If a “concept inventor program” comes up with some property of numbers that mathematicians have not so far given any thought to, it could mean EITHER
that the program has spotted something interesting that mathematicians have missed, OR
that the program has spotted something that isn’t actually mathematically interesting.
I think it’s too early to say whether this notion of “refactorable numbers” is actually useful in number theory, or whether it’s a mere curiosity that doesn’t go anywhere. My money’s on the latter.
Too early? It was discovered 16 years ago. Now it has it’s own Wikipedia article and some other hits, so someone must have found it interesting.
But yes this is the biggest problem with automated discovery, that there is no definition of “interesting”. Automated discovery systems tend to produce random garbage after enough time. One paper defined it as things which are difficult to prove, and so it discards trivial and obvious stuff (EDIT: better paper on this subject.)
Speaking as a mathematician who wrote one of the papers cited there, the concept isn’t very interesting at all. It is the sort of recreational mathematics that we enjoy playing with and is fairly natural but it isn’t the sort of thing that is going to lead to deep insights or structural mathematics. The bar of being interesting enough to have papers written on it is really low.
More like 25, actually; the first paper on these numbers was published in 1990, long before Colton’s program. So far as I can tell, in those 25 years no one has found anything else (in pure mathematics or elsewhere) that is made easier, or understood better, as a result of having the notion of “refactorable numbers” available. So perhaps rather than “too early to say”, I should say that they’ve been discovered and found to be nothing more than a curiosity.
But I don’t think I should. In mathematics, sometimes quite a long time passes between when a thing is first thought of and when it’s first found to be useful. Maybe refactorable numbers will turn out to be a key concept in the proof of the Riemann Hypothesis in 2137. I wouldn’t bet on it, though.
Lest I be misunderstood, that doesn’t mean I think there’s anything wrong with being interested in refactorable numbers or studying their properties for their own sake. But if they’re just a curiosity, as seems to be the case, then I’m not so impressed by HR’s achievement in finding them. It seems like one could do about equally well by picking (say) ten simply-defined properties of a positive integer (say: number of divisors, sum of divisors, sum of squares of divisors, number of prime factors, sum of prime factors, Euler totient function, number of 1s in binary expansion, Ramanujan tau function, floor of square root, floor of base-2 logarithm), considering all pairs of them and putting a few things like “equals”, “divides”, “equals square of” in between. I bet that at least 10% of the results will define sets of positive integers that are neither much less new, nor much less interesting, than the refactorable numbers were when HR found them.
Embarrassingly: no, not offhand. I have a general impression that this happens sometimes, rather than specific examples. Maybe some of the specialfunctionology used by Louis de Branges to prove the Bieberbach conjecture?
Supposedly it did discover some interesting things from it. Like if the sum of the divisors of an integer is prime, then the number of divisors must be prime. That seems interesting, but I’m not a mathematician.
Well, let’s see. Sum of divisors is product of 1+p+...+p^k so if that’s prime then the number must be a prime power so only one p, and then if the number of terms in that sum is composite you get an obvious nontrivial factor for the sum of divisors, and we’re done. I reckon something I can prove in two minutes without making use of the notion of refactorable numbers probably isn’t a great argument for the importance of that notion.
Perhaps the point is that this theorem was discovered while thinking about refactorable numbers. I rather doubt it, but in any case the theorem itself seems like a cute curiosity rather than something any number theorist would care much about.
(I am a mathematician, though I’ve been out of academia for years and was never a number theorist.)
I expect that may be the case. Colton and a few others were working on similar stuff in the late 1990s and early 2000s but there’s been less interest in the last few years. I expect that will change soon.
Neat—can you give me some pointers on things to read that caused you to become more ?optimistic?. I want to be optimistic, too!
I made a list of AI developments in 2014.
In the case you haven’t seen the JoshuaZ’s link, it’s a good starter:
https://cs.uwaterloo.ca/journals/JIS/colton/joisol.html
The “concept inventor program HR”, was able to invent “those numbers, which the numbers of divisors also divides that number”.
One million or so human mathematicians in the last 300 years in the field of Number theory all failed to spot it before.
