Mathematical Quotes - Page 35

• When I look out the window at my backyard, I can't think of anything interesting to ask. I mean, it's green, it's growing-but nothing occurs to me that any concentrated effort of thought could possibly enlighten. Whereas in economic, statistical, or mathematical kinds of things, I can think lots of questions.

  Lawrence Summers

  Mean, Thinking, Interesting

  "Inside the Crisis" by Ryan Lizza, www.newyorker.com. October 12, 2009.

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• Nonmathematical people sometimes ask me, “You know math, huh? Tell me something I’ve always wondered, What is infinity divided by infinity?” I can only reply, “The words you just uttered do not make sense. That was not a mathematical sentence. You spoke of ‘infinity’ as if it were a number. It’s not. You may as well ask, 'What is truth divided by beauty?’ I have no clue. I only know how to divide numbers. ‘Infinity,’ ‘truth,’ ‘beauty’—those are not numbers.

  John Derbyshire

  Beauty, Math, Numbers

  John Derbyshire (2003). “Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics”, p.21, Joseph Henry Press

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• It is a mathematical fact that if a line be not perfectly directed towards a point, it will actually go further away from it as it comes nearer to it.

  Gilbert K. Chesterton

  Lines, Facts, Not Perfect

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• The physicist may be satisfied when he has the mathematical scheme and knows how to use for the interpretation of the experiments. But he has to speak about his results also to non-physicists who will not be satisfied unless some explanation is given in plain language. Even for the physicist the description in plain language will be the criterion of the degree of understanding that has been reached.

  Werner Heisenberg

  Philosophy, Plain Language, Understanding

  "Physics and Philosophy: The Revolution in Modern Science". Book by Werner Heisenberg, 1958.

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• By and large it is uniformly true in mathematics that there is a time lapse between a mathematical discovery and the moment when it is useful; and that this lapse of time can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful.

  John von Neumann

  Years, Discovery, Desire

  "The Role of Mathematics in the Sciences and in Society". John von Neumann's address to Princeton alumni (1954) as quoted in "John von Neumann: Collected Works" edited by A. H. Taub, 1963.

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• Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of
  our science and at the secrets of its development during future centuries? What particular goals will there be toward
  which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the
  wide and rich field of mathematical thought will the new centuries disclose?

  David Hilbert

  Lying, Goal, Secret

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• Music, also, the architect ought to understand so that he may have knowledge of the canonical and mathematical theory, and besides be able to tune ballistae , catapultae, and scorpiones to the proper key. For to the right and left in the beams are the holes in the frames through which the strings of twisted sinew are stretched by means of windlasses and bars, and these strings must not be clamped and made fast until they give the same correct note to the ear of the skilled workman.

  Marcus Vitruvius Pollio

  Mean, Keys, Giving

  "De architectura". Book by Vitruvius. Book I, Chapter I, Section 8,

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• Remembering that Eratosthenes of Cyrene, employing mathematical theories and geometrical methods, discovered from the course of the sun, the shadows cast by an equinoctial gnomon, and the inclination of the heaven that the circumference of the earth is two hundred and fifty-two thousand stadia, that is, thirty-one million five hundred thousand paces.

  Marcus Vitruvius Pollio

  Two, Heaven, Shadow

  "De architectura". Book by Vitruvius. Book I, Chapter VI, Section 9,

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• In accordance with the foregoing investigations on mathematical principles, let bronze vessels be made, proportionate to the size of the theatre, and let them be so fashioned that, when touched, they may produce with one another the notes of the fourth, the fifth, and so on up the double octave.

  Marcus Vitruvius Pollio

  Theatre, Principles, May

  "De architectura". Book by Vitruvius. Book V, Chapter V, Section 1,

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• I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in a certain order, or that that order has been, to a great extent, necessary -- being determined, either by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for their evolution had arrived.

  George Boole

  Order, Ideas, Doubt

  George Boole (1872). “A Treatise on Differential Equations”, p.5

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• The mathematical fraternity is a little like a self-perpetuating priesthood. The mathematicians of today teach the mathematicians of tomorrow and, in effect, decide whom to admit to the priesthood.

  Paul Halmos

  Self, Littles, Today

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• Perspective is an Art Mathematical which demonstrates the manner and properties of all radiations direct, broken and reflected.

