9 ms·

The Two Cultures of Mathematics (2000) [pdf]

145 points by vector_spaces 5y ago

throwaway17_17 5y ago
I feel like I could sit down and rewrite this, in a couple of hours, to be about Computer Science, but the practical field and the educational as well. I mean, I could feasibly include this sentence as a quote, depending on which aspect of CS I was talking about (academic CS and language design in particular): ‘It is that the subjects that appeal to theory-builders are, at the moment, much more fashionable than the ones that appeal to problem-solvers.’ It is an interesting article and I will look forward to any discussion about it, especially the correlation to CS and development.
- graycat 5y ago
  From my experience, among academic computer science professors, the attitude is that computer science is about the "fundamentals" of computing. E.g., once I was talking with such a computer science professor and listing features I wanted in a better programming language, and immediately his reaction was that for a professor developing such a language would be "academic suicide".
  - throwaway17_17 5y ago
    That matches my experience as well. It also illustrates the divide in educational tracks, for lack of a better term, that I probably should have expanded on in my original post. In addition to individual developers and development teams have the potential to be either the problem-solving or the theory building types, the path a person take to programming can go either of those directions. It is most apparent in the difference between a BS in CS from somewhere like Carnegie-Mellon (a bastion of high level theoretical CS education) and any code boot camp or online web-dev centric training course.
  - rocqua 5y ago
    My university had an amazing C++ programming course. Essentially all STEM majors could take credits from the course. CS did not allow credits from this course because "It is just programming" it teaches nothing fundamental. Meanwhile, this course was more difficult that most other courses, taught me to code much better, which helped most other courses, and taught me basic principles that I easily generalized and re-applied in other CS courses. I think that a lot of people going to do Academic work would have been better of for having taken this course. Even if they never were to touch a C++ like language again. The fact that, when talking about inheritance and sub-typing in C++, the terms "variance" or "co-variance" never came up, does not mean that this course taught me little on type-theory.
  - bigbillheck 5y ago
    There are several ways to interpret that person's reaction. 1. It would be academic suicide to develop a new programming language. 2. Developing a new programming language is fine, but developing one designed to be better for practitioners is academic suicide. 3. Developing a new programming language for whatever reason is fine, but the particular ideas you were suggesting were of such a nature (perhaps "generally acknowledged as bad" or "not an improvement over accepted practice") as to be academic suicide. (1) seems unlikely at best, there's a whole section on arxiv for programming languages (https://arxiv.org/list/cs.PL/recent https://arxiv.org/list/cs.PL/recent), and just within the last decade we have languages like Julia and Elm coming out of academia. (2) also seems unlikely, all the examples of recent academic-derived programming languages I can find are designed to make somebody's experience better. (And who would bother designing a programming language if they weren't at least hoping to improve something?) Without knowing further details, I won't comment any further on (3).
