"Mathematics, like dialectics, is an instrument of the inner higher sense, while in practice it is an art like rhetoric." (Johann Wolfgang von Goethe, "Wilhelm Meisters Wanderjahre" ["Reflections in the Spirit of the Wanderers"], 1821)
"Mathematics is based upon nothing, and, consequently, arises out of nothing." (Lorenz Oken, "Elements of Physiophilosophy", 1847)
"Mathematics is an experimental science, and definitions do not come first but later on." (Oliver Heaviside, "On Operators in Physical Mathematics", Proceedings of the Royal Society of London, Series A Vol. 54, 1854)
"Mathematics is peculiarly and preeminently the science of relations, and whether quantity or direction may severally form its object, these are never contemplated in characters purely absolute, but invariably in comparison with other objects like themselves; and it is hence that relations once established by the unerring theorems of the science, we are enabled, disregarding magnitude in itself, to pass indifferently from the finite to the infinite, from the limited regions of sense to those of conception, and with all the assurance and all the certainty that even the geometry of the ancients could confer." (John H W Waugh, Mathematical Essays", 1854)
"Mathematics is the science of the functional laws and transformations which enable us to convert figured extension and rated motion into number." (George H Howison, "The Departments of Mathematics, and Their Mutual Relations", Journal of Speculative Philosophy Vol. 5, 1871)
"Mathematics is that form of intelligence in which we bring the objects of the phenomenal world under the control of the conception of quantity." (George H Howison, "The Departments of Mathematics, and Their Mutual Relations", Journal of Speculative Philosophy Vol. 5, 1871)
"Mathematics is a science of Observation, dealing with reals, precisely as all other sciences deal with reals." (George H Lewes, "Problems of Life and Mind: The Method of Science and its Application", 1874)
"Mathematics is an instrument of thought. It is a sort of machine by which the crude and imperfect results of thought are taken and disentangled, and arranged in such a way that the thought can act upon them most readily." (Charles C Everett, "The Science of Thought" II, 1890)
"Mathematics is the predominant science of our time; its conquests grow daily, though without noise; he who does not employ it for himself, will some day find it employed against himself." (Johann F Herbart, Werke, 1890)
"Mathematics is the instrument by which the engineer tunnels out mountains, bridges our rivers, constructs our aqueducts, erects out factories and makes them musical by the busy hum of spindles." (Edward Brooks, "Mental Science and Culture", 1891)
"Mathematics is the most abstract of all the sciences. [...] The whole science of mathematics is a science of hypotheses; so that nothing could be more completely abstracted from concrete reality." (Charles S Peirce, "The Regenerated Logic", The Monist Vol. 7 (1), 1896)
"The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; [...]" (Bertrand Russell, "Principles of Mathematics", 1903
"The new mathematics is a sort of supplement to language, affording a means of thought about form and quantity and a means of expression, more exact, compact, and ready than ordinary language." (Herbert G Wells, "Mankind in the Making", 1903)
"Mathematics is the language of definiteness, the necessary vocabulary of those who know." (William F White, "A Scrap-book of Elementary Mathematics", 1908)
"Symbolic Logic is Mathematics, Mathematics is Symbolic Logic, the twain are one." (Cassius J Keyser, "Lectures on Science, Philosophy and Art", 1908)
"Mathematics is merely a shorthand method of recording physical intuition and physical reasoning, but it should not be a formalism leading from nowhere to nowhere, as it is likely to be made by one who does not realize its purpose as a tool." (Charles P Steinmetz, Transactions of the American Institute of Electrical Engineers, 1909)
"Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition." (Graham D Fitch, "The Fourth Dimension simply Explained", 1910)
"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)
"Science is reduction. Mathematics is its ideal, its form par excellence, for it is in mathematics that assimilation, identification, is most perfectly realized." (Émile Boutroux, "Natural law in Science and Philosophy", 1914)
"The mathematics is a voluntary and intelligent adaptation of thought to things, it represents the forms that will allow of qualitative diversity being surmounted, the moulds into which reality must enter in order to become as intelligible as possible." (Émile Boutroux, "Natural Law in Science and Philosophy", 1914)