  John Dee

  Art, Broken, Perspective

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• The art of Navigation demonstrates how, by the shortest good way, by the aptest direction, and in the shortest time, a sufficient ship, between any two places (in passage navigable) assigned, may be conducted; and in all storms and natural disturbances chancing, how to use the best possible means, whereby to recover the place first assigned.
  Mathematical Preface

  John Dee

  Art, Mean, Two

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• Mathematical Analysis is as extensive as nature herself.

  Joseph Fourier

  Analysis, Mathematical Analysis

  "Fourier Analysis" by T. W. Korner quoted in "Mathematical Analysis" by Alladi Sitaram, Alladi Sitaram, (p. 130), 2001.

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• I have presented principles of philosophy that are not, however, philosophical but strictly mathematical-that is, those on which the study of philosophy can be based. These principles are the laws and conditions of motions and of forces, which especially relate to philosophy.

  Isaac Newton

  Philosophy, Philosophical, Law

  Isaac Newton, I. Bernard Cohen, Anne Whitman (1999). “The Principia: Mathematical Principles of Natural Philosophy”, p.793, Univ of California Press

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• Conventions of generality and mathematical elegance may be just as much barriers to the attainment and diffusion of knowledge as may contentment with particularity and literary vagueness... It may well be that the slovenly and literary borderland between economics and sociology will be the most fruitful building ground during the years to come and that mathematical economics will remain too flawless in its perfection to be very fruitful.

  Kenneth E. Boulding

  Years, Perfection, Contentment

  "Samuelson's Foundations: The Role of Mathematics in Economics". Journal of Political Economy, Volume 56, June 1948.

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• It is clear that the building of models is not a purely mechanical process but requires skill of a high order - not merely mathematical skill but a sensitivity to the relative importance of different factors and a critical, almost an artistic, faculty in the selection of behaviour equations which are reasonable, tentative hypotheses in explaining the behaviour of actual economies.

  Kenneth E. Boulding

  Order, Skills, Different

  "The Skills of the Economist". Book by Kenneth Ewart Boulding, pp. 16-17, 1958.

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• One can prove or refute anything at all with words. Soon people will perfect language technology to such an extent that they'll be proving with mathematical precision that twice two is seven.

  Anton Chekhov

  Technology, Two, People

  "Lights". Short story by Anton Chekhov, 1888.

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• She saw every personal religion as a pair of intersecting circles. . . . Probably perfection is reached when the area of the two outer crescents, added together, is exactly equal to that of the leaf-shaped piece in the middle. On paper there must be some neat mathematical formula for arriving at this; in life, none.

  Jan Struther

  Women, Circles, Two

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• The material which a scientist actually has at his disposal, his laws, his experimental results, his mathematical techniques, his epistemological prejudices, his attitude towards the absurd consequences of the theories which he accepts, is indeterminate in many ways, ambiguous, and never fully separated from the historical background . This material is always contaminated by principles which he does not know and which, if known, would be extremely hard to test.

  Paul Feyerabend

  Attitude, Law, Historical

  Paul Feyerabend (1993). “Against Method”, p.51, Verso

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• Mathematical Reasoning is not only exact; it has its own criteria of reality.

  Paul Feyerabend

  Reality, Criteria, Reasoning

  "Science in a Free Society". Book by Paul Feyerabend, 1978.

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• The attempt to apply rational arithmetic to a problem in geometry resulted in the first crisis in the history of mathematics. The two relatively simple problems -- the determination of the diagonal of a square and that of the circumference of a circle -- revealed the existence of new mathematical beings for which no place could be found within the rational domain.

  David van Dantzig

  Determination, Simple, Squares

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• [Lord Brougham's writings on the bee's cell contain] as striking examples of bad reasoning as are often to be met with in writings related to mathematical subjects.

  James Whitbread Lee Glaisher

  Writing, Cells, Example

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• Can the existence of a mathematical entity be proved without defining it ?

  Jacques Hadamard

  Defining, Entity, Existence

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• ... there are those who believe that mathematics can sustain itself and grow without any further contact with anything outside itself, and those who believe that nature is still and always will be one of the main (if not the main) sources of mathematical inspiration. The first group is identified as "pure mathematicians" (though "purist" would be more adequate) while the second is, with equal inadequacy, referred to as "applied".

  Mark Kac

  Believe, Inspiration, Groups

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