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I was watching a video on Andrew Wiles and his proof of Fermat's Last Theorem and I quite liked the video, especially the complexity of the proof only to prove a simple concept which can be understood by most people. I also liked the graphics they used to illustrate elliptic curves and modular forms.

But then Andrew Wiles said that he never uses a computer, he only uses pen and paper and I also heard of other mathematicians that don't use computers.

Do they not use computers because there are problems only a mathematicians can solve or are there other motives? Wouldn't the proof have taken him less time if he used a computer to assist him?

Dan Rust
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please delete me
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    You can't delegate the thinking to a computer. If your work involves mostly thinking and little computing, you can use a computer mostly for emails, there simply isn't much where a computer could help. – Daniel Fischer Aug 11 '13 at 16:20
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    There are areas of mathematics where the use of computers is not so useful. The emphasis is not on numerical computations or modelling of large systems, for example. And the theorem provers we have are not developed enough to be useful in all areas. – Andrés E. Caicedo Aug 11 '13 at 16:20
  • I know this is a more general question but for Fermat's Last Theorem, you should prove it for all numbers, so computers obviously can't do it. – aeyalcinoglu Aug 11 '13 at 16:21
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    @DanielFischer: ... and for typing up your papers. ;) – Rasmus Aug 11 '13 at 16:23
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    @Rasmus Isn't that what assistants/students (or even secretaries) are for? – Daniel Fischer Aug 11 '13 at 16:24
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    @KonformistLiberal It is not obvious. It is in fact reasonable to assume that you can reduce a problem of Fermat type to verifying properties of certain explicit curves, and these properties could in principle be verified with the assistant of explicit computations. – Andrés E. Caicedo Aug 11 '13 at 16:24
  • It's been interesting to observe how computers have been playing an important role in the twin primes polymath project. – littleO Aug 12 '13 at 02:20
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    I vote not to close. – Alexander Gruber Aug 12 '13 at 03:08
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    I recently heard something along these lines: "Mathematics asks 'what?' and 'why?', while computer science asks 'how?'" I use computers a lot in my mathematical work, but that is because many of my mathematical questions either a) have to do with computation, or b) can be helped to some extent by having an unbiased source confirm that I'm not doing something stupid. (One example of b would be using Monte Carlo sanity checks for complex integration proofs.) It's easy to see why someone working on sufficiently non-constructible mathematics, e.g. the continuum hypothesis, would have no use for b. – Alexander Gruber Aug 12 '13 at 03:14
  • We can also discuss this question with respect to the future of computing, i.e. quantum computers and the impact that more powerful computers can have on mathematics and whether more mathematicians will start using them. Currently it seems like the main use of computers in mathematics is computation but with more advanced artificial intelligence computers can probably prove conjectures etc. but this discussion is off topic as it should probably be discussed on on a computing site or physics site. – please delete me Aug 12 '13 at 03:17
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    @user2357 I'm not quite sure what the difference is between computation and a computer-based AI solving a problem. The latter is still computation, but with a (I guess only implied) more sophisticated algorithm. – Dan Rust Aug 12 '13 at 03:38
  • @DanielRust may be I should have said calculations instead of computation. I am not sure exactly what they are used for right now and what their mathematical limitations are. – please delete me Aug 12 '13 at 05:30
  • Maybe related question: How to properly use technology for back-of-the-envelope calculations? – Tobias Kienzler Oct 07 '14 at 09:45
  • I think they somehow consider the computer as a rival to their mind, fearing it might blunt their minds. – Mehdi Haghgoo Sep 25 '15 at 12:34

1 Answers1

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I work in software and I have an amateur interest in mathematics, and from what I can tell many theoretical mathematicians don't have much use for computers because of the domains they work in. In order for a mathematician to utilize a computer, the problem must either be something that requires a lot of computation or is already formalized enough for a proof assistant or theorem prover to attack.

If you need one closed form solution to an easy differential equation, just using the stuff you learned in calculus is easier than figuring out how to use software that does integration for you, much as you wouldn't turn to a calculator to multiply six times nine. Computers can be useful for areas of mathematics where you need to come up with lots of closed form solutions (computer algebra systems) numerical integration or integral transforms, or running other intensive computing tasks in other domains like computing the class number (or other properties) for a large number of number fields.

Many of these problems are tedious and not particularly interesting to theorists, even if they're quite useful for applied math; Further, solving the problem often requires writing software to do it, given theorists often work in unexplored areas. For theorists, you often want to prove a statement of some sort, and that requires a proof assistant of some sort; While they can be useful, the foundations for most graduate level mathematics aren't formalized in first order logic, and where they are they're usually inaccessibly unwieldy compared to more informal reasoning.

Further complicating the picture is the traditional method of formalizing mathematics is built on the foundation of set theory in first order logic, which is often incompatible (or at least unwieldy) with metamathimatical reasoning about categories, nonstandard models, and other foundational issues. So if mathematicians who worked in this field tried to use computers, they'd spend more time writing software and formalizing existing math than doing new work.

As an example, very little of Wiles proof has been formalized in a form that can be verified by mechanized reasoning, because most of the branches of mathematics that it rests on have yet to be formalized. This may change in coming decades, as theorem provers and proof assistants get more advanced, but for the time being computers are useful for the most mature, formalized areas of mathematics that is largely the domain of physics and applied math.

dezakin
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