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> Holding the field back from what? If the goal of the practitioners of the field is to seek mathematically beauty, then well, that is what they will focus on. Holding the field back from answering questions about the behavior of simple deterministic systems that arguably have an outsized influence on science, machine learning, and mathematics itself. I don't disagree that "beauty" is certainly subjective, but the mathematics research that gets funded by universities and research labs is only tangentially related to such an aesthetic notion. > In the vast majority of cases, the difficulty in finding a proof of a statement is not that the statement isn't provable under a given formal system, it's that we simply can't find it Finding a proof is an inherently algorithmic process. Presumably, the human brain uses heuristics (intuition) that have thus far managed to beat simple algorithmic approaches. But it's hard to believe that's going to last much longer. As an extreme example, simply knowing more values of the Busy Beaver function allows one to find proofs more easily. For example, encode the veracity of the statement corresponding to Fermat's Last Theorem into a program (a k-state Turing machine) that dovetails enumeration over all possible values for x, y, z, and n > 2. This program halts if the statement is satisfied by some set of natural numbers and never halts otherwise. But if we know which k-state Turing machine corresponds to BB(k), then we just run the two programs in parallel and if the Fermat program hasn't halted by the time the program for BB(k) has, then the conjecture is true (necessarily false otherwise). Scott Aaronson has written some articles and blog posts about this. Now the argument I'm making doesn't necessarily depend on trying to determine more values of the uncomputable busy beaver function. You don't have to know the fastest growing computable sequence in order to derive some benefit from greater knowledge of the halting sequence. If we presume that an interesting proof exists within ZFC, then more axioms (that constitute a more powerful system) can lead to much shorter proofs. This is a result obtained from the work of Gödel, Feferman, Jean-Yves Girard, et al. So while you're right that a conjecture we are interested in might have a proof that "exists" within ZFC, whether the minimal such proof can be written down in a human lifespan is a different question. Considering there are many famous conjectures that have been open for centuries now, perhaps the next step isn't having more people stare harder at the problem but rather enriching our foundational formal systems with additional axioms. |