Timothy Gowers

(Original poster here.) If you are interested in this sort of thinking, I recommend checking out the page I found it on: https://www.dpmms.cam.ac.uk/~wtg10/vsipage.html , including the last link on the page that delves a bit into the philosophy behind mathematics and its implications. That page in turn was found at the end of the author's book 'Mathematics: a Very Short Introduction', which is pretty good exploration of what goes into mathematical definitions and how modern mathematical thinking works, addressed towards the layman but without excessive watering down of the subject.

There is also a follow-up "The Two Cultures of Mathematics" by Timothy Gowers. I found it a crystal clear explanation of the differences between the schools of "problem solving" and "theory building" - and particularly of what really motivates "combinatorics". https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf‎

Direct link here: http://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf Again, anything Tim Gowers writes is worth reading - highly recommended.

A series of statements that are valid are logical. The conclusions can be wrong, as "anything follows from a false premise" (about the only actual statement I recall from logic lectures; see eg http://www.dpmms.cam.ac.uk/~wtg10/implication.html). Occam's Razor is unsound. The truth is not necessarily simple.

There are two kinds of mathematicians, or one can say, two cultures of mathematics: (a) problem solving, proving conjectures, etc (b) theories. Alexander Grothendieck belongs to (b). Paul Erdos to (a). You can read Gower's paper on "The two cultures of Mathematics" at: https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

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… 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.

This unsigned essay is at the web site of the well-known mathematician Tim Gowers, but I couldn't find any link to it via his home page [1], which does link to many other fascinating things. 1. https://www.dpmms.cam.ac.uk/~wtg10/

I wonder if you're approaching him from the other culture of mathematics. https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

This article reminds me of this: https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf I especially like the Atiyah quote.

> It's possible to do quite a lot without infinities in calculus and trigonometry This is like Stacey King saying "I'll always remember this as the night that Michael Jordan and I combined for 70 points." Pretty much the entire field of Analysis (of which calculus is a part) relies on 'infinities' of some kind. Even if you try to restrict to the rationals, you're still typically working with infinite series of them. You can do some analysis over the rationals, but often this takes the form of Cauchy sequences of rationals which might be cheating. [0] https://math.stackexchange.com/questions/387234/how-far-can-one-get-in… [1] https://www.dpmms.cam.ac.uk/~wtg10/reals.html

Being "optimally useful" is not the goal of mathematics. See for example, https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

He's a combinatorist. He's trolling mathematicians who are specialized in more abstract fields who often hold views of contempt for computational, concrete fields like combinatorics. One of the main themes of modern mathematics is the use of algebraic, numerical, and combinatoric invariants to characterize abstract mathematical objects such as spaces. Doron is dishing out their own medicine in this post. This is an essay that touches on similar themes with less snark and more context for people who aren't in the know: https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

Most mathematical results would be unreachable by amateurs due to the sheer amount of background knowledge, especially of existing literature, required to make progress on a problem (and in some cases to even understand the statement of the problem, e.g. https://en.m.wikipedia.org/wiki/Hodge_conjecture). Elementary graph theory and combinatorics are somewhat outliers in this regard, as there is not so much "theory" per se that one has to build on or work with. Tim Gowers's "Two Cultures" essay is an interesting read on this topic: https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf

Also known as the tensor product of linear maps in a particular basis: https://jeremykun.com/2014/01/17/how-to-conquer-tensorphobia/ http://www.math.harvard.edu/archive/25b_spring_05/tensor.pdf https://www.dpmms.cam.ac.uk/~wtg10/tensors3.html

https://www.dpmms.cam.ac.uk/~wtg10/decimals.html You can construct a complete ordered field from infinite decimals

For anyone interested in learning more about this cultural divide within mathematics -- theory building versus problem solving -- Fields medalist Tim Gowers has an interesting article: https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf. People I talk to are often surprised to learn that there is quite a bit of snobbery within mathematics on this issue. Mostly people in theory-heavy subjects looking down on areas that revolve more around problem solving. Of course, it's academia, so it surprises me that anyone is surprised that there are turf wars and snobbery.

Timothy Gowers has a few thoughts on this: https://www.dpmms.cam.ac.uk/~wtg10/commutative.html

I'll note for the record that there are many other very interesting articles about mathematics living at that site, http://www.dpmms.cam.ac.uk/~wtg10/ the academic home page for Timothy Gowers.

Gowers has a few dozen short pieces of this sort, listed at http://www.dpmms.cam.ac.uk/~wtg10/mathsindex.html. Another one that might be of interest is "why study finite-dimensional vector spaces in the abstract if they are all isomorphic to R^n?" (http://www.dpmms.cam.ac.uk/~wtg10/vspaces.html) The answer, roughly, is that the notation is less ugly if you don't have to keep track of coordinates, and that the abstract theory carries over better to the infinite-dimensional case.

I used to read Tim's expository writings religiously. The old ones can be found at: http://www.dpmms.cam.ac.uk/~wtg10/mathsindex.html and the newer stuff is on his blog: http://gowers.wordpress.com/

Not what I was expecting. Here's some beautiful (and more technical) explanations about mathematical topics that consider the motivation behind them, and are thus very natural summaries: http://www.dpmms.cam.ac.uk/~wtg10/mathsindex.html

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

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.

Transcript: http://www.dpmms.cam.ac.uk/~wtg10/importance.pdf