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I'm excited about seeing what the current learning algorithms can do in Math. Games like Chess, Go, Starcraft as "easy" (note the quotes) compared to mathemati

I'm excited about seeing what the current learning algorithms can do in Math. Games like Chess, Go, Starcraft as "easy" (note the quotes) compared to mathematical theorem proving, and using the bag of tricks (known theorems and axioms) to solve new problems. Heck, even understanding a problem is a challenge in itself. Just look at the sample question provided: Let n be an integer greater than or equal to 3. Prove that there is a set of n points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area. 99.9% of the people would have no idea what to do here.

riku_iki 6y ago
with AI system it is also hard, you need to have some knowledge graph which would explain what is plane, distance, etc, and map it on some set of axioms, theorems, predicates and functions.
- currymj 6y ago
  this is what Lean and other theorem provers are trying to build. which would be useful whether you have a human operator generating the proofs or some kind of automated system.
est31 6y ago
One shouldn't use human concepts for "easy" vs "hard" when applied to AI capabilities. Machines aren't humans. Coal mining is hard for humans, but easy for a Bagger 288. Similarily, computers can do millions of floating point operations per second, where even a single operation would be very hard for humans, especially when larger numbers are involved. Doing the laundry, cleaning the toilet, etc. are all "simple" tasks but we don't have any humanoid robots yet that can do them. Also important to note: human capabilities are no natural constant. Once you reach human performance in some area, it's not guaranteed that you'll plateau, but instead likely that you'll outstrip human capabilities, especially in areas where there hasn't been billions of years of evolution led engineering like abstract thinking.
- ChuckNorris89 6y ago
  >Doing the laundry, cleaning the toilet, etc. are all "simple" tasks but we don't have any humanoid robots yet that can do them. That's because we have millions of unfortunate underpaid workers on the market with no better options that will do these chores cheaper than it costs to design, build and maintain a robot that can do such things. Hell, Atlas from Boston Dynamics can already do parkour so I'm sure they can program it to scrub toilets and do laundry but who would buy it for that when the minimum wage is what it is? The tech is nearly here, the business case isn't until the price comes orders of magnitude down.
  - est31 6y ago
    This is what the 3rd place winner of the World Robot Summit 2018 looks like cleaning a toilet: https://www.youtube.com/watch?v=vx3QdaPD8bw https://www.youtube.com/watch?v=vx3QdaPD8bw Your point about the workers is true, but there are economies of scale here. I don't think it's going to be different in robots than in other areas. As for the human competition, we often had cases where technology has put humans out of their job. They provide a baseline price that you need to undercut. Today we might use robots to clean nuclear waste sites. Tomorrow we use mass produce them and put them into people's homes. As long as you don't make a larger mess at the toilet than Fukushima you should be fine! :)
Invictus0 6y ago
I'm not a mathematician. Can I prove this by just providing an example? My points are (0,0), (0,sqrt(2)), (sqrt(8),0). This should give irrational side lengths of sqrt(2), sqrt(8), and sqrt(10). The area of this triangle is 2, which is rational. I solved this by guessing and checking the square roots of positive integers.
- im3w1l 6y ago
  The question asks you to prove that it works for every n>=3. So you have to show it works also for n=4, n=5 etc. You are on the right track with square roots though.
- hugodutka 6y ago
  You have proved it for n = 3, but you haven't for n >= 4, which the problem also requires.
  - kosievdmerwe 6y ago
    Yeah, an alternate phrasing could be "For all integers n >= 3, show that there exists a set of n points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area." For only having to find a specific n the phrasing would be something like "Find an integer n >= 3, such that ..." But in general a good heuristic is that if your solution is really simple odds are you've misunderstood the problem. IMO problems aren't easy. Even the "easy" ones take a bit of work.
otoburb 6y ago
Daniel would agree with you that this is an underappreciated and overlooked area of AI research. A 2018 blog post[1] gave an indication that he'd started to turn his attention to this particular area (mathematics) more forcefully. [1] https://dselsam.github.io/posts/2018-06-24-mathematics-our-overlooked-ability.html https://dselsam.github.io/posts/2018-06-24-mathematics-our-o...
karmakaze 6y ago
The difference between being better than humans at simple games and harder games like Go is scale. We thought this wouldn't happen for generations but here we are. I don't see the how the sorts of problems presented as being qualitatively different as ones we can already solve. The problem is scale and having the problem formally defined. We'll separate the english-to-machine readable form as a separate step as I don't think that's the 'math' part of the challenge--specifically excluding math word problems. We will develop better representations and search strategies and meta-strategy searching abilities and make advances in improving their runtimes as hardware devices continue to get faster allowing research to accelerate. For the example question, a machine could have picked a special case, right triangles where a^2 + b^2 = c^2 and the area a*b/2. Now choose a^2, b^2 as whole numbers, say 2 and 3 ==> c = root(5) with area 3. As noted in other comments, it would then have to take this premise and algebraically extend to all n>3 which I believe we have inductive provers as one example can do.
- deleted 6y ago
  [deleted]

riku_iki 6y ago

with AI system it is also hard, you need to have some knowledge graph which would explain what is plane, distance, etc, and map it on some set of axioms, theorems, predicates and functions.

