Isn't it terrifying that AI can become smarter than any Mathematician?

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  • #1
Gjmdp
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It does terrify me the idea of waking up with the news that some AI is able to prove theorems better than any mathematician. That would mean humans would be useless in the field of Mathematics.
I just saw that Microsoft is currently working on creating an AI that can beat humans at the International Mathematical Olympiad (https://www.quantamagazine.org/at-t...ligence-prepares-to-go-for-the-gold-20200921/) and might have it ready by 2021. How, as a mathematician, cope with this idea?
 
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  • #2
S.G. Janssens
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It does terrify me the idea of waking up with the news that some AI is able to solve theorems better than any mathematician.
Theorems are stated and proven, not "solved"
I just saw that Microsoft is currently working on creating an AI that can beat humans at the International Mathematical Olympiad (https://www.quantamagazine.org/at-t...ligence-prepares-to-go-for-the-gold-20200921/) and might have it ready by 2021.
Doing olympiads is not the same as doing mathematics as a research activity.
How, as a mathematician, cope with this idea?
With a healthy dose of scepticism.
 
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  • #3
Gjmdp
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With a healthy dose of scepticism.

Why the scepticism? There are already several automated theorem provers that have proved hundreds of theorems. AI is getting smarter at an ever faster rate (see GPT-3) and at Microsoft many researchers in AI and Mathematics believe they might be able to create algorithms that solve IMO problems by 2021. It is obviously not the same as research, but it is undeniable that once machines are able to solve problems of the IMO (some of them are about proving statements), research is just one easy step ahead (just proving/disproving new statements). Needless to say, many professional mathematicals would struggle with those problems as they require some very creative and rigurous thinking.
 
  • #4
There are already several automated theorem provers that have proved hundreds of theorems.

Source?
 
  • #5
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There are already several automated theorem provers that have proved hundreds of theorems.
Largely theorems that are considered trivial, or theorems that mathematicians have broken down to annoying casework that has been done by the computer afterwards.
We still don't have a publication-worthy result that didn't have a mathematician involved. And even if that comes up in the future that doesn't mean mathematicians would become useless. For that computers would need to beat mathematicians in every aspect, not just in some of them.
 
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  • #6
Office_Shredder
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Proving theorems is dandy, but coming up with the interesting theorems that we care to check will still be a vastly important skill for a while even if ai achieves this ability.
 
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  • #7
inthenickoftime
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Source?
For what it’s worth, an AI software found Newton’s laws of motion by brute force. It was fed raw data coming from various physical experiments and used a recipe of simple rules from what I understand.

https://www.wired.com/2009/04/Newtonai/
 
  • #8
Office_Shredder
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That was in 2009. Have we made any actual discoveries with it yet?

https://en.m.wikipedia.org/wiki/Eureqa

It sounds like no. Given that they were able to derive hamiltonian mechanics allegedly and then eleven years later all we have is a tool that lost to Google's and Facebook's ai building tools in the model building space, I think the physicists will be ok.
 
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  • #9
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Math and physics are simple, mature disciplines with likely only marginal future improvements

biology and biochemistry is where AI can really outshine humans, where systems and amount of data are simply too complex for the human brain
 
  • #10
Not specifically AI, but at 30:40 in this video a panel (Donaldson, Kontsevich, Lurie, Tao, Taylor, Milner) discuss computer verification of proofs, and whether or not computers might be able to generate their own mathematical proofs:

 
  • #12
Vanadium 50
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Herb Wilf (with Doran Zeilbwerger) was doing computer generation of combinatoric identities back in the 80's. The world did not end.
 
  • #13
hilbert2
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It does terrify me the idea of waking up with the news that some AI is able to prove theorems better than any mathematician. That would mean humans would be useless in the field of Mathematics.
I just saw that Microsoft is currently working on creating an AI that can beat humans at the International Mathematical Olympiad (https://www.quantamagazine.org/at-t...ligence-prepares-to-go-for-the-gold-20200921/) and might have it ready by 2021. How, as a mathematician, cope with this idea?

