6 ms·

When I was in grad school, I had the opportunity to take a course from my adviser in which he discussed his current research and some open questions. It was a

by Dove 2mo ago

When I was in grad school, I had the opportunity to take a course from my adviser in which he discussed his current research and some open questions. It was a relatively accessible subject area and the questions were sometimes easy enough that we could meaningfully contribute. On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it. It was the sort of thing that he really wanted to be true; he liked things smooth and beautiful. I, on the other hand, hoped it was false as I like the weird and exceptional in mathematics. It was also the case that I had absolutely no command of the sort of machinery that one would use to prove such a thing, but I could certainly look for a counterexample. I learned on Monday that he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour. My single (quite small) contribution to mathematical research was a counterexample because it was all I could do. The story does illustrate that it can be helpful to have people with different tools, hopes, and motivations working on a problem, though. I was not, and will never be, even a shadow of that great mathematiciam I studied under, but on that occasion, I had reason to look in a different direction than he did.

bananaflag 2mo ago
> It was also the case that I had absolutely no command of the sort of machinery that one would use to prove such a thing, but I could certainly look for a counterexample. Hm, as a mathematician, my experience feels opposite. A proof would be an adaptation of a proof I know, some tweaking it here and there. A counterexample would require some deep understanding of the structure of the objects involved, which frequently is beyond my comprehension. But probably this is because I think of quite abstract objects which are harder to grasp. For numbers or polynomials, this would be the other way round.
- Dove 2mo ago
  We were studying geometry - my adviser was the great Branko Grünbaum: https://en.wikipedia.org/wiki/Branko_Gr%C3%BCnbaum https://en.wikipedia.org/wiki/Branko_Gr%C3%BCnbaum The conjecture had to do with whether one convex polygon could be continuously deformed into another while remaining convex, under certain conditions and constraints. The answer turns out to be no, but surprise and disappointment are understandable reactions to that outcome. It was indeed much more practical for a young grad student to look for a clever misbehaving polygon than to try to prove something about all of them at once.
  - bananaflag 2mo ago
    Thanks! It makes sense.
  - jobigoud 2mo ago
    How interesting I was just reading yesterday his paper "An enduring error" about how we have been miscounting the Archimedean solids for two thousand years. But also, for this conjecture to be wrong is quite surprising to me. Intuitively I would think any convex polygon to be topologically equivalent to a circle, and any convex n-gon should be deformable into its regular version, then back to the other one…
    - Dove 2mo ago
      He was an incredibly great man, and it remains a privilege to have learned from him. While I left mathematics for engineering, his audacious asking of the right questions around the philosophical foundations of an endeavor remains a large influence on me and has become a hallmark of my engineering work. It makes me smile that you are familiar with him as well. :) --- Here's the full puzzle, as best I remember it: Suppose you have two convex polygons with the following very specific relationship: one has been created from the other by making one side stretchy, moving an adjacent side on a hinge, and keeping the remaining sides fixed. For example, imagine a square with a top side made of rubber, and a rigid right side hinged at the lower right corner. You can make a series of convex polygons in a continuous fashion by rotating that right side on its hinge. The question is, if you have two convex polygons that differ only by this one stretchy side and this adjacent hinged side, can you guarantee that you can always smoothly deform the one into the other by this hinging method while keeping the whole thing convex? That is to say, if you are deforming one polygon into another by this hinge and rubber band method, if your starting polygon and ending polygon are both convex, are all the middle ploygons guaranteed to be? The answer is intuitively obviously yes, but in point of fact, it is no. --- To bring this back to the original story, the question was a small step in a larger constructive proof he was working on. The overall result was already known - in fact, we had just discussed it in class - but the proof had this distressingly jerky, discrete movement to it, and he was hoping to construct a more pleasing and smooth algorithm as a more satisfying proof. As for me, I would not have known how to begin to prove even the smaller question... but I sure could doodle a counterexample. ;) I therefore looked for a one with all the gusto of a young grad student hoping against all odds to do something helpful. You may look with all the confidence of knowing there is something to find, which is also a tremendous help. I only know his side of the story because he started Monday's class with this line: "I spent the entire weekend trying to prove the result, without success, and it was a good thing too, as there was a counterexample in my box this morning." He did seem genuinely frustrated, but I also wouldn't have put it past him to have exaggerated that part for the laugh. Anyway. Asking an AI to find counterexamples under such circumstances seems to me similarly reasonable to asking grad students. In my engineering work, I find there is a balance between using the AI to improve and augment your work (especially to call on the diverse perspectives in its training set), and using the AI to avoid your work. I do think the best experience and results are found in that balance. I would expect the same to be true in mathematics.
