It is a common observation that a science first begins to be exact when it is quantitatively treated. What are called the exact sciences are no others than the mathematical ones.
Charles Sanders PeirceAnother characteristic of mathematical thought is that it can have no success where it cannot generalize.
Charles Sanders PeirceOne will meet, for example, the virtual assumption that what is relative to thought cannot be real. But why not, exactly? Red is relative to sight, but the fact that this or that is in that relation to vision that we call being red is not itself relative to sight; it is a real fact.
Charles Sanders PeirceKepler's discovery would not have been possible without the doctrine of conics. Now contemporaries of Kepler-such penetrating minds as Descartes and Pascal-were abandoning the study of geometry ... because they said it was so UTTERLY USELESS. There was the future of the human race almost trembling in the balance; for had not the geometry of conic sections already been worked out in large measure, and had their opinion that only sciences apparently useful ought to be pursued, the nineteenth century would have had none of those characters which distinguish it from the ancien régime.
Charles Sanders PeirceBad reasoning as well as good reasoning is possible; and this fact is the foundation of the practical side of logic.
Charles Sanders PeirceIt is terrible to see how a single unclear idea, a single formula without meaning, lurking in a young man's head, will sometimes act like an obstruction of inert matter in an artery, hindering the nutrition of the brain and condemning its victim to pine away in the fullness of his intellectual vigor and in the midst of intellectual plenty.
Charles Sanders Peirce