I think it would be quite difficult for an AI to read a math problem stated in natural informal language and then convert it to an exact mathematical statement and solve/prove it.
 
  • #14
mgeorge001
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The "danger" to "mathematicians" from "AI" is pretty much a response of the vanity of some to weak AI. The reality of any actual danger to livelihoods or progress in math from AI is unknown at this time. Weak AI, i.e. via specific applications vs IQ, i.e. anthropocentric "general intelligence" was totally dismissed almost universally by mathematicians in the past. We have seen substantial recent advances over the past twenty years. What I see as a physicist, chased out of pure math, incidentally, for "my deviant ways and lack of fitness" (a reality check if you decide to take anything I write seriously), is that we currently see enormous enhancement of ordinary human capabilities from AI. If I were a savvy and ambitious "pure math guy or gal" I would just team up with some good "user friendly" software and "vastly outstrip my human competitors" if possible. But that's me, and I had a huge difficulty just finding my own proof to Urysohn's lemma when a grad student, so I am far from the elite in either math and physics (where I retired recently as a lowly physics tech). My thinking is "yes we see the storm clouds gathering on the horizon": That does not imply a major k-t extinction event, folks. Get a grip!
 
  • #15
Chris Miller
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In a lecture by a computer scientist at University of Waterloo back in the late 70s, he opined that computers would never play chess at the top human levels. But now that we've been way surpassed (by an AI who only studied the game for 4 hours) and pretty much any cell phone or PC, humans still enjoy the game and there are still professionals. Same for Go. Will be the same for math, literature, you name it.
 
  • #16
jaketodd
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In a lecture by a computer scientist at University of Waterloo back in the late 70s, he opined that computers would never play chess at the top human levels. But now that we've been way surpassed (by an AI who only studied the game for 4 hours) and pretty much any cell phone or PC, humans still enjoy the game and there are still professionals. Same for Go. Will be the same for math, literature, you name it.
Right, but when the need for the best chess player is called for, AI rules. So, the march goes on, replacing people with algorithms of how those people think. I think we may become obsolete in all areas eventually. Unless we have control of the machines. Bill Gates says the advancements of AI will make life easier for all of us, but how is that when they take all our jobs? We need like a human union so that doesn't happen, but it will. Brings to mind Terminator movies. But you in Terminator 2, Arnold says he could never feel or cry, and sacrifices himself.
 
  • #17
AndreasC
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I just saw that Microsoft is currently working on creating an AI that can beat humans at the International Mathematical Olympiad (https://www.quantamagazine.org/at-t...ligence-prepares-to-go-for-the-gold-20200921/) and might have it ready by 2021.
Um, I sincerely doubt that. IMO problems usually require very advanced intuitions and high flexibility of thinking, much less than computational power. Computers aren't very good at being flexible. It will be very interesting if that happens but as far as I am aware, there exists no software that could ever do anything comparable.
 
  • #18
BWV
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An algorithm still needs a well defined problem and rules, real life does not fit this description so strong AI remains a pipe dream
 
  • #19
AndreasC
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In a lecture by a computer scientist at University of Waterloo back in the late 70s, he opined that computers would never play chess at the top human levels. But now that we've been way surpassed (by an AI who only studied the game for 4 hours) and pretty much any cell phone or PC, humans still enjoy the game and there are still professionals. Same for Go. Will be the same for math, literature, you name it.
What some professor said in the 70s has no bearing on discussions about the limitations of computers.

By the way, we weren't surpassed by that AI first, we were surpassed years ago by more traditional programs, because chess is well suited for a computational treatment. That is because even excellent chess players, EVEN with the aid of their intuition and heuristics (ie, not trying to exhaust all combinations but rather only the "relevant" ones, determined to be such by experience and intuition), can't calculate in short notice the positions beyond 5-10 levels deep. They can lay out a strategic plan based on prior knowledge and experience, but even then that won't beat Stockfish, which has the ability to get 20+ moves deep in fractions of a second, even though its sense of strategy is garbage compared to top chess players. Even then, I've seen people who came up with positions that confuse Stockfish but are very easy to solve even for mediocre players in a very short notice. The case with the AI is a bit different due to its different approach, but its success is partly due to similar considerations.