- veunes 2mo ago
  I think the asymmetry depends on the representation
codemog 2mo ago
There’s a story in How to Solve It that’s basically the same.
parl_match 2mo ago
> I learned on Monday that he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour. He spent an entire weekend before having the wisdom to pause, and let someone else contribute their time to finding a counter.
- Dove 2mo ago
  This was back when the internet was mostly chain emails and personal web pages, being unreachable once you went home for the weekend was perfectly normal and expected, and automatically thinking the worst of people was not a common form of public performance art. ;)
- jibal 2mo ago
  > having the wisdom Ahem. > On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it. It was a parallel effort ... we don't know how many people were working on it that weekend. And since the professor wanted it to be true and presumably believed that it was true, why the heck should he wait for students of unknown number and ability to find a counterexample that he didn't think existed?
- lotharbot 2mo ago
  [Background: I am Dove's husband, and remember the incident described] What actually happened was the professor posed the question in class as an open research question, which he regularly did, with no particular expectation that any students would work on it or make meaningful progress. My wife happened to sit down in her office and come up with a counterexample, which she wrote on paper and left in the professor's physical mailbox on campus fairly late in the evening. He had almost certainly already left campus for the weekend by that time. This wasn't a case of some sort of arrogance or lack of wisdom, just a case of non-instantaneous asynchronous analog communication methods. This particular professor was extremely good at letting others contribute. As soon as the contribution actually reached him, he assessed it and used it.
veunes 2mo ago
This is probably part of why machines are doing so well at counterexamples. They have no aesthetic commitment to the conjecture and no embarrassment about producing something ugly
- OscarCunningham 2mo ago
  They're trained on human data. I would expect them to emulate human biases as closely as possible.
  - SiempreViernes 2mo ago
    Is it? I'd expect most of the training set to be synthetic data extrapolated from a small set of human authored texts.
    - TeMPOraL 2mo ago
      Most of the training set is half of the Internet. LLMs are pre-trained on general set of human biases and patterns of thinking.
  - 2b3a51 2mo ago
    Your comment stopped me in my tracks a little bit. Is a 'bias' in a piece of writing generally a property of word to word choice and sentence to sentence construction or is it something more nebulous? Especially in terms of the appreciation of mathematics and someone's hesitance about publishing a mathematical argument they think is ugly or brute forced in some way.
    - sebastiennight 2mo ago
      > Is a 'bias' in a piece of writing generally a property of word to word choice and sentence to sentence construction or is it something more nebulous? You might be fascinated when you read the story of Golden Gate Claude: https://www.anthropic.com/news/golden-gate-claude https://www.anthropic.com/news/golden-gate-claude
  - 2mo ago
kqr 2mo ago
> On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it. This kind of professor/researcher/teacher needs more praise. One of the first engineering courses I took when I started out in higher education was taught by such a person. Maybe it's just me, but I never felt so welcomed and included during my time in higher education as when that lecturer told a bunch of first-year students "here are some things we haven't figured out which you can help with, let me know if you come up with something". It was inspiring and a great introduction to what's otherwise a rather dull first couple of years of academia.
- jibal 2mo ago
  https://en.wikipedia.org/wiki/George_Dantzig https://en.wikipedia.org/wiki/George_Dantzig > During his study in 1939, Dantzig solved two unsolved problems in statistics due to a misunderstanding. Near the beginning of a class, Professor Neyman wrote two problems on the blackboard. Dantzig arrived late and assumed that they were a homework assignment. According to Dantzig, they "seemed to be a little harder than usual", but a few days later he handed in completed solutions for both problems, still believing that they were an assignment that was overdue.[4][6] Six weeks later, an excited Neyman eagerly told him that the problems he had solved were two of the most famous unsolved problems in statistics.[2][4] He had prepared one of Dantzig's solutions for publication in a mathematical journal.[7] This story spread and was used as a motivational lesson demonstrating the power of positive thinking. Over time, some facts were altered, but the basic story persisted in the form of an urban legend and as an introductory scene in the 1997 film Good Will Hunting.[6]
lou1306 2mo ago
> he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour. For a more extreme (although somewhat inverted) version of this, see Zeeman. He spent years trying to find a knotted sphere in a 5D space. Then realised this was impossible and got a proof for it in a few hours. [1] [1] https://ima.org.uk/28009/sir-erik-christopher-zeeman-the-mathematician-who-did-everything/ https://ima.org.uk/28009/sir-erik-christopher-zeeman-the-mat...
- gowld 2mo ago
  Trying and failing to prove something tells you quite a lot about what a counterexample would look like.
ITVerticals 2mo ago
[flagged]

