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It seems to me now that mathematics is capable of an artistic excellence as great as that of any music, perhaps greater; not because the pleasure it gives (although very pure) is comparable, either in intensity or in the number of people who feel it, to that of music, but because it gives in absolute perfection that combination, characteristic of great art, of godlike freedom, with the sense of inevitable destiny; because, in fact, it constructs an ideal world where everything is perfect and yet true.
Mathematics, like dialectics, is an instrument of the inner higher sense, while in practice it is an art like rhetoric. For both of these, nothing has value but form; content is immaterial. Whether mathematics is adding up pennies or guineas, whether rhetoric is defending truth or falsehood, makes no difference to either.
I should consider that I know nothing about physics if I were able to explain only how things might be, and were unable to demonstrate that they could not be otherwise. For, having reduced physics to mathematics, the demonstration is now possible, and I think that I can do it within the small compass of my knowledge.
But my understanding is there are many villages you go to where there's really no schools, as a practical matter, and many of the schools still teach just how to memorize certain things rather than how to think critically about problems. And every country at this point, if it wants to succeed, needs to put in place free, compulsory education for its young people — because they just can't succeed unless they have some basic skills. They have to be able to read. They have to be able to do mathematics. They have to have some familiarity with computers. They have to be able to understand basic principles of science. If you don't have those basic tools, then it's very hard to find a decent job in today's economy.
Whereas young people become accomplished in geometry and mathematics, and wise within these limits, prudent young people do not seem to be found. The reason is that prudence is concerned with particulars as well as universals, and particulars become known from experience, but a young person lacks experience, since some length of time is needed to produce it.
Pure mathematics can never deal with the possibility, that is to say, with the possibility of an intuition answering to the conceptions of the things. Hence it cannot touch the question of cause and effect, and consequently, all the finality there observed must always be regarded simply as formal, and never as a physical end.