    - graycat 5y ago
      Your points are well taken. But the more likely explanation for the professor's reaction was just that doing anything in programming language design would be "academic suicide". The date of his remark was about 1974, and I was suggesting improvements in PL/I that I'd been using for about 4 years. I believe his view was that academic research for programming languages in practice was over with -- e.g., LISP, APL, Algol 68, and PL/I were all implemented by 1974. So, your examples show that he was wrong: Long after 1974 others in academics did work in programming languages without academic death via suicide or otherwise. Sooo, in 1974, views of the academic research content of programming language design varied -- such variations are with us frequently, i.e., make horse races. But that encounter in 1974 was just one example. I had another one from a computer science professor in the 1990s. Also, with some irony, in the 1980s and 1990s, I was in the group at IBM's Watson lab that designed and implemented the artificial intelligence rule-based language YES/L1. We published lots of papers in academic conferences. Moreover, beyond just design, I was our lead on joint work with GM Research and was one of our two presenters of our paper at the AAAI (American Association for Artificial Intelligence) IAAI (Innovative Applications of Artificial Intelligence) at Stanford; so the work, design, implementation, and application of YES/L1 was regarded as worth of publication in academics. Indeed, MIT published a book with paper from the conference, and our paper was one of the ones published. So, with my work, I, too, showed that the comments of those two professors I mentioned were not correct. Still, it remains, with my two examples from professors, there long has been an attitude and belief that work in programming languages is not sufficiently "fundamental" to be in computer science and would be "academic suicide". That has been a common attitude; now both of us have shown that the attitude was not correct, but it remains that that attitude has long existed -- that was my claim, that there was such an attitude. I didn't claim that the attitude was universal or correct. Given that the attitude has been in academic computer
graycat 5y ago
It appears that roughly the point of the OP (original post) is that in math there are two cultures (1) people who want to develop new fields of math with definitions, theorems, and proofs and (2) people who want to use math, all or nearly all old, to solve problems usually from outside math. Apparently the OP is from the UK (United Kingdom). Here I attempt to provide a view and explanation of those two cultures from and for the US. Math got taken very seriously due heavily to WWII and The Bomb. There The Bomb was seen as heavily from Einstein's E = mc^2. So, suddenly the US (especially Congress) concluded that for US national security the US had to lead in science and math. Over the years after WWII, various events, e.g., Sputnik, reinforced this conclusion. One result was that the US NSF had funds for math research for culture (1) in the US research universities. The emphasis was on what was really new; that is, if there was to be another E = mc^2 result from math and science, then the US wanted to be the first to discover that result. But the NSF was not much interested in funding math in culture (2). Then there was some irony: For US national security, the NSF was funding math in culture (1) while also for US national security, especially the Cold War and the Space Race, the US DoD and NASA were heavily funding applications of math via culture (2). In those years, there were good culture (2) careers, especially near DC, for people with comparatively good backgrounds in math and computing. But in the profit seeking, practical, commercial US, away from the motivation of the Cold War and the Space Race, math from either culture was ignored, laughed at, or in rare but overwhelming cases terribly feared. The there was and is help for culture (2): The US research universities also commonly have schools of engineering, and there are journals eager to publish applications of math. E.g., the usual criteria for publication are "new, correct, and significant", and an application of some math that is a new and correct solution for a significant practical problem can be seen to satisfy these criteria and qualify for publication. So, net, currently, an application of math -- maybe all or nearly all old math -- that is powerful and valuable in practice -- i.e., some secret sauce -- can count on little or no competition.
- dan-robertson 5y ago
  You seem to imply that the two cultures are pure vs applied but that’s not really right. Something like Sturm-Liouville theory is generally very applied and useful but falls into culture 1. The sort of mathematics that Erdős did was pure but fell more into culture 2.
alberto_ol 5y ago
previous submissions https://hn.algolia.com/?dateRange=all&page=0&prefix=false&query=https%3A%2F%2Fwww.dpmms.cam.ac.uk%2F~wtg10%2F2cultures.pdf&sort=byPopularity&type=story https://hn.algolia.com/?dateRange=all&page=0&prefix=false&qu...
sdenton4 5y ago
Just to also a couple existing comments here, a problem solver is not necessarily an applied person or an engineer. It's someone who looks for interesting unsolved questions, and invents what needs inventing to get a solution. By contrast, an engineer uses well established techniques (with creativity!) to build bridges that they know can be built. Diving into unsolved mathematical problems - especially the ones that have picked up a reputation - is inherently a riskier endeavor. As the text says: 'the interesting problems tend to be open precisely because the established techniques cannot easily be applied.'
dan-robertson 5y ago
In case it is not familiar, the title of this alludes to https://en.m.wikipedia.org/wiki/The_Two_Cultures https://en.m.wikipedia.org/wiki/The_Two_Cultures
- jonsen 5y ago
  Thanks. I didn’t know this. So it’s a recursive phenomenon and as culture continues developing we can expect quite a mess of a fractal. That would explain the noticeable decline in ability of holistic judgement and the rise of kompetencelessness.
thomasahle 5y ago
Unfortunately the article quickly descends into a long defence of the merits of Combinatorics, rather than a description of the two cultures. Since combinatorics are so ubiquitious today in computer science, algorithms in particular, it would be more interesting for me to better understand the "algebraic number theory and differential geometry" world thst the author continously refers to. I guess I should just read the Michael Atiyah interview that this article sometimes feel like a direct reply to.
