A Quote by Andrew Wiles
I tried to fit it in with some previous broad conceptual understanding of some part of mathematics that would clarify the particular problem I was thinking about.
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A work of art can start you thinking about some aesthetic or philosophical problem; it can suggest some new method, some fresh approach to fiction.
Thinking - in particular abstract thinking, which most of us are introduced to through the study of mathematics and literature - helps us learn that we can become problem solvers.
It is impossible to overstate the imporance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps... Every new discovery in mathematics, results from an attempt to solve some problem.
The question you raise, 'How can such a formulation lead to computations?' doesn't bother me in the least! Throughout my whole life as a mathematician, the possibility of making explicit, elegant computations has always come out by itself, as a byproduct of a thorough conceptual understanding of what was going on. Thus I never bothered about whether what would come out would be suitable for this or that, but just tried to understand - and it always turned out that understanding was all that mattered.
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.
One of the nice things about our marriage, at least to my way of thinking, is that my wife and I no longer have to argue every thing through. We each know what the other will say, and so the saying becomes an unnecessary formality. No doubt some marriage counselor would explain to us that our problem is a failure to communicate, but to my way of thinking we've worked long and hard to achieve this silence, Lily's and mine, so fraught with mutual understanding.
There is nothing mysterious, as some have tried to maintain, about the applicability of mathematics. What we get by abstraction from something can be returned.
I'd like to see education play a larger role in our daily lives, have people come to a larger understanding - a "bigger picture" understanding - of how we fit into the world, and how we fit into the universe. Not necessarily thinking of ourselves, but thinking of others.
There is a big misunderstanding about the idea of naturalness. Most people who come to us believing in some freedom or naturalness, but their understanding is what we call [heretical naturalness] ... a kind of "let-alone policy" or sloppiness... For a plant or stone to be natural is no problem. But for us there is some problem, indeed a big problem. To be natural is something we must work on.
I am part of a broad social movement. Ten years ago, it would have unimaginable for some random Dutch historian to go viral when talking about taxes. Yet here we are.
All our surest statements about the nature of the world are mathematical statements, yet we do not know what mathematics "is"... and so we find that we have adapted a religion strikingly similar to many traditional faiths. Change "mathematics" to "God" and little else might seem to change. The problem of human contact with some spiritual realm, of timelessness, of our inability to capture all with language and symbol-all have their counterparts in the quest for the nature of Platonic mathematics.
When we are not engaged in thinking about some definite problem, we usually spend about 95 percent of our time thinking about ourselves. Now, if we stop thinking about ourselves for a while and begin to think of the other person's good points, we won't have to resort to flattery so cheap and false that it can be spotted almost before it is out of the mouth.
What's the best part of being a mathematician? I'm not a religious man, but it's almost like being in touch with God when you're thinking about mathematics. God is keeping secrets from us, and it's fun to try to learn some of the secrets.
Some people feel like you need to have a very specialized understanding of music to have the authority to talk about it. They are such good directors that it's perfectly possible to have conceptual and directorial and storytelling conversations about music without needing to know all the technical pieces.
Some people ask: "Why the word feminist? Why not just say you are a believer in human rights, or something like that?" Because that would be dishonest. Feminism is, of course, part of human rights in general-but to choose to use the vague expression human rights is to deny the specific and particular problem of gender. It would be a way of pretending that it was not women who have, for centuries, been excluded. It would be a way of denying that the problem of gender targets women.
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.
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