"The mathematics is a voluntary and intelligent adaptation of thought to things, it represents the forms that will allow of qualitative diversity being surmounted, the moulds into which reality must enter in order to become as intelligible as possible." (Émile Boutroux, "Natural Law in Science and Philosophy", 1914)
"Mathematics is the most exact science, and its conclusions are capable of absolute proof. But this is so only because mathematics does not attempt to draw absolute conclusions. All mathematical truths are relative, conditional." (Charles P Steinmetz, 1923)
"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, 1925)
"Mathematics is the foe of nature and of insight into nature." (Alfred Döblin, "Die abscheuliche Relativitätslehre" ["The Abominable Relativity Theory"], Berliner Tageblatt. 1923)
"Mathematics is the supreme arbiter. From its decisions there is no appeal." (Tobias Danzig, "Number: The Language of Science", 1930)
"Mathematics is the tool specially suited for dealing with abstract concepts of any kind and there is no limit to its power in this field." (Paul A M Dirac, "The Principles of Quantum Mechanics", 1930)
"Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is finite, means." (Hermann Weyl, "The Open World: Three Lectures In the Metaphysical Implications of Science", 1932)
"Mathematics is the science of the infinite, its goal the symbolic comprehension of the infinite with human, that is, finite means." (Hermann Weyl, "Mind and Nature", 1934)
"Mathematics is the science of number and space." (E Russell Stabler, "An Interpretation and Comparison of Three Schools of Thought in the Foundations of Mathematics", The Mathematics Teacher, Vol 26, 1935)
"[…] mathematics is like music, freely exploring the possibilities of form." (George Santayana, "The Realm of Truth: Book Third of Realms of Being", 1937)
"Mathematics is an infinity of flexibles forcing pure thought into a cosmos. It is an arc of austerity cutting realms of reason with geodesic grandeur. Mathematics is crystallized clarity, precision personified, beauty distilled and rigorously sublimated." (Cletus O Oakley, "Mathematics", The American Mathematical Monthly, 1949)
"Mathematics is one component of any plan for liberal education. Mother of all the sciences, it is a builder of the imagination, a weaver of patterns of sheer thought, an intuitive dreamer, a poet." (Cletus O Oakley, "Mathematics", The American Mathematical Monthly, 1949)
"Mathematics is not only the model along the lines of which the exact sciences are striving to design their structure; mathematics is the cement which holds the structure together." (Tobias Dantzig, "Number: The Language of Science" 4th Ed, 1954)
"Mathematics is a model of exact reasoning, an absorbing challenge to the mind, an esthetic experience for creators and some students, a nightmarish experience to other students, and an outlet for the egotistic display of mental power." (Morris Kline, "Mathematics and the Physical World", 1959)
"Mathematics is neither a description of nature nor an explanation of its operation; it is not concerned with physical motion or with the metaphysical generation of quantities. It is merely the symbolic logic of possible relations, and as such is concerned with neither approximate nor absolute truth, but only with hypothetical truth." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)
"Not only are the problems of mathematics infinite and hence inexhaustible, but mathematics itself is inexhaustible." (Constance Reid, "Introduction to Higher Mathematics for the General Reader", 1959)
"Somebody once said that philosophy is the misuse a terminology which was invented just for this purpose. In the same vein, I would say that mathematics is the science of skillful operations with concepts and rules invented just for this purpose." (Eugene Wigner, "The of Mathematics in the Natural Sciences," Communications on Pure Applied Mathematics 13 (2), 1960)
"Mathematics is a creation of the mind." (Hubert S Wall, "Creative Mathematics", 1963)
"Mathematics is a body of knowledge, but it contains no truths." (Morris Kline, “Mathematics in Western Culture”, 1964)
"Mathematics is a vast adventure of ideas [...]." (Dirk J Struik, "A Concise History of Mathematics", 1967)
"[…] mathematics is a science whose concepts are too breakable, too dry, too precisely limited." (Grigore C Moisil, 1968)
"Mathematics is much more than a language for dealing with the physical world. It is a source of models and abstractions which will enable us to obtain amazing new insights into the way in which nature operates." (Melvin Schwartz, "Principles of Electrodynamics", 1972)
"[...] mathematics is an escape from reality." (Stanislaw Ulam, “Adventures of a Mathematician”, 1976)
"Mathematics is a way of finding out, step by step, facts which are inherent in the statement of the problem but which are not immediately obvious."