- QuesnayJr 5y ago
  It's a real phenomenon that you see at elite departments. I got into an argument here in the comment section with a mathematician on whether graph theory was a "core" area of mathematics. I don't even like graph theory or combinatorics.
josephg 5y ago
I wrote a blog post a few years ago[1] making a similar claim about computer science. I said there’s 3 main camps of programmers: - People who enjoy programming because it’s mathematically beautiful (eg Haskell programmers) - People who enjoy programming because they like reasoning about machines, and like mechanical sympathy (eg C programmers) - And people who like programming because it can solve real problems for their users. I got pushback here and elsewhere that lots of people fit into multiple camps - which makes sense. Maybe they’re better described as an ecology of values. But I also think there’s something real here. I worked in a consulting shop a few years ago and my boss came back from some front end development conference gushing about a talk he’d seen that I had to watch. The talk was a layman’s description of FP’s immutability approach applied to react. My boss had never heard of immutability before because it had never come up in front end web development circles. There’s lots of opportunities for our software to improve by cross contaminating all our best ideas. [1] https://josephg.com/blog/3-tribes/ https://josephg.com/blog/3-tribes/
- Karrot_Kream 5y ago
  Fantastic! Thanks for this. The FP vs non-FP split has always just felt like a less rigorous version of the "analysis vs algebra" or "problem-solving vs theoretical" discussion and I'm glad I'm not alone in thinking this. This is also what rubs me the wrong way about the simplistic level with which FP languages (like Haskell or Scala+monads) are either cargo culted or hated. The truth is, the two perspectives really go hand-in-hand, and we're all the worse for not realizing this.
  - Kalium 5y ago
    In my opinion, the FP/IP split is fueled in no small part by a wildly divergent set of vocabularies. A dozen different <greek letter> transforms are inscrutable to anyone who has not spent much time studying Lambda calculus, and the concept of a monad is famously difficult to explain. You wind up with two camps that cannot communicate. It's often made worse by the functional programmers - who are mostly also imperative programmers - doing a poor job of enabling translation.
  - ukj 5y ago
dang 5y ago
Some past threads: The two cultures of mathematics and biology - https://news.ycombinator.com/item?id=8819811 https://news.ycombinator.com/item?id=8819811 - Dec 2014 (69 comments) The Two Cultures of Mathematics (2000) [pdf] - https://news.ycombinator.com/item?id=7970284 https://news.ycombinator.com/item?id=7970284 - July 2014 (28 comments) "Problem Solvers" vs "Theory Developers": The Two Cultures of Mathematics. [pdf] - https://news.ycombinator.com/item?id=682913 https://news.ycombinator.com/item?id=682913 - July 2009 (5 comments)
- alberto_ol 5y ago
  the first thred "The two cultures of mathematics and biology" is not the same essay
  - dang 5y ago
    Ah yes, sorry - I noticed that and meant to take it out but evidently just imagined that I did!
jorgenveisdal 5y ago
Sylvia Nasar's take on John Forbes Nash, Jr.'s type: "He was a mathematician who viewed mathematics not as a grand scheme, but as a collection of challenging problems. In the taxonomy of mathematicians, there are problem solvers and theoreticians, and, by temperament, Nash belonged to the first group. He was not a game theorist, analyst, algebraist, geometer, topologist, or mathematical physicist. But he zeroed in on areas in these fields where essentially nobody had achieved anything. The thing was to find an interesting question that he could say something about."
