On Karl Weierstrass

"[Up to that time] one would have said that a continuous function is essentially capable of being represented by a curve, and that a curve has always a tangent. Such reasoning has no mathematical value whatever; it is founded on intuition, or rather on a visible representation. But such representation is crude and misleading. We think we can figure to ourselves a curve without thickness; but we only figure a stroke of small thickness. In like manner we see the tangent as a straight band of small thickness, and when we say that it touches the curve, we wish merely to say that these two bands coincide without crossing. If that is what we call a curve and a tangent, it is clear that every curve has a tangent; but this has nothing to do with the theory of functions. We see to what error we are led by a foolish confidence in what we take to be visual evidence. By the discovery of this striking example Weierstrass has accordingly given us a useful reminder, and has taught us better to appreciate the faultless and purely arithmetical methods with which he more than any one has enriched our science." Henri Poincaré, "L'oeuvre Mathématique de Weierstrass",Acta Mathematica Vol.22, 1901)

"It is not surprising, in view of the polydynamic constitution of the genuinely mathematical mind, that many of the major heros of the science, men like Desargues and Pascal, Descartes and Leibnitz, Newton, Gauss and Bolzano, Helmholtz and Clifford, Riemann and Salmon and Plücker and Poincaré, have attained to high distinction in other fields not only of science but of philosophy and letters too. And when we reflect that the very greatest mathematical achievements have been due, not alone to the peering, microscopic, histologic vision of men like Weierstrass, illuminating the hidden recesses, the minute and intimate structure of logical reality, but to the larger vision also of men like Klein who survey the kingdoms of geometry and analysis for the endless variety of things that flourish there, as the eye of Darwin ranged over the flora and fauna of the world, or as a commercial monarch contemplates its industry, or as a statesman beholds an empire; when we reflect not only that the Calculus of Probability is a creation of mathematics but that the master mathematician is constantly required to exercise judgment. - judgment, that is, in matters not admitting of certainty - balancing probabilities not yet reduced nor even reducible perhaps to calculation; when we reflect that he is called upon to exercise a function analogous to that of the comparative anatomist like Cuvier, comparing theories and doctrines of every degree of similarity and dissimilarity of structure; when, finally, we reflect that he seldom deals with a single idea at a tune, but is for the most part engaged in wielding organized hosts of them, as a general wields at once the division of an army or as a great civil administrator directs from his central office diverse and scattered but related groups of interests and operations; then, I say, the current opinion that devotion to mathematics unfits the devotee for practical affairs should be known for false on a priori grounds. And one should be thus prepared to find that as a fact Gaspard Monge, creator of descriptive geometry, author of the classic Applications de l’analyse à la géométrie; Lazare Carnot, author of the celebrated works, Géométrie de position, and Réflections sur la Métaphysique du Calcul infinitesimal; Fourier, immortal creator of the Théorie analytique de la chaleur; Arago, rightful inheritor of Monge’s chair of geometry; Poncelet, creator of pure projective geometry; one should not be surprised, I say, to find that these and other mathematicians in a land sagacious enough to invoke their aid, rendered, alike in peace and in war, eminent public service." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)

"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)

"Inspiration plays no less a role in science than it does in the realm of art. It is a childish notion to think that a mathematician attains any scientifically valuable results by sitting at his desk with a ruler, calculating machines or other mechanical means. The mathematical imagination of a Weierstrass is naturally quite differently oriented in meaning and result than is the imagination of an artist, and differs basically in quality. But the psychological processes do not differ. Both are frenzy (in the sense of Plato’s 'mania') and 'inspiration'." (Max Weber, [speech (1918) presented at Munich University] 1919) 

"The arithmetization of mathematics […] which began with Weierstrass […] had for its object the separation of purely mathematical concepts, such as number and correspondence and aggregate, from intuitional ideas, which mathematics had acquired from long association with geometry and mechanics. These latter, in the opinion of the formalists, are so firmly entrenched in mathematical thought that in spite of the most careful circumspection in the choice of words, the meaning concealed behind these words, may influence our reasoning. For the trouble with human words is that they possess content, whereas the purpose of mathematics is to construct pure thought. But how can we avoid the use of human language? The […] symbol. Only by using a symbolic language not yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason-only thus may we hope to build mathematics on the solid foundation of logic." (Tobias Danzig & Joseph Mazur (ed.), "Number: The Language of Science", 1930

"Men of science belong to two different types - the logical and the intuitive. Science owes its progress to both forms of minds. Mathematics, although a purely logical structure, nevertheless makes use of intuition. Among the mathematicians there are intuitives and logicians, analysts and geometricians. Hermite and Weierstrass were intuitives. Riemann and Bertrand, logicians. The discoveries of intuition have always to be developed by logic." (Alexis Carrel,"Man the Unknown", 1935)

"Objections […] inspired Kronecker and others to attack Weierstrass’ 'sequential' definition of irrationals. Nevertheless, right or wrong, Weierstrass and his school made the theory work. The most useful results they obtained have not yet been questioned, at least on the ground of their great utility in mathematical analysis and its implications, by any competent judge in his right mind. This does not mean that objections cannot be well taken: it merely calls attention to the fact that in mathematics, as in everything else, this earth is not yet to be confused with the Kingdom of Heaven, that perfection is a chimaera, and that, in the words of Crelle, we can only hope for closer and closer approximations to mathematical truth - whatever that may be, if anything - precisely as in the Weierstrassian theory of convergent sequences of rationals defining irrationals." (Eric T Bell, "Men of Mathematics", 1937) 

"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)