"[...] mathematics is […] a whole world of invention and discovery - an art." (George F J Temple, "100 Years of Mathematics: a Personal Viewpoint", 1981)
"To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples [...]" (John B Conway, "Subnormal Operators", 1981)
"Mathematics is more than doing calculations, more than solving equations, more than proving theorems, more than doing algebra, geometry or calculus, more than a way of thinking. Mathematics is the design of a snowflake, the curve of a palm frond, the shape of a building, the joy of a game, the frustration of a puzzle, the crest of a wave, the spiral of a spider's web. It is ancient and yet new. Mathematics is linked to so many ideas and aspects of the universe." (Theoni Pappas, "More Joy of Mathematics: Exploring Mathematics All Around You", 1986)
"Mathematics is good if it enriches the subject, if it opens up new vistas, if it solves old problems, if it fills gaps, fitting snugly and satisfyingly into what is already known, or if it forges new links between previously unconnected parts of the subject It is bad if it is trivial, overelaborate, or lacks any definable mathematical purpose or direction. It is pure if its methods are pure - that is, if it doesn't cheat and tackle one problem while pretending to tackle another, and if there are no gaping holes in its logic It is applied if it leads to useful insights outside mathematics." (Ian Stewart, "The Problems of Mathematics", 1987)
"Mathematics is infinitely wide, while the language that describes it is finite." (Doron Zeilberger, 1988)
"Mathematics is full of unanswered questions, which far outnumber known theorems and results." (Ivars Peterson, "Islands of Truth: A Mathematical Mystery Cruise", 1990)
"Mathematics is the study of analogies between analogies." (Gian-Carlo Rota, "Indiscrete Thoughts", 1997)
"[...] modern mathematics is rich with vivid images and provocative ideas." (Ivars Peterson, "The Mathematical Tourist", 1988)
"[...] mathematics is a science of pattern and order." (National Research Council, "Everybody Counts", 1989)
"Mathematics is one of the surest ways for a man to feel the power of thought and the magic of the spirit. Mathematics is one of the eternal truths and, as such, raises the spirit to the same level on which we feel the presence of God." (Malba Tahan & Patricia R Baquero,"The Man Who Counted", 1993)
"Mathematics is [...] is a language that allows one to think about extraordinary questions." (James O Bullock, "Literacy in the Language of Mathematics", The American Mathematical Monthly Vol. 101 (8), 1994)
"Mathematics is a way of thinking that can help make muddy relationships clear. It is a language that allows us to translate the complexity of the world into manageable patterns."
"[...] mathematics is [...] a way of thought with inexhaustible possibilities." (Karma V Mital, "Understanding Mathematics And Computers" , 1997)
"Mathematics is a product - a discovery - of the human mind." (Keith Devlin, "Life By the Numbers", 1998)
"Mathematics is an exploratory science that seeks to understand every kind of pattern - patterns that occur in nature, patterns invented by the human mind, and even patterns created by other patterns." (Lynn A Steen, "The Future of Mathematics Education", 1998)
"[...] mathematics is about ideas and about thought." (Silvio Levy, "The Eightfold Way: The Beauty of Klein’s Quartic Curve", 1999)
"Mathematics, in one view, is the science of infinity." (Phillip J Davis & Reuben Hersh, “The Mathematical Experience”, 1999)
"[...] mathematics is a source of delight, excitement, and even controversy which are hard to share with non-mathematicians." (Gil Kalai, "Combinatorics with a Geometric Flavor: Some Examples", 2000)
"Mathematics is an activity about activity." (Robert Kaplan, "The Nothing that Is: A Natural History of Zero", 2000)
"Mathematics is a mental creation that evolved to study objects in the world. Given that objects in the world have these properties, it is no surprise that mathematical entities should inherit them. Thus, mathematics, too, is universal, precise, consistent within each subject matter, stable over time, generalizable, and discoverable. The view that mathematics is a product of embodied cognition - mind as it arises through interaction with the world - explains why mathematics has these properties." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)
"Mathematics is an objective feature of the universe; mathematical objects are real; mathematical truth is universal, absolute, and certain." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)
"Mathematics is effective in characterizing and making predictions about certain aspects of the real world as we experience it. We have evolved so that everyday cognition can, by and large, fit the world as we experience it. Mathematics is a systematic extension of the mechanisms of everyday cognition. Any fit between mathematics and the world is mediated by, and made possible by, human cognitive capacities. Any such 'fit' occurs in the human mind, where we cognize both the world and mathematics." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)
"Mathematics is not purely literal; it is an imaginative, profoundly metaphorical enterprise." (George Lakoff & Rafael E Nuñez, "Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, 2000)
"Mathematics is the ultimate in technology transfer." (Ian Stewart, "Does God Play Dice: The New Mathematics of Chaos", 2002)
"[...] mathematics is based on a strict set of definitions and rules that have been instated and to which meaning has been given." (Christopher Tremblay, "Mathematics for Game Developers", 2004)