I, myself, recently build an efficient machine for mass producing this:
http://www.critticall.com/cubus_maximus/test.html
Too hard for intelligent humans, too big task for a supercomputers with a brute force approach.
And for the hardware we’ll also need.
http://www.sciencedaily.com/releases/2015/02/150202160711.htm
If a “concept inventor program” comes up with some property of numbers that mathematicians have not so far given any thought to, it could mean EITHER
that the program has spotted something interesting that mathematicians have missed, OR
that the program has spotted something that isn’t actually mathematically interesting.
I think it’s too early to say whether this notion of “refactorable numbers” is actually useful in number theory, or whether it’s a mere curiosity that doesn’t go anywhere. My money’s on the latter.
Too early? It was discovered 16 years ago. Now it has it’s own Wikipedia article and some other hits, so someone must have found it interesting.
But yes this is the biggest problem with automated discovery, that there is no definition of “interesting”. Automated discovery systems tend to produce random garbage after enough time. One paper defined it as things which are difficult to prove, and so it discards trivial and obvious stuff (EDIT: better paper on this subject.)
Speaking as a mathematician who wrote one of the papers cited there, the concept isn’t very interesting at all. It is the sort of recreational mathematics that we enjoy playing with and is fairly natural but it isn’t the sort of thing that is going to lead to deep insights or structural mathematics. The bar of being interesting enough to have papers written on it is really low.
More like 25, actually; the first paper on these numbers was published in 1990, long before Colton’s program. So far as I can tell, in those 25 years no one has found anything else (in pure mathematics or elsewhere) that is made easier, or understood better, as a result of having the notion of “refactorable numbers” available. So perhaps rather than “too early to say”, I should say that they’ve been discovered and found to be nothing more than a curiosity.
But I don’t think I should. In mathematics, sometimes quite a long time passes between when a thing is first thought of and when it’s first found to be useful. Maybe refactorable numbers will turn out to be a key concept in the proof of the Riemann Hypothesis in 2137. I wouldn’t bet on it, though.
Lest I be misunderstood, that doesn’t mean I think there’s anything wrong with being interested in refactorable numbers or studying their properties for their own sake. But if they’re just a curiosity, as seems to be the case, then I’m not so impressed by HR’s achievement in finding them. It seems like one could do about equally well by picking (say) ten simply-defined properties of a positive integer (say: number of divisors, sum of divisors, sum of squares of divisors, number of prime factors, sum of prime factors, Euler totient function, number of 1s in binary expansion, Ramanujan tau function, floor of square root, floor of base-2 logarithm), considering all pairs of them and putting a few things like “equals”, “divides”, “equals square of” in between. I bet that at least 10% of the results will define sets of positive integers that are neither much less new, nor much less interesting, than the refactorable numbers were when HR found them.
Can you give an example of something in mathematics that was invented, condemned as boring, and a decade or more later found to be useful?
Embarrassingly: no, not offhand. I have a general impression that this happens sometimes, rather than specific examples. Maybe some of the specialfunctionology used by Louis de Branges to prove the Bieberbach conjecture?
Supposedly it did discover some interesting things from it. Like if the sum of the divisors of an integer is prime, then the number of divisors must be prime. That seems interesting, but I’m not a mathematician.
Well, let’s see. Sum of divisors is product of 1+p+...+p^k so if that’s prime then the number must be a prime power so only one p, and then if the number of terms in that sum is composite you get an obvious nontrivial factor for the sum of divisors, and we’re done. I reckon something I can prove in two minutes without making use of the notion of refactorable numbers probably isn’t a great argument for the importance of that notion.
Perhaps the point is that this theorem was discovered while thinking about refactorable numbers. I rather doubt it, but in any case the theorem itself seems like a cute curiosity rather than something any number theorist would care much about.
(I am a mathematician, though I’ve been out of academia for years and was never a number theorist.)
Minor note: the Colton article I linked to is from 1999.
I have noticed this. And it ain’t good, at all!
Until now, it should be a lot of artificially discovered concepts around, but which are nowhere to be seen!
Perhaps it’s a low hanging fruit?
I expect that may be the case. Colton and a few others were working on similar stuff in the late 1990s and early 2000s but there’s been less interest in the last few years. I expect that will change soon.