by Dove 2mo ago

bananaflag 2mo ago

> It was also the case that I had absolutely no command of the sort of machinery that one would use to prove such a thing, but I could certainly look for a counterexample. Hm, as a mathematician, my experience feels opposite. A proof would be an adaptation of a proof I know, some tweaking it here and there. A counterexample would require some deep understanding of the structure of the objects involved, which frequently is beyond my comprehension. But probably this is because I think of quite abstract objects which are harder to grasp. For numbers or polynomials, this would be the other way round.

Dove 2mo ago
We were studying geometry - my adviser was the great Branko Grünbaum: https://en.wikipedia.org/wiki/Branko_Gr%C3%BCnbaum https://en.wikipedia.org/wiki/Branko_Gr%C3%BCnbaum The conjecture had to do with whether one convex polygon could be continuously deformed into another while remaining convex, under certain conditions and constraints. The answer turns out to be no, but surprise and disappointment are understandable reactions to that outcome. It was indeed much more practical for a young grad student to look for a clever misbehaving polygon than to try to prove something about all of them at once.
- bananaflag 2mo ago
  Thanks! It makes sense.
- jobigoud 2mo ago
  How interesting I was just reading yesterday his paper "An enduring error" about how we have been miscounting the Archimedean solids for two thousand years. But also, for this conjecture to be wrong is quite surprising to me. Intuitively I would think any convex polygon to be topologically equivalent to a circle, and any convex n-gon should be deformable into its regular version, then back to the other one…
  - Dove 2mo ago
    He was an incredibly great man, and it remains a privilege to have learned from him. While I left mathematics for engineering, his audacious asking of the right questions around the philosophical foundations of an endeavor remains a large influence on me and has become a hallmark of my engineering work. It makes me smile that you are familiar with him as well. :) --- Here's the full puzzle, as best I remember it: Suppose you have two convex polygons with the following very specific relationship: one has been created from the other by making one side stretchy, moving an adjacent side on a hinge, and keeping the remaining sides fixed. For example, imagine a square with a top side made of rubber, and a rigid right side hinged at the lower right corner. You can make a series of convex polygons in a continuous fashion by rotating that right side on its hinge. The question is, if you have two convex polygons that differ only by this one stretchy side and this adjacent hinged side, can you guarantee that you can always smoothly deform the one into the other by this hinging method while keeping the whole thing convex? That is to say, if you are deforming one polygon into another by this hinge and rubber band method, if your starting polygon and ending polygon are both convex, are all the middle ploygons guaranteed to be? The answer is intuitively obviously yes, but in point of fact, it is no. --- To bring this back to the original story, the question was a small step in a larger constructive proof he was working on. The overall result was already known - in fact, we had just discussed it in class - but the proof had this distressingly jerky, discrete movement to it, and he was hoping to construct a more pleasing and smooth algorithm as a more satisfying proof. As for me, I would not have known how to begin to prove even the smaller question... but I sure could doodle a counterexample. ;) I therefore looked for a one with all the gusto of a young grad student hoping against all odds to do something helpful. You may look with all the confidence of knowing there is something to find, which is also a tremendous help. I only know his side of the story because he started Monday's class with this line: "I spent the entire weekend trying to prove the result, without success, and it was a good thing too, as there was a counterexample in my box this morning." He did seem genuinely frustrated, but I also wouldn't have put it past him to have exaggerated that part for the laugh. Anyway. Asking an AI to find counterexamples under such circumstances seems to me similarly reasonable to asking grad students. In my engineering work, I find there is a balance between using the AI to improve and augment your work (especially to call on the diverse perspectives in its training set), and using the AI to avoid your work. I do think the best experience and results are found in that balance. I would expect the same to be true in mathematics.
veunes 2mo ago
I think the asymmetry depends on the representation

codemog 2mo ago

There’s a story in How to Solve It that’s basically the same.

parl_match 2mo ago

> I learned on Monday that he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour. He spent an entire weekend before having the wisdom to pause, and let someone else contribute their time to finding a counter.

