Discover the most popular and inspiring quotes and sayings on the topic of Theorems. Share them with your friends on social media platforms like Facebook, Twitter, or your personal blogs, and let the world be inspired by their powerful messages. Here are the Top 100 Theorems Quotes And Sayings by 87 Authors including Martin Gardner,David Mumford,Isaac Newton,Albert Einstein,Bertrand Russell for you to enjoy and share.
In many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance.
I think that mathematics can benefit by acknowledging that the creation of good models is just as important as proving deep theorems.
In experimental philosophy, propositions gathered from phenomena by induction should be considered either exactly or very nearly true notwithstanding any contrary hypotheses, until yet other phenomena make such propositions either more exact or liable to exceptions.
A theory is more impressive the greater the simplicity of its premises, the
more different are the kinds of things it relates, and the more extended its range of applicability.
One of the chief triumphs of modern mathematics consists in having discovered what mathematics really is.
But in the present century, thanks in good part to the influence of Hilbert, we have come to see that the unproved postulates with which we start are purely arbitrary. They must be consistent, they had better lead to something interesting.
The axiomatic method is very powerful
For hundreds of pages the closely-reasoned arguments unroll, axioms and theorems interlock. And what remains with us in the end? A general sense that the world can be expressed in closely-reasoned arguments, in interlocking axioms and theorems.
The whole of mathematics consists in the organization of a series of aids to the imagination in the process of reasoning.
Logic issues in tautologies, mathematics in identities, philosophy in definitions; all trivial, but all part of the vital work of clarifying and organising our thought.
Mathematical discoveries, like springtime violets in the woods, have their season which no man can hasten or retard.
There is a certain way of searching for the truth in mathematics that Plato is said first to have discovered. Theon called this analysis.
It gives me the same pleasure when someone else proves a good theorem as when I do it myself.
Mathematics may be the only exception in the sciences that leaves no room for skepicism. But, if mathematical results are exact as no empirical law can ever be, philosophers have discovered that they are not absolutely novel - instead, they are tautological.
Mathematics takes us into the region of absolute necessity, to which not only the actual word, but every possible word, must conform.
In completing one discovery we never fail to get an imperfect knowledge of others of which we could have no idea before, so that we cannot solve one doubt without creating several new ones.
Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. Ambiguity of language is philosophy's main source of problems. That is why it is of the utmost importance to examine attentively the very words we use.
A reply to Olbers' attempt in 1816 to entice him to work on Fermat's Theorem. I confess that Fermat's Theorem as an isolated proposition has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of. []
The most distinctive characteristic which differentiates mathematics from the various branches of empirical science, and which accounts for its fame as the queen of the sciences, is no doubt the peculiar certainty and necessity of its results.
The higher arithmetic presents us with an inexhaustible store of interesting truths - of truths, too, which are not isolated, but stand in a close internal connection, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties.
The greatest mathematics has the simplicity and inevitableness of supreme poetry and music, standing on the borderland of all that is wonderful in Science, and all that is beautiful in Art.
One may characterize physics as the doctrine of the repeatable, be it a succession in time or the co-existence in space. The validity of physical theorems is founded on this repeatability.
Mathematics is not a careful march down a well cleared highway, but a journey into a strange wilderness, where the explorers often get lost.
In [great mathematics] there is a very high degree of unexpectedness, combined with inevitability and economy.
Mathematicians come to the solution of a problem by the simple arrangement of the data, and reducing the reasoning to such simple operations, to judgments so brief, that they never lose sight of the evidence that serves as their guide.
The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it.
But my relief that David Auburn's Proof is less about its ballyhooed higher mathematics than the fragility of life and love was matched by my delight in his fine and tender play. ( ... ) Proof surprises us with its aliveness and intelligent modesty, and we have not met these characters before.
As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention.
It is the nature of an hypothesis, when once a man has conceived it, that it assimulates every thing to itself as proper nourishment; and, from the first moment of your begetting it, it generally grows the stronger by every thing you see, hear, read, or understand.
The fact that such objective and enduring knowledge exists (and moreover, belongs to all of us) is nothing short of a miracle. It suggests that mathematical concepts exist in a world separate from the physical and mental worlds
I have the vagary of taking a lively interest in mathematical subjects only where I may anticipate ingenious association of ideas and results recommending themselves by elegance or generality.
What is best in mathematics deserves not merely to be learnt as a task, but to assimilated as a part of daily thought, and brought again and again before the mind with ever-renewed encouragement.
