Post #2920378

2026-05-12 11:38 UTC

@tao@mathstodon.xyz I think I have learned that the big rift between pro AI and anti AI sensibilities in mathematics (and programming actually) is mostly due to the rift between 'problem solvers' and 'theory builders' in Gower's words. To this end, the cooking analogy is a good one, but leaves out important things for the 'theory builder'. In the food case, chewing vs eating through a tube are functionally equivalent when it comes to essentially all strictly utilitarian considerations with regards to food. This may be true for the problem solver, but it is not the case for the theory builder. When we do the equivalent of putting mathematics through a blender, we actually change the 'theory being built'. I.e. in AI's case, the optimization you are describing is a *local* optimization, not a *global* optimization. And we are leaving *global* optimization entirely on the floor by using the AI only approach. In other words, it is not just enjoyment that we are missing, it is also longer term utilitarian benefits, like theory building, and paradigm shifting. 'Theory builders' will see little benefit from AI. A good theory builder will find relatively easy small proofs, that an AI probably could find, but not much time is saved in the process, because the bulk of the time for a theory builder is taken up by figuring out what we *should* prove, which an AI is entirely incapable of without some motivating problem, which a theory builder is not really interested in. Theory builders use problems as a testing board, not a motivator. On the other hand, 'problem solvers' will see immense benefit from AI. Both sides have trouble seeing the other's perspective.

Replies (1)

@tao@mathstodon.xyz 2026-05-13 12:33

@wikiemol@mathstodon.xyz Problem solving and theory building are in fact highly synergistic. I can continue the food analogy with this quote from Erdos, often regarded as a quintessential "problem solver": "A well-chosen problem can isolate an essential difficulty in a particular area, serving as a benchmark against which progress in this area can be measured. It might be like a 'marshmallow', serving as a tasty tidbit supplying a few moments of fleeting enjoyment. Or it might be like an 'acorn', requiring deep and subtle new insights from which a mighty oak can develop." To continue the analogy further, current levels of AI technology are like blenders that can now produce significant amounts of paste consisting of both ground up marshmallows and ground up acorns, which can score well on such benchmark metrics as edibility, nutrient value, and digestibility, yet remain distinctly unappetizing. As the technology improves, the flavor and acorn content of this paste may get better, but - as you say - this is still insufficient to generate oaks. Nevertheless, I do not believe that the solution to this issue is to ban blenders and food processors, or decry them as evil, but to view them as highly situational tools for cooking - very useful in certain scenarios, when used responsibly, but wholly inappropriate for use in others. Furthermore, the culture of racing to be the first to produce an edible substance will need to be de-emphasized in favor of a more holistic approach to the broader objectives of sustainable food production, preparation, and cultivation.

Open ##2920381