Will computers make mathematicians obsolete?

[Mr.Watson]

I mean if we someday has quantum computers etc. wouldn't it be able to solve all math problems just by heave number crunching and doing so, wouldn't that meant that every mathematician would be out of job? So is math really bad career for future?

 

[rudolfstr]

not really, it you can't crunch most of problems in math just by checking a finite amount of solutions. Most problems have infinite amount of posible solutions, so checking them all is imposible.

 

[Mentallic]

Quantum computers will supposedly only be able to do whatever our current computers can do, just faster.

 

[micromass]

I mean if we someday has quantum computers etc. wouldn't it be able to solve all math problems just by heave number crunching and doing so, wouldn't that meant that every mathematician would be out of job? So is math really bad career for future?

Most mathematics problems there is an infinite cases that need to be checked. No matter how fast your computer is, you can never check infinite cases.
What a computer might possibly do is to randomly come up with theorems. So you start with axioms, and you apply the logical rules on that to come up with new true statements. Given enough time, the compute might (or not) come up with a proof for mathematical statements. But the numbers involved are extremely large here and I don't see this happening any time soon.

 

[Mr.Watson]

Most mathematics problems there is an infinite cases that need to be checked. No matter how fast your computer is, you can never check infinite cases.
What a computer might possibly do is to randomly come up with theorems. So you start with axioms, and you apply the logical rules on that to come up with new true statements. Given enough time, the compute might (or not) come up with a proof for mathematical statements. But the numbers involved are extremely large here and I don't see this happening any time soon.

Well then, will that kind of theorem finding make mathematicians obsolete? Because although you say that not anytime soon, but if somebody would make quantum computer the computing capacity would be unimaginable.

 

[Number Nine]

Well then, will that kind of theorem finding make mathematicians obsolete? Because although you say that not anytime soon, but if somebody would make quantum computer the computing capacity would be unimaginable.

It would be perfectly imaginable; it's being imagined right now in any number of academic journals. It can be quantified precisely. Quantum computers are not miracle devices, they're just very useful.

Part of the problem seems to be that you imagine mathematics to be a collection of calculations or equations, which is not true. Many branches of mathematics don't concern numbers at all, and proofs in these areas involve an extremely long process of logical deduction that often involves techniques from many different fields. It's not as simple as telling a computer (quantum or not) to "Prove the Hodge Conjecture" and then coming back in a week when it's done.

 

[micromass]

Well then, will that kind of theorem finding make mathematicians obsolete? Because although you say that not anytime soon, but if somebody would make quantum computer the computing capacity would be unimaginable.

Unimaginable?? I think you greatly overestimate the power of quantum computing.

 

[Jimmy Snyder]

Quantum computing is probabilistic so they are useful for executing probabilistic algorithms. An example is Shor's algorithm for factoring certain composite numbers. I believe the number 15 has been factored this way. Maybe not, but anyway, I changed my public key to 77 just in case. I don't know if such a computer can be used to execute deterministic algorithms.

 

[Borek]

I believe the number 15 has been factored this way. Maybe not, but anyway, I changed my public key to 77 just in case.

 

[micromass]

Quantum computing is probabilistic so they are useful for executing probabilistic algorithms. An example is Shor's algorithm for factoring certain composite numbers. I believe the number 15 has been factored this way. Maybe not, but anyway, I changed my public key to 77 just in case. I don't know if such a computer can be used to execute deterministic algorithms.

According to wiki, they factored 21 already!

 

[AlephZero]

I believe the number 15 has been factored this way.

IMO computers will be start to be able to prove math theorems when they can add to this list of jokes: http://www.gdargaud.net/Humor/OddPrime.html

 

[Patrick Kale]

I was also worried by an idea similar to this, given my knowledge of mathematics and things like that.

 

[Jimmy Snyder]

IMO computers will be start to be able to prove math theorems when they can add to this list of jokes: http://www.gdargaud.net/Humor/OddPrime.html

I must be a quantum computer. They left out priest: All the odd numbers are prime.

