A Quote by Oliver Heaviside
The best result of mathematics is to be able to do without it.
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The sciences, even the best,-mathematics and astronomy,-are like sportsmen, who seize whatever prey offers, even without being able to make any use of it.
But there is another reason for the high repute of mathematics: it is mathematics that offers the exact natural sciences a certain measure of security which, without mathematics, they could not attain.
Thers is this wonderful iconoclast at Rutgers, Doron Zeilberger, who says that our mathematics is the result of a random walk, by which he means what WE call mathematics. Likewise, I think, for the sciences.
Imagine life without any algorithms at all, you wouldn't be able to do anything. This is already completely encompassing. We have a habit of over-trusting what mathematics or computer scientists tell us to do, without questioning it, too much faith in the magical power of analysis.
I'd like to be able to design as easily as if I was using Photoshop. I'd like to be able to create a multicolumn layout and control source order without having to do advanced mathematics or hire Eric Meyer or Dan Cederholm to figure out the CSS, because I can't.
One cannot inquire into the foundations and nature of mathematics without delving into the question of the operations by which the mathematical activity of the mind is conducted. If one failed to take that into account, then one would be left studying only the language in which mathematics is represented rather than the essence of mathematics.
I don't think that everyone should become a mathematician, but I do believe that many students don't give mathematics a real chance. I did poorly in math for a couple of years in middle school; I was just not interested in thinking about it. I can see that without being excited mathematics can look pointless and cold. The beauty of mathematics only shows itself to more patient followers.
In order to be able to make it, you have to put aside the fear of failing and the desire of succeeding. You have to do these things completely and purely without fear, without desire. Because things that we do without lust of result are the purest actions we shall ever take.
Mystery is an inescapable ingredient of mathematics. Mathematics is full of unanswered questions, which far outnumber known theorems and results. It's the nature of mathematics to pose more problems than it can solve. Indeed, mathematics itself may be built on small islands of truth comprising the pieces of mathematics that can be validated by relatively short proofs. All else is speculation.
Srinivasa Ramanujan was the strangest man in all of mathematics, probably in the entire history of science. He has been compared to a bursting supernova, illuminating the darkest, most profound corners of mathematics, before being tragically struck down by tuberculosis at the age of 33, like Riemann before him. Working in total isolation from the main currents of his field, he was able to rederive 100 years' worth of Western mathematics on his own. The tragedy of his life is that much of his work was wasted rediscovering known mathematics.
We know that nature is described by the best of all possible mathematics because God created it. So there is a chance that the best of all possible mathematics will be created out of physicists' attempts to describe nature.
Maturity: Be able to stick with a job until it is finished. Be able to bear an injustice without having to get even. Be able to carry money without spending it. Do your duty without being supervised.
A chess problem is genuine mathematics, but it is in some way "trivial" mathematics. However, ingenious and intricate, however original and surprising the moves, there is something essential lacking. Chess problems are unimportant. The best mathematics is serious as well as beautiful-"important" if you like, but the word is very ambiguous, and "serious" expresses what I mean much better.
What is mathematics? Ask this question of person chosen at random, and you are likely to receive the answer "Mathematics is the study of number." With a bit of prodding as to what kind of study they mean, you may be able to induce them to come up with the description "the science of numbers." But that is about as far as you will get. And with that you will have obtained a description of mathematics that ceased to be accurate some two and a half thousand years ago!
I do not think the division of the subject into two parts - into applied mathematics and experimental physics a good one, for natural philosophy without experiment is merely mathematical exercise, while experiment without mathematics will neither sufficiently discipline the mind or sufficiently extend our knowledge in a subject like physics.
The principles of logic and mathematics are true simply because we never allow them to be anything else. And the reason for this is that we cannot abandon them without contradicting ourselves, without sinning against the rules which govern the use of language, and so making our utterances self-stultifying. In other words, the truths of logic and mathematics are analytic propositions or tautologies.
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