Top 1200 Mathematical Proof Quotes & Sayings - Page 2

It is not conclusive proof of a doctrine's correctness that its adversaries use the police, the hangman, and violent mobs to fight it. But it is a proof of the fact that those taking recourse to violent oppression are in their subconscious convinced of the untenability of their own doctrines.

When Pococke inquired of Grotius, where the proof was of that story of the pigeon, trained to pick peas from Mahomet's (Muhammad's) ear, and pass for an angel dictating to him? Grotius answered that there was no proof!

The Phoenicians are entitled to be commemorated in history by the side of the Hellenic and Latin nations; but their case affords a fresh proof, and perhaps the strongest proof of all, that the development of national energies in antiquity was of a one-sided character.

According to PhD astrophysicists and qualified American science professors, the odds of the more than 2,500 prophecies found in the Bible being fulfilled by chance are 1 with 2,000 zeros after it. According to the mathematical science of probability, if a number has more than 50 zeros after it, the odds of that happening by chance is virtually impossible. This is irrefutable proof that the Bible is inspired by God!

Not thinking critically, I assumed that the successful prayers were proof that God answers prayer while the failures were proof that there was something wrong with me.

There was, I think, a feeling that the best science was that done in the simplest way. In experimental work, as in mathematics, there was "style" and a result obtained with simple equipment was more elegant than one obtained with complicated apparatus, just as a mathematical proof derived neatly was better than one involving laborious calculations. Rutherford's first disintegration experiment, and Chadwick's discovery of the neutron had a "style" that is different from that of experiments made with giant accelerators.

For generations, field guides to plants and animals have sharpened the pleasure of seeing by opening our minds to understanding. Now John Adam has filled a gap in that venerable genre with his painstaking but simple mathematical descriptions of familiar, mundane physical phenomena. This is nothing less than a mathematical field guide to inanimate nature.

It can be shown that a mathematical web of some kind can be woven about any universe containing several objects. The fact that our universe lends itself to mathematical treatment is not a fact of any great philosophical significance.

The proof of the depth and embodiment of your realization will be seen in your love relationship. That's where the proof is in the pudding. If it all collapses in your relationship, you have some work to do. And people do have a lot of difficulties in their relationships.

All writing seems to me worse in the state of proof than in any other form. In manuscript one's own wisdom is rather remarkable to one, but in proof it has the effect of one's private furniture repeated in the shop windows. And then there is the sense that the worst errors will go to press unnoticed!

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.

One can never really give a proof of the reality of anything; reality is not something open to proof, it is something established. It is established just because proof is not enough. It is this characteristic of language, at once indispensable and inadequate, which shows the reality of the external world. Most people hardly ever realize this, because it is rare that the very same man thinks and puts his thought into action.

The proof given by Wright, that non-adaptive differentiation will occur in small populations owing to "drift," or the chance fixation of some new mutation or recombination, is one of the most important results of mathematical analysis applied to the facts of neo-mendelism. It gives accident as well as adaptation a place in evolution, and at one stroke explains many facts which puzzled earlier selectionists, notably the much greater degree of divergence shown by island than mainland forms, by forms in isolated lakes than in continuous river-systems.

I felt like I had proof that not all days are the same length, not all time has the same weight. Proof that there are worlds and worlds and worlds on top of worlds, if you want them to be there.

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.

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.

The Atheist waits for proof of God. Till that proof comes he remains, as his name implies, without God. His mind is open to every new truth, after it has passed the warder Reason at the gate.

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.

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.

The proof of battle is action, proof of words, debate. No time for speeches now, it's time to fight.

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.

I read in the proof sheets of Hardy on Ramanujan: "As someone said, each of the positive integers was one of his personal friends." My reaction was, "I wonder who said that; I wish I had." In the next proof-sheets I read (what now stands), "It was Littlewood who said..."

The result of the mathematician's creative work is demonstrative reasoning, a proof; but the proof is discovered by plausible reasoning, by guessing.

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.

The only possible proof of the existence of water, the most convincing and the most intimately true proof, is thirst.

Look at us! Are we not proof that there is no good, no evil, no truth, no reason? Are we not proof that the universe is a drooling idiot with no fashion sense - Mr Nobody on the fundamental philosophy of the Brotherhood of DADA

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.

