“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.”

"Creative aspect of language use"
Quotes 2000s, 2007-09, (3rd ed., 2009)

Mont Saint Michel and Chartres (1904)
Context: God, as Descartes justly said, we know! but what is man? The schools answered:— Man is a rational animal! So was apparently a dog, or a bee, or a beaver, none of which seemed to need churches. Modern science, with infinite effort, has discovered and announced that man is a bewildering complex of energies, which helps little to explain his relations with the ultimate Substance or Energy or Prime Motor whose existence both Science and Schoolmen admit; which Science studies in laboratories and Religion worships in churches. The Man whom God created to fill his Church, must be an energy independent of God; otherwise God filled his own Church with his own energy.

Letter to Frénicle (1657) Oeuvres de Fermat Vol.II as quoted by Edward Everett Whitford, The Pell Equation http://books.google.com/books?id=L6QKAAAAYAAJ (1912)
Context: There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? This certainly is indicated by many works ancient and modern. Diophantus himself also indicates this. But he has freed himself from geometry a little more than others have, in that he limits his analysis to rational numbers only; nevertheless the Zetcica of Vieta, in which the methods of Diophantus are extended to continuous magnitude and therefore to geometry, witness the insufficient separation of arithmetic from geometry. Now arithmetic has a special domain of its own, the theory of numbers. This was touched upon but only to a slight degree by Euclid in his Elements, and by those who followed him it has not been sufficiently extended, unless perchance it lies hid in those books of Diophantus which the ravages of time have destroyed. Arithmeticians have now to develop or restore it. To these, that I may lead the way, I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof: Given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square.

Methodical Realism

Source: The Gutenberg Galaxy (1962), p. 280

The Philosophy of Space and Time (1928, tr. 1957)

Les Loix du Mouvement et du Repos, déduites d'un Principe Métaphysique (1746)

Geometry as a Branch of Physics (1949)

Methodical Realism