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Geometry is knowledge of the eternally existent. Pythagoras

is to be found in the humming of the harp strings just as there is music in the spacing of the spheres.

There is geometry in the humming of the strings, there is music in the spacing of the spheres. Pythagoras

is the purest realization of human reason; but Euclid's axioms cannot be proved. He who does not believe in them sees the whole building crash.

Geometry is not true, it is advantageous.

is irrefutable and -- God can make an ass with three tails, but not a triangle with four sides.

God can make an ass with three tails, but not a triangle with four sides.

is the truth about the universe, but algebra offers the science of method.

Descartes... complained that Greek geometry was so much tied to figures "that is can exercise the understanding only on condition of greatly fatiguing the imagination." Descartes also deplored that the methods of Euclidean geometry were exceedingly diverse and specialized and did not allow for general applicability. Each theorem required a new kind of proof... What impressed Descartes especially was that algebra enables man to reason efficiently. It mechanizes thought, and hence produces almost automatically results that may otherwise be difficult to establish. ...historically it was Descartes who clearly perceived and called attention to this feature. Whereas geometry contained the truth about the universe, algebra offered the science of method. It is... paradoxical that great thinkers should be enamored with ideas that mechanize thought. Of course, their goal is to get at more difficult problems, as indeed they do.

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