Discovery in Mathematics (1850-1899)

"Isolated, so-called ‘pretty theorems’ have even less value in the eyes of a modern mathematician than the discovery of a new ‘pretty flower’ has to the scientific botanist, though the layman finds in these the chief charm of the respective Sciences." (Hermann Hankel, "Die Entwickelung der Mathematik in den letzten Jahrhunderten", 1869)

"Nor do I know any study which can compete with mathematics in general in furnishing matter for severe and continued thought. Metaphysical problems may be even more difficult; but then they are far less definite, and, as they rarely lead to any precise conclusion, we miss the power of checking our own operations, and of discovering whether we are thinking and reasoning or merely fancying and dreaming." (Isaac Todhunter, "Conflict of Studies and Other Essays", 1873)

"Mathematics is a science of Observation, dealing with reals, precisely as all other sciences deal with reals. It would be easy to show that its Method is the same: that, like other sciences, having observed or discovered properties, which it classifies, generalises, co-ordinates and subordinates, it proceeds to extend discoveries by means of Hypothesis, Induction, Experiment and Deduction." (George H Lewes, "Problems of Life and Mind: The Method of Science and its Application", 1874)

"Mathematics is not the discoverer of laws, for it is not induction; neither is it the framer of theories, for it is not hypothesis; but it is the judge over both, and it is the arbiter to which each must refer its claims; and neither law can rule nor theory explain without the sanction of mathematics." (Benjamin Peirce, "Linear Associative Algebra", American Journal of Mathematics Vol. 4, 1881)

"There cannot be a greater mistake than that of looking superciliously upon the practical applications of science. The life and soul of science is its practical application; and just as the great advances in mathematics have been made through the desire of discovering the solution of problems which were of a highly practical kind in mathematical science, so in physical science many of the greatest advances that have been made from the beginning of the world to the present time have been made in earnest desire to turn the knowledge of the properties of matter to some purpose useful to mankind." (William T Kelvin, "Electrical Units of Measurement", 1883)

"In addition to this it [mathematics] provides its disciples with pleasures similar to painting and music. They admire the delicate harmony of the numbers and the forms; they marvel when a new discovery opens up to them an unexpected vista; and does the joy that they feel not have an aesthetic character even if the senses are not involved at all? […] For this reason I do not hesitate to say that mathematics deserves to be cultivated for its own sake, and I mean the theories which cannot be applied to physics just as much as the others." (Henri Poincaré, 1897)