Timothy Gowers

I agree with Dyson that the main difference between 'birds' and 'frogs' is that while frogs tend to focus on their particular domain solving problems, birds try to find and bring in connections between areas which can open up a much bigger new area of problems to solve. However to me this way of looking at things is almost dependent on your philosophical outlook on mathematics. Do you view mathematics as fundamentally one big area similar to how fundamental physics has their theory of everything? Or is it just collections of various problem solving strategies grouped by likeness? I find Tim Gowers' equally famous essay on the "Two Cultures" of mathematics[0] quite relevant here. Particularly of relevance is how some areas of mathematics, especially those of the algebraic kind, are more "structural" compared to a subject like combinatorics (on the surface this would make obvious sense because algebra broadly studies structure). I can't speak to being an expert on the past 50 years in mathematics but it would seem to me that these structural areas of mathematics have 'dominated' so to say pure mathematics in that time period. One thing however I would like to point out is going into the future how is the split between so called birds and frogs going to be? Most mathematicians or physicists or even academics more broadly will always be frogs because to be a bird you need to have frog like depth in several areas before you can start making connections between them. However as…

I'm not sure what you mean by "real world problems". Most mathematicians do mathematics because they find it beautiful, rewarding, fun, and for various other reasons that have little to do with creating things in the physical world. Mathematics is extremely interconnected and some minor discovery in a certain field can later become extremely important in some seemingly unrelated field, similarly - real world applications often arise unexpectedly out of things which at first seem completely theoretical (famous example: public key cryptography), and we cannot foresee in advanced what will lead to what.

Timothy Gowers - 1998 Fields medalist - gave a lecture in 2000 titled "The Importance of Mathematics". I advice everyone to watch/read it:

Video:
https://www.youtube.com/view_play_list?p=3641A12A6ADDB4B1

PDF:
https://www.dpmms.cam.ac.uk/~wtg10/importance.pdf

Edit: more philosophically, this question always seems puzzling to me, because we can always ask: whats so important about "real world applications" anyway? We can start from any activity and ask "why is it important?", and then ask the same about the answer, etc'. This chain of "why"s eventually comes to an end: we can still ask the question, but there is no meaningful answer other than the realization that the things which are important are the things people find important - whether it is to get faster from point A to point B, have a baby, go to a party, or to solve mathematical problems. Something isn't more…

I grew to like Gardner's writings somewhat now. They however, indirectly, had almost
devastating effect on my earlier life, promoting sort of mathematics I never
could do.

There is "bag of tricks" culture of mathematics
(http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf) where one often stands
no chance against a "tricky" problem without prior acquaintance with "tricky"
sort of shortcut principle. I was always too slow and never grokked or scored
this sort of mathematics. I barely found anything profound or beautiful in Gardner's
SciAm columns. They were scary like black magic more than they were interesting. My teachers all from this school just bashed me and didn't told
me there is richer structured, theoretical and constructive side to mathematics. The same
year I qualified to the International Physics Olympiad I got non passing grade
in mathematics, at which point I dropped out of HS, went to do what unexpectedly
turned out mathematical physics and now I'm for a degree in pure mathematics, several
tantalizing years later than if my teachers were not so enchanted by Gardner's
tricks.

It's more about derivation of theorems than calculations.

Computation can only use rationals, and of course can get arbitrarily close to an answer because they are dense in the reals.

However, the entire edifice of analysis rests on the completeness axiom of the reals. The extreme value theorem, for example, is equivalent to the completeness axiom; the useful properties of continuous functions break down without it; the fundamental theorem of calculus doesn't work without it; Etc. So if the maths used in your physics (the structure of the theory, not just the calculations you perform with it) relies on these things at all, you're relying on the reals for confidence that the maths is sound.

Now you could argue that we don't need mathematical rigour for physics, that real analysis is a preoccupation of mathematicians, while physicists should be fine with informal calculus. I'm not going to argue that point. I'm just pointing out what the real numbers bring to the table.

Here's Tim Gowers on the subject: https://www.dpmms.cam.ac.uk/~wtg10/reals.html

Related articles in math and stats communities:

- The Two Cultures of Mathematics by Tim Gowers:
https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

Discusses theory builders (think algebra) vs. problem solvers (think combinatorics) in mathematics, and how the latter have been sort of marginalized and under-appreciated in the community.

- Statistical Modeling: The Two Cultures by the legendary Leo Breiman

https://projecteuclid.org/journals/statistical-science/volume-16/issue-3/Statistical-Modeling--The-Two-Cultures-with-comments-and-a/10.1214/ss/1009213726.full

Discusses predictive modeling (think random forests and NNs) vs. statistical inference (think null hypothesis significance testing), and how the former have been under-appreciated in the stats community, to the detriment of statisticians.

Both very fun reads by two giants.

I think this is written from the perspective of a continuous mathematician or physicist, and the examples of discrete maths they have have in mind are numerical simulations, which are certainly harder to reason about than their continuous counterparts.

I also suspect that this is a fairly orthodox attitude among mathematicians - in "The two cultures of mathematics" (https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf) Tim Gowers says with more authority than I could:

> the subjects that appeal to theory-builders are, at the moment, much more fashionable than the ones that appeal to problem-solvers.

This isn't precisely the discrete/continuous split, but it's mostly aligned and the article puts combinatorics is firmly in the latter category.