- 276 5y ago
  let me try an example... ^^ '12 x 23 = 276' Most often listed is the number '2'. The highest numbers together make '7' and if you defer 'the multiplication one position to the right', you have '2 x 3' -making '6'. 276 (-;
Double_Cast 5y ago
To me, the distinction maps onto the Effectual vs Causal distinction. For some path "A to B", some people prefer to consider their present tools/resources (point A) and work forward opportunistically. While others prefer to consider the end-goal (point B) and work backwards recursively. If the analogy is unclear: a theory is a tool; a specific problem is an end-goal.
hatmatrix 5y ago
It sounds like the distinction is between pure and applied mathematics, though the writer says that the battle is within pure mathematics. I've also read about two other categorizations of mathematicians: active and passive. The active are out trying to prove new theorems while passive mathematicians are trying to collect and generalize past theorems. Finding generalizations also requires new theorems and could be argued to be an active task, but this sounded like a distinction between researcher and educator, and has stuck with me.
isolli 5y ago
Only tangentially related, but a friend of mine had to interrupt his doctoral thesis in mathematics because the set he was studying turned out to be empty. This gave a lot of ammunition to our Russian colleagues, who routinely make fun of French mathematicians for being far too ensconced in theory.
- dTal 5y ago
  That still sounds like an interesting and publishable result, if unfortunate if you're trying to spin an entire PhD out of it.
piadista 5y ago
Actually the late Prof Leo Breiman expanded this metaphor into the statistical modelling world. The two cultures being the explanatory and predictive modelling folk. The explanatory modelling culture (that he calls the data modelling culture) being those that first come up with a guess of how the data is generated and then try to test that hypothesis using goodness of fit measures, and the predictive modelling culture (he calls algorithmic modelling culture) being the modern machine learning researchers that are purely interested in predictive power. The full essay is here: https://projecteuclid.org/journals/statistical-science/volume-16/issue-3/Statistical-Modeling--The-Two-Cultures-with-comments-and-a/10.1214/ss/1009213726.full https://projecteuclid.org/journals/statistical-science/volum...
ukj 5y ago
This paper makes a ton of brilliant points all of which strongly resonate with me: * The results that will last are the ones that can be organized coherently and explained economically to future generations (yes! effective compression!) * How effectively a result can be communicated to another mathematician (and perhaps even s/mathematician/person/) 90% of my time spend 'studying mathematics' is spent lexing the notation. What do those symbols even mean? I can't even copy-paste this into Google to get any meaningful results!!!!!!!!!!!!@#!$@#$@#$@#$#@!$@#$ Have you noticed how we don't have this problem in Computer Science? Because the source code gives you the context in which to interpret the meaning of the grammar! Homoiconicity is the panacea of formal languages.
- theflyinghorse 5y ago
  I remember in our intro to AI course our prof would sometime run out of English letters for symbols, then start introducing German ones, and eventually start using triangles and squares for stuff
- frakt0x90 5y ago
  I do think there needs to be a better search system for latex/math symbols. That would be amazing. As far as using the notation, I forget where I read this but I remember seeing that one excuse for the use of abstract symbols is keep the ideas abstract so as to not narrow your mind into just what you're working on. So many areas of math cross over so keeping things abstract could aid in that recognition.
yummypaint 5y ago
There is a similar kind of division of labor in physics between theorists and experimentalists. The cultural difference is closer to that between mathemeticians and engineers, but of course both groups are entirely dependent on each other and most successful collaborations have both. The days of solo publishing are mostly over.

145 points by vector_spaces 5y ago

I feel like I could sit down and rewrite this, in a couple of hours, to be about Computer Science, but the practical field and the educational as well. I mean, I could feasibly include this sentence as a quote, depending on which aspect of CS I was talking about (academic CS and language design in particular): ‘It is that the subjects that appeal to theory-builders are, at the moment, much more fashionable than the ones that appeal to problem-solvers.’ It is an interesting article and I will look forward to any discussion about it, especially the correlation to CS and development.

graycat 5y ago
From my experience, among academic computer science professors, the attitude is that computer science is about the "fundamentals" of computing. E.g., once I was talking with such a computer science professor and listing features I wanted in a better programming language, and immediately his reaction was that for a professor developing such a language would be "academic suicide".
