DailyDirt: Computers Are Really Good At Math, So When Will Shalosh B. Ekhad Get Tenure?
from the urls-we-dig-up dept
There are a lot of math problems that can be more easily solved with a computer because humans are prone to errors and get tired... and have lives outside of math. There are already several examples of computer programs that have helped to prove some important mathematical conjectures, but sometimes the resulting proof is too hard for humans to double-check. So we just have to write more programs to check our programs. (And hope that the computers don't conspire against us.)- The era of being able to publish a significant mathematics paper completely without the aid of a computer is coming to a close. Checking programs for bugs and trying to find errors in a proof that is too long for a single human to read in a lifetime... is a problem. [url]
- Shalosh B. Ekhad is the name of math software that has been a co-author on several published papers since the late 1980s. Unfortunately, not all human mathematicians are appreciated for their use of computers, but at least one human (Doron Zeilberger) gives credit to both machines and the people who use them. [url]
- Thomas Hales wrote a program that solved the Kepler conjecture and produced a 300-page proof in 1998. Due to the difficulty of humans being able to check a 300-page proof for errors, Hales wrote two more programs to verify every part of the proof, and Hales has announced that the 1998 proof has been verified to be free of logic errors. [url]
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Filed Under: ai, algorithms, artificial intelligence, doron zeilberger, kepler conjecture, logic errors, mathematics, proof, shalosh b. ekhad, thomas hales
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Is Mathematics Becoming A Science?
Except that now you have to run computer programs to test out proofs, those programs can be considered to be actual practical experiments in mathematics.
So now that maths has both a theoretical and a practical side, doesn’t that make it very much just another “science”?
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Re: Is Mathematics Becoming A Science?
Of course, in the upper reaches of Modern Physics, there is an increasing quantity of material for which empirical testing is pragmatically impossible, and this material is no longer Science, but has become Mathematics or Theology.
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Pi
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Re: π
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The first item above is stupidly just a rehash of the second item. Both are just vintage Zeilberger. Quoting him about the future of mathematics is kind of like quoting Alexander Abian in all seriousness about NASA's forthcoming missions.
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"The percentage of mathematics that computers are able to help us with is extremely tiny, perhaps only 1%."
Citation, please. I've worked with a large number of mathematicians and physicists (who are, at heart, applied mathematicians). You know what they all used to help with their math work? Computers. Admittedly a tiny sample, but remarkably consistent.
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I am a mathematician, and have done decades of both pure and applied work. And I am also an experienced programmer, going back to the end of the punched card era.
The estimate I made is pretty obvious: just look through the journals. Unlike you, I do not count millions of cash registers in action and millions of taxpayers running an app as distinct mathematical activity.
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What you call pointless flailin, I call highlighting a pointless and uncalled-for personal attack.
"I am also an experienced programmer, going back to the end of the punched card era."
Welcome to the club! Not sure how either of our bona fides is relevant to this discussion, though.
"Unlike you, I do not count millions of cash registers in action and millions of taxpayers running an app as distinct mathematical activity."
I don't understand what you're saying here. I specifically said I was talking about advanced mathematics.
Why are you so hostile? I find it baffling unless you're just trolling. I don't think I said anything that could reasonably be considered offensive to anybody.
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Unlike you, I don't count thousands of physicists running the same suite of numerical PDE solvers as distinct mathematical activity. For the same reason I don't count millions of cash registers, etc.
Look through the journals. Seriously. I've been doing so my whole life. You'll see about 1% computer-based mathematics. In some fields the proportion gets higher. For example, theoretical computer science seems to be about 10-20% computer-based. Numerical analysis is about 80-90% computer-based. In some fields, the proportion has jumped significantly in recent years, for example, the rise of Grobner base computations in commutative algebra and algebraic geometry.
But overall, the proportion is remarkably low.
As for hostility, well, 2 out of the 3 "news" items were quoting Doron Zeilberger engaged in his typical rant. And it's pretty obvious you have no real experience with "real" mathematics, just some low end computational stuff. There's a tiny number of stereotyped problems that the computers do amazing stuff with (including many kinds brilliantly discovered by Zeilberger), there's a larger number where computers are able to make doable an otherwise impossible proof (four-color theorem, new Fields Medalist Bhargava's proof of the 290 theorem), there's a goodly number where computers, especially graphics, have proved indispensable to discovery and understanding (Birch-Swinnerton-Dyer conjecture, minimal surfaces, fractals), yet these are still a tiny proportion of all the mathematics people do.
And all you've got is a childish sci-fi claim about "brain-enhanced".
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"just look through the journals"
That's nonresponsive. Looking through the journals wouldn't answer this question.
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