Felix Klein > Quotes

“Undoubtedly, the capstone of every mathematical theory is a convincing proof of all of its assertions. Undoubtedly, rnathematics inculpates itself when it foregoes convincing proofs. But the mystery of brilliant productivity will always be the posing of new questions, the anticipation of new theorems that make accessible valuable results and connections. Without the creation of new viewpoints, without the statement of new aims, mathematics would soou exhaust itself in the rigor of its logical proofs and begin to stagnate as its substance vanishes. Thus, in a sense, mathematics has been most advanced by those who distinguished thernselves by intuition ratber than by rigorous proofs.”
― Felix Klein

“Regarding the fundamental investigations of mathematics, there is no final ending ... no first beginning.”
― Felix Klein

“In mathematics, however, as everywhere else, men are inclined to form parties, so that there arose schools of pure synthesists and schools of pure analysts, who placed chief emphasis upon absolute “purity of method,” and who were thus more one-sided than the nature of the subject demanded. Thus the analytic geometricians often lost themselves in blind calculations, devoid of any geometric representation, The synthesists, on the other hand, saw salvation in an artificial avoidance of all formulas, and thus they accomplished nothing more, finally, than to develop their own peculiar language formulas, different from ordinary formulas.”
― Felix Klein

“The teacher manages to get along still with the cumbersome algebraic analysis, in spite of its difficulties and imperfections, and avoids the smooth infinitesimal calculus, although the eighteenth century shyness toward it had long lost all point.”
― Felix Klein, Elementary Mathematics from an Advanced Standpoint: Geometry

“Synthetic geometry is that which studies figures as such, without recourse to formulas, whereas analytic geometry consistently makes use of such formulas as can be written down after the adoption of an appropriate system of coordinates. Rightly understood, there exists only a difference of gradation between these two kinds of geometry, according as one gives more prominence to the figures or to the formulas. Analytic geometry which dispenses entirely with geometric representation can hardly be called geometry; synthetic geometry does not get very far unless it makes use of a suitable language of formulas to give precise expression to its results.”
― Felix Klein

“The distinction between modern synthesis and modern analytic geometry must no longer be regarded as essential, inasmuch as both subject-matter and methods of reasoning have gradually taken a similar form in both. We choose therefore in the text as common designation of them both the term projective geometry. Although the synthetic method has more to do with space-perception and thereby imparts a rare charm to its first simple developments, the realm of space-perception is nevertheless not closed to the analytic method, and the formulae of analytic geometry can be looked upon as a precise and perspicuous statement of geometrical relations. On the other hand, the advantage to original research of a well formulated analysis should not be underestimated, — an advantage due to its moving, so to speak, in advance of the thought. But it should always be insisted that a mathematical subject is not to be considered exhausted until it has become intuitively evident, and the progress made by the aid of analysis is only a first, though a very important, step.”
― Felix Klein

“Regarding the fundamental investigations of mathematics, there is no final ending... no first beginning.”
― Felix Klein