I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our "creations," are simply the notes of our observations.— G. H. Hardy
Impressive Theorem quotations
No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it also gives the impression of being beautiful.
A mathematician is a device for turning coffee into theorems.
If only I had the Theorems! Then I should find the proofs easily enough.
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.
There are three signs of senility. The first sign is that a man forgets his theorems. The second sign is that he forgets to zip up. The third sign is that he forgets to zip down.
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.
I am persuaded that this method [for calculating the volume of a sphere] will be of no little service to mathematics. For I foresee that once it is understood and established, it will be used to discover other theorems which have not yet occurred to me, by other mathematicians, now living or yet unborn.
Murphy's Law, that brash proletarian restatement of Godel's 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.
God has the Big Book, the beautiful proofs of mathematical theorems are listed here.
The development of mathematics towards 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.
Do not say a little in many words but a great deal in a few.
I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems while the root drives into the depths.
The product of mathematics is clarity and understanding.
Not theorems, by themselves. ... In short, mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new.
One geometry cannot be more true than another; it can only be more convenient.
Bells theorem dealt a shattering blow to Einsteins position by showing that the conception of reality as consisting of separate parts, joined by local connections, is incompatible with quantum theory... Bells theorem demonstrates that the universe is fundamentally interconnected, interdependent, and inseparable.
Mathematics is not a deductive science - that's a cliché.
When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.
Theorems are fun especially when you are the prover, but then the pleasure fades. What keeps us going are the unsolved problems.
There are very few theorems in advanced analysis which have been demonstrated in a logically tenable manner. Everywhere one finds this miserable way of concluding from the special to the general and it is extremely peculiar that such a procedure has led to so few of the so-called paradoxes.
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.
Mathematics is not a deductive science - that's a cliché.
.. What you do is trial and error, experimentation, guesswork.
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.
What exactly is mathematics? Many have tried but nobody has really succeeded in defining mathematics; it is always something else. Roughly speaking, people know that it deals with numbers, figures, with relations, operations, and that its formal procedures involving axioms, proofs, lemmas, theorems have not changed since the time of Archimedes.
Mathematics is not arithmetic. Though mathematics may have arisen from the practices of counting and measuring it really deals with logical reasoning in which theorems-general and specific statements-can be deduced from the starting assumptions. It is, perhaps, the purest and most rigorous of intellectual activities, and is often thought of as queen of the sciences.
Share prices follow the theorem: hope divided by fear minus greed.
The missing piece in his stomach hurt so much-and eventually he stopped thinking about the Theorem and wondered only how something that isn't there can hurt you.
Never call yourself a philosopher, nor talk a great deal among the unlearned about theorems, but act conformably to them. Thus, at an entertainment, don't talk how persons ought to eat, but eat as you ought. For remember that in this manner Socrates also universally avoided all ostentation.
The analysis of variance is not a mathematical theorem, but rather a convenient method of arranging the arithmetic.
One cannot really argue with a mathematical theorem.
Mathematics does not grow through a monotonous increase of the number of indubitably established theorems but through the incessant improvement of guesses by speculation and criticism, by the logic of proofs and refutations.
Theorems often tell us complex truths about the simple things, but only rarely tell us simple truths about the complex ones. To believe otherwise is wishful thinking or "mathematics envy."
A mathematician is a person who can find analogies between theorems;
a better mathematician is one who can see analogies between proofs and the best mathematician can notice analogies between theories.
Less depends upon the choice of words than upon this, that their introduction shall be justified by pregnant theorems.
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.