Popular quotes about Mathematics! Wisdom and inspiration are here! | page 25
Mathematics has two faces: it is the rigorous science of Euclid, but it is also something else. Mathematics presented in the Euclidean way appears as a systematic, deductive science; but mathematics in the making appears as an experimental, inductive science. Both aspects are as old as the science of mathematics itself.
George PolyaThe point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction. . . . Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about.
Alfred North WhiteheadBut there is another reason for the high repute of mathematics: it is mathematics that offers the exact natural sciences a certain measure of security which, without mathematics, they could not attain.
Albert EinsteinLike a stool which needs three legs to be stable, mathematics education needs three components: good problems, with many of them being multi-step ones, a lot of technical skill, and then a broader view which contains the abstract nature of mathematics and proofs. One does not get all of these at once, but a good mathematics program has them as goals and makes incremental steps toward them at all levels.
Richard AskeyMathematics is often defined as the science of space and number . . . it was not until the recent resonance of computers and mathematics that a more apt definition became fully evident: mathematics is the science of patterns.
Lynn SteenOne may say that mathematics talks about the things which are of no concern to men. Mathematics has the inhuman quality of starlight - brilliant, sharp but cold ... thus we are clearest where knowledge matters least: in mathematics, especially number theory.
Hermann WeylMathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country.
David HilbertIt is indubitable that a 50-year-old mathematician knows the mathematics he learned at 20 or 30, but has only notions, often rather vague, of the mathematics of his epoch, i.e. the period of time when he is 50.
Jean Dieudonne...the source of all great mathematics is the special case, the concrete example. It is frequent in mathematics that every instance of a concept of seemingly generality is, in essence, the same as a small and concrete special case.
Paul HalmosThe proof of Fermat's Last Theorem underscores how stable mathematics is through the centuries - how mathematics is one of humanity's long continuous conversations with itself.
Barry MazurIt has been a fortunate fact in the modern history of physical science that the scientist constructing a new theoretical system has nearly always found that the mathematics. . . required. . . had already been worked out by pure mathematicians for their own amusement. . . . The moral for statesmen would seem to be that, for proper scientific "planning", pure mathematics should be endowed fifty years ahead of scientists.
R. B. BraithwaiteTo exist (in mathematics), said Henri Poincarรฉ, is to be free from contradiction. But mere existence does not guarantee survival. To survive in mathematics requires a kind of vitality that cannot be described in purely logical terms.
Mark KacThe most rewarding part of my work is the "Aha" moment, the excitement of discovery and enjoyment of understanding something new - the feeling of being on top of a hill and having a clear view. But most of the time, doing mathematics for me is like being on a long hike with no trail and no end in sight. I find discussing mathematics with colleagues of different backgrounds one of the most productive ways of making progress.
Maryam MirzakhaniMathematics is the art of explanation. If you deny students the opportunity to engage in this activity-- to pose their own problems, to make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs-- you deny them mathematics itself.
Paul LockhartMathematics without natural history is sterile, but natural history without mathematics is muddled.
John Maynard SmithI tell you that studying humanities in high school is more important than mathematics - mathematics is too sharp an instrument, no good for kids.
Stefan BanachI didn't feel comfortable at first with pure mathematics, or as a professor of pure mathematics. I wanted to do a little bit of everything and explore the world.
Benoit MandelbrotMathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work for yourself
George PolyaWomen have a passion for mathematics. They divide their age in half, double the price of their clothes, and always add at least five years to the age of their best friend.
Marcel AchardEveryone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions.
Felix KleinI think math is a hugely creative field, because there are some very well-defined operations that you have to work within. You are, in a sense, straightjacketed by the rules of the mathematics. But within that constrained environment, it's up to you what you do with the symbols.
Brian GreeneThat is, you can have nothingness, absolute nothingness for maybe a tiny fraction of a second, if a second can be defined in that arena, but then it falls apart into a something and an anti-something. And that something is then what we call the universe. But can we really understand that or put rigorous mathematics or testable experiments against that? Not yet. So one of the big holy grail of physics is to understand why there is something rather than nothing.
Brian GreeneTaking mathematics from the beginning of the world to the time when Newton lived, what he had done was much the better half.
Gottfried LeibnizI went to Princeton from Amherst, where I split my interests between mathematics and philosophy.
