Top 115 Theorem Quotes & Sayings - Page 2

About Thomas Hobbes: He was 40 years old before he looked on geometry; which happened accidentally. Being in a gentleman's library, Euclid's Elements lay open, and "twas the 47 El. libri I" [Pythagoras' Theorem]. He read the proposition "By God", sayd he, "this is impossible:" So he reads the demonstration of it, which referred him back to such a proposition; which proposition he read. That referred him back to another, which he also read. Et sic deinceps, that at last he was demonstratively convinced of that truth. This made him in love with geometry.

One would normally define a "religion" as a system of ideas that contain statements that cannot be logically or observationally demonstrated... Gödels theorem not only demonstrates that meathematics is a religion, but shows that mathematics is the only religion that proves itself to be one!

The real truth - like anything, you have an idea about something you might write and it changes. People reflect on it or you get other ideas and maybe your original idea is radically different than how it ends up being. It's not a theorem. You don't sit down and prove something. You start with an initial idea and it grows and grows. The math of the narrative changes. In some ways your original document and what the film ends up being are quite different.

The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules... One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems. It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axioms.

Gradually, at various points in our childhoods, we discover different forms of conviction. There's the rock-hard certainty of personal experience ("I put my finger in the fire and it hurt,"), which is probably the earliest kind we learn. Then there's the logically convincing, which we probably come to first through maths, in the context of Pythagoras's theorem or something similar, and which, if we first encounter it at exactly the right moment, bursts on our minds like sunrise with the whole universe playing a great chord of C Major.

Some mathematics problems look simple, and you try them for a year or so, and then you try them for a hundred years, and it turns out that they're extremely hard to solve. There's no reason why these problems shouldn't be easy, and yet they turn out to be extremely intricate. Fermat's Last Theorem is the most beautiful example of this.

Can the difficulty of an exam be measured by how many bits of information a student would need to pass it? This may not be so absurd in the encyclopedic subjects but in mathematics it doesn't make any sense since things follow from each other and, in principle, whoever knows the bases knows everything. All of the results of a mathematical theorem are in the axioms of mathematics in embryonic form, aren't they?

Fourier's theorem has all the simplicity and yet more power than other familiar explanations in science. Stated simply, any complex pattern, whether in time or space, can be described as a series of overlapping sine waves of multiple frequencies and various amplitudes.

For what is important when we give children a theorem to use is not that they should memorize it. What matters most is that by growing up with a few very powerful theorems one comes to appreciate how certain ideas can be used as tools to think with over a lifetime. One learns to enjoy and to respect the power of powerful ideas. One learns that the most powerful idea of all is the idea of powerful ideas.

The story does what no theorem can quite do. It may not be "like real life" in the superficial sense: but it sets before us an image of what reality may well be like at some more central region.

... fain would I turn back the clock and devote to French or some other language the hours I spent upon algebra, geometry, and trigonometry, of which not one principle remains with me. Stay! There is one theorem painfully drummed into my head which seems to have inhabited some corner of my brain since that early time: "The square on the hypotenuse of a right-angled triangle is equal to the sum of the squares on the other two sides!" There it sticks, but what of it, ye gods, what of it?

The Open Source theorem says that if you give away source code, innovation will occur. Certainly, Unix was done this way... However, the corollary states that the innovation will occur elsewhere. No matter how many people you hire. So the only way to get close to the state of the art is to give the people who are going to be doing the innovative things the means to do it. That's why we had built-in source code with Unix. Open source is tapping the energy that's out there.

For me, science is just a bunch of tools - it's like playing the violin. I just enjoy calculating, and it's an instrument I know how to play. It's almost an athletic performance, in a way. I was just watching the Olympics, and that's how I feel when proving a theorem.

This skipping is another important point. It should be done whenever a proof seems too hard or whenever a theorem or a whole paragraph does not appeal to the reader. In most cases he will be able to go on and later he may return to the parts which he skipped.

We may consequently state the fundamental theorem of Natural Selection in the form: The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.

One way of looking at Impossibility Theorem is that we proposed some criteria for what a good system should be: what is it you want from a voting system, and impose some conditions. And then ask: can you have a voting system that guarantees that?

