Replying separately so it isn’t missed. I wonder also how much of these issues is the two cultures problem that Gowers talks about. The top people conception seems to at least lean heavily into the theory-builder side.
I agree with you, and there are strong problem solver types who conceptualize mathematical value in a different way from the people who I’ve quoted and from myself.
Still, there are some situations where one has apples-to-apples comparisons.
There’s a large body of work giving unconditional proofs of theorems that would follow from the Riemann hypothesis and its generalizations. Many problem solvers would agree that a proof of the Riemann hypothesis and its generalizations would be more valuable than all of this work combined.
We don’t yet know how or when the Riemann hypothesis will be proved. But suppose that it turns out that Alain’s Connes’ approach using noncommutative geometry (which seems most promising right now, though I don’t know how promising) turns out to be possible to implement over the next 40 years or so. In this hypothetical. What attitude do you think that problem solvers would take to the prior unconditional proofs of consequences of RH?
I agree with you, and there are strong problem solver types who conceptualize mathematical value in a different way from the people who I’ve quoted and from myself.
Yes, but at the same time (against my earlier point) the best problem solvers are finding novel techniques that can be then applied to a variety of different problems- that’s essentially what Gowers seems to be focusing on.
There’s a large body of work giving unconditional proofs of theorems that would follow from the Riemann hypothesis and its generalizations. Many problem solvers would agree that a proof of the Riemann hypothesis and its generalizations would be more valuable than all of this work combined.
I’m not sure I’d agree with that, but I feel like I’m in the middle of the two camps so maybe I’m not relevant? All those other results tell us what to believe well before we actually have a proof of RH. So that’s at least got to count for something. It may be true that a proof of GRH would be that much more useful, but GRH is a much broader idea. Note also that part of the point of proving things under RH is to then try and prove the same statements with weaker or no assumptions, and that’s a successful process.
What attitude do you think that problem solvers would take to the prior unconditional proofs of consequences of RH?
I’m not sure. Can you expand on what you think would be their attitude?
Replying separately so it isn’t missed. I wonder also how much of these issues is the two cultures problem that Gowers talks about. The top people conception seems to at least lean heavily into the theory-builder side.
I agree with you, and there are strong problem solver types who conceptualize mathematical value in a different way from the people who I’ve quoted and from myself.
Still, there are some situations where one has apples-to-apples comparisons.
There’s a large body of work giving unconditional proofs of theorems that would follow from the Riemann hypothesis and its generalizations. Many problem solvers would agree that a proof of the Riemann hypothesis and its generalizations would be more valuable than all of this work combined.
We don’t yet know how or when the Riemann hypothesis will be proved. But suppose that it turns out that Alain’s Connes’ approach using noncommutative geometry (which seems most promising right now, though I don’t know how promising) turns out to be possible to implement over the next 40 years or so. In this hypothetical. What attitude do you think that problem solvers would take to the prior unconditional proofs of consequences of RH?
Yes, but at the same time (against my earlier point) the best problem solvers are finding novel techniques that can be then applied to a variety of different problems- that’s essentially what Gowers seems to be focusing on.
I’m not sure I’d agree with that, but I feel like I’m in the middle of the two camps so maybe I’m not relevant? All those other results tell us what to believe well before we actually have a proof of RH. So that’s at least got to count for something. It may be true that a proof of GRH would be that much more useful, but GRH is a much broader idea. Note also that part of the point of proving things under RH is to then try and prove the same statements with weaker or no assumptions, and that’s a successful process.
I’m not sure. Can you expand on what you think would be their attitude?