"Mathematics is insight and invention and the flash of something grasped at once, but it is also something salt-cleaned and stout as a Gothic cathedral." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"Mathematics is like a game. It has rules, and to enjoy playing or watching it, you have to know and understand the rules. Mathematicians make up the rules as they go along." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)
"What is mathematics? It is the shared social construct created by people who are aware of certain opportunities, and we call those people mathematicians." (Ian Stewart, "Letters to a Young Mathematician", 2006)
"Mathematics is about truth: discovering the truth, knowing the truth, and communicating the truth to others." (William Byers, "How Mathematicians Think", 2007)
"Mathematics is more than a tool and language for science. It is also an end in itself, and as such, it has, over the centuries, affected our worldview in its own right." (Stephen Hawking, "God Created the Integers", 2007)
"Mathematics is created through slow, incremental progress, large leaps, missteps, corrections, and connections." (Richard S Richeson, "Eulers Gem: The Polyhedron Formula and the birth of Topology", 2008)
"Mathematics is the art of explanation." (Paul Lockhart, "A Mathematician's Lament", 2009)
"Mathematics is the music of reason." (Paul Lockhart, "A Mathematician's Lament", 2009)
"Mathematics is the purest of the arts, as well as the most misunderstood." (Paul Lockhart, "A Mathematician's Lament", 2009)
"Mathematics is a model of exact reasoning, the most precise branch of human knowledge." (Paul Hartal, "Mathematics and Reality", 2010)
"Just as music is not about reaching the final chord, mathematics is about more than just the result. It is the journey that excites the mathematician." (Marcus du Sautoy, "Listen by numbers: music and maths", 2011)
"Mathematics is both abstract and concrete, revealing much of the mental experiment, working with unobserved abstractions and objects […]" (Octavian Stanasila, Metabolism of Mathematics and Computer Science No. 8, 2015)
"Mathematics is a fascinating discipline that calls for
creativity, imagination, and the mastery of rigorous standards of proof." (John Meier
& Derek Smith, "Exploring Mathematics: An Engaging Introduction to Proof", 2017)
"Mathematics is particularly good at making things out of itself, like how higher-dimensional spaces are built up from lower-dimensional spaces. This is because mathematics deals with abstract ideas like space and dimensions and infinity, and is itself an abstract idea. […] Mathematics is abstract enough that we can always make more mathematics out of mathematics." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)
"Mathematics is the eternal, necessary system of rational, analytic tautology." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)
"Mathematics is existence itself, existence in itself, existence for itself. […] Mathematics is the basis of everything. Mathematics is the true God, and it creates the universe from itself, using, naturally, mathematics." (Thomas Stark, "God Is Mathematics: The Proofs of the Eternal Existence of Mathematics", 2018)
"All Mathematics is, properly speaking, an equation on a great scale for the other sciences." (Friederich von Hardenberg [Novalis])
"Besides language and music, mathematics is one of the primary manifestations of the free creative power of the human mind." (Hermann Weyl)
"Mathematics is a form of poetry which transcends poetry in that it proclaims the truth, a form of reasoning which transcends reasoning in that it wants to accomplish the truth it proclaims, a form of action, ritual behavior, which finds fulfillment in the act, but must proclaim and elaborate a poetic form of truth." (Salomon Bochner)
"Mathematics is a model for thinking, for developing scientific structure, for drawing conclusions and for solving problems." (Kehinde E Adenegan)
"Mathematics is a way of expressing natural laws, it is the easiest and best way to describe a general law or the flow of a phenomenon, it is the most perfect language in which one can narrate a natural phenomenon." (Gheorghe Ţiţeica)
"Mathematics is, as it were, a sensuous logic, and relates to philosophy as do the arts, music, and plastic art to poetry." (Friedrich von Schlegel)
"Mathematics is a small, often comfortable, important in practice, and clean auxiliary discipline." (Alfred Döblin, "Insight into Nature, not Natural Science")
"Mathematics is not only real, but it is the only reality." (Martin Gardner)
"Mathematics is pure language - the language of science. It is unique among languages in its ability to provide precise expression for every thought or concept that can be formulated in its terms. [...] It is also an art - the most intellectual and classical of the arts." (Alfred Adler)
"Mathematics is pure poetry." (Immanuel Kant, "Opus Postumum")
"Mathematics is the art of giving the same name to different things." (Henri Poincare)
"Mathematics is the only good metaphysics." (Lord William T Kelvin)
"Mathematics is the only infinite human activity." (Paul Erdos)
"[...] mathematics is the science of patterns." (Lynn A Steen)
"Mathematics is the science that yields the best opportunity to observe the working of the mind. Its study is the best training of our abilities as it develops both the power and the precision of our thinking. Mathematics is valuable on account of the number and variety of its applications. And it is equally valuable in another respect: By cultivating it, we acquire the habit of a method of reasoning which can be applied afterwards to the study of any subject and can guide us in life's great and little problems." (Nicolas de Condorcet)
“Mathematics is the summit of human thinking." (Meir Shalev)
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