss did not publish anything about hyperbolic geometry out of fear of the "uproar of the [[Boeotia]]ns" (stereotyped as dullards by the ancient Athenians<ref>{{cite journal | last = Wood | first = Donald | date = April 1959 | doi = 10.1177/030639685900100207 | issue = 2 | journal = Race | pages = 65β71 | title = Some Greek stereotypes of other peoples | volume = 1}}</ref>), which would ruin his status as ''princeps mathematicorum'' (Latin, "the Prince of Mathematicians").<ref>{{cite book|last1=Torretti|first1=Roberto|title=Philosophy of Geometry from Riemann to Poincare|date=1978|publisher=Reidel|location=Dordrecht Holland|page=255}}</ref> The "uproar of the Boeotians" came and went, and gave an impetus to great improvements in [[mathematical rigour]], [[analytical philosophy]] and [[logic]]. Hyperbolic geometry was finally proved consistent and is therefore another valid geometry.
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