> "What is the point of insight? I thought its point was that it led to results."
The 'results' are a substitute product for the real social benefit, which is an availability of mathematically educated and educable society. There has been no 'waste' of public money except when the results of publicly funded research is published by for profit corporations and held from the public's access behind paywalls.
It is unfortunate that one outcome of this situation is apparently a public which perceives that the final 'result' of a math research project as the actual product of value, when it has repeatedly been shown by history that the most significant value is within the multiple alternate potential paths explored by other researchers working toward the same result.
Most of these do not lead to the specific result, and many may instead demonstrate that a specific approach conclusively does not lead to the initial objective result. This also has value for humanity, in many cases leading to new paradigms of thinking about similar or unrelated problems which may later provide foundational insight for new approaches to solving different problems in a manner not yet known or discovered. The value of the system is not enclosed within the single results, but in the collective search and expansion of human consciousness and ingenuity that the search for all of these results entails.
This point circles back to my original. When we cheat on a math test or homework, thinking that the objective is to get the answers right, we are only cheating ourselves out of the learning that would have enabled us to develop the correct answers on not only that one test, but on the unknowable challenges which will later arise that require the new solutions to build upon that learning.
The 'prize money' for solving the biggest hurdles in math is less about those specific problems and their specific answers, but on encouraging many people, who do not get 'first place' and win the prize, but who do work toward it in their own unique ways and in turn provide an uplift to the general capability of humanity to solve hard, as yet undefined problems large and small.
Having an LLM give us the answer is entirely missing the point of the challenge in the first place, and robs humanity of the opportunity to improve its collective capability through making the effort. It also does not find any unique perspectives, which new and unique perspectives have formed the basis for civilization scale improvements since the dawn of time.
A good example of this difference comes from Bruce Schnier, who teaches public policy at the Harvard Kennedy School and the Munk School at the University of Toronto from an article originally published in The Guardian, which compares LLM use in education to having a forklift at the gym. It may be the right tool in a warehouse to get heavy things across the floor quickly, but using one to do your workout is both overkill for the amount of weight you need for arm curls and bench presses, and accomplishes nothing in the development of your strength or cardiovascular health.
If the objective is to program another widget, and you can get it done in a fraction of the time with an LLM, sure, why not. But, if the objective is to expand the corpus of human knowledge, which is the most beneficial and desired outcome of the study of mathematics, then outsourcing that task to electrons through semiconducting silicon is both overkill which results in a 'proof' larger than the entire MatLab code base, and does not accomplish the stated objective. Humanity is not made any more capable through this process.
For a deeper insight into what the measurable benefits for individuals and society, an in depth study of the neurobiology results from the study of complex topics; such as linguistics, mathematics, and music theory might help with the comprehension of the less direct value of the process. Challenge yourself to discover what the effects of later-in-life study of foreign language have on the factors leading to senility and mental decline, and see if a study of mathematical theories has any observable effects on the brain's electrochemical development across different ages. Identify how these kinds of study can have collateral improvements for other fields of study, such as material science, medicine, or applied physics.
So, yes. I am saying that. In addition to looking for the result, because it's cool, I am saying that mathematics is indeed searching for the inherent uplift to humanity which are available through many distributed states of psychological insight that mathematics prizes can serve to inspire. The results are simply one of the points of this search, and if all of these searches are cleared off the board mechanistically, we may lose the greater race against our own potential for growth in the trade-off.