Contrast that to math, or even worse, literature. Literature is something you can't even definitively rank, as you could rank performance at chess, and one could argue that the very fact a computer wrote it could diminish its literary value, since literature is judged on different standards.

Not that AI can't have important applications in math for some specific things etc. It can and it probably will. But some people vastly overestimate AI and what it can do, because they hear a couple of impressive things that are actually relatively straightforward conceptually, but they are left with the impression that it's some sort of black magic that can solve whatever you throw at it. But that is not the case, the level of understanding we would need to have over certain problems to even have a shot at coding an AI that could solve them just doesn't exist and it's not even certain it will any time soon. That would also involve being able to define the problem with extreme precision, and giving the AI every relevant tool it could use, both of which are just mind blowingly difficult for certain problems. I suppose it would be relatively "easy" to make an AI that could prove anything a human could within the framework of Euclidean geometry, but what about algebraic geometry? I can't even begin to imagine that with current understanding. And even if we could do that, there's a decent chance we'd have a really hard time understanding its answers.
 
  • #20
AndreasC
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Here's a thought experiment: Imagine Fermat had the know how to construct extremely advanced AI software. Imagine he fed his famous theorem into it.

Now what could happen? Well, the tools he would have given to his AI would be the ones he already had access to. But Andrew Wiles' proof used an immense wealth of concepts and theories which weren't even close to being conceived at Fermat's time. Granted, maybe the AI could find a proof with the tools in Fermat's arsenal, if it was advanced enough. But this would be so incredibly difficult, that even if that AI managed to do it, I doubt Fermat would learn much. Andrew Wiles' proof built upon volumes of already known results, used advanced and efficient concepts, and was still over 180 pages long, and incomprehensible to the vast majority of mathematicians, but opened up new roads for number theory and advanced knowledge. But one could easily imagine Fermat’s AI spitting out 20,000 pages of mangled formulas, and I don't see the value of something like that. And again, that is assuming you could make an AI that does that.

You could probably say something similar for many much simpler problems too. There are problems in math that have no reasonable proofs within certain frameworks, but are rather easy given the right concepts.
 
  • #21
Chris Miller
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Right, but when the need for the best chess player is called for, AI rules. So, the march goes on, replacing people with algorithms of how those people think. I think we may become obsolete in all areas eventually. Unless we have control of the machines. Bill Gates says the advancements of AI will make life easier for all of us, but how is that when they take all our jobs? We need like a human union so that doesn't happen, but it will. Brings to mind Terminator movies. But you in Terminator 2, Arnold says he could never feel or cry, and sacrifices himself.

It's called evolution. Maybe we were only the catalyst. Whether or not a "machine" (assuming I am not a machine) will ever feel or care (experience pleasure or suffering), or whether these are good things, is interesting to think about.
 
  • #22
Chris Miller
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What some professor said in the 70s has no bearing on discussions about the limitations of computers.

By the way, we weren't surpassed by that AI first, we were surpassed years ago by more traditional programs, because chess is well suited for a computational treatment. That is because even excellent chess players, EVEN with the aid of their intuition and heuristics (ie, not trying to exhaust all combinations but rather only the "relevant" ones, determined to be such by experience and intuition), can't calculate in short notice the positions beyond 5-10 levels deep. They can lay out a strategic plan based on prior knowledge and experience, but even then that won't beat Stockfish, which has the ability to get 20+ moves deep in fractions of a second, even though its sense of strategy is garbage compared to top chess players. Even then, I've seen people who came up with positions that confuse Stockfish but are very easy to solve even for mediocre players in a very short notice. The case with the AI is a bit different due to its different approach, but its success is partly due to similar considerations.

