"It [mathematics] is in the inner world of pure thought, where all entia dwell, where is every type of order and manner of correlation and variety of relationship, it is in this infinite ensemble of eternal verities whence, if there be one cosmos or many of them, each derives its character and mode of being, - it is there that the spirit of mathesis has its home and its life. [/] Is it a restricted home, a narrow life, static and cold and grey with logic, without artistic interest, devoid of emotion and mood and sentiment? That world, it is true, is not a world of solar light, not clad in the colours that liven and glorify the things of sense, but it is an illuminated world, and over it all and everywhere throughout are hues and tints transcending sense, painted there by radiant pencils of psychic light, the light in which it lies. It is a silent world, and, nevertheless, in respect to the highest principle of art - the interpenetration of content and form, the perfect fusion of mode and meaning - it even surpasses music. In a sense, it is a static world, but so, too, are the worlds of the sculptor and the architect. The figures, however, which reason constructs and the mathematic vision beholds, transcend the temple and the statue, alike in simplicity and in intricacy, in delicacy and in grace, in symmetry and in poise. Not only are this home and this life thus rich in aesthetic interests, really controlled and sustained by motives of a sublimed and supersensuous art, but the religious aspiration, too, finds there, especially in the beautiful doctrine of invariants, the most perfect symbols of what it seeks - the changeless in the midst of change, abiding things hi a world of flux, configurations that remain the same despite the swirl and stress of countless hosts of curious transformations." (Cassius J Keyser, "The Universe and Beyond", Hibbert Journal, 1904-1906)
"For scientific theories are, each and all of them, and they will continue to be, built upon and about notions which, however sublimated, are nevertheless derived from common sense." (Cassius J Keyser, "Mathematics", 1907)
"It seems indeed as if the entire surface of the world of human consciousness were predestined to be covered over, in varying degrees of luxuriance, by the flora of mathematic science." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)
"Mathematics is no more the art of reckoning and computation than architecture is the art of making bricks or hewing wood, no more than painting is the art of mixing colors on a palette, no more than the science of geology is the art of breaking rocks, or the science of anatomy the art of butchering." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)
"Science does not seek emancipation in order to become a drudge, she consents to serve indeed but her service aims at freedom as an end." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)
"Symbolic Logic is Mathematics, Mathematics is Symbolic Logic, the twain are one." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)
"The great mathematician, like the great poet or great naturalist or great administrator, is born." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)
"The belief that mathematics, because it is abstract, because it is static and cold and gray, is detached from life, is a mistaken belief. Mathematics, even in its purest and most abstract estate, is not detached from life. It is just the ideal handling of the problems of life, as sculpture may idealize a human figure or as poetry or painting may idealize a figure or a scene. Mathematics is precisely the ideal handling of the problems of life, and the central ideas of the science, the great concepts about which its stately doctrines have been built up, are precisely the chief ideas with which life must always deal and which, as it tumbles and rolls about them through time and space, give it its interests and problems, and its order and rationality. " (Cassius J Keyser, "The Humanization of the Teaching of Mathematics", 1912)
"Curiosity is the aspect of the universe seeking to realise itself, and the fruit of such activity is new reality, stimulating to new research." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)
"Mathematics, like any other cardinal activity of the human spirit, has an individuality of its own." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)
"Knowledge - a kind of proliferating sphere, expanding along divergent lines by the outward-seeking of an inner life of wonder." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)
"Projective Geometry: a boundless domain of countless fields where reals and imaginaries, finites and infinites, enter on equal terms, where the spirit delights in the artistic balance and symmetric interplay of a kind of conceptual and logical counterpoint - an enchanted realm where thought is double and flows throughout in parallel streams." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)
"The domain of mathematics is the sole domain of certainty." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)
"The ideal of thought is rigor; mathematics is the name that usage employs to designate, not attainment of the ideal, for it cannot be attained, but its devoted pursuit and close approximation." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)
"The rigor of mathematics is not absolute - absolute rigor is an ideal, to be, like other ideals, aspired unto, forever approached, but never quite attained, for such attainment would mean that every possibility of error or indetermination, however slight, had been eliminated from idea, from symbol, and from argumentation." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)
"To humanize the teaching of mathematics means so to present the subject, so to interpret its ideas and doctrines, that they shall appeal, not merely to the computatory faculty or to the logical faculty but to all the great powers and interests of the human mind." (Cassius J Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1916)
"Absolute certainty is a privilege of uneducated minds and fanatics. - It is, for scientific folk, an unattainable ideal." (Cassius J Keyser, "Mathematical Philosophy: A Study of Fate and Freedom", 1922)
"It can, you see, be said, with the same approximation to truth, that the whole of science, including mathematics, consists in the study of transformations or in the study of relations." (Cassius J Keyser. "Mathematical Philosophy: A Study of Fate and Freedom", 1922)
"Mathematics is, in many ways, the most precious response that the human spirit has made to the call of the infinite." (Cassius J. Keyser, "The Human Worth of Rigorous Thinking: Essays and Addresses", 1925)
"Every major concern among the intellectual concerns of man is a concern of mathematics." (Cassius J Keyser, "Mole Philosophy and Other Essays", 1927)
"It is customary to speak of mathematics, of pure mathematics, and of applied mathematics, as if the first were a genus owning the other two as species. The custom is unfortunate because it is misleading." (Cassius J Keyser, "Mole Philosophy and Other Essay", 1927)
"The theory of numbers, more than any other branch of mathematics, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations.” (Godfrey H Hardy)
"The validity of mathematical propositions is independent of the actual world - the world of existing subject-matters - is logically prior to it, and would remain unaffected were it to vanish from being." (Cassius J Keyser, "The Pastures of Wonder: The Realm of Mathematics and the Realm of Science", 1929)
"The validity of mathematical propositions is independent of the actual world - the world of existing subject-matters - is logically prior to it, and would remain unaffected were it to vanish from being." (Cassius J Keyser, "The Pastures of Wonder: The Realm of Mathematics and the Realm of Science", 1929)
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