If the system exhibits a structure which can be represented by a mathematical equivalent, called a mathematical model, and if the objective can be also so quantified, then some computational method may be evolved for choosing the best schedule of actions among alternatives. Such use of mathematical models is termed mathematical programming.

**George Dantzig**

if-system-exhibits-structure-which-can-be-represented-by-mathematical-equivalent-called-mathematical-model-if-objective-can-be-also-quantified-george-dantzig

The 'seriousness' of a mathematical theorem lies, not in its practical consequences, which are usually negligible, but in the significance of the mathematical ideas which it connects. We may say, roughly, that a mathematical idea is 'significant' if it can be connected, in a natural and illuminating way, with a large complex of other mathematical ideas. Thus a serious mathematical theorem, a theorem which connects significant ideas, is likely to lead to important advances in mathematics itself and even in other sciences.

**G.H. Hardy**

the-seriousness-mathematical-theorem-lies-not-in-its-practical-consequences-which-are-usually-negligible-but-in-significance-mathematical-ideas-which-it-connects-we-may-say-rough

I imagine that whenever the mind perceives a mathematical idea, it makes contact with Plato's world of mathematical concepts... When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through the process of 'seeing'.

**Roger Penrose**

i-imagine-that-whenever-mind-perceives-mathematical-idea-it-makes-contact-with-platos-world-mathematical-concepts-when-mathematicians-communicate-roger-penrose

An old French mathematician said: A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street. This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us.

**David Hilbert**

an-old-french-mathematician-said-a-mathematical-theory-is-not-to-be-considered-complete-until-you-have-made-it-clear-that-you-can-explain-it-to-david-hilbert

Mathematical knowledge is unlike any other knowledge. While our perception of the physical world can always be distorted, our perception of mathematical truths can't be. They are objective, persistent, necessary truths. A mathematical formula or theorem means the same thing to anyone anywhere - no matter what gender, religion, or skin color; it will mean the same thing to anyone a thousand years from now. And what's also amazing is that we own all of them. No one can patent a mathematical formula, it's ours to share. There is nothing in this world that is so deep and exquisite and yet so readily available to all. That such a reservoir of knowledge really exists is nearly unbelievable. It's too precious to be given away to the 'initiated few.' It belongs to all of us.

**Edward Frenkel**

mathematical-knowledge-is-unlike-any-other-knowledge-while-our-perception-physical-world-can-always-be-distorted-our-perception-mathematical-truths-cant-be-they-are-objective-per

To do any important work in physics a very good mathematical ability and aptitude are required. Some work in applications can be done without this, but it will not be very inspired. If you must satisfy your "personal curiosity concerning the mysteries of nature" what will happen if these mysteries turn out to be laws expressed in mathematical terms (as they do turn out to be)? You cannot understand the physical world in any deep or satisfying way without using mathematical reasoning with facility.

**Richard P. Feynman**

to-do-any-important-work-in-physics-good-mathematical-ability-aptitude-are-required-some-work-in-applications-can-be-done-without-this-but-it-will-richard-p-feynman

The constructs of the mathematical mind are at the same time free and necessary. The individual mathematician feels free to define his notions and set up his axioms as he pleases. But the question is will he get his fellow mathematician interested in the constructs of his imagination. We cannot help the feeling that certain mathematical structures which have evolved through the combined efforts of the mathematical community bear the stamp of a necessity not affected by the accidents of their historical birth.

**Hermann Weyl**

the-constructs-mathematical-mind-are-at-same-time-free-necessary-the-individual-mathematician-feels-free-to-define-his-notions-set-up-his-axioms-hermann-weyl

my-reasons-are-same-as-for-any-mathematical-conjecture-1-it-is-legitimate-mathematical-possibility-2-i-do-not-know-jack-edmonds

One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?

**Stephen Hawking**

one-might-think-this-means-that-imaginary-numbers-are-just-mathematical-game-having-nothing-to-do-with-real-world-from-viewpoint-positivist-stephen-hawking

The mathematical is that evident aspect of things within which we are always already moving and according to which we experience them as things at all, and as such things. The mathematical is this fundamental position we take toward things by which we take up things as already given to us, and as they must and should be given. Therefore, the mathematical is the fundamental presupposition of the knowledge of things.