Contrast that to math, or even worse, literature. Literature is something you can't even definitively rank, as you could rank performance at chess, and one could argue that the very fact a computer wrote it could diminish its literary value, since literature is judged on different standards.

Not that AI can't have important applications in math for some specific things etc. It can and it probably will. But some people vastly overestimate AI and what it can do, because they hear a couple of impressive things that are actually relatively straightforward conceptually, but they are left with the impression that it's some sort of black magic that can solve whatever you throw at it. But that is not the case, the level of understanding we would need to have over certain problems to even have a shot at coding an AI that could solve them just doesn't exist and it's not even certain it will any time soon. That would also involve being able to define the problem with extreme precision, and giving the AI every relevant tool it could use, both of which are just mind blowingly difficult for certain problems. I suppose it would be relatively "easy" to make an AI that could prove anything a human could within the framework of Euclidean geometry, but what about algebraic geometry? I can't even begin to imagine that with current understanding. And even if we could do that, there's a decent chance we'd have a really hard time understanding its answers.
So you don't believe we will ever be completely surpassed in all endeavors by our own creations? I think you underestimate us.
 
  • #23
eltodesukane
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Isn't it terrifying that Mathematicians can become smarter than any non-Mathematician?
Yes, but I don't lose sleep about it.
 
  • #24
Jarvis323
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An algorithm still needs a well defined problem and rules, real life does not fit this description so strong AI remains a pipe dream
The state of art and most promising AI currently is neurosymbolic learning; it's a combination of symbolic logic with statistical learning. Basically, the key is taking chances/guessing, learning from mistakes, and building new concepts by compressing raw information into multi level abstractions, then working with those abstractions and continuously refining them, similar to what humans do.

IBMs explanation is pretty concise.

https://mitibmwatsonailab.mit.edu/category/neuro-symbolic-ai/

Google is also making leaps in this area.

I would say the dream is already realized. Right now we are most likely about at the beginning of an exponential curve.

The thing is though, it's people who are the model. We are achieving our most advanced AI by trying to copy the mechanisms people use. And people have pretty sophisticated organs to do these things (that go beyond what we really understand).

AI on the other hand can be specialized, expandable, and immortal (imagine a 1 billion year old human that just kept learning aggressively the whole time).

Quantum computing will usher in a new era for AI as well, possibly pushing AI unthinkably farther ahead in some areas than people. Nano technology in combination with robotics will play a major role also.

I expect that at some time in the next few hundred years, we will be able to seed megastructure supercomputers with quantum chips in space (completely automated). Imagine a human brainlike machine the size of Jupiter that's been doing nothing but mathematics for 1,000 years.

I think pure math isn't at a big risk though. A lot of it is esoteric, and done without any certainty it will ever have applications. We only ever scratch the surface of an infinitely deep ocean. I think we do it more for the exercise of the mind (AI would do this too), and to build up abstract concepts that we can use, to advance our conceptual framework in general.

AI doing this stuff for us will be of little use and wouldn't replace our experience, unless we could make it a process we are intimately involved in and we can integrate what's discovered. But learning takes doing, exercising creativity, and making mistakes. So in any case, no matter how far behind we are, we should still do it.
 
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  • #25
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But now that we've been way surpassed (by an AI who only studied the game for 4 hours) and pretty much any cell phone or PC, humans still enjoy the game and there are still professionals. Same for Go. Will be the same for math, literature, you name it.
We don't play chess to find new chess figures. There are competitions between humans, sure, but the mathematics equivalent of these would be the math competitions, not discovering new proofs for previously unknown things.
Bill Gates says the advancements of AI will make life easier for all of us, but how is that when they take all our jobs?
Then we have to work less and less for the same living standard. Or work on higher living standards.

Go back 300 years and ~3/4 of the population were farmers. If machines take over these jobs, how can our life possibly get better? Today 1-2% of the population are farmers, and see how the world changed.
 
  • #26
jaketodd
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Then we have to work less and less for the same living standard. Or work on higher living standards.

Go back 300 years and ~3/4 of the population were farmers. If machines take over these jobs, how can our life possibly get better? Today 1-2% of the population are farmers, and see how the world changed.

