Computer experiments are transforming mathematics

Science News ^ | April 24, 2004 | Erica Klarreich

[js1138]

"If experimental discoveries indeed flood in faster than they can be proved, could that change the very nature of mathematics? In their book Mathematics by Experiment (2003, A K Peters), Bailey and Jonathan Borwein advance the controversial thesis that mathematics should move toward a more empirical approach. In it, formal proof would not be the only acceptable way to establish mathematical knowledge."

[js1138]

ping

[js1138]

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[js1138]

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[Southack]

Wait until they figure out how such results have been and are being applied to simplifying the breaking of encryption algorythms!

[Dr. Faust]

I recall my excitement when I first applied Lotus 123 to the solution of some problems in chemical kinetics using linear differential equations.

It was only later that I discovered physicists had been doing the same thing from the time the program was developed!

Oh brave new world---

[krb]

Over the centuries, mathematicians have found many amazingly simple ways to express pi as an infinite sum, for instance, 1 – 1/3 + 1/5 – 1/7 + 1/9 . . . .

I don't see how that's a formula for pi.

[js1138]

I don't see how that's a formula for pi.

It's not. Go back and read the next few paragraphs.

The formula is not published in the article.

[blowfish]

It's actually a formula for pi/4

[PatrickHenry]

Science list PING. [This list is for the evolution side of evolution threads, and some other science topics like cosmology. Long-time list members get all pings, but can request evo-only status. New additions will be evo-only, but can request all pings. FReepmail me to be added or dropped. Specify all pings or you'll get evo-pings only.]

[SpaceBar]

Specifically it is the taylor expansion of arctan evaluated at one which is pi/4.

[longshadow]

In it, formal proof would not be the only acceptable way to establish mathematical knowledge."

Slippery slope (no pun intended), IMHO.

[RightWingAtheist]

I wrote a review for the movie Colossus: The Forbin Project, in which I tut-tutted the film makers for assuming that a computer could formulate a new theory of gravity all by itself. Looks like I'll have to revise it again.

[js1138]

I don't think the authors are saying the computer found a formula. I believe it found a matching relationship from a list supplied by the authors. Still, it tooks weeks of computation, making the method impractical to do by hand or by intuition.

[Physicist]

I tut-tutted the film makers for assuming that a computer could formulate a new theory of gravity all by itself.

I recommend you read The Age of Spiritual Machines by Raymond Kurzweil. (Bear in mind that I consider Kurzweil to be something of a pessimist regarding the future capabilities of machines.)

[RightWingAtheist]

I've read an excerpt from it which was published in Maclean's, but I never got around to reading the book; there was always a long wait list for it at the library. The (great) Canadian rock band Out Lady Peace actually released an album based on Kurzweil's writings titles(what else?) Spiritual Machines.

I also recommend the book The Cognitive Basis of Science to those interested in asking the question if a machine can do science. I've done it before, but might as well do it again.

[Rocky]

The answer is 42. Now, what's the question?

[Doctor Stochastic]

One of the new uses of computers is in functional iteration. This is rather hard by hand. (Poincaré did seem to do it though.)

One can study things like the fixed points of Ax(1-x) for example.

Of course, proofs are necessary at some point.

[Constantine XIII]

bump!

[jennyp]

Wait until they figure out how such results have been and are being applied to simplifying the breaking of encryption algorythms!

Yeah, that's what I was thinking. There's a whole class of computations that are extremely hard. ("NP-complete", is it?) I suppose if they cracked, say, the factoring of the product of two primes, that would sink RSA & similar public key encryption systems. But there's still elliptical equations, etc. etc.

(The preceding paragraph probably has several subtle factual errors. :-)