Dove 2mo ago
This was back when the internet was mostly chain emails and personal web pages, being unreachable once you went home for the weekend was perfectly normal and expected, and automatically thinking the worst of people was not a common form of public performance art. ;)
jibal 2mo ago
> having the wisdom Ahem. > On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it. It was a parallel effort ... we don't know how many people were working on it that weekend. And since the professor wanted it to be true and presumably believed that it was true, why the heck should he wait for students of unknown number and ability to find a counterexample that he didn't think existed?
lotharbot 2mo ago
[Background: I am Dove's husband, and remember the incident described] What actually happened was the professor posed the question in class as an open research question, which he regularly did, with no particular expectation that any students would work on it or make meaningful progress. My wife happened to sit down in her office and come up with a counterexample, which she wrote on paper and left in the professor's physical mailbox on campus fairly late in the evening. He had almost certainly already left campus for the weekend by that time. This wasn't a case of some sort of arrogance or lack of wisdom, just a case of non-instantaneous asynchronous analog communication methods. This particular professor was extremely good at letting others contribute. As soon as the contribution actually reached him, he assessed it and used it.

veunes 2mo ago

This is probably part of why machines are doing so well at counterexamples. They have no aesthetic commitment to the conjecture and no embarrassment about producing something ugly

OscarCunningham 2mo ago
They're trained on human data. I would expect them to emulate human biases as closely as possible.
- SiempreViernes 2mo ago
  Is it? I'd expect most of the training set to be synthetic data extrapolated from a small set of human authored texts.
  - TeMPOraL 2mo ago
    Most of the training set is half of the Internet. LLMs are pre-trained on general set of human biases and patterns of thinking.
- 2b3a51 2mo ago
  Your comment stopped me in my tracks a little bit. Is a 'bias' in a piece of writing generally a property of word to word choice and sentence to sentence construction or is it something more nebulous? Especially in terms of the appreciation of mathematics and someone's hesitance about publishing a mathematical argument they think is ugly or brute forced in some way.
  - sebastiennight 2mo ago
    > Is a 'bias' in a piece of writing generally a property of word to word choice and sentence to sentence construction or is it something more nebulous? You might be fascinated when you read the story of Golden Gate Claude: https://www.anthropic.com/news/golden-gate-claude https://www.anthropic.com/news/golden-gate-claude
- 2mo ago

kqr 2mo ago

> On one particular Friday afternoon, he stated a conjecture that he hoped was true, and invited us to try to help him prove or disprove it. This kind of professor/researcher/teacher needs more praise. One of the first engineering courses I took when I started out in higher education was taught by such a person. Maybe it's just me, but I never felt so welcomed and included during my time in higher education as when that lecturer told a bunch of first-year students "here are some things we haven't figured out which you can help with, let me know if you come up with something". It was inspiring and a great introduction to what's otherwise a rather dull first couple of years of academia.

jibal 2mo ago
https://en.wikipedia.org/wiki/George_Dantzig https://en.wikipedia.org/wiki/George_Dantzig > During his study in 1939, Dantzig solved two unsolved problems in statistics due to a misunderstanding. Near the beginning of a class, Professor Neyman wrote two problems on the blackboard. Dantzig arrived late and assumed that they were a homework assignment. According to Dantzig, they "seemed to be a little harder than usual", but a few days later he handed in completed solutions for both problems, still believing that they were an assignment that was overdue.[4][6] Six weeks later, an excited Neyman eagerly told him that the problems he had solved were two of the most famous unsolved problems in statistics.[2][4] He had prepared one of Dantzig's solutions for publication in a mathematical journal.[7] This story spread and was used as a motivational lesson demonstrating the power of positive thinking. Over time, some facts were altered, but the basic story persisted in the form of an urban legend and as an introductory scene in the 1997 film Good Will Hunting.[6]

lou1306 2mo ago

> he had spent the entire weekend trying and failing to prove it. I, on the other hand, had put all my energy into finding a counterexample and had one within an hour. For a more extreme (although somewhat inverted) version of this, see Zeeman. He spent years trying to find a knotted sphere in a 5D space. Then realised this was impossible and got a proof for it in a few hours. [1] [1] https://ima.org.uk/28009/sir-erik-christopher-zeeman-the-mathematician-who-did-everything/ https://ima.org.uk/28009/sir-erik-christopher-zeeman-the-mat...

gowld 2mo ago
Trying and failing to prove something tells you quite a lot about what a counterexample would look like.

ITVerticals 2mo ago

[flagged]