How would this model treat mathematicians working on hard open problems? P vs NP might be counter factual just because no one else is smart enough or has the right advantage to solve it. Insofar as central problems of a field have been identified but not solved, I’m not sure your model gives good advice.
I visited Mikhail Khovanov once in New York to give a seminar talk, and after it was all over and I was wandering around seeing the sights, he gave me a call and offered a long string of general advice on how to be the kind of person who does truly novel things (he’s famous for this, you can read about Khovanov homology). One thing he said was “look for things that aren’t there” haha. It’s actually very practical advice, which I think about often and attempt to live up to!
I’m ashamed to say I don’t remember. That was the highlight. I think I have some notes on the conversation somewhere and I’ll try to remember to post here if I ever find it.
I can spell out the content of his Koan a little, if it wasn’t clear. It’s probably more like: look for things that are (not there). If you spend enough time in a particular landscape of ideas, you can (if you’re quiet and pay attention and aren’t busy jumping on bandwagons) get an idea of a hole, which you’re able to walk around but can’t directly see. In this way new ideas appear as something like residues from circumnavigating these holes. It’s my understanding that Khovanov homology was discovered like that, and this is not unusual in mathematics.
By the way, that’s partly why I think the prospect of AIs being creative mathematicians in the short term should not be discounted; if you see all the things you see all the holes.
For those who might not have noticed Dan’s clever double entendre: (Khovanov) homology is literally about counting/measuring holes in weird high-dimensional spaces—designing a new homology theory is in a very real sense about looking for holes that are not (yet) there.
There’s plenty, including a line of work by Carina Curto, Katrin Hess and others that is taken seriously by a number of mathematically inclined neuroscience people (Tom Burns if he’s reading can comment further). As far as I know this kind of work is the closest to breaking through into the mainstream. At some level you can think of homology as a natural way of preserving information in noisy systems, for reasons similar to why (co)homology of tori was a useful way for Kitaev to formulate his surface code. Whether or not real brains/NNs have some emergent computation that makes use of this is a separate question, I’m not aware of really compelling evidence.
There is more speculative but definitely interesting work by Matilde Marcolli. I believe Manin has thought about this (because he’s thought about everything) and if you have twenty years to acquire the prerequisites (gamma spaces!) you can gaze into deep pools by reading that too.
Though my understanding is this is used in interp, not so much because people necessarily expect deep connections to homology, but because its just another way to look for structure in your data.
As someone who does both data analysis and algebraic topology, my take is that TDA showed promise but ultimately there’s something missing such that it’s not at full capacity. Either the formalism isn’t developed enough or it’s being consistently used on the wrong kinds of datasets. Which is kind of a shame, because it’s the kind of thing that should work beautifully and in some cases even does!
I thought it might be “look for things that might not even be there as hard as you would if they are there.” Then the koan form takes it closer to “the thereness of something just has little relevance on how hard you look for it.” But it needs to get closer to the “biological” part of your brain, where you’re not faking it with all your mental and bodily systems, like when your blood pressure rises from “truly believing” a lion is around the corner but wouldn’t if you “fake believe” it.
How would this model treat mathematicians working on hard open problems? P vs NP might be counter factual just because no one else is smart enough or has the right advantage to solve it. Insofar as central problems of a field have been identified but not solved, I’m not sure your model gives good advice.
I visited Mikhail Khovanov once in New York to give a seminar talk, and after it was all over and I was wandering around seeing the sights, he gave me a call and offered a long string of general advice on how to be the kind of person who does truly novel things (he’s famous for this, you can read about Khovanov homology). One thing he said was “look for things that aren’t there” haha. It’s actually very practical advice, which I think about often and attempt to live up to!
What else did he say? (I’d love to hear even the “obvious” things he said.)
I’m ashamed to say I don’t remember. That was the highlight. I think I have some notes on the conversation somewhere and I’ll try to remember to post here if I ever find it.
I can spell out the content of his Koan a little, if it wasn’t clear. It’s probably more like: look for things that are (not there). If you spend enough time in a particular landscape of ideas, you can (if you’re quiet and pay attention and aren’t busy jumping on bandwagons) get an idea of a hole, which you’re able to walk around but can’t directly see. In this way new ideas appear as something like residues from circumnavigating these holes. It’s my understanding that Khovanov homology was discovered like that, and this is not unusual in mathematics.
By the way, that’s partly why I think the prospect of AIs being creative mathematicians in the short term should not be discounted; if you see all the things you see all the holes.
For those who might not have noticed Dan’s clever double entendre: (Khovanov) homology is literally about counting/measuring holes in weird high-dimensional spaces—designing a new homology theory is in a very real sense about looking for holes that are not (yet) there.
Are there any examples yet, of homology or cohomology being applied to cognition, whether human or AI?
There’s plenty, including a line of work by Carina Curto, Katrin Hess and others that is taken seriously by a number of mathematically inclined neuroscience people (Tom Burns if he’s reading can comment further). As far as I know this kind of work is the closest to breaking through into the mainstream. At some level you can think of homology as a natural way of preserving information in noisy systems, for reasons similar to why (co)homology of tori was a useful way for Kitaev to formulate his surface code. Whether or not real brains/NNs have some emergent computation that makes use of this is a separate question, I’m not aware of really compelling evidence.
There is more speculative but definitely interesting work by Matilde Marcolli. I believe Manin has thought about this (because he’s thought about everything) and if you have twenty years to acquire the prerequisites (gamma spaces!) you can gaze into deep pools by reading that too.
Topological data analysis comes closest, and there are some people who try to use it for ML, eg.
Though my understanding is this is used in interp, not so much because people necessarily expect deep connections to homology, but because its just another way to look for structure in your data.
TDA itself is also a relatively shallow tool too.
As someone who does both data analysis and algebraic topology, my take is that TDA showed promise but ultimately there’s something missing such that it’s not at full capacity. Either the formalism isn’t developed enough or it’s being consistently used on the wrong kinds of datasets. Which is kind of a shame, because it’s the kind of thing that should work beautifully and in some cases even does!
No.
I thought it might be “look for things that might not even be there as hard as you would if they are there.” Then the koan form takes it closer to “the thereness of something just has little relevance on how hard you look for it.” But it needs to get closer to the “biological” part of your brain, where you’re not faking it with all your mental and bodily systems, like when your blood pressure rises from “truly believing” a lion is around the corner but wouldn’t if you “fake believe” it.
I imagine it’s something like “look for things that are notably absent, when you would expect them to have been found if there”?
Some things even withdraw. https://tsvibt.blogspot.com/2023/05/the-possible-shared-craft-of-deliberate.html#aside-on-withdrawal-and-the-leap https://tsvibt.blogspot.com/2023/09/a-hermeneutic-net-for-agency.html#withdrawal