48 replies to this topic

[MarcZ]

So I couldn't really find a location to discuss mathematics trends and the future of mathematics, I also found there was a bit of a lack of mathematics related stuff in the timeline (if you don't include computing).

So what does everyone think the future of mathematics looks like, it seems it's a lot harder to predict as it's very hard to guess when someone will have the Eureka moment to make a groundbreaking proof like the Poincare Conjecture which was solved in 2003. Also how long do you think it is until we resolve the remaining 6 millennium problems (Poincare was the 7th and to date only solved one), especially the notorious P versus NP problem which could rock the entire foundations of computing theory? (And to a lesser degree the remaining Hilbert's problems)

Also does everyone think the population will become much more mathematically literate in the future being able to deal with mathematics of much greater complexity? How will education evolve to optimize public mathematical education?

For anyone unfamiliar with the 7 millennium problems, please see here: http://en.wikipedia...._Prize_Problems

Anyone is free to comment, I'm not a huge expert therefore all us amateurs in mathematics can discuss.

[RCM88x]

AI will solve all our problems.

[MarcZ]

AI will solve all our problems.

I don't think so, AI will be based upon mathematical knowledge we already know, and machine learning is based off mathematics that we have developed as well. I don't think it likely a machine will be able to discover a type of mathematics that a human hasn't, proofs are something that I don't think AI will ever be doing.

The P versus NP problem is a good illustration of my argument I think.

Edited by MarcZ, 04 May 2012 - 01:00 AM.

[Alric]

As computers get more and more powerful, it seems likely they will be able to solve any problem. Even if they can't figure a problem out, they will just brute force it by doing guess and check on trillions of possible numbers at a time. They will be so powerful they can go through those trillions of numbers to find patterns that we can't as well.

Guess and check is extremely slow and tedious, but for a computer it is trivial.

[MarcZ]

As computers get more and more powerful, it seems likely they will be able to solve any problem. Even if they can't figure a problem out, they will just brute force it by doing guess and check on trillions of possible numbers at a time. They will be so powerful they can go through those trillions of numbers to find patterns that we can't as well.

Guess and check is extremely slow and tedious, but for a computer it is trivial.

Computers can already do this for petty Calculus and Differential Equations, but computers cannot solve high level mathematics problems which are not simple computation but exercises in theory, logic and human creativity. Although math is used to find patterns, finding new types of mathematics, and proving a conjecture to be true or false is not the same thing as performing a bajillion different kinds of already established operations until we get the result.

[Alric]

I agree with that for the most part, but I don't think it takes creativity. For humans it does but for a computer they can just try every possible thing. They can use brute force methods in place of creativity, and I am not talking about just stuff we know. You can run a computer to check new things as well.

Also that is kind of the worst case scenario, computer get thousands of times faster but the software remains kind of stupid. I think the software will get far better, and we will get computers that can learn and program them self. When we get to that point, there is no reason they shouldn't be better able to solve math.

[MarcZ]

I agree with that for the most part, but I don't think it takes creativity. For humans it does but for a computer they can just try every possible thing. They can use brute force methods in place of creativity, and I am not talking about just stuff we know. You can run a computer to check new things as well.

Also that is kind of the worst case scenario, computer get thousands of times faster but the software remains kind of stupid. I think the software will get far better, and we will get computers that can learn and program them self. When we get to that point, there is no reason they shouldn't be better able to solve math.

Perhaps a computer can try every possible thing, but the problem is it probably won't understand when it has proven something, because it will be restricted to mathematical principles humans have taught it and most likely unable to "get the bigger picture", and really humans will have to learn it before the computer in my opinion. And yes, solving and proving are two different beasts though.

Edited by MarcZ, 04 May 2012 - 02:46 AM.

[SG-1]

Alric thinks that computers are already conscious, so I can see why he would believe that a machine could come up with original ideas within our lifetime. Our lifetime being 80 years, disregarding the shot at immortality (I believe I should refer to immortality pessimistically until it is proven, but I have the highest hope for it).