 

[FreeMitya]

Unimaginable?? I think you greatly overestimate the power of quantum computing.

I think pop science has distorted people's views on QC.

 

[micromass]

I think pop science has distorted people's views on QC.

Not only on Quantum Computing, sadly enough.

 

[kaos]

An interesting perspective on this issue is the notion of NP completeness in computational complexity. Theorem proving is a NP complete problem (a problem that requires an exponential amount of steps to solve((exponential to the size of the inputs )) but can be verified in a polynomial amount of steps).

Quantum computers are not believed to be able to solve NP complete problems efficiently (quantum computers are able to solve BQP complete problems and NP complete is a harder class). The reason is while a QC can represent an exponential number of states in a superposition, it is not clear how to determine which particular state represents the correct answer. There is paper on it by Bernstein and Vazirani (BBBV theorem) but i can't find a source.

There is a good blog for the computer science aspects of QCs here
http://www.scottaaronson.com/blog/?cat=17

But i am not a computer scientist and my view might be mistaken.

 

[Dembadon]

No, computers will not replace mathematicians. Computers need humans to tell them what to do.

 

[doubled5]

The mind is not made up of fairy dust and unicorns. It is itself a machine. Any and all insights you or anyone have towards math can be replicated by a computer. Super computers of today are not imperceptible relative to the brain in terms of raw computing power; so the jump to quantum computing technologies might not even be a requirement. What you need are good algorithms to simulate thought processes.

 

[symbolipoint]

NO! Computers will never make mathematicians obsolete. This is a major point made in the movie, 2001: A Space Odyssey.

 

[Containment]

Don't worry about if it'll be useless or not just do what you love. In this current age nobody can really tell you accurately what jobs you might be able to find 5-10 years down the road or what skills it'll take to get the job.

 

[collinsmark]

I was about to say that computers will never replace mathematicians, but I'll hold back so as not to risk displeasing our future, robot overlords.

 

[Mr.Watson]

It would be perfectly imaginable; it's being imagined right now in any number of academic journals. It can be quantified precisely. Quantum computers are not miracle devices, they're just very useful.

Part of the problem seems to be that you imagine mathematics to be a collection of calculations or equations, which is not true. Many branches of mathematics don't concern numbers at all, and proofs in these areas involve an extremely long process of logical deduction that often involves techniques from many different fields. It's not as simple as telling a computer (quantum or not) to "Prove the Hodge Conjecture" and then coming back in a week when it's done.

Yeah but never mind computers, I mean even some other super-computer could do that kind of theorem solving that micro mass was talking about:

"What a computer might possibly do is to randomly come up with theorems. So you start with axioms, and you apply the logical rules on that to come up with new true statements. Given enough time, the compute might (or not) come up with a proof for mathematical statements. But the numbers involved are extremely large here and I don't see this happening any time soon."

So wouldn't that kind of computers at least replace mathematicians in creating new theorems?

 

[Pythagorean]

Computers will make mathematicians who cannot use computers obsolete.

Computers make less demand for researchers to know mathematics... but only because of mathematicians writing computer programs.

 

[micromass]

Computers will make mathematicians who cannot use computers obsolete.

Computers make less demand for researchers to know mathematics... but only because of mathematicians writing computer programs.

Huh, can you expand?? How does a computer allow a mathematician to know less mathematics?? And how exactly are computers essential in pure math research?

 

[Pythagorean]

I didn't say mathematician. I'm talking about science research (including biology). I never learned the Adams mueller bashforth method (or whatever) but I can still solve differential equations using it.

I'm saying mathematicians supply us the benefits from computers so they won't be obsolete.

 

[AlephZero]

I never learned the Adams mueller bashforth method (or whatever) but I can still solve differential equations using it.

I don't doubt that you can set up a computer model and run it. but if you want ME to believe the answers, you have to show me why the numbers you get back look anything like the real solution of the DE (and I'm not talking about trivia like rounding errors, or numerical results that blow up and go to infinity)

If you didn't even know it was possible that a finite difference approximation to a DE has completely different solutions from the underlying DE, maybe you need to learn some more math before you outsource everything to a computer.

