Richard Feynman: The Ralation of Mathematics and Physics
Mathematic is a tool for reasoning. It's in fact a big collection of the results of some person's careful thought and reasoning.
By mathematics it is possible to connect one statement to another.
Mathematics, then, is a way of going from one statements to another.
It's evidently useful in physics, because we have all these different ways that we could speak of things, and it permits us to develop consequences, and analyze situations, and change the laws in different ways, and to connect all the various statements.
So that, as a matter of fact, the total amount that a physicist knows is very little: he has only to remember the rules for getting from one place to another, and he's able to do it then.
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Now, an interesting question comes up: is there some pattern to it? Is there a place to begin-fundamental principles-and deduce the whole works? Or, is there some particular pattern, or order in nature, in which we can understand that these are more fundamental statements, and these are more consequential statements? There are two kinds of ways of looking at mathematics, which for the purpose of this lecture I will call the Babylonian tradition, and the Greek tradition.
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巴比伦式和希腊式（归纳和演绎）
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In Babylonian schools in mathematics, the student would learn something by doing a large number of examples, until he caught onto the general rule.
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But Euclid discovered that there was a way in which all of the theorems of geometry could be ordered from a set of axioms that were particularly simple-and you're all familiar with that much geometry.
But the Babylonian attitude was-if I make my way of talking what I call Babylonian mathematics-is that you know all these various theorems and many of the connections in between, but you've never really realized that it could all come up from a bunch of axioms.
The mathematical tradition of today is to start with some particular ones, whicha are chosen by some kind of convention to be axioms, and then to build up the structure from there.
But if you have to remember a few things in the geometry, you can always get somewhere else; it's much more efficient to do it the other way: what the best axioms are, are not exactly the same-in fact, are not ever the same-as the most efficient way of getting around in the territory.
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