A Quote by William Thurston
Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.
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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.
The power of equations lies in the philosophically difficult correspondence between mathematics, a collective creation of human minds, and an external physical reality. Equations model deep patterns in the outside world. By learning to value equations, and to read the stories they tell, we can uncover vital features of the world around us.
A mathematician either has a feeling for equations and an understanding and delight in it, not only in the purity of it, but in its beauty as well. I don't think that's something that you learn at school. I think you can get better in mathematics on a school level, but when you're talking about being a mathematician, I think that's definitely a gift of genes or whatever, you know? Whatever your pool is.
In the Pythagorean system, thinking about numbers, or doing mathematics, was an inherently masculine task. Mathematics was associated with the gods, and with transcendence from the material world; women, by their nature, were supposedly rooted in this latter, baser realm.
The theory of numbers, more than any other branch of mathematics, began by being an experimental science. Its most famous theorems have all been conjectured, sometimes a hundred years or more before they were proved; and they have been suggested by the evidence of a mass of computations.
Mathematics doesn’t care about those beyond the numbers.
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!
Mathematics my foot! Algorithms are mathematics too, and often more interesting and definitely more useful.
The product of mathematics is clarity and understanding. Not theorems, by themselves. ... In short, mathematics only exists in a living community of mathematicians that spreads understanding and breathes life into ideas both old and new.
It took me 1057 pages to describe the hundreds of mathematical equations, algorithms and programming techniques that I invented and used.
While physics and mathematics may tell us how the universe began, they are not much use in predicting human behavior because there are far too many equations to solve. I'm no better than anyone else at understanding what makes people tick, particularly women.
In my own research when I'm working with equations, I never feel like I really understand what I'm doing if I'm solely relying on the mathematics for my understanding. I need to have a visual picture in my mind. I'm constantly translating from the math to some intuitive mind's-eye picture.
Well can I just make a point about the numbers because people talk a lot about police numbers as if police numbers are the holy grail. But actually what matters is what those police are doing. It's about how those police are deployed.
In deep learning, the algorithms we use now are versions of the algorithms we were developing in the 1980s, the 1990s. People were very optimistic about them, but it turns out they didn't work too well.
Poetry is a sort of inspired mathematics, which gives us equations, not for abstract figures, triangles, squares, and the like, but for the human emotions. If one has a mind which inclines to magic rather than science, one will prefer to speak of these equations as spells or incantations; it sounds more arcane, mysterious, recondite.
Einstein had looked at the numbers and dimensions that went into his equations for gravitational waves and said, essentially, 'This is so tiny that it will never have any influence on anything, and nobody can measure it.' And when you think about the times and the technology in 1916, he was probably right.
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