This smacks me as a very combinatorist view of theorem proving (no coincidence there.. the author is a combinatorist). Very famously, combinatorics is a field overrepresented by problem solvers as opposed to theory builders, mostly for the following simple reason: the body corpus of combinatorics theory is mostly the techniques and tools that can be used to solve combinatorial problems, and the field does not possess as much of a broad structured general theory of combinatorics compared to other fields. Because of that, solving problems in combinatorics becomes theory building.
To study a combinatorial conjecture all your life but fail to produce a proof of anything is almost definitionally a failure in that field, because if you failed to prove anything, no new techniques were developed which could be leveraged to solve other similar or different combinatorial problems.
The same is not true of other, more theoretical areas of mathematics, where the balance between problem solving and theory building is more on the theory side. In such areas, a lot of valuable work can come out of pursuing conjectures without achieving the ultimate glory of proving them.
In such parts of mathematics, the most famous conjectures are usually upheld because of the volume of circumstantial or theoretical evidence in their favour. Most of the time a disproof of such a conjecture is far more likely to be "for trivial reasons" rather than revealing some fundamental failure of the theory. Take Yang-Mills existence and mass gap: what would it even mean to mathematics for that famous conjecture to be "disproved"? Well, our universe exists and appears to be mathematical, so (unless the disproof was really remarkable) it cannot mean that no such theory can exist, it just means that the axioms declared need to be tinkered with.
It is still absolutely the case that a disproof which reveals something fundamental or interesting would be held up as a remarkable result. But there are many areas of mathematics where the deck has been stacked so favourably for the positive case (because the conjecture is in essence "obviously true" but in practice we do not know the right statement of the conjecture, and constructing the statement requires developing theory, this is the Grothendieck nut-crack quote at the conjecture-forming level) that a non-trivial disproof is almost impossible to imagine. The Clay problems which do not have a significant reward for a disproof are all of this form.
A disproof of the Jacobian conjecture which is just an oracle giving us a polynomial which is not injective is not such a fundamental or interesting result. An oracle giving us a non-trivial zero of the Riemann zeta function not on the critical line would be similar, and so on.
In other words, for many parts of mathematics, there is a big difference between conjectures being false for trivial reasons [1] or false for non-trivial reasons.
To me, conjectures seem less useful than a meteorologist offering weather predictions.
Aside from mere intellectual entertainment, they are worthless in the practical sense. They cannot be used in proofs, nor aid in solutions of other problems. They can’t even suggest if I would take an umbrella.
They are little more than the type of ramblings from people who drink heavily after a day’s work.
Go tell me how many conjectures have been proven over time....
It's only useless to study a conjectured if their never solved, and since we can't look into the infinite future it's only solvable if we learn how to send messages back to the distant past we can never tell if it gets solved or not.
With that said, theirs a lot of conjectures that wouldn't help anyone if solved, so maybe pick the most useful ones at first like P=NP.
This smacks me as a very combinatorist view of theorem proving (no coincidence there.. the author is a combinatorist). Very famously, combinatorics is a field overrepresented by problem solvers as opposed to theory builders, mostly for the following simple reason: the body corpus of combinatorics theory is mostly the techniques and tools that can be used to solve combinatorial problems, and the field does not possess as much of a broad structured general theory of combinatorics compared to other fields. Because of that, solving problems in combinatorics becomes theory building.
To study a combinatorial conjecture all your life but fail to produce a proof of anything is almost definitionally a failure in that field, because if you failed to prove anything, no new techniques were developed which could be leveraged to solve other similar or different combinatorial problems.
The same is not true of other, more theoretical areas of mathematics, where the balance between problem solving and theory building is more on the theory side. In such areas, a lot of valuable work can come out of pursuing conjectures without achieving the ultimate glory of proving them.
In such parts of mathematics, the most famous conjectures are usually upheld because of the volume of circumstantial or theoretical evidence in their favour. Most of the time a disproof of such a conjecture is far more likely to be "for trivial reasons" rather than revealing some fundamental failure of the theory. Take Yang-Mills existence and mass gap: what would it even mean to mathematics for that famous conjecture to be "disproved"? Well, our universe exists and appears to be mathematical, so (unless the disproof was really remarkable) it cannot mean that no such theory can exist, it just means that the axioms declared need to be tinkered with.
It is still absolutely the case that a disproof which reveals something fundamental or interesting would be held up as a remarkable result. But there are many areas of mathematics where the deck has been stacked so favourably for the positive case (because the conjecture is in essence "obviously true" but in practice we do not know the right statement of the conjecture, and constructing the statement requires developing theory, this is the Grothendieck nut-crack quote at the conjecture-forming level) that a non-trivial disproof is almost impossible to imagine. The Clay problems which do not have a significant reward for a disproof are all of this form.
A disproof of the Jacobian conjecture which is just an oracle giving us a polynomial which is not injective is not such a fundamental or interesting result. An oracle giving us a non-trivial zero of the Riemann zeta function not on the critical line would be similar, and so on.
In other words, for many parts of mathematics, there is a big difference between conjectures being false for trivial reasons [1] or false for non-trivial reasons.
[1] https://www.sciencedirect.com/science/article/pii/0040938369...
To me, conjectures seem less useful than a meteorologist offering weather predictions.
Aside from mere intellectual entertainment, they are worthless in the practical sense. They cannot be used in proofs, nor aid in solutions of other problems. They can’t even suggest if I would take an umbrella.
They are little more than the type of ramblings from people who drink heavily after a day’s work.
Go tell me how many conjectures have been proven over time....
It's only useless to study a conjectured if their never solved, and since we can't look into the infinite future it's only solvable if we learn how to send messages back to the distant past we can never tell if it gets solved or not.
With that said, theirs a lot of conjectures that wouldn't help anyone if solved, so maybe pick the most useful ones at first like P=NP.