Yeah, there’s something less legible about combinatorics compared to most other fields of mathematics. People like Erdos know lots of important principles and meta-principles for solving combinatorial problems but it’s a tremendous chore to state those principles explicitly in terms of theorems and nobody really does it (the closest thing I’ve seen is Flajolet and Sedgewick—by the way, amazing book, highly recommended). A concrete example here is the exponential formula, which is orders of magnitude more complicated to state precisely than it is to understand and use.
(Ray has been suggesting to me in person that an important chunk of the current big LW debate is not about S1/S2 but about illegibility and that sounds right to me.)
I really like the phrasing “become the grand meta-theorem.”
I think I heard the “become grand meta-theorem” phrasing originally from Alon & Spencer. I actually bought the Flajolet and Sedgewick book a couple months ago (only got through the first chapter), but it was mind-boggling that something like this could be done for combinatorics.
Of course reality is self-similar, so it’s not surprising that there’s currently a big divide in combinatorics between what I would call the “algebraic/enumerative” style of Richard Stanley containing the Flajolet and Sedgewick stuff, characterized by fancy algebra/explicit formulae/crystalline structures and the “analytic/extremal” style of Erdos, characterized by asymptotic formulae and less legibility. It’s surprisingly rare to see a combinatorialist bridge this gap.
I went through most of the first half of Flajolet and Sedgewick when I was 18 or so and was blown away, then recently went through the second half and was blown away in a completely different way. It’s really wild. Take a look. It’s where I learned the argument in this blog post about the asymptotics of the partition function.
Yeah, there’s something less legible about combinatorics compared to most other fields of mathematics. People like Erdos know lots of important principles and meta-principles for solving combinatorial problems but it’s a tremendous chore to state those principles explicitly in terms of theorems and nobody really does it (the closest thing I’ve seen is Flajolet and Sedgewick—by the way, amazing book, highly recommended). A concrete example here is the exponential formula, which is orders of magnitude more complicated to state precisely than it is to understand and use.
(Ray has been suggesting to me in person that an important chunk of the current big LW debate is not about S1/S2 but about illegibility and that sounds right to me.)
I really like the phrasing “become the grand meta-theorem.”
I think I heard the “become grand meta-theorem” phrasing originally from Alon & Spencer. I actually bought the Flajolet and Sedgewick book a couple months ago (only got through the first chapter), but it was mind-boggling that something like this could be done for combinatorics.
Of course reality is self-similar, so it’s not surprising that there’s currently a big divide in combinatorics between what I would call the “algebraic/enumerative” style of Richard Stanley containing the Flajolet and Sedgewick stuff, characterized by fancy algebra/explicit formulae/crystalline structures and the “analytic/extremal” style of Erdos, characterized by asymptotic formulae and less legibility. It’s surprisingly rare to see a combinatorialist bridge this gap.
I went through most of the first half of Flajolet and Sedgewick when I was 18 or so and was blown away, then recently went through the second half and was blown away in a completely different way. It’s really wild. Take a look. It’s where I learned the argument in this blog post about the asymptotics of the partition function.