A Quote by George Polya
The world is anxious to admire that apex and culmination of modern mathematics: a theorem so perfectly general that no particular application of it is feasible.
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The principle is so perfectly general that no particular application of it is possible.
Money cannot be applied to the *general welfare*, otherwise than by an application of it to some *particular* measure conducive to the general welfare. Whenever, therefore, money has been raised by the general authority, and is to be applied to a particular measure, a question arises whether the particular measure be within the enumerated authorities vested in Congress. If it be, the money requisite for it may be applied to it; if it be not, no such application can be made.
If you have to prove a theorem, do not rush. First of all, understand fully what the theorem says, try to see clearly what it means. Then check the theorem; it could be false. Examine the consequences, verify as many particular instances as are needed to convince yourself of the truth. When you have satisfied yourself that the theorem is true, you can start proving it.
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.
To most outsiders, modern mathematics is unknown territory. Its borders are protected by dense thickets of technical terms; its landscapes are a mass of indecipherable equations and incomprehensible concepts. Few realize that the world of modern mathematics is rich with vivid images and provocative ideas.
There is a great difference, whether the poet seeks the particular for the sake of the general or sees the general in the particular. From the former procedure there ensues allegory, in which the particular serves only as illustration, as example of the general. The latter procedure, however, is genuinely the nature of poetry; it expresses something particular, without thinking of the general or pointing to it.
General rules are dangerous of application in particular instances.
I think that if your tenure case depends on your proving what you thought was a mathematical theorem and the proposed theorem turns out to be false just before your tenure decision, and you want to get tenure very badly, there is a sense in which it's perfectly understandable and reasonable of you to wish the proposed theorem were true and provable, even if it's logically impossible for it to be.
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.
The point of mathematics is that in it we have always got rid of the particular instance, and even of any particular sorts of entities. So that for example, no mathematical truths apply merely to fish, or merely to stones, or merely to colours. So long as you are dealing with pure mathematics, you are in the realm of complete and absolute abstraction. . . . Mathematics is thought moving in the sphere of complete abstraction from any particular instance of what it is talking about.
A good deal of my research in physics has consisted in not setting out to solve some particular problem, but simply examining mathematical equations of a kind that physicists use and trying to fit them together in an interesting way, regardless of any application that the work may have. It is simply a search for pretty mathematics. It may turn out later to have an application. Then one has good luck. At age 78.
A man long accustomed to admire his wife in general, seldom pauses to admire her in a particular gown or attitude, unless his attention is directed to her by the appreciative gaze of another man.
One is a great deal less anxious if one feels perfectly free to be anxious, and the same may be said of guilt.
To criticize mathematics for its abstraction is to miss the point entirely. Abstraction is what makes mathematics work. If you concentrate too closely on too limited an application of a mathematical idea, you rob the mathematician of his most important tools: analogy, generality, and simplicity. Mathematics is the ultimate in technology transfer.
One of the most amazing things about mathematics is the people who do math aren't usually interested in application, because mathematics itself is truly a beautiful art form. It's structures and patterns, and that's what we love, and that's what we get off on.
The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.
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