Stephen Cole KleeneMathematics is not yet capable of coping with the naรฏvetรฉ of the mathematician himself.
Abraham KaplanWhat would mathematics have amounted to without the imagination of its devotees-its giants and their followers? There never was a discovery made without the urge of imagination-of imagination which broke the roadway through the forest in order that cold logic might follow.
David Eugene SmithOutside of mathematics and logic, there are common sense truths, such as that it is snowing that normal observers, in a specified context can agree on, subject to vagueness considerations, and theoretical truths, such as that snow is crystallised water vapour, and maybe in-between truths.
Catherine WilsonOne of my colleagues likes to say that, mathematics is the - he thinks about the only subject that he knows in academia or in the real world where if two people disagree about something - if people are studying some mathematical object and there's supposed to be a proof and they disagree about whether this proof or not, the will go into a room, sit down and talk about it and fairly quickly or at the end of the day one of them will admit they're wrong.
Peter WoitI read one or two other books which gave me a background in mathematics other than logic.
Stephen Cole KleeneMathematics is as much an aspect of culture as it is a collection of algorithms.
Carl Benjamin BoyerThe science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience
Immanuel KantThere are fields of scientific work...which have been explored from the different sides of pure mathematics, statistics, electrical engineering, and neurophysiology...in which every single notion receives a separate and different name from each group, and in which important work has been triplicated or quadruplicated, while still other important work is delayed by the unavailability in one field of results that may have already become classical in the next field.
Norbert WienerPerhaps the most surprising thing about mathematics is that it is so surprising.
Edward Charles TitchmarshOf the properties of mathematics, as a language, the most peculiar one is that by playing formal games with an input mathematical text, one can get an output text which seemingly carries new knowledge. The basic examples are furnished by scientific or technological calculations: general laws plus initial conditions produce predictions, often only after time-consuming and computer-aided work. One can say that the input contains an implicit knowledge which is thereby made explicit.
IU?. I. ManinWhat is especially striking and remarkable is that in fundamental physics, a beautiful or elegant theory is more likely to be right than a theory that is inelegant. A theory appears to be beautiful or elegant (or simple, if you prefer) when it can be expressed concisely in terms of mathematics we already have. Symmetry exhibits the simplicity. The Foundamental Law is such that the different skins of the onion resemble one another and therefore the math for one skin allows you to express beautifully and simply the phenomenon of the next skin.
Murray Gell-MannAmong the minor, yet striking characteristics of mathematics, may be mentioned the fleshless and skeletal build of its propositions; the peculiar difficulty, complication, and stress of its reasonings; the perfect exactitude of its results; their broad universality; their practical infallibility.
Charles Sanders PeirceThe so-called mysteries of quantum mechanics are in its philosophical interpretation, not in its mathematics.
Victor J. StengerMathematics, in its widest significance, is the development of all types of formal, necessary, deductive reasoning.
Alfred North WhiteheadIn any case, do you really think kids even want something that is relevant to their daily lives? You think something practical like compound interest is going to get them excited? People enjoy fantasy, and that is just what mathematics can provide - a relief from daily life, an anodyne to the practical workaday world.
Paul LockhartRemote from human passions, remote even from the pitiful facts of nature, the generations have gradually created an ordered cosmos [mathematics], where pure thought can dwell in its natural home.
Bertrand RussellMan is full of desires: he loves only those who can satisfy them all. "This man is a good mathematician," someone will say. But I have no concern for mathematics; he would take me for a proposition. "That one is a good soldier." He would take me for a besieged town. I need, that is to say, a decent man who can accommodate himself to all my desires in a general sort of way.
Blaise PascalEven when I was studying mathematics, physics, and computer science, it always seemed that the problem of consciousness was about the most interesting problem out there for science to come to grips with.
David ChalmersAs one reads mathematics, one needs to have an active mind, asking questions, forming mental connections between the current topic and other ideas from other contexts, so as to develop a sense of the structure, not just familiarity with a particular tour through the structure.
William ThurstonThere are two generic and invariable features that characterize utopias. One is the content: the authors of utopias paint what they consider to be ideal societies; translating this into the language of mathematics, we might say that utopias bear a + sign. The other feature, organically growing out of the content, is to be found in the form: a utopia is always static; it is always descriptive and has no, of almost no, plot dynamics.
Yevgeny Zamyatin