 

[Pythagorean]

I knew that... and I had to write my own fourth order runge-kutta method before being turned loose with a computer in the first place.

But... I didn't have to write my own adams moulton bashform predictor-corrector to use it. I've never even studied predictor-correctors.

I've actually had problems with them though... the interpolation step acts like a perturbation which sensitive systems can't handle, so I can actually get non-deterministic results with the predictor-corrector method, depending on where interpolation is interrupted by my conditions detector. They just don't work with systems that have a positive Lyapunov exponent. In this case, I had to go back to runge-kutta, as there's a fixed step option with it.

Anyway, I wouldn't mention any of this in my publication, so you wouldn't know it. I verify the system with common sense and by verifying that it behaves correctly in already-determined parameter regimes.

 

[Mr.Watson]

What a computer might possibly do is to randomly come up with theorems. So you start with axioms, and you apply the logical rules on that to come up with new true statements. Given enough time, the compute might (or not) come up with a proof for mathematical statements. But the numbers involved are extremely large here and I don't see this happening any time soon.

I would still like to hear your thoughts about this kind of theorem finding. Is it feasible and could it replace human mathematicians?

 

[micromass]

I would still like to hear your thoughts about this kind of theorem finding. Is it feasible and could it replace human mathematicians?

If you mean the process I described: no, it is not feasible. The numbers involved are huuuuuge. Even calculating all the possible games in chess is probably more doable.

That said, there have been programs which have been able to find good conjectures and proof them. I'll search for a link, but I remember that some people programmed a computer to find conjectures in number theory and the computer came up with the fundamental theorem of arithmetic (any nonzero, noninvertible number can be expressed in an essantially unique way as the product of prime numbers). I don't remember if the program was able to prove it, but it's a nice feat nevertheless. But of course, modern mathematics is faaaaar more advanced than simple theorems like that.

 

[Whovian]

Could a computer replace Newton and come up with calculus? Probably not.

The mind is not made up of fairy dust and unicorns. It is itself a machine. Any and all insights you or anyone have towards math can be replicated by a computer. Super computers of today are not imperceptible relative to the brain in terms of raw computing power; so the jump to quantum computing technologies might not even be a requirement. What you need are good algorithms to simulate thought processes.

Yea, but it's going to be almost impossible to create such algorithms.

 

[M Quack]

There is already a sizable branch of mathematics close to computer sciences that deals with developing algorithms for computers so that they can carry out math.

The more powerful the computers, the wider the scope for exotic math to be carried out by them, so if anything, the progress in computers has increased the need for mathematicians.

As to the mind being just another machine that can be replaced by a computer - this probably is true in theory, but we are still a far cry away from that. "Artificial intelligence" can so far only solve very few, very very well defined problems. Last time I checked, you could not tell a computer "sit in that car, learn to drive, drive yourself to that Chess tournament, learn the game, win, come back and do my homework on partial differential equations".

PS: I did not want to give the impression that I could do that :)

 

[ImaLooser]

The mind is not made up of fairy dust and unicorns. It is itself a machine. Any and all insights you or anyone have towards math can be replicated by a computer.

That's the theory, but no one knows how to do it. So one may or may not believe that.

 

[ImaLooser]

I would still like to hear your thoughts about this kind of theorem finding. Is it feasible and could it replace human mathematicians?

I looked into it in about 1995. It's called automatic theorem proving. I found that everyone in the field had given up, and had abandoned their graduate students.

The most positive results were Doug Lenat's from CMU. It so happens that I worked for a former CMU professor who told me that the consensus there was that Lenat's results were bogus.

The basic problem is that mathematicians do not know how they prove difficult theorems. Brute force search, such as used in chess, seems entirely unfeasible even in theory.

So I gave up on that.

 

[Kholdstare]

To OP: No. We have programs which simulate physics very accurately. But it did not make physicists obsolete.