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Old 2010-01-06, 17:27   #1
ewmayer
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Default The Future of Computational Mathematics

While searching for something on a related topic on the web, I came across this interesting 2005 paper from Jonathan Bailey and Peter Borwein (it's a pdf), summarizing the state of the art computational mathematics:

The Future of Computer -Assisted Mathematics
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Old 2010-01-07, 04:53   #2
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I'm reminded of the history of chess-playing computers.

In the 1970s, they were pretty bad -- I, a class B player at roughly the 65th percentile of USCF players, could easily and consistently beat the ones marketed in games shops. Then there were steady improvements, so that the top chess-playing computers (either just software running on ordinary general-purpose hardware, or some cases of special-purpose hardware optimized for investigating future chess moves) ranked as class A, then Expert, then Master-level.

It became necessary to add tournament rules governing how chess-playing computers could participate as players in tournaments. Tournament organizers were given the right to exclude all computer participation if they wished.

Top computers began to challenge (or, rather, their human handlers issued challenges on their behalf) grandmasters, to the extent of winning a significant proportion of games played in international tournament conditions. Then they challenged the world champion and made a decent score in matches.

Then they beat the human world champion -- by a margin similar to that by which the human champion had beaten his human predecessors.

In the early years, it had been thought that the most effective way to improve chess computers was to employ grandmasters to codify their knowledge and techniques into the software. That is, making computers play more and more like humans was thought to be the road to success. The application of more and more sheer brute speed in looking ahead at the possibilities in a game seemed to be ineffective.

But then the brute-speed method began to surpass the expert-knowledge method. The strongest chess computers began to be the ones that could look ahead and evaluate the largest number (plies) of future move candidates in the game time available for each move.

It now appears that a certain number of brute-speed plies is sufficient to surpass the look-ahead capabilities and intuition of the best human players.

Computers now sometimes win games with eerie move combinations that no human player could imagine discovering, because there is something about those combinations that was just too strange for humans to conceive -- even when the depth of look-ahead is the same as for combinations more readily found by grandmasters. It seems that chess computers have revealed that there are certain types of chess move combinations to which humans are "blind".

The fact that this was revealed by the brute-speed method, rather than the expert-knowledge method, is disturbing to many.

This too-strange-for-humans-to-imagine phenomenon may occur in computational mathematics, too. Once the math computers have advanced sufficiently, they may find things that will seem too strange for humans to imagine ever discovering by themselves, even though the computers discover these things via a brute-speed approach.

(I don't mean, for example, finding that the four-color theorem is proven by considering thousands of cases. That, I think, was readily conceivable by humans, even if they thought it unlikely. I mean stuff that is too strange for anyone to have imagined beforehand.)

Last fiddled with by cheesehead on 2010-01-07 at 04:54
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Old 2010-01-07, 06:32   #3
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Quote:
Originally Posted by cheesehead View Post
I mean stuff that is too strange for anyone to have imagined beforehand.
Like the BBP formulas for pi and other constants?
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Old 2010-01-07, 06:33   #4
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Quote:
Originally Posted by ewmayer View Post
While searching for something on a related topic on the web, I came across this interesting 2005 paper from Jonathan Bailey and Peter Borwein (it's a pdf), summarizing the state of the art computational mathematics:

The Future of Computer -Assisted Mathematics
I noticed you were mentioned in the article

Last fiddled with by flouran on 2010-01-07 at 06:50
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Old 2010-01-07, 13:51   #5
fivemack
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I am probably too ignorant of theta-functions, hypergeometric sequences, polylogarithms, modular equations and the works of Ramanujan validly to make this statement, but I don't think the expressions in the paper would have shocked Ramanujan.
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Old 2010-01-07, 13:58   #6
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Quote:
Originally Posted by fivemack View Post
I am probably too ignorant of theta-functions, hypergeometric sequences, polylogarithms, modular equations and the works of Ramanujan validly to make this statement, but I don't think the expressions in the paper would have shocked Ramanujan.
Yes, the statement should be amended to "computers may find things that will seem too strange for humans (other than Ramanujan) to imagine".
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Old 2010-01-07, 20:54   #7
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A relevant essay dated the same year 2005:

Brian Davies Whither Mathematics?

And yet another paper on the topic:

Towards Computer Aided Mathematics

.

Last fiddled with by maxal on 2010-01-07 at 20:59
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Old 2010-01-07, 22:31   #8
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Borwein and Bailey published some books about experimental math: http://www.experimentalmath.info/books/
I wonder if they are good or not.
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Old 2010-01-07, 22:56   #9
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about eing careful about approximations...
I remember that i have read somewhere that there is an infinite sum which converges to some value very close but not pi. (There were similar examples in this paper). The sum is accurate to some billion digits, but after that is wrong. Does anybody know which equation is meant?
I don't remember that equation or the place i have read it.
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Old 2010-01-07, 23:23   #10
fivemack
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Sums of exp(-a*x^2) get accidentally very close to sqrt(pi), for reasons related to the fact that the Fourier transform of exp(a*x^2) is the very fast-shrinking (if a is small) function exp(1/a * x^2):

so sum(t=-400,400,exp(-t^2/100))

is within 10^-420 of 10*sqrt(Pi)

I think this may be what you're thinking of

Last fiddled with by fivemack on 2010-01-07 at 23:23
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Old 2010-01-08, 03:44   #11
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http://mersenneforum.org/showpost.ph...0&postcount=30

may be of interest/relevance.
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