The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry.

Russell, Bertrand Arthur William Russell, 3rd EarlMathematics is the door and key to the sciences.

1267 Opus Majus, pt.4, ch.1 (translated by Robert Belle Burke, 1928).Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about.

Alfred North WhiteheadAny author who uses mathematics should always express in ordinary language the meaning of the assumptions he admits, as well as the significance of the results obtained. The more abstract his theory, the more imperative this obligation.In fact, mathematics are and can only be a tool to explore reality. In this exploration, mathematics do not constitute an end in itself, they are and can only be a means.

maurice allaisAs a fraction of the lifespan of the universe as measured from the beginning to the evaporation of the last black hole, life as we know it is only possible for 1/10^30 of a percent. And that's why, for me, the most astonishing wonder of the universe isn't a star or a planet or a galaxy. It isn't a thing at all. It's an instant in time. And that time is now. Humans have walked the earth for just the shortest fraction of that briefest of moments in deep time. But in our 200,000 years on this planet we've made remarkable progress. It was only 2,500 years ago that we believed that the sun was a god and measured its orbit with stone towers built on the top of a hill. Today the language of curiosity is not sun gods, but science. And we have observatories that are almost infinitely more sophisticated than those towers, that can gaze out deep into the universe. And perhaps even more remarkably through theoretical physics and mathematics we can calculate what the universe will look like in the distant future. And we can even make concrete predictions about its end. And I believe that it's only by continuing our exploration of the cosmos and the laws of nature that govern it that we can truly understand ourselves and our place in this universe of wonders.

brian coxThe sciences are differentiated precisely on the basis of the particular contradictions inherent in their respective objects of study. Thus the contradiction peculiar to a certain field of phenomena constitutes the object of study for a specific branch of science . For example, positive and negative numbers in mathematics ; action and reaction in mechanics ; positive and negative electricity in physics ; dissociation and combination in chemistry ; forces of production and relations of production, classes and class struggle, in social science; offence and defence in military science; idealism and materialism , the metaphysical outlook and the dialectical outlook, in philosophy ; and so on--all these are the objects of study of different branches of science precisely because each branch has its own particular contradiction and particular essence .

Mao Tse-tung, in "Selected Works of Mao Tse-tung On Contradiction".A trivial philosophy of mathematics holds that every well-formed mathematical formula, in any language of mathematics is true (and its negation is true), and any philosophical sentence about mathematics is also true.

Koji Tanaka, et al., in "Paraconsistency: Logic and Applications: Logic and Applications", p.296There is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world.

Nikolai Ivanovich Lobachevsky,As quoted in George Edward Martin,Mathematics is the queen of the sciences.

carl friedrich gaussI am obliged to interpolate some remarks on a very difficult subject: proof and its importance in mathematics. All physicists, and a good many quite respectable mathematicians, are contemptuous about proof. I have heard Professor Eddington, for example, maintain that proof, as pure mathematicians understand it, is really quite uninteresting and unimportant, and that no one who is really certain that he has found something good should waste his time looking for proof.

g. h. hardyMathematics is the language in which God wrote the universe.

Attributed to Galileo Galilei inSwift objects to Enlightenment because it encourages a hypertrophic development of mathematics, physics and astronomy, thus returning to the pre-Socratic philosophy that Aristophanes had criticized for being unselfconscious or unable to understand man. But, unlike pre-Socratic philosophy, which had no interest in politics at all, this science wished to rule and could rule. The new science had indeed generated sufficient power to rule, but in order to do so had had to lose the human perspective. In other words, Swift denied that modern science had actually established a human or political science. All to the contrary, it had destroyed it.

Allan Bloom,The temptation to use mathematics is irresistible for economists. It appears to convey the appropriate air of scientific authority and precision to economists' musings.

paul ormerodThe object of pure Physic[s] is the unfolding of the laws of the intelligible world; the object of pure Mathematic[s] that of unfolding the laws of human intelligence.

james joseph sylvesterWhat exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else.

stanislaw ulamComputer science research is different from these more traditional disciplines. Philosophically it differs from the physical sciences because it seeks not to discover, explain, or exploit the natural world, but instead to study the properties of machines of human creation. In this it is analogous to mathematics, and indeed the "science" part of computer science is, for the most part mathematical in spirit. But an inevitable aspect of computer science is the creation of computer programs: objects that, though intangible, are subject to commercial exchange.

