I honest don't understand why so many mathematicians think that math is doomed. I don't deny that many of their careers will be impacted. They spend their career studying certain problems yet now they may have to pursue a whole new direction. That can be very painful. But math, math itself has never been better. What will really change, is our definition of "hard problems". And I don't buy Prof. Tao's "math is contaminated" argument just because we knew that some problems are likely solved. If we suddenly got a few more Ramanujans and their manuscripts, would math be contaminated? Last I checked, Ramanujan's manuals are treasured as absolute gems.
Take the Quasi-RH proof as an example. At its core, OAI's proof relies on classical tools. It’s essentially a proof by contradiction, constructing conflicting upper and lower bounds. The high-level strategy is standard in my eyes. So why couldn't we humans crack Quasi-RH for so long? My understanding is that we human simply don't have the capacity to grind the solution out: the proof uses 277 estimates with a max depth of 12 layers, plus 64 lemmas. For context: Yau’s proof of the Calabi conjecture used a dozen or so estimates, with the key ones going only 3 layers deep, and that alone took peak Yau two to three years of intense regular work.
A human mathematician's absolute peak only lasts about a decade. It’s basically impossible for a human to pull off a chain of estimates so long, so deep, and so tightly cross-dependent as a machine can, especially when every single estimate has to be razor-sharp and mutually consistent without any conflicts.
But the real question is, why do we think that it's the apocalypse if a question like quasi-RH is solved (if it is solved)? Would it only mean that quasi-RH may not be as hard as we assessed? Or more generally, we thought that some problems were deeply complex and therefore required entire paradigm shift, but it turned out there was a computationally massive solution. So what? It's not like all the problems can be solved through sheer computational search, right? It can even be evaluable for someone to tackle a solved problem from a different angle too, like how Lomonosov solved the invariant surface problem with a new framework, or like how Newton generalized the binomial expansion.
I honest don't understand why so many mathematicians think that math is doomed. I don't deny that many of their careers will be impacted. They spend their career studying certain problems yet now they may have to pursue a whole new direction. That can be very painful. But math, math itself has never been better. What will really change, is our definition of "hard problems". And I don't buy Prof. Tao's "math is contaminated" argument just because we knew that some problems are likely solved. If we suddenly got a few more Ramanujans and their manuscripts, would math be contaminated? Last I checked, Ramanujan's manuals are treasured as absolute gems.
Take the Quasi-RH proof as an example. At its core, OAI's proof relies on classical tools. It’s essentially a proof by contradiction, constructing conflicting upper and lower bounds. The high-level strategy is standard in my eyes. So why couldn't we humans crack Quasi-RH for so long? My understanding is that we human simply don't have the capacity to grind the solution out: the proof uses 277 estimates with a max depth of 12 layers, plus 64 lemmas. For context: Yau’s proof of the Calabi conjecture used a dozen or so estimates, with the key ones going only 3 layers deep, and that alone took peak Yau two to three years of intense regular work.
A human mathematician's absolute peak only lasts about a decade. It’s basically impossible for a human to pull off a chain of estimates so long, so deep, and so tightly cross-dependent as a machine can, especially when every single estimate has to be razor-sharp and mutually consistent without any conflicts.
But the real question is, why do we think that it's the apocalypse if a question like quasi-RH is solved (if it is solved)? Would it only mean that quasi-RH may not be as hard as we assessed? Or more generally, we thought that some problems were deeply complex and therefore required entire paradigm shift, but it turned out there was a computationally massive solution. So what? It's not like all the problems can be solved through sheer computational search, right? It can even be evaluable for someone to tackle a solved problem from a different angle too, like how Lomonosov solved the invariant surface problem with a new framework, or like how Newton generalized the binomial expansion.