"Cauchy is mad, and there is no way of being on good terms with him, although at present he is the only man who knows how mathematics should be treated. What he does is excellent, but very confused…" (Niels H Abel, "Oeuvres", 1826)
"Ultima se tangunt. How expressive, how nicely characterizing withal is mathematics! As the musician recognizes Mozart, Beethoven, Schubert in the first chords, so the mathematician would distinguish his Cauchy, Gauss, Jacobi, Helmholtz in a few pages." (Ludwig Bolzmann, [ceremonial speech] 1887)
"The mathematical talent of Cayley was characterized by clearness and extreme elegance of analytical form; it was re-enforced by an incomparable capacity for work which has caused the distinguished scholar to be compared with Cauchy." (Charles Hermite, "Comptes Rendus", 1895)
"The natural development of this work soon led the geometers in their studies to embrace imaginary as well as real values of the variable. The theory of Taylor series, that of elliptic functions, the vast field of Cauchy analysis, caused a burst of productivity derived from this generalization. It came to appear that, between two truths of the real domain, the easiest and shortest path quite often passes through the complex domain." (Paul Painlevé, "Analyse des travaux scientifiques", 1900)
"Thought-economy is most highly developed in mathematics, that science which has reached the highest formal development, and on which natural science so frequently calls for assistance. Strange as it may seem, the strength of mathematics lies in the avoidance of all unnecessary thoughts, in the utmost economy of thought-operations. The symbols of order, which we call numbers, form already a system of wonderful simplicity and economy. When in the multiplication of a number with several digits we employ the multiplication table and thus make use of previously accomplished results rather than to repeat them each time, when by the use of tables of logarithms we avoid new numerical calculations by replacing them by others long since performed, when we employ determinants instead of carrying through from the beginning the solution of a system of equations, when we decompose new integral expressions into others that are familiar, - we see in all this but a faint reflection of the intellectual activity of a Lagrange or Cauchy, who with the keen discernment of a military commander marshalls a whole troop of completed operations in the execution of a new one." (Ernst Mach,/ "Populär-wissenschafliche Vorlesungen", 1903)
"Who has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, Sophus Lie, and Weierstrass, can doubt that a great mathematician is a great artist? The faculties possessed by such men, varying greatly in kind and degree with the individual, are analogous with those requisite for constructive art. Not every mathematician possesses in a specially high degree that critical faculty which finds its employment in the perfection of form, in conformity with the ideal of logical completeness; but every great mathematician possesses the rarer faculty of constructive imagination." (Ernest W Hobson, "Presidential Address British Association for the Advancement of Science", Nature, 1910)
"Some of the most important results (e.g. Cauchy’s theorem) are so surprising at first sight that nothing short of a proof can make them credible." (Harold Jeffreys et al, "Methods of Mathematical Physics", 1946)
"An unbelievably large literature tried to establish a transcendental ‘law of logistic growth’. Lengthy tables, complete with chi-square tests, supported this thesis for human populations, for bacterial colonies, development of railroads. etc. Both height and weight of plants and animals were found to follow the logistic even though it is theoretically clear these two variables cannot subject to the same distribution. […] The only trouble with the theory is that not only the logistic distribution, but also the normal, the Cauchy, and other distributions can be fitted to the material with the same or better goodness of fit. In thig competition the logistic distribution plays no distinguished role whatever; most theoretical models can be supported by the same observational material. Theories of this nature are short-lived because they open no new ways, and new confirmations of the same old thing soon grow boring. But the naïve reasoning has not been superseded by sense." (William A Feller, "An Introduction to Probability Theory and Its Applications" Vol. 2, 1950)
"One of the ironies of mathematics is that the ratio of two Gaussian quantities gives a Cauchy quantity. So you get Cauchy noise if you divide one Gaussian white noise process by another. [...] There is a still deeper relationship between the Cauchy and Gaussian bell curves. Both belong to a special family of probability curves called stable distributions [...] Gaussian quantities [...] are closed under addition. If you add two Gaussian noises then the result is still a Gaussian noise. This 'stable' property is not true for most noise or probability types. It is true for Cauchy processes." (Bart Kosko, "Noise", 2006)
"Dirichlet alone, not I, nor Cauchy, nor Gauss knows what a completely rigorous mathematical proof is. Rather we learn it first from him. When Gauss says that he has proved something, it is very clear; when Cauchy says it, one can wager as much pro as con; when Dirichlet says it, it is certain." (Carl G J Jacobi)
"The splendid creations of this theory have excited the admiration of mathematicians mainly because they have enriched our science in an almost unparalleled way with an abundance of new ideas and opened up heretofore wholly unknown fields to research. The Cauchy integral formula, the Riemann mapping theorem and the Weierstrass power series calculus not only laid the groundwork for a new branch of mathematics but at the same time they furnished the first and till now the most fruitful example of the intimate connections between analysis and algebra. But it isn't just the wealth of novel ideas and discoveries which the new theory furnishes; of equal importance on the other hand are the boldness and profundity of the methods by which the greatest of difficulties are overcome and the most recondite of truths, the mysteria functiorum, are exposed tothe brightest." (Richard Dedekind)
No comments:
Post a Comment