**Martin Heidegger**

the-mathematical-is-that-evident-aspect-things-within-which-we-are-always-already-moving-according-to-which-we-experience-them-as-things-at-all-as-martin-heidegger

Mathematical demonstrations being built upon the impregnable Foundations of Geometry and Arithmetick are the only truths that can sink into the Mind of Man, void of all Uncertainty; and all other Discourses participate more or less of Truth according as their Subjects are more or less capable of Mathematical Demonstration.

**Christopher Wren**

mathematical-demonstrations-being-built-upon-impregnable-foundations-geometry-arithmetick-are-only-truths-that-can-sink-into-mind-man-void-all-christopher-wren

What a mathematical proof actually does is show that certain conclusions, such as the irrationality of , follow from certain premises, such as the principle of mathematical induction. The validity of these premises is an entirely independent matter which can safely be left to philosophers.

**Timothy Gowers**

what-mathematical-proof-actually-does-is-show-that-certain-conclusions-such-as-irrationality-follow-from-certain-premises-such-as-principle-timothy-gowers

The problem with cinema nowadays is that it's a math problem. People can read a film mathematically; they know when this comes or that comes; in about 30 minutes, it's going to be over and have an ending. So film has become a mathematical solution. And that is boring, because art is not mathematical.

**Nicolas Winding Refn**

the-problem-with-cinema-nowadays-is-that-its-math-problem-people-can-read-film-mathematically-they-know-when-this-comes-that-comes-in-about-30-nicolas-winding-refn

The classes of problems which are respectively known and not known to have good algorithms are of great theoretical interest. [...] I conjecture that there is no good algorithm for the traveling salesman problem. My reasons are the same as for any mathematical conjecture: (1) It is a legitimate mathematical possibility, and (2) I do not know.

**Jack Edmonds**

the-classes-problems-which-are-respectively-known-not-known-to-have-good-algorithms-are-great-theoretical-interest-i-conjecture-that-there-is-no-jack-edmonds

[Before the time of Benjamin Peirce it never occurred to anyone that mathematical research] was one of the things for which a mathematical department existed. Today it is a commonplace in all the leading universities. Peirce stood alone-a mountain peak whose absolute height might be hard to measure, but which towered above all the surrounding country.

**Julian Coolidge**

before-time-benjamin-peirce-it-never-occurred-to-anyone-that-mathematical-research-was-one-things-for-which-mathematical-department-existed-today-julian-coolidge

{Before the time of Benjamin Peirce it never occurred to anyone that mathematical research} was one of the things for which a mathematical department existed. Today it is a commonplace in all the leading universities. Peirce stood alone-a mountain peak whose absolute height might be hard to measure, but which towered above all the surrounding country.

**Julian Coolidge**

before-time-benjamin-peirce-it-never-occurred-to-anyone-that-mathematical-research-was-one-things-for-which-mathematical-department-existed-today-it-is-commonplace-in-all-leading

The mathematical question is "Why?" It's always why. And the only way we know how to answer such questions is to come up, from scratch, with these narrative arguments that explain it. So what I want to do with this book is open up this world of mathematical reality, the creatures that we build there, the questions that we ask there, the ways in which we poke and prod (known as problems), and how we can possibly craft these elegant reason-poems.

**Paul Lockhart**

the-mathematical-question-is-why-its-always-why-and-only-way-we-know-how-to-answer-such-questions-is-to-come-up-from-scratch-with-these-narrative-paul-lockhart

This common and unfortunate fact of the lack of adequate presentation of basic ideas and motivations of almost any mathematical theory is probably due to the binary nature of mathematical perception. Either you have no inkling of an idea, or, once you have understood it, the very idea appears so embarrassingly obvious that you feel reluctant to say it aloud...

**Mikhail Leonidovich Gromov**

this-common-unfortunate-fact-lack-adequate-presentation-basic-ideas-motivations-almost-any-mathematical-theory-is-probably-due-to-binary-mikhail-leonidovich-gromov

We feel certain that the extraterrestrial message is a mathematical code of some kind. Probably a number code. Mathematics is the one language we might conceivably have in common with other forms of intelligent life in the universe. As I understand it, there is no reality more independent of our perception and more true to itself than mathematical reality.