Without the concepts, methods and results found and developed by previous generations right down to Greek antiquity one cannot understand either the aims or achievements of mathematics in the last 50 years. [Said in 1950]
The science of mathematics presents the most brilliant example of how pure reason may successfully enlarge its domain without the aid of experience
[I was advised] to read Jordan's 'Cours d'analyse'; and I shall never forget the astonishment with which I read that remarkable work, the first inspiration for so many mathematicians of my generation, and learnt for the first time as I read it what mathematics really meant.
In disquisitions of every kind there are certain primary truths, or first principles, upon which all subsequent reasoning must depend.
As time goes on, it becomes increasingly evident that the rules which the mathematician finds interesting are the same as those which Nature has chosen.
Mathematics, which most of us see as the most factual of all sciences, constitutes the most colossal metaphor imaginable, and must be judged, aesthetically as well as intellectually in terms of the success of this metaphor.
[All phenomena] are equally susceptible of being calculated, and all that is necessary, to reduce the whole of nature to laws similar to those which Newton discovered with the aid of the calculus, is to have a sufficient number of observations and a mathematics that is complex enough.
Mathematics is a vast adventure; its history reflects some of the noblest thoughts of countless generations.
Thus not only the mental and the material, but the theoretical and the practical in the mathematical world, are brought into more intimate and effective connection with each other.
One cannot really argue with a mathematical theorem.
[In mathematics] There are two kinds of mistakes. There are fatal mistakes that destroy a theory, but there are also contingent ones, which are useful in testing the stability of a theory.
Mathematics, however, is, as it were, its own explanation; this, although it may seem hard to accept, is nevertheless true, for the recognition that a fact is so is the cause upon which we base the proof.
We can ... treat only the geometrical aspects of mathematics and shall be satisfied in having shown that there is no problem of the truth of geometrical axioms and that no special geometrical visualization exists in mathematics.
Theorem: Consider the set of all sets that have never been considered. Hey! They're all gone!! Oh, well, never mind...
Many of the proofs in mathematics are very long and intricate. Others, though not long, are very ingeniously constructed.
A modern mathematical proof is not very different from a modern machine, or a modern test setup: the simple fundamental principles are hidden and almost invisible under a mass of technical details.
Paul Erdos has a theory that God has a book containing all the theorems of mathematics with their absolutely most beautiful proofs, and when he wants to express particular appreciation of a proof he exclaims, "This is from the book!"
Every hypothesis is a construction, and because of this it is an authentic theory. In so far as they merit that exigent name, ideas are never a mere reception of presumed realities, but they are constructions of possibilities; therefore they are pure bits of imagination, or fine ideas of our own ...
All mathematicians share ... a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics.
One cannot escape the feeling that these mathematical formulas have an independent existence and an intelligence of their own, that they are wiser than we are, wiser even than their discoverers ...
When you come up with a theory, you fall in love with the beauty the simplicity and elegance of it. But then you have to get a sheet of paper and pencil and crack out all the details. Hundreds and hundreds of pages. Because you have to prove it.
The rules of scientific investigation always require us, when we enter the domains of conjecture, to adopt that hypothesis by which the greatest number of known facts and phenomena may be reconciled.
Those who have racked their brains to discover new proofs have perhaps been induced to do so by a compulsion they could not quite explain to themselves. Instead of giving us their new proofs they should have explained to us the motivation that constrained them to search for them.
Theory is continually the precursor of truth; we must pass through the twilight and its shade, to arrive at the full and perfect day.
The perfection of mathematical beauty is such ... that whatsoever is most beautiful and regular is also found to be most useful and excellent.
Not that the propositions of geometry are only approximately true, but that they remain absolutely true in regard to that Euclidean space which has been so long regarded as being the physical space of our experience.
Concepts are vindicated by the constant accrual of data and independent verification of data. No prize, not even a Nobel Prize, can make something true that is not true.
It is however pretty evident, on general principles, that in devising for mathematical truths a new form in which to record and throw themselves out for actual use, views are likely to be induced, which should again react on the more theoretical phase of the subject.
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.
If we wish to foresee the future of mathematics, our proper course is to study the history and present condition of the science.
The faith of scientists in the power and truth of mathematics is so implicit that their work has gradually become less and less observation, and more and more calculation ... But the facts which are accepted by virtue of these tests are not actually observed at all.
The better to understand the nature, manner, and extent of our knowledge, one thing is carefully to be observed concerning the ideas we have; and that is, that some of them are simple and some complex.
Wherever Mathematics is mixed up with anything, which is outside its field, you will find attempts to demonstrate these merely conventional propositions a priori, and it will be your task to find out the false deduction in each case.
