Very pleased to see all of these dichotomies collected in one place. The natural question is whether these divides can be integrated to a useful picture with more pieces.
My take on the “Two Cultures” model of problem-solvers and theory-builders: theory-building fields of mathematics like algebraic topology (say) are those where the goal is to articulate grand meta-theorems that are bigger than any particular application. This was the work of a Grothendieck.
Meanwhile, concrete problem-solving fields of mathematics like combinatorics are those where the goal is to become the grand meta-theorem that contains more understanding than any particular theorem you can prove. This was the style of an Erdos. The inarticulate grand meta-theorems lived in his cognitive strategies so that the theorems he actually proved are individually only faint impressions thereof.
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.
Do you think that theory-building and problem-solving maps at all to your hammers and nails dichotomy? One would be about becoming a hammer that can hit all the nails, and the other is more about really understanding each particular nail.
I think it’s more about the nature of the hammers. Theory-building hammers are legible: they’re big theorems, or maybe big messes of definitions and then theorems (the term of art for this is “machinery”). Problem-solving hammers are illegible: they’re a bunch of tacit knowledge sitting inside some mathematician’s head.
I mostly agree. By the end of the hammers and nails post I realized the real dichotomy was between systematic (including both hammers and nails) and haphazard, and this is a different dichotomy from all the others mentioned in this thread because I will actually make a value judgment that systematic is just better.
Then again, these are just two stages of development and you can extrapolate there’s some third stage that’s even better than systematic that looks like haphazard genius from the outside.
Very pleased to see all of these dichotomies collected in one place. The natural question is whether these divides can be integrated to a useful picture with more pieces.
My take on the “Two Cultures” model of problem-solvers and theory-builders: theory-building fields of mathematics like algebraic topology (say) are those where the goal is to articulate grand meta-theorems that are bigger than any particular application. This was the work of a Grothendieck.
Meanwhile, concrete problem-solving fields of mathematics like combinatorics are those where the goal is to become the grand meta-theorem that contains more understanding than any particular theorem you can prove. This was the style of an Erdos. The inarticulate grand meta-theorems lived in his cognitive strategies so that the theorems he actually proved are individually only faint impressions thereof.
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.
Do you think that theory-building and problem-solving maps at all to your hammers and nails dichotomy? One would be about becoming a hammer that can hit all the nails, and the other is more about really understanding each particular nail.
I think it’s more about the nature of the hammers. Theory-building hammers are legible: they’re big theorems, or maybe big messes of definitions and then theorems (the term of art for this is “machinery”). Problem-solving hammers are illegible: they’re a bunch of tacit knowledge sitting inside some mathematician’s head.
I mostly agree. By the end of the hammers and nails post I realized the real dichotomy was between systematic (including both hammers and nails) and haphazard, and this is a different dichotomy from all the others mentioned in this thread because I will actually make a value judgment that systematic is just better.
Then again, these are just two stages of development and you can extrapolate there’s some third stage that’s even better than systematic that looks like haphazard genius from the outside.