**Don DeLillo**

we-feel-certain-that-extraterrestrial-message-is-mathematical-code-some-kind-probably-number-code-mathematics-is-one-language-we-might-don-delillo

Every attempt to employ mathematical methods in the study of chemical questions must be considered profoundly irrational and contrary to the spirit of chemistry.... if mathematical analysis should ever hold a prominent place in chemistry -- an aberration which is happily almost impossible -- it would occasion a rapid and widespread degeneration of that science.

**Auguste Comte**

every-attempt-to-employ-mathematical-methods-in-study-chemical-questions-must-be-considered-profoundly-irrational-contrary-to-spirit-chemistry-if-auguste-comte

The research worker, in his efforts to express the fundamental laws of Nature in mathematical form, should strive mainly for mathematical beauty. He should take simplicity into consideration in a subordinate way to beauty ... It often happens that the requirements of simplicity and beauty are the same, but where they clash, the latter must take precedence.

**Paul Dirac**

the-research-worker-in-his-efforts-to-express-fundamental-laws-nature-in-mathematical-form-should-strive-mainly-for-mathematical-beauty-he-should-paul-dirac

String theory has had a long and wonderful history. It originated as a technique to try to understand the strong force. It was a calculational mechanism, a way of approaching a mathematical problem that was too difficult, and it was a promising way, but it was only a technique. It was a mathematical technique rather than a theory in itself.

**Sheldon Lee Glashow**

string-theory-has-had-long-wonderful-history-it-originated-as-technique-to-try-to-understand-strong-force-it-was-calculational-mechanism-way-approaching-mathematical-problem-that

Not only in geometry, but to a still more astonishing degree in physics, has it become more and more evident that as soon as we have succeeded in unraveling fully the natural laws which govern reality, we find them to be expressible by mathematical relations of surprising simplicity and architectonic perfection. It seems to me to be one of the chief objects of mathematical instruction to develop the faculty of perceiving this simplicity and harmony.

**Hermann Weyl**

not-only-in-geometry-but-to-still-more-astonishing-degree-in-physics-has-it-become-more-more-evident-that-as-soon-as-we-have-succeeded-in-unraveling-hermann-weyl

Nature seems to take advantage of the simple mathematical representations of the symmetry laws. When one pauses to consider the elegance and the beautiful perfection of the mathematical reasoning involved and contrast it with the complex and far-reaching physical consequences, a deep sense of respect for the power of the symmetry laws never fails to develop.

**Chen Ning Yang**

nature-seems-to-take-advantage-simple-mathematical-representations-symmetry-laws-when-one-pauses-to-consider-elegance-beautiful-perfection-mathematical-reasoning-involved-contras

The Reader may here observe the Force of Numbers, which can be successfully applied, even to those things, which one would imagine are subject to no Rules. There are very few things which we know, which are not capable of being reduc'd to a Mathematical Reasoning, and when they cannot, it's a sign our Knowledge of them is very small and confus'd; and where a mathematical reasoning can be had, it's as great folly to make use of any other, as to grope for a thing in the dark when you have a Candle standing by you.

**John Arbuthnot**

the-reader-may-here-observe-force-numbers-which-can-be-successfully-applied-even-to-those-things-which-one-would-imagine-are-subject-to-no-rules-john-arbuthnot

Those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a great deal about them; if they do not expressly mention them, but prove attributes which are their results or definitions, it is not true that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree.

**Aristotle**

those-who-assert-that-mathematical-sciences-say-nothing-beautiful-good-are-in-error-for-these-sciences-say-prove-great-deal-about-them-if-they-aristotle

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**

conventions-generality-mathematical-elegance-may-be-just-as-much-barriers-to-attainment-diffusion-knowledge-as-may-contentment-with-particularity-kenneth-e-boulding

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 ofour science and at the secrets of its development during future centuries? What particular goals will there be towardwhich the leading mathematical spirits of coming generations will strive? What new methods and new facts in thewide and rich field of mathematical thought will the new centuries disclose?