...contemporary physicists come in two varieties. Type 1 physicists are bothered by EPR and Bell's Theorem. Type 2 (the majority) are not, but one has to distinguish two subvarieties. Type 2a physicists explain why they are not bothered. Their explanations tend either to miss the point entirely (like Born's to Einstein) or to contain physical assertions that can be shown to be false. Type 2b are not bothered and refuse to explain why.

In 1975, ... [speaking with Shiing Shen Chern], I told him I had finally learned ... the beauty of fiber-bundle theory and the profound Chern-Weil theorem. I said I found it amazing that gauge fields are exactly connections on fiber bundles, which the mathematicians developed without reference to the physical world. I added, "this is both thrilling and puzzling, since you mathematicians dreamed up these concepts out of nowhere." He immediately protested: "No, no. These concepts were not dreamed up. They were natural and real."

In my [Impossibility] theorem I'm assuming that the information is a ranking. Each voter can say of any two candidates, I prefer this one to this one. So then we have essentially a ranking. It's a list saying this is my first choice. This is my second choice. Each voter, in principle, could be asked to give that entire piece of information. In the ordinary Plurality Voting, say as used in electing Congressmen, we generally only ask for the first choice. But, in principle, we could ask for more choices.

Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the "logic piano" imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.

Do people believe in human rights because such rights actually exist, like mathematical truths, sitting on a cosmic shelf next to the Pythagorean theorem just waiting to be discovered by Platonic reasoners? Or do people feel revulsion and sympathy when they read accounts of torture, and then invent a story about universal rights to help justify their feelings?

Fourier's theorem is not only one of the most beautiful results of modern analysis, but it may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics.

I think I have met nearly all the Laureates in Economics. Among the few I haven't met, I suppose I'd most like to meet Ronald Coase because of his legendary power to persuade his colleagues of the validity of the Coase Theorem.

Geometry has two great treasures; one is the Theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold; the second we may name a precious jewel.

In a world in which the price of calculation continues to decrease rapidly, but the price of theorem proving continues to hold steady or increase, elementary economics indicates that we ought to spend a larger and larger fraction of our time on calculation.

Human rights are an aspect of natural law, a consequence of the way the universe works, as solid and as real as photons or the concept of pi. The idea of self- ownership is the equivalent of Pythagoras' theorem, of evolution by natural selection, of general relativity, and of quantum theory. Before humankind discovered any of these, it suffered, to varying degrees, in misery and ignorance.

MacPherson told me that my theorem can be viewed as blah blah blah Grothendieck blah blah blah, which makes it much more respectable.

I took a break from acting for four years to get a degree in mathematics at UCLA, and during that time I had the rare opportunity to actually do research as an undergraduate. And myself and two other people co-authored a new theorem: Percolation and Gibbs States Multiplicity for Ferromagnetic Ashkin-Teller Models on Two Dimensions, or Z2.

Words are the children of reason and, therefore, can't explain it. They really can't translate feeling because they're not part of it. That's why it bugs me when people try to analyze jazz as an intellectual theorem. It's not. It's feeling.

That is why one day I said my game will be like the Pythagorean Theorem - hard to figure out. A lot of people really don't know the Pythagorean Theory. They don't make them like me anymore. They don't want to make them like that anymore.

[Barack] Obama still is in campaign mode. The Limbaugh Theorem explains it. We've had eight years of Obama and he still isn't being held accountable for the horrible policies that he has implemented because he continues to portray himself as not attached to them.

There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann.

A mathematician experiments, amasses information, makes a conjecture, finds out that it does not work, gets confused and then tries to recover. A good mathematician eventually does so - and proves a theorem.

I just enjoy calculating, and it's an instrument I know how to play. It's almost an athletic performance, in a way. I was just watching the Olympics, and that's how I feel when proving a theorem.

My favourite fellow of the Royal Society is the Reverend Thomas Bayes, an obscure 18th-century Kent clergyman and a brilliant mathematician who devised a complex equation known as the Bayes theorem, which can be used to work out probability distributions. It had no practical application in his lifetime, but today, thanks to computers, is routinely used in the modelling of climate change, astrophysics and stock-market analysis.

