Quotes of All Topics . Occasions . Authors
I think that mathematics can benefit by acknowledging that the creation of good models is just as important as proving deep theorems.
That language is an instrument of human reason, and not merely a medium for the expression of thought, is a truth generally admitted.
Solving problems is a practical art, like swimming, or skiing, or playing the piano: you can learn it only by imitation and practice.
To doubt everything, or, to believe everything, are two equally convenient solutions; both dispense with the necessity of reflection.
The full beauty of the subject of generating functions emerges only from tuning in on both channels: the discrete and the continuous.
Knowing something for oneself or for communication to an expert colleague is not the same as knowing it for explanation to a student.
The value of a problem is not so much coming up with the answer as in the ideas and attempted ideas it forces on the would be solver.
I've been giving this lecture to first-year classes for over twenty-five years. You'd think they would begin to understand it by now.
Alekhine is a poet who creates a work of art out of something that would hardly inspire another man to send home a picture post card.
I have said that the modern man, and especially the modern American, however much 'know-how' he may have, has very little 'know-what'
The importance of the "New Mathematics" lies mainly in the fact that it has taught us the difference between the disc and the circle.
... the notions category and functor were not formulated or put in print until the idea of a natural transformation was also at hand.
It was traditional to not actually cash the prizes that Erdos did award while he was alive. People usually framed the cheque instead.
I think one nice thing about mathematics is that we don't really have one prize that dominates all the others, like the Nobel prizes.
When an action is once done, it is right or wrong for ever; no accidental failure of its good or evil fruits can possibly alter that.
The learned tradition is not concerned with truth, but with the learned adjustment of learned statements of antecedent learned people.
It has pleased God that divine verities should not enter the heart through the understanding, but the understanding through the heart.
Arc, amplitude, and curvature sustain a similar relation to each other as time, motion, and velocity, or as volume, mass, and density.
If we look at the fact, we shall find that the great inventions of the age are not, with us at least, always produced in universities.
I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems while the root drives into the depths.
Sin is never at a stay; if we do not retreat from it, we shall advance in it; and the farther on we go, the more we have to come back.
To put it simply, we first explain what we are talking about, and then explain why what we are saying is true (pace Bertrand Russell).
Practical application is found by not looking for it, and one can say that the whole progress of civilization rests on that principle.
A mathematical idea should not be petrified in a formalised axiomatic setting, but should be considered instead as flowing as a river.
The present author confesses that, to him, geometry is nothing at all, if not a branch of art ... A Treatise on Algebraic Plane Curves
The only way to learn mathematics is to do mathematics. That tenet is the foundation of the do-it-yourself, Socratic, or Texas method.
When a student comes and asks, "Should I become a mathematician?" the answer should be no. If you have to ask, you shouldn't even ask.
A clever graduate student could teach Fourier something new, but surely no one claims that he could teach Archimedes to reason better.
Being a mathematician is a bit like being a manic depressive: you spend your life alternating between giddy elation and black despair.
Can you make fancy patterns of water that actually have some computation power? I'm betting that fluids are complex enough to do this.
This attitude [the abstract method in mathematics] can be encapsulated in the following slogan: a mathematical object is what it does.
What is morality in any given time or place? It is what the majority then and there happen to like and immorality is what they dislike.
The heart has its reasons which reason knows nothing of... We know the truth not only by the reason, but by the heart." - Blaise Pascal
How I hate this folly of not believing in the Eucharist, etc.! If the gospel be true, if Jesus Christ be God, what difficulty is there?
Montaigne is wrong in declaring that custom ought to be followed simply because it is custom, and not because it is reasonable or just.
There are many degrees of Probable, some nearer Truth than others, in the determining of which lies the chief exercise of our Judgment.
Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts.
Mathematics education is much more complicated than you expected, even though you expected it to be more complicated than you expected.
Thus, in a sense, mathematics has been most advanced by those who distinguished themselves by intuition rather than by rigorous proofs.
It is the simple hypotheses of which one must be most wary; because these are the ones that have the most chances of passing unnoticed.
The object of mathematical rigor is to sanction and legitimize the conquests of intuition, and there was never any other object for it.
One of the chief duties of the mathematician in acting as an advisor ... is to discourage ... from expecting too much from mathematics.
Most writers on the subject seem to agree that the typical working mathematician is a Platonist on weekdays and a formalist on Sundays.
Thought is powerless, except it make something outside of itself: the thought which conquers the world is not contemplative but active.
It is impossible to meditate on time and the mystery of nature without an overwhelming emotion at the limitations of human intelligence.
Vigorous societies harbor a certain extravagance of objectives, so that men wander beyond the safe provision of personal gratifications.
It is natural for the mind to believe and for the will to love; so that, for want of true objects, they must attach themselves to false.
[On Cantor's work:] The finest product of mathematical genius and one of the supreme achievements of purely intellectual human activity.
Mathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work for yourself
Nothing is more important than to see the sources of invention which are, in my opinion more interesting than the inventions themselves.