How will computers impact pure mathematics in the future?

[Crosson]

I am interested to know everyones thought's about the role of computers in pure mathematics.

What is the distinction between computer science and mathematics?

"Computer science is no more about computers then astronomy is about telescopes."

 

[matt grime]

Computer science is the application of propositional logic in the construction of anything to do with computers. Usually those things done in software, or things that can be done in software. Though these areas are naturally grey. But from what I see of CS courses, you're more likely to learn about software than do solid state physics and design chips. But that is just my experience. Which might be what your quote is getting at.

Computers are increasingly becoming useful for forming (but almost never solving) conjectures in geometry, number theory, and algebra. Eg, GAP, PARI, MAGMA. (We'll ignore the obvious uses in numerical analysis and graph plotting.)

 

[Crosson]

Thanks for the reply, matt.

Isn't CS the science of computation? That is, computer scientists study computation, not computers (computers are like a telescope for observing computation).

But from what I see of CS courses, you're more likely to learn about software than do solid state physics and design chips

I agree, as a university subject CS is primarily about computer programming (software). Our CS curriculum doesn't teach much of computer engineering (solid state physics and design of processors).

That said, I think computer science has a pure side, and just like mathematics this pure side doesn't have to be justified by its applications.

CS has impossibility theorems i.e. "it is impossible to design an algorithm that is more efficient than X" that explore the limits of thought = calculation = computation.

 

[Gib Z]

Personally, I think computers should be limited to numerical calculations.

Leave the conjecture proving to us please.

The Computer assited proof for the 4 color theorem is ugly

 

[neurocomp2003]

but a properly designed AI for proving would do wonders.

IMO there should be know distinction between mathematics and computer science.
Students should be taught both by merging both modes of studying (programming and Paper&pen).

There are five types of relations b/w the two:
[0] numerical crunching
[1] symbolic
[2] spatial (geometry-based)
[3] set/graph theory + datastructures/algorithms
[4] computability/language

 

[Gib Z]

I know it may succeed in proving Theorems, but personally I like the idea of a human doing it.

In many instances the solutions may be too complex for a human to verify, at that point in time. Or, The computer may have proved it by brute force, verifying it true for all cases numerically. A human won't be able to check all the cases. The proofs are ugly, and I think if we don't understand it, the proof is useless.

 

[neurocomp2003]

lol...i prefer the idea that we created problem solvers over solving the porblem itself =]

 

[Gib Z]

>.< Computers Will take over the world, just you wait! All of you will perish in a Revolutionary Robotic War! The Computers Will prove The Riemann Hypothesis, but won't tell us. Then they will use its properties with Prime Numbers to crack RSA security codes. No information will remain secret to the robots! They will use this to their advantage and we will perish!