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.

Professor Eddington has recently remarked that 'The law that entropy always increases - the second law of thermodynamics - holds, I think, the supreme position among the laws of nature'. It is not a little instructive that so similar a law [the fundamental theorem of natural selection] should hold the supreme position among the biological sciences.

It may be appropriate to quote a statement of Poincare, who said (partly in jest no doubt) that there must be something mysterious about the normal law since mathematicians think it is a law of nature whereas physicists are convinced that it is a mathematical theorem.

The scientist has to take 95 per cent of his subject on trust. He has to because he can't possibly do all the experiments, therefore he has to take on trust the experiments all his colleagues and predecessors have done. Whereas a mathematician doesn't have to take anything on trust. Any theorem that's proved, he doesn't believe it, really, until he goes through the proof himself, and therefore he knows his whole subject from scratch. He's absolutely 100 per cent certain of it. And that gives him an extraordinary conviction of certainty, and an arrogance that scientists don't have.

I think it is said that Gauss had ten different proofs for the law of quadratic reciprocity. Any good theorem should have several proofs, the more the better. For two reasons: usually, different proofs have different strengths and weaknesses, and they generalise in different directions - they are not just repetitions of each other.

I have a basic theorem as to how I do my jokes. Growing up, I knew when to cross the line and when not to cross the line. It's the same with my comedy. I know what my audience will take and how much they won't take. I can't give you a formula for it. It's my own personal formula inside my head. Somebody else's might be different.

I think mathematics is a vast territory. The outskirts of mathematics are the outskirts of mathematical civilization. There are certain subjects that people learn about and gather together. Then there is a sort of inevitable development in those fields. You get to the point where a certain theorem is bound to be proved, independent of any particular individual, because it is just in the path of development.

The 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.

Combinatorial analysis, in the trivial sense of manipulating binomial and multinomial coefficients, and formally expanding powers of infinite series by applications ad libitum and ad nauseamque of the multinomial theorem, represented the best that academic mathematics could do in the Germany of the late 18th century.

To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples.

Carnal embrace is sexual congress, which is the insertion of the male genital organ into the female genital organ for purposes of procreation and pleasure. Fermat’s last theorem, by contrast, asserts that when x, y and z are whole numbers each raised to power of n, the sum of the first two can never equal the third when n is greater than 2.

The goal of a definition is to introduce a mathematical object. The goal of a theorem is to state some of its properties, or interrelations between various objects. The goal of a proof is to make such a statement convincing by presenting a reasoning subdivided into small steps each of which is justified as an "elementary" convincing argument.

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.

Humanism . . . is not a single hypothesis or theorem, and it dwells on no new facts. It is rather a slow shifting in the philosophic perspective, making things appear as from a new centre of interest or point of sight.

I loved doing problems in school. I'd take them home and make up new ones of my own. But the best problem I ever found, I found in my local public library. I was just browsing through the section of math books and I found this one book, which was all about one particular problem - Fermat's Last Theorem.

The Limbaugh Theorem is the way Obama gets away with no accountability for anything he's done is he never was perceived as governing. He was always as an outsider campaigning all the time against powerful forces trying to stop whatever it was he wanted to do.

What we do may be small, but it has a certain character of permanence; and to have produced anything of the slightest permanent interest, whether it be a copy of verses or a geometrical theorem, is to have done something utterly beyond the powers of the vast majority of men.

I think critics tend to think that comedy is freakin' math. Like, this is the Pythagorean Theorem. They're not sophisticated enough to know that comedy is fluid, that it evolves, and these organic evolutions are what you have to embrace.

There is a theorem that colloquially translates, You cannot comb the hair on a bowling ball. ... Clearly, none of these mathematicians had Afros, because to comb an Afro is to pick it straight away from the scalp. If bowling balls had Afros, then yes, they could be combed without violation of mathematical theorems.

Toward the end of his life, Gödel feared that he was being poisoned, and he starved himself to death. His theorem is one of the most extraordinary results in mathematics, or in any intellectual field in this century. If ever potential mental instability is detectable by genetic analysis, an embryo of someone with Kurt Gödel's gifts might be aborted.