I similarly felt in the past that by the time computers were pareto-better than I at math, there would already be mass-layoffs. I no longer believe this to be the case at all, and have been thinking about how I should orient myself in the future. I was very fortunate to land an offer for an applied-math research job in the next few months, but my plan is to devote a lot more energy to networking + building people skills while I’m there instead of just hyperfocusing on learning the relevant fields.
o1 (standard, not pro) is still not the best at math reasoning though. I occasionally give it linear algebra lemmas that I suspect it to be able to help with, but it always has major errors. Here are some examples:
I have a finite-dimensional real vector space V equipped with a symmetric bilinear form (⋅,⋅) which is not necessarily non-degenerate. Let n be the dimension of V, K be the subspace of V with (K,V)=0, and k be the dimension of K. Let W1 and W2 be n+k dimensional real vector spaces that contain V and are equipped with symmetric non-degenerate bilinear forms that extend (⋅,⋅). Show that there exists an isometry from W1 to W2. To its credit, it gave me some references that helped me prove this, but its argument was completely bogus.
Let V be a real finite-dimensinoal vector space equipped with a symmetric non-degenerate bilinear form (⋅,⋅) and let σ be an isometry of V. Prove or disprove that the restriction of (⋅,⋅) to the fixed-point subspace of σ on V is non-degenerate. (Here it sort of had the right idea but its counter-examples were never right).
Does there exist a symmetric irreducible square matrix with diagonal entries 2 and non-positive integer off-diagonal entries such that the corank is more than 1? Here it gave a completely wrong proof of “no” and, no matter how many times I corrected its errors, kept gaslighting me into believing that the general idea must work and that it’s a standard result in the field that it follows from a book that I happened to actually have read. It kept insisting this, no matter how many times I corrected its errors, until I presented with an example of a corank-1 matrix that made it clear that its idea was unfixable.
I have a strong suspicion that o3 will be much better than o1 though.
Thank you for your insight. Out of idle curiosity, I tried putting your last query into Gemini 2 Flash Thinking Experimental and it told me yes first-shot.
Here’s the final output, it’s absolutely beyond my ability to evaluate, so I’m curious if you think it went about it correctly. I can also share the full COT if you’d like, but it’s lengthy:
corank has to be more than 1, not equal to 1. I’m not sure if such a matrix exists; the reason I was able to change its mind by supplying a corank-1 matrix was that its kernel behaved in a way that significantly violated its intuition.
I similarly felt in the past that by the time computers were pareto-better than I at math, there would already be mass-layoffs. I no longer believe this to be the case at all, and have been thinking about how I should orient myself in the future. I was very fortunate to land an offer for an applied-math research job in the next few months, but my plan is to devote a lot more energy to networking + building people skills while I’m there instead of just hyperfocusing on learning the relevant fields.
o1 (standard, not pro) is still not the best at math reasoning though. I occasionally give it linear algebra lemmas that I suspect it to be able to help with, but it always has major errors. Here are some examples:
I have a finite-dimensional real vector space V equipped with a symmetric bilinear form (⋅,⋅) which is not necessarily non-degenerate. Let n be the dimension of V, K be the subspace of V with (K,V)=0, and k be the dimension of K. Let W1 and W2 be n+k dimensional real vector spaces that contain V and are equipped with symmetric non-degenerate bilinear forms that extend (⋅,⋅). Show that there exists an isometry from W1 to W2. To its credit, it gave me some references that helped me prove this, but its argument was completely bogus.
Let V be a real finite-dimensinoal vector space equipped with a symmetric non-degenerate bilinear form (⋅,⋅) and let σ be an isometry of V. Prove or disprove that the restriction of (⋅,⋅) to the fixed-point subspace of σ on V is non-degenerate. (Here it sort of had the right idea but its counter-examples were never right).
Does there exist a symmetric irreducible square matrix with diagonal entries 2 and non-positive integer off-diagonal entries such that the corank is more than 1? Here it gave a completely wrong proof of “no” and, no matter how many times I corrected its errors, kept gaslighting me into believing that the general idea must work and that it’s a standard result in the field that it follows from a book that I happened to actually have read. It kept insisting this, no matter how many times I corrected its errors, until I presented with an example of a corank-1 matrix that made it clear that its idea was unfixable.
I have a strong suspicion that o3 will be much better than o1 though.
Thank you for your insight. Out of idle curiosity, I tried putting your last query into Gemini 2 Flash Thinking Experimental and it told me yes first-shot.
Here’s the final output, it’s absolutely beyond my ability to evaluate, so I’m curious if you think it went about it correctly. I can also share the full COT if you’d like, but it’s lengthy:
https://ibb.co/album/rx5Dy1
(Image since even copying the markdown renders it ugly here)
corank has to be more than 1, not equal to 1. I’m not sure if such a matrix exists; the reason I was able to change its mind by supplying a corank-1 matrix was that its kernel behaved in a way that significantly violated its intuition.