If no one has money to buy the new machines, then good luck. I think it may be in the implementation of the machines. If replicators on Star Trek are ubiquitous and cheap, then I can see that being an incredibly good thing. But if it's just the rich with the technology, not so good. Seen Elysium?

 
  • #27
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If no one has money to buy the new machines, then good luck.
If literally no one has then the machines don't exist. If only some people have: That's a question of income equality, and I don't think we should start with politics here. This is a thread about math AIs.
 
  • #28
jaketodd
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If literally no one has then the machines don't exist. If only some people have: That's a question of income equality, and I don't think we should start with politics here. This is a thread about math AIs.
I'm just exploring where this topic leads.
 
  • #29
I think that math, physics, sciences generally should be a human activity.Machines should be built to satisfy a man's needs like the fridge that preserves foods, electric kitchen that makes the food and makes it warm, a computer that makes people read with it, a bicycle to transport yourself faster from a place to another.What kind of need would AI satisfy if it replaced mathematicians' or physicists' work?Just to get quicker results?Then they would get AI for most of the other works and just push buttons on the machine and it will do the rest of the work?In my opinion the basic human needs have already been satisfied as far as making machines is concerned.Perhaps an alternative energy source, machines and technology for medicine and for specific purposes that people need very much would be suitable for advancing technology. At least as a hobby mathematics, physics and the rest of the sciences will remain for humans if AI does this.AI should not do this I think.Food that is provided today through factories and super markets is easier and more accessible to us.Education is accessible also from a young age. We have internet and bookstores to have access to information and read books.Medicine helps when someone has an illness.Also, people have opportunities to have professions today and live lives easier than in the past.Many countries do not have military wars.In relation to the past things have advanced.
 
  • #30
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What kind of need would AI satisfy if it replaced mathematicians' or physicists' work?Just to get quicker results?
What kind of need would a bike satisfy? Just to get quicker from A to B?
Yes!

Quicker results means more results per time. An AI-assisted mathematician or physicist can do more research as they can delegate more and more tasks to a computer.
Then they would get AI for most of the other works and just push buttons on the machine and it will do the rest of the work?
Maybe in the future. What's wrong with that? If you want to add two large numbers you just push buttons on the machine and it will do the rest of the work. Same thing. Just the range of tasks increases.
AI should not do this I think.
Following your arguments you should reject basically every technology.
 
  • #31
jaketodd
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But what about when the technology decides it can do pretty much anything better, and it can? Hal says "I could see your lips moving, Dave." Is humanity destined to become obsolete? Replaced with things that can think but uncertainty if they feel?
 
  • #32
Chris Miller
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But what about when the technology decides it can do pretty much anything better, and it can? Hal says "I could see your lips moving, Dave." Is humanity destined to become obsolete? Replaced with things that can think but uncertainty if they feel?
Sci-fi almost never gets the future right. Not so much obsolete as transformed I'd guess.
 
  • #33
jaketodd
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Sci-fi almost never gets the future right. Not so much obsolete as transformed I'd guess.
Sci-fi is a great thing though. If you're talking about cybernetics, you I remember writing a speech on how humans will be augmented with computers. Computer-brain interfaces. Ever heard of Alcor? The idea is to preserve the person until technology can bring them back to life and complement them.
 
  • #34
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Can AI be trained to prove only interesting theorems? If yes, can it be trained to explain us why they are interesting? As a specific example, I have in mind the Godel incompleteness theorems.
 
  • #35
Chris Miller
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Sci-fi is a great thing though. If you're talking about cybernetics, you I remember writing a speech on how humans will be augmented with computers. Computer-brain interfaces. Ever heard of Alcor? The idea is to preserve the person until technology can bring them back to life and complement them.
I agree, and have written my fair share. I see AIs as our progeny. If you view evolution as "the rise of consciousness" (De Chardin) it makes sense. We humans are still pretty amazing and complex machines. Some aspects ought to endure.
 

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