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There's a difference between the training you need to be a world-class mathematician and what the rest of the world thinks of as "mathematics." One of the key

by j2kun 12y ago

There's a difference between the training you need to be a world-class mathematician and what the rest of the world thinks of as "mathematics." One of the key differences (being a mathematician myself) is that the big leaps of progress often come at the high level, often talking with others, and ignoring computations. Then when you have three hours to sit down and calculate, you go back and make sure your high-level ideas pan out. And you worry that you're making mistakes the whole time :) The point is that you don't just sit and compute for its own sake, nor is there a time crunch, nor is there even a single "right" way to get your computations to go through. Often you can choose one of many routes to bound some quantity, or many different proof techniques that involve very different computations.

jordigh 12y ago
I think if you're already doing computations all the time as a matter of course, there's no need to specifically set time aside to do them. But would you agree that it's necessary to know how to do them? Perhaps when working in an unfamiliar area, you take some time to learn how to compute some things in this field?
- j2kun 12y ago
  The computations just get more technical. So in research it's rare to "know" a computation in the sense that I think you mean because it's different almost every time you apply it. A common example is rounding linear programs to integer solutions for the purpose of approximation (or exact solutions, as the case may be). There is almost no way to tell in advance if it will work, and most problems require significantly different versions of this technique. There are many textbooks showcasing the many variations. So you might need days or weeks before you figure out if it will work on your problem, and then it's a "standard calculation" only in retrospect. People really don't compute things "on the fly" in mathematics. When you hear stories about Gauss these are folk legends about people with extraordinary practiced abilities, and the majority of the mathematical world doesn't work like that. People do the grunt calculations offline so they can spend their time in talks/discussions doing actual work.

by j2kun 12y ago

jordigh 12y ago

I think if you're already doing computations all the time as a matter of course, there's no need to specifically set time aside to do them. But would you agree that it's necessary to know how to do them? Perhaps when working in an unfamiliar area, you take some time to learn how to compute some things in this field?

j2kun 12y ago
The computations just get more technical. So in research it's rare to "know" a computation in the sense that I think you mean because it's different almost every time you apply it. A common example is rounding linear programs to integer solutions for the purpose of approximation (or exact solutions, as the case may be). There is almost no way to tell in advance if it will work, and most problems require significantly different versions of this technique. There are many textbooks showcasing the many variations. So you might need days or weeks before you figure out if it will work on your problem, and then it's a "standard calculation" only in retrospect. People really don't compute things "on the fly" in mathematics. When you hear stories about Gauss these are folk legends about people with extraordinary practiced abilities, and the majority of the mathematical world doesn't work like that. People do the grunt calculations offline so they can spend their time in talks/discussions doing actual work.