Economists often like startling theorems, results which seem to run counter to conventional wisdom.
A mathematician," he liked to say, "is a machine for turning coffee into theorems.
Mathematics was born and nurtured in a cultural environment. Without the perspective which the cultural background affords, a proper appreciation of the content and state of present-day mathematics is hardly possible.
The most vitally characteristic fact about mathematics is, in my opinion, its quite peculiar relationship to the natural sciences, or more generally, to any science which interprets experience on a higher than purely descriptive level.
The world of conceptualized ideas is quite wonderful, even when it's - like Aristotle's Physics - an outmoded book. The physics is not true. But the reasoning is dazzling.
The mathematics are distinguished by a particular privilege, that is, in the course of ages, they may always advance and can never recede.
As a teacher, Tengo pounded into his students' heads how voraciously mathematics demanded logic. Here things that could not be proven had no meaning, but once you had succeeded in proving something, the world's riddles settled into the palm of your hand like a tender oyster.
The moving power of mathematical invention is not reasoning but imagination.
Frege has the merit of ... finding a third assertion by recognising the world of logic which is neither mental nor physical.
The discoveries of yesterday are the truisms of tomorrow, because we can add to our knowledge but cannot subtract from it.
What promotes math progress even more than new ideas are new technical tools and habits of thought that encapsulate existing ideas, so that insights of one generation become the instincts of the next.
The more reasonable a student was in mathematics, the more unreasonable she was in the affairs of real life, concerning which fewtrustworthy postulates have yet been ascertained.
This is not mathematics, it is theology.
(On being exposed to Hilbert's work in invariant theory.)
Mathematicians are like lovers. Grant a mathematician the least principle, and he will draw from it a consequence which you must also grant him, and from this consequence another.
What philosophy worthy of the name has truly been able to avoid the link between poem and theorem?
Mathematics is entirely free in its development, and its concepts are only linked by the necessity of being consistent, and are co-ordinated with concepts introduced previously by means of precise definitions.
The art of discovering the causes of phenomena, or true hypothesis, is like the art of decyphering, in which an ingenious conjecture greatly shortens the road.
Mathematics takes us still further from what is human into the region of absolute necessity, to which not only the actual world, but ever possible world, must conform.
Experience has shown repeatedly that a mathematical theory with a rich internal structure generally turns out to have significant implications for the understanding of the real world, often in ways no one could have envisioned before the theory was developed.
The history of mathematics, lacking the guidance of philosophy, [is] blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics, is empty.
A proof tells us where to concentrate our doubts.
Once a day, especially in the early years of life and study, call yourselves to an account what new ideas, what new proposition or truth you have gained, what further confirmation of known truths, and what advances you have made in any part of knowledge.
Special emphasis should be laid on this intimate interrelation of general statements about empirical fact with the logical elements and structure of theoretical systems.
The heart of mathematics consists of concrete examples and concrete problems. Big general theories are usually afterthoughts based on small but profound insights; the insights themselves come from concrete special cases.
If I found any new truths in the sciences, I can say that they follow from, or depend on, five or six principal problems which I succeeded in solving and which I regard as so many battles where the fortunes of war were on my side.
Very often in mathematics the crucial problem is to recognize and discover what are the relevant concepts; once this is accomplished the job may be more than half done.
If a concept or principle finds its place in an explanatory theory, it cannot be excluded on methodological grounds.
Great mathematics is achieved by solving difficult problems not by fabricating elaborate theories in search of a problem.
Mathematics and logic have been proved to be one; a fact from which it seems to follow that mathematics may successfully deal with non-quantitative problems in a much broader sense than was suspected to be possible.
Some of the greatest advances in mathematics have been due to the invention of symbols, which it afterwards became necessary to explain; from the minus sign proceeded the whole theory of negative quantities.
It has never yet been supposed, that all the facts of nature, and all the means of acquiring precision in the computation and analysis of those facts, and all the connections of objects with each other, and all the possible combinations of ideas, can be exhausted by the human mind.
Mathematics, in the development of its ideas, has only to take account of the immanent reality of its concepts and has absolutely no obligation to examine their transient reality.
A math lecture without a proof is like a movie without a love scene. This talk has two proofs.
Analytic It is clear that the definition of "logic" or "mathematics" must be sought by trying to give a new definition of the old notion of "analytic" propositions.
The result of the mathematician's creative work is demonstrative reasoning, a proof, but the proof is discovered by plausible reasoning, by GUESSING.