currymj 6y ago
this is what Lean and other theorem provers are trying to build. which would be useful whether you have a human operator generating the proofs or some kind of automated system.

est31 6y ago

One shouldn't use human concepts for "easy" vs "hard" when applied to AI capabilities. Machines aren't humans. Coal mining is hard for humans, but easy for a Bagger 288. Similarily, computers can do millions of floating point operations per second, where even a single operation would be very hard for humans, especially when larger numbers are involved. Doing the laundry, cleaning the toilet, etc. are all "simple" tasks but we don't have any humanoid robots yet that can do them. Also important to note: human capabilities are no natural constant. Once you reach human performance in some area, it's not guaranteed that you'll plateau, but instead likely that you'll outstrip human capabilities, especially in areas where there hasn't been billions of years of evolution led engineering like abstract thinking.

ChuckNorris89 6y ago
>Doing the laundry, cleaning the toilet, etc. are all "simple" tasks but we don't have any humanoid robots yet that can do them. That's because we have millions of unfortunate underpaid workers on the market with no better options that will do these chores cheaper than it costs to design, build and maintain a robot that can do such things. Hell, Atlas from Boston Dynamics can already do parkour so I'm sure they can program it to scrub toilets and do laundry but who would buy it for that when the minimum wage is what it is? The tech is nearly here, the business case isn't until the price comes orders of magnitude down.
- est31 6y ago
  This is what the 3rd place winner of the World Robot Summit 2018 looks like cleaning a toilet: https://www.youtube.com/watch?v=vx3QdaPD8bw https://www.youtube.com/watch?v=vx3QdaPD8bw Your point about the workers is true, but there are economies of scale here. I don't think it's going to be different in robots than in other areas. As for the human competition, we often had cases where technology has put humans out of their job. They provide a baseline price that you need to undercut. Today we might use robots to clean nuclear waste sites. Tomorrow we use mass produce them and put them into people's homes. As long as you don't make a larger mess at the toilet than Fukushima you should be fine! :)

Invictus0 6y ago

I'm not a mathematician. Can I prove this by just providing an example? My points are (0,0), (0,sqrt(2)), (sqrt(8),0). This should give irrational side lengths of sqrt(2), sqrt(8), and sqrt(10). The area of this triangle is 2, which is rational. I solved this by guessing and checking the square roots of positive integers.

im3w1l 6y ago
The question asks you to prove that it works for every n>=3. So you have to show it works also for n=4, n=5 etc. You are on the right track with square roots though.
hugodutka 6y ago
You have proved it for n = 3, but you haven't for n >= 4, which the problem also requires.
- kosievdmerwe 6y ago
  Yeah, an alternate phrasing could be "For all integers n >= 3, show that there exists a set of n points in the plane such that the distance between any two points is irrational and each set of three points determines a non-degenerate triangle with rational area." For only having to find a specific n the phrasing would be something like "Find an integer n >= 3, such that ..." But in general a good heuristic is that if your solution is really simple odds are you've misunderstood the problem. IMO problems aren't easy. Even the "easy" ones take a bit of work.

otoburb 6y ago

Daniel would agree with you that this is an underappreciated and overlooked area of AI research. A 2018 blog post[1] gave an indication that he'd started to turn his attention to this particular area (mathematics) more forcefully. [1] https://dselsam.github.io/posts/2018-06-24-mathematics-our-overlooked-ability.html https://dselsam.github.io/posts/2018-06-24-mathematics-our-o...

karmakaze 6y ago

The difference between being better than humans at simple games and harder games like Go is scale. We thought this wouldn't happen for generations but here we are. I don't see the how the sorts of problems presented as being qualitatively different as ones we can already solve. The problem is scale and having the problem formally defined. We'll separate the english-to-machine readable form as a separate step as I don't think that's the 'math' part of the challenge--specifically excluding math word problems. We will develop better representations and search strategies and meta-strategy searching abilities and make advances in improving their runtimes as hardware devices continue to get faster allowing research to accelerate. For the example question, a machine could have picked a special case, right triangles where a^2 + b^2 = c^2 and the area a*b/2. Now choose a^2, b^2 as whole numbers, say 2 and 3 ==> c = root(5) with area 3. As noted in other comments, it would then have to take this premise and algebraically extend to all n>3 which I believe we have inductive provers as one example can do.

deleted 6y ago
[deleted]