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.

As it is natural to believe many things without proof, so, despite all proof, is it natural to disbelieve others.

Act on your inner guidance, and give up your need for 'proof' that your inner guidance is authentic. The more you ask for proof, the less likely you are to receive any.

It is not proof that I sought. I, of all men, know that proof is but a fallacy invented by man to justify to himself and his fellows his own crass lust and folly.

For those who believe, no proof is necessary. For those who disbelieve, no amount of proof is sufficient.

The rationalist imagines an imbecile-free society; the empiricist and imbecile-proof one, or even better, a rationalist-proof one.

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.

Not to be a socialist at twenty is proof of want of heart; to be one at thirty is proof of want of head.

I've had people say to me, 'Look at the sky, the fields, the ocean, the beautiful sunset. Isn't that proof positive of God?' Following that line of thought, look at the magnificent rainbows after a big rainstorm. Isn't that proof positive that God is gay?

The sciences do not try to explain, they hardly even try to interpret, they mainly make models. By a model is meant a mathematical construct which, with the addition of certain verbal interpretations, describes observed phenomena. The justification of such a mathematical construct is solely and precisely that it is expected to work-that is, correctly to describe phenomena from a reasonably wide area.

We have here a proof that there is in the atom a fundamental quantity, which increases by regular steps as one passes from one element to the next. This quantity can only be the charge on the central positive nucleus, of the existence of which we already have definite proof.

Wisdom is not finally tested by the schools, Wisdom cannot be pass'd from one having it to another not having it, Wisdom is of the soul, is not susceptible of proof, is its own proof.

To execute a man we don't need proof of his guilt. We only need proof that it's necessary to execute him. It's that simple.

It seems perfectly clear that Economy, if it is to be a science at all, must be a mathematical science. There exists much prejudice against attempts to introduce the methods and language of mathematics into any branch of the moral sciences. Most persons appear to hold that the physical sciences form the proper sphere of mathematical method, and that the moral sciences demand some other method-I know not what.

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.

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.

In mathematical analysis we call x the undetermined part of line a: the rest we don't call y, as we do in common life, but a-x. Hence mathematical language has great advantages over the common language.

I would not read the proof of one of my books for any fair & reasonable sum whatever, if I could get out of it. The proof-reading on the P & Pauper cost me the last rags of my religion.

The proof that the little prince existed is that he was charming, that he laughed, and that he was looking for a sheep. If anybody wants a sheep, that is a proof that he exists.

... each of the 24 modes in the Ramanujan function corresponds to a physical vibration of a string. Whenever the string executes its complex motions in space-time by splitting and recombining, a large number of highly sophisticated mathematical identities must be satisfied. These are precisely the mathematical identities discovered by Ramanujan.

But what of faith? What of fidelity and loyalty? Complete trust? Faith is not granted by tangible proof. It comes from the heart and the soul. If a person needs proof of god's existence, then the very notion of spirituality is diminished into sensuality and we have reduced what is holy into what is logical.-Drizzt Do'urden

I have no proof for any proof.

The photograph is kind of a proof - a proof that I actually met these people, that they actually have lives, and that they're worth considering.

The perfectly natural thing to do with an unreadable book is to give it away; and the publication, for more than a quarter of a century, of volumes which fulfilled this one purpose and no other is a pleasant proof, if proof were needed, of the business principles which underlay the enlightened activity of publishers.

From a mathematical standpoint it is possible to have infinite space. In a mathematical sense space is manifoldness, or combinations of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, the 10-dimension system.

[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.

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.

My proof convinces the ignorant, and the wise man's proof convinces me. But he whose reasoning falls between wisdom and ignorance, I neither can convince him, nor can he convince me.

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.

There are such beings as vampires, some of us have evidence that they exist. Even had we not the proof of our own unhappy experience, the teachings and the records of the past give proof enough for sane peoples.

I don't believe anything till I have seen the proof. For anything without proof, I think we should believe the theory that gives us peace. It doesn't matter whether the theory is true or not.

I don't believe Fermat had a proof. I think he fooled himself into thinking he had a proof.

Love can never be a sin. It can be only a blessing. Even if you're not loved in return -- though I can't imagine that -- to love is a proof of life -- indeed, it's the only proof, for once you can't love another human being, you're not alive.