**David Hilbert**

who-us-would-not-be-glad-to-lift-veil-behind-which-future-lies-hidden-to-cast-glance-at-next-advances-ofour-science-at-secrets-its-development-david-hilbert

We shall see that the mathematical treatment of the subject [of electricity] has been greatly developed by writers who express themselves in terms of the 'Two Fluids' theory. Their results, however, have been deduced entirely from data which can be proved by experiment, and which must therefore be true, whether we adopt the theory of two fluids or not. The experimental verification of the mathematical results therefore is no evidence for or against the peculiar doctrines of this theory.

**James Clerk Maxwell**

we-shall-see-that-mathematical-treatment-subject-electricity-has-been-greatly-developed-by-writers-who-express-themselves-in-terms-two-fluids-james-clerk-maxwell

It is almost as if ideas set in mathematical form melt and become liquid and just as rivers can, from the most humble beginnings, flow for thousands of miles, through the most varied topography bringing nourishment and life with them wherever they go, so too can ideas cast in mathematical form flow far from their original sources, along well-defined paths, electrifying and dramatically affecting much of what they touch. pp. xii - xiii.

**G. Arnell Williams**

it-is-almost-as-if-ideas-set-in-mathematical-form-melt-become-liquid-just-as-rivers-can-from-most-humble-beginnings-flow-for-thousands-miles-through-most-varied-topography-bringi

Kant, discussing the various modes of perception by which the human mind apprehends nature, concluded that it is specially prone to see nature through mathematical spectacles. Just as a man wearing blue spectacles would see only a blue world, so Kant thought that, with our mental bias, we tend to see only a mathematical world.

**James Jeans**

kant-discussing-various-modes-perception-by-which-human-mind-apprehends-nature-concluded-that-it-is-specially-prone-to-see-nature-through-james-jeans

Here is a quilted book about mathematical practice, each patch wonderfully prepared. Part invitation to number theory, part autobiography, part sociology of mathematical training, Mathematics without Apologies brings us into contemporary mathematics as a living, active inquiry by real people. Anyone wanting a varied, cultured, and penetrating view of today's mathematics could find no better place to engage.

**Peter Galison**

here-is-quilted-book-about-mathematical-practice-each-patch-wonderfully-prepared-part-invitation-to-number-theory-part-autobiography-part-sociology-peter-galison

But as the work proceeded I was continually reminded of the fable about the elephant and the tortoise. Having constructed an elephant upon which the mathematical world could rest, I found the elephant tottering, and proceeded to construct a tortoise to keep the elephant from falling. But the tortoise was not more secure than the elephant, and after some twenty years of very arduous toil, I came to the conclusion that there was nothing more that I could do in the way of making mathematical knowledge indubitable.

**Bertrand Russell**

but-as-work-proceeded-i-was-continually-reminded-fable-about-elephant-tortoise-having-constructed-elephant-upon-which-mathematical-world-could-bertrand-russell

There is no such thing as beauty, especially in the human face... what we call the physiognomy. It's all a mathematical and imagined alignment of features. Like, if the nose doesn't stick out too much, the sides are in fashion, if the earlobes aren't too large, if the hair is long... It's kind of a mirage of generalization. People think of certain faces as beautiful, but, truly, in the final measure, they are not. It's a mathematical equation of zero. 'True beauty' comes, of course, of character. Not through how the eyebrows are shaped. So many women that I'm told are beautiful... hell, it's like looking into a soup bowl.

**Charles Bukowski**

there-is-no-such-thing-as-beauty-especially-in-human-face-what-we-call-physiognomy-its-all-mathematical-imagined-alignment-features-like-if-nose-doesnt-stick-out-too-much-sides-a

We humans have a wide range of abilities that help us perceive and analyze mathematical content. We perceive abstract notions not just through seeing but also by hearing, by feeling, by our sense of body motion and position. Our geometric and spatial skills are highly trainable, just as in other high-performance activities. In mathematics we can use the modules of our minds in flexible ways - even metaphorically. A whole-mind approach to mathematical thinking is vastly more effective than the common approach that manipulates only symbols.

**William Thurston**

we-humans-have-wide-range-abilities-that-help-us-perceive-analyze-mathematical-content-we-perceive-abstract-notions-not-just-through-seeing-but-william-thurston