Well, if, e.g. you’re working on a special case of an unsolved problem using an ad hoc method with applicability that’s clearly limited to that case, and you think that the problem will probably be solved in full generality with a more illuminating solution within the next 50 years, then you have good reason to believe that work along these lines has no lasting significance.
It is really hard to tell when an ad hoc method will turn out many years later to be a special case of some more broad technique. It may also be that the special case will still need to be done if some later method uses it for bootstrapping.
the original contributions in math are more densely concentrated in a smaller number of people than one would guess from the outside,”
I’m not sure about this at all. Have you tried talking to people who aren’t already in academia about this? As far as I can tell, they think that there are a tiny number of very smart people who are mathematicians and are surprised to find out how many there are.
It is really hard to tell when an ad hoc method will turn out many years later to be a special case of some more broad technique. It may also be that the special case will still need to be done if some later method uses it for bootstrapping.
There are questions of quantitative effect sizes. Feel free to give some examples that you find compelling.
I’m not sure about this at all. Have you tried talking to people who aren’t already in academia about this? As far as I can tell, they think that there are a tiny number of very smart people who are mathematicians and are surprised to find out how many there are.
By “from the outside” I mean “from the outside of a field” (except to the extent that you’re able to extrapolate from your own field.)
Feel free to give some examples that you find compelling.
Fermat’s Last Theorem. The proof assumed that p >=11 and so the ad hoc cases from the 19th century were necessary to round it out. Moreover, the attempt to extend those ad hoc methods lead to the entire branch of algebraic number theory.
Primes in arithmetic progressions: much of what Tao and Greenberg did here extended earlier methods in a deep systematic way that were previously somewhat ad hoc. In fact, one can see a large fraction of modern work that touches on sieves as taking essentially ad hoc sieve techniques and generalizing them.
The proof assumed that p >=11 and so the ad hoc cases from the 19th century were necessary to round it out. Moreover, the attempt to extend those ad hoc methods lead to the entire branch of algebraic number theory.
I don’t recall Wiles’ proof assuming that p >= 11 – can you give a reference? I can’t find one quickly.
The n = 3 and 4 cases were proved by Euler and Fermat. It’s prima facie evident that Euler’s proof (which introduced a new number system with no historical analog) points to the existence of an entire field of math. I find this less so of Fermat’s proof as he stated it, but Fermat is also famous for the obscurity of his writings.
I don’t know the history around the n = 5 and n = 7 cases, and so don’t know whether they were important to the development of algebraic number theory, but exploring them is a natural extension of the exploration of new kinds of number systems that Euler had initiated.
They were subsumed by Kummer’s work, which I understand to have been motivated more by a desire to understand algebraic number fields and reciprocity laws than by Fermat’s last theorem in particular. For this, he developed the theory of ideal numbers, which is very general.
Primes in arithmetic progressions: much of what Tao and Greenberg did here extended earlier methods in a deep systematic way that were previously somewhat ad hoc. In fact, one can see a large fraction of modern work that touches on sieves as taking essentially ad hoc sieve techniques and generalizing them.
Ben Green, not Greenberg :-).
Sure, but the ultimate significance of the work remains to be seen. Of course, tastes vary, and there’s an element of subjectivity, but I think that we can agree that even if there’s a case for the proof being something that people will find interesting in 50 years, that the prior in favor of it is much weaker than the prior in favor of this being the case of, e.g. the Gross-Zagier formula.
I don’t recall Wiles’ proof assuming that p >= 11 – can you give a reference? I can’t find one quickly.
I think this is in the original paper that modularity implies FLT, but I’m on vacation and don’t have a copy available to check. Does this suffice as a reference?
Ben Green, not Greenberg
Yes, thank you.
They were subsumed by Kummer’s work, which I understand to have been motivated more by a desire to understand algebraic number fields and reciprocity laws than by Fermat’s last theorem in particular. For this, he developed the theory of ideal numbers, which is very general.
Sure, but Kummer was aware of the literature before him, and almost certainly used their results to guide him.
Sure, but the ultimate significance of the work remains to be seen. Of course, tastes vary, and there’s an element of subjectivity, but I think that we can agree that even if there’s a case for the proof being something that people will find interesting in 50 years, that the prior in favor of it is much weaker than the prior in favor of this being the case of, e.g. the Gross-Zagier formula.
Agreement may there depend very strongly on how you unpack “much weaker” but I’d be inclined to agree at least weaker without the much.
It is really hard to tell when an ad hoc method will turn out many years later to be a special case of some more broad technique. It may also be that the special case will still need to be done if some later method uses it for bootstrapping.
I’m not sure about this at all. Have you tried talking to people who aren’t already in academia about this? As far as I can tell, they think that there are a tiny number of very smart people who are mathematicians and are surprised to find out how many there are.
There are questions of quantitative effect sizes. Feel free to give some examples that you find compelling.
By “from the outside” I mean “from the outside of a field” (except to the extent that you’re able to extrapolate from your own field.)
Yes, and I’m not sure how to measure that.
Fermat’s Last Theorem. The proof assumed that p >=11 and so the ad hoc cases from the 19th century were necessary to round it out. Moreover, the attempt to extend those ad hoc methods lead to the entire branch of algebraic number theory.
Primes in arithmetic progressions: much of what Tao and Greenberg did here extended earlier methods in a deep systematic way that were previously somewhat ad hoc. In fact, one can see a large fraction of modern work that touches on sieves as taking essentially ad hoc sieve techniques and generalizing them.
I don’t recall Wiles’ proof assuming that p >= 11 – can you give a reference? I can’t find one quickly.
The n = 3 and 4 cases were proved by Euler and Fermat. It’s prima facie evident that Euler’s proof (which introduced a new number system with no historical analog) points to the existence of an entire field of math. I find this less so of Fermat’s proof as he stated it, but Fermat is also famous for the obscurity of his writings.
I don’t know the history around the n = 5 and n = 7 cases, and so don’t know whether they were important to the development of algebraic number theory, but exploring them is a natural extension of the exploration of new kinds of number systems that Euler had initiated.
They were subsumed by Kummer’s work, which I understand to have been motivated more by a desire to understand algebraic number fields and reciprocity laws than by Fermat’s last theorem in particular. For this, he developed the theory of ideal numbers, which is very general.
Ben Green, not Greenberg :-).
Sure, but the ultimate significance of the work remains to be seen. Of course, tastes vary, and there’s an element of subjectivity, but I think that we can agree that even if there’s a case for the proof being something that people will find interesting in 50 years, that the prior in favor of it is much weaker than the prior in favor of this being the case of, e.g. the Gross-Zagier formula.
I think this is in the original paper that modularity implies FLT, but I’m on vacation and don’t have a copy available to check. Does this suffice as a reference?
Yes, thank you.
Sure, but Kummer was aware of the literature before him, and almost certainly used their results to guide him.
Agreement may there depend very strongly on how you unpack “much weaker” but I’d be inclined to agree at least weaker without the much.