I'm not a mathematician, but I do think that quantum computers will be able to create codes that will be unbreakable and they will be able to crunch anything thrown at it. Making a new math or finding solutions that we would not understand if we saw the answer to would not happen by the 2070s which is around where all of our lifespans are on this forum.

[MarcZ]

Alric thinks that computers are already conscious, so I can see why he would believe that a machine could come up with original ideas within our lifetime. Our lifetime being 80 years, disregarding the shot at immortality (I believe I should refer to immortality pessimistically until it is proven, but I have the highest hope for it).

I'm not a mathematician, but I do think that quantum computers will be able to create codes that will be unbreakable and they will be able to crunch anything thrown at it. Making a new math or finding solutions that we would not understand if we saw the answer to would not happen by the 2070s which is around where all of our lifespans are on this forum.

That's the main objective of the P versus NP problem, is can we prove that computers can solve or crack any formula. Most mathematicians believe the answer is no. I don't think it's likely this problem will be resolved this century if not the next few centuries, most of the research into it right now is simply finding that practically all our modern mathematical tools can't solve it, we will need to create a new set of mathematical tools before it can be solved, as for infinitely powerful computers that can solve any equation, that will only be true if the P versus NP proof turns out to be true.

Mind you if it was found to be true say tomorrow, all of global cybersecurity would immediately be ineffective, destroying the global cyber economy as all our encryption techniques that provide transaction and financial security would be useless. In fact for everyone wanting computers that can solve anything if it were to occur soon it would actually be apocalyptic and a major cybersecurity threat. But I will go with the math experts who think this scenario is unlikely simply because they don't believe P versus NP is true.

Edited by MarcZ, 04 May 2012 - 03:40 AM.

[Alric]

I do not think computers are conscious. What I believe is that consciousness comes from simple beginnings and that humans are at their core just like computers. What a computer is doing isn't much different from what humans are doing. Humans are far more complex but the complexity in our brain comes from simple things repeated a great many times.

Also I plan on being alive well past 2070. Even at current technology it is possible for me to still be alive then, and I am expecting a great deal of health technology to help improve my odds.

Back to the topic at hand, going by the timeline on this site, in 2057 we will have desktop computers that have more processing power than all humans who have ever existed. That is just a desk top computer, the super computers of that time will be insanely powerful. It is hard to believe that those computers will not be able to solve our current problems. I suspect they will be working on even harder problems around that time.

[MarcZ]

I do not think computers are conscious. What I believe is that consciousness comes from simple beginnings and that humans are at their core just like computers. What a computer is doing isn't much different from what humans are doing. Humans are far more complex but the complexity in our brain comes from simple things repeated a great many times.

Also I plan on being alive well past 2070. Even at current technology it is possible for me to still be alive then, and I am expecting a great deal of health technology to help improve my odds.

Back to the topic at hand, going by the timeline on this site, in 2057 we will have desktop computers that have more processing power than all humans who have ever existed. That is just a desk top computer, the super computers of that time will be insanely powerful. It is hard to believe that those computers will not be able to solve our current problems. I suspect they will be working on even harder problems around that time.

See here is where people are getting confused. Yes I have no doubt computers will be extremely powerful, that however does not translate into them being able to perform mathematical proofs, I don't think in all of history a computer has ever proved anything, data has simply been input into conditions given to the computer and the answer is output, however proofs often involve discovering and making conditions with mathematical tools that a computer can't make or discover on it's own because all it's programming would be based off the mathematical guidelines and abilities we GAVE it or TAUGHT it.