Dennis Ritchie (1984)The Art of Sight Recognition, being no longer needed, was no longer practised; and the studies of Geometry, Statics, Kinetics, and other kindred subjects, came soon to be considered superfluous, and fell into disrespect and neglect even at our University. The inferior Art of Feeling speedily experienced the same fate at our Elementary Schools.... Year by year the Soldiers and Artisans began more vehemently to assert and with increasing truth that there was no great difference between them and the very highest class of Polygons, now that they were raised to an equality with the latter, and enabled to grapple with all the difficulties and solve all the problems of life, whether Statical or Kinetical, by the simple process of Colour Recognition. Not content with the natural neglect into which Sight Recognition was falling, they began boldly to demand the legal prohibition of all "monopolizing and aristocratic Arts" and the consequent abolition of all endowments for the studies of Sight Recognition, Mathematics, and Feeling. Soon, they began to insist that inasmuch as Colour, which was a second Nature, had destroyed the need of aristocratic distinctions, the Law should follow in the same path, and that henceforth all individuals and all classes should be recognized as absolutely equal and entitled to equal rights.

edwin abbotHe has made contributions to many areas of science; among them are agronomy , anthropology, astronomy, bacteriology, botany, economics, forestry, meteorology, psychology, public health, and-above all-genetics, in which he is recognized as one of the leaders. Out of this varied scientific research and his skill in mathematics, he has evolved systematic principles for the interpretation of empirical data; and he has founded a science of experimental design. On the foundations he has laid down, there has been erected a structure of statistical techniques that are used whenever men attempt to learn about nature from experiment and observation.

ronald fisherScience and mathematics Run parallel to reality , they symbolize it, they squint at it, They never touch it: consider what an explosion Would rock the bones of men into little white fragments and unsky the world If any mind for a moment touch truth .

robinson jeffersBeing a language, mathematics may be used not only to inform but also, among other things, to seduce.

benoît mandelbrotPure Mathematics is the class of all propositions of the form “ p implies q ,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that , the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth.

bertrand russellI trust the clergy are not involved in this problem. They‘re notoriously unscientific. One can‘t attack them with mathematics.

s. s. van dineDuring the 1950s and 1960s most of the work which was called cybernetics tended to focus on control systems in engineering or on applications of the concept of feedback in fields ranging from mathematics to sociology. At the 1970 meeting of the American Society for Cybernetics in Philadelphia Heinz von Foerster sought to redirect attention to the original interests which had led to the founding of the field of cybernetics. In a paper titled "Cybernetics of Cybernetics" he made a distinction between first order cybernetics, the cybernetics of observed systems, and second order cybernetics, the cybernetics of observing systems.

Stuart A. Umpleby (1991) "Strategies for Winning Acceptance of Second Order Cybernetics." In George E. Lasker, et al. (eds.) Advances in Human Systems and Information Technologies. Windsor, Canada: International Institute for Advanced Studies in Systems Research and Cybernetics, 1992. pp. 97-196. (paper)But one will say, 'What is the principal behind these scales, and what, if not acoustic phenomena and the laws of mathematics, has set the order of their tones?' I respond that this principle is purely metaphysical [anthropological]. We conceive this order and the melodic and harmonic phenomena that spring from it out of our conformation and education. (p.249)

Hugo Riemann, cited in Gurlitt, W. (1950). "Hugo Riemann (1849-1919)".The Ionic school lasted over one hundred years. The progress of mathematics during that period was slow, as compared with its growth in a later epoch of Greek history. A new impetus to its progress was given by Pythagoras .

p. 19 (The Greeks)The doctrine of Proportion, in the Fifth Book of Euclid's Elements , is obscure, and unintelligible to most readers. It is not taught either in foreign or American colleges, and is now become obsolete. It has therefore been omitted in this edition of Euclid's Elements , and a different method of treating Proportion has been substituted for it. This is the common algebraical method, which is concise, simple, and perspicuous ; and is sufficient for all useful purposes in practical mathematics. The method is clear and intelligible to all persons who know the first principles of algebra. The rudiments of algebra ought to be taught before geometry, because algebra may be applied to geometry in certain cases, and facilitates the study of it.

Francis Nichols (1829)He is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics , kinematics and spherics , handed over the three sciences to the learned world .

Bhaskara I, quoted in: J J O'Connor and E F Robertson "Aryabhata the Elder".Mathematics is the gate and key of the sciences... Neglect of mathematics works injury to all knowledge , since he who is ignorant of it cannot know the other sciences or the things of this world. And what is worse, men who are thus Ignorant are unable to perceive their own ignorance and so do not seek a remedy.

cited in: Morris Kline (1969)Logic and mathematics seem to be the only domains where self-evidence manages to rise above triviality; and this it does, in those domains, by a linking of self-evidence on to self-evidence in the chain reaction known as proof.

Willard van Orman Quine (1978),There were found some famous men, such as Robert, Bishop of Lincoln, and Adam Marsh, and several others, who were aware that the power of mathematics is capable of unfolding the causes of all things, and of giving a sufficient explanation of human and divine phenomena; and the assurance of this fact is to be found in the writings of those great men, as, for instance, in their works on the impressions on the rainbow and the comets, on the generation of heat, on the investigation of geography, on the sphere, and on other questions appertaining both to theology and to natural philosophy.

Roger Bacon,