[Alric]

I can't say I am an expert on advanced math since I am still studying math in collage, and I am only at calculus level. I am great at googling though. I found there is a wiki on this, and there has been some cases where computers have proved things that a human couldn't.

http://en.wikipedia....-assisted_proof

[Zeitgeist123]

the views exchanged by both of you just made me think: if alric is right and that computers cannot invent a higher mathematics that hasnt been discovered or implemented yet or may discovered it but cannot distinguish that it made a breakthrough in the field of mathematics, then we dont have to worry about AI taking over the planet. it can also mean that AI, would only be very useful if its merged with the human brain.

[Alric]

That is actually where I think the real cool technology is at, is humans merging with it. It may be that this topic is totally irrelevant, because humans might have computers in their head and you get the best of both humans and computers.

[MarcZ]

I can't say I am an expert on advanced math since I am still studying math in collage, and I am only at calculus level. I am great at googling though. I found there is a wiki on this, and there has been some cases where computers have proved things that a human couldn't.

http://en.wikipedia....-assisted_proof

Yeah I'm at about the same level as you. The Wikipedia article is interesting, but it's a very particular kind of proof which is via exhaustion which means it only works for cases where you have a general idea of the solution but not the means to test them all until correct. It would not have been useful for a proof of something like the Poincare Conjecture where the general idea or road to a solution was not known and had to be developed by Perelman when he strung together different theories from across topology like Ricci Flow and Thurston's Geometrization Conjecture.

[MarcZ]

the views exchanged by both of you just made me think: if alric is right and that computers cannot invent a higher mathematics that hasnt been discovered or implemented yet or may discovered it but cannot distinguish that it made a breakthrough in the field of mathematics, then we dont have to worry about AI taking over the planet. it can also mean that AI, would only be very useful if its merged with the human brain.

I'm afraid this question really won't be answered until we can see if P versus NP is true or false, and see what the limitations of machine learning are.

Just an update that this explains what NP-complete problems are, with current computing power some math problems would take billions of years to perform, and without a P = NP which is unlikely scientists can only use approximations to make a best guess as opposed to an absolute answer, so the ultimate power of computers very much rests with the P = NP or P =/= NP proof. See: http://en.wikipedia....iki/NP-complete. It may be that computers have a limit to realistic computing ability, just as light is the universal speed limit to all other things.

Edited by MarcZ, 04 May 2012 - 06:20 AM.

[Alric]

Well when you look at something and say it would take a billion years to perform using current computers, I say well that is doable by future computers. A billion years is a long time, but really it isn't. In 50-60 years if current trends continue we will have computers a billion times faster than the ones today. Meaning if something will take a billion years to calculate today, it would only take a single year in the future.

Now we can't say for sure that computers will continue to increase power at the exponential rate that they are now, but I doubt they would really have to. Since software and stuff is always improving as well, I don't think the computers of 50 years for now will be limited to solely brute forcing it.

[MarcZ]

Well when you look at something and say it would take a billion years to perform using current computers, I say well that is doable by future computers. A billion years is a long time, but really it isn't. In 50-60 years if current trends continue we will have computers a billion times faster than the ones today. Meaning if something will take a billion years to calculate today, it would only take a single year in the future.

Now we can't say for sure that computers will continue to increase power at the exponential rate that they are now, but I doubt they would really have to. Since software and stuff is always improving as well, I don't think the computers of 50 years for now will be limited to solely brute forcing it.

Perhaps but I think P versus NP specifically said those billion year problems were for only moderate sized questions, meaning it would be virtually impossible even in the future to deal with the big hard major optimization questions, unless humans find new tools first, thus I still hold to my belief that humans need to discover new mathematical tools before computers can increase their abilities beyond simply doing stuff faster.

[Shimmy]

If my degree and masters in maths taught me anything it's that all mathematical problems can be solved via induction.
And I've proven that statement is always true, using induction.

[MarcZ]

If my degree and masters in maths taught me anything it's that all mathematical problems can be solved via induction.
And I've proven that statement is always true, using induction.

Maybe so, but what do you think of a computers ability to solve complex math problems via induction?