- throwaway17_17 5y ago
  That matches my experience as well. It also illustrates the divide in educational tracks, for lack of a better term, that I probably should have expanded on in my original post. In addition to individual developers and development teams have the potential to be either the problem-solving or the theory building types, the path a person take to programming can go either of those directions. It is most apparent in the difference between a BS in CS from somewhere like Carnegie-Mellon (a bastion of high level theoretical CS education) and any code boot camp or online web-dev centric training course.
- rocqua 5y ago
  My university had an amazing C++ programming course. Essentially all STEM majors could take credits from the course. CS did not allow credits from this course because "It is just programming" it teaches nothing fundamental. Meanwhile, this course was more difficult that most other courses, taught me to code much better, which helped most other courses, and taught me basic principles that I easily generalized and re-applied in other CS courses. I think that a lot of people going to do Academic work would have been better of for having taken this course. Even if they never were to touch a C++ like language again. The fact that, when talking about inheritance and sub-typing in C++, the terms "variance" or "co-variance" never came up, does not mean that this course taught me little on type-theory.
- bigbillheck 5y ago
  There are several ways to interpret that person's reaction. 1. It would be academic suicide to develop a new programming language. 2. Developing a new programming language is fine, but developing one designed to be better for practitioners is academic suicide. 3. Developing a new programming language for whatever reason is fine, but the particular ideas you were suggesting were of such a nature (perhaps "generally acknowledged as bad" or "not an improvement over accepted practice") as to be academic suicide. (1) seems unlikely at best, there's a whole section on arxiv for programming languages (https://arxiv.org/list/cs.PL/recent https://arxiv.org/list/cs.PL/recent), and just within the last decade we have languages like Julia and Elm coming out of academia. (2) also seems unlikely, all the examples of recent academic-derived programming languages I can find are designed to make somebody's experience better. (And who would bother designing a programming language if they weren't at least hoping to improve something?) Without knowing further details, I won't comment any further on (3).
  - graycat 5y ago
    Your points are well taken. But the more likely explanation for the professor's reaction was just that doing anything in programming language design would be "academic suicide". The date of his remark was about 1974, and I was suggesting improvements in PL/I that I'd been using for about 4 years. I believe his view was that academic research for programming languages in practice was over with -- e.g., LISP, APL, Algol 68, and PL/I were all implemented by 1974. So, your examples show that he was wrong: Long after 1974 others in academics did work in programming languages without academic death via suicide or otherwise. Sooo, in 1974, views of the academic research content of programming language design varied -- such variations are with us frequently, i.e., make horse races. But that encounter in 1974 was just one example. I had another one from a computer science professor in the 1990s. Also, with some irony, in the 1980s and 1990s, I was in the group at IBM's Watson lab that designed and implemented the artificial intelligence rule-based language YES/L1. We published lots of papers in academic conferences. Moreover, beyond just design, I was our lead on joint work with GM Research and was one of our two presenters of our paper at the AAAI (American Association for Artificial Intelligence) IAAI (Innovative Applications of Artificial Intelligence) at Stanford; so the work, design, implementation, and application of YES/L1 was regarded as worth of publication in academics. Indeed, MIT published a book with paper from the conference, and our paper was one of the ones published. So, with my work, I, too, showed that the comments of those two professors I mentioned were not correct. Still, it remains, with my two examples from professors, there long has been an attitude and belief that work in programming languages is not sufficiently "fundamental" to be in computer science and would be "academic suicide". That has been a common attitude; now both of us have shown that the attitude was not correct, but it remains that that attitude has long existed -- that was my claim, that there was such an attitude. I didn't claim that the attitude was universal or correct. Given that the attitude has been in academic computer

graycat 5y ago

It appears that roughly the point of the OP (original post) is that in math there are two cultures (1) people who want to develop new fields of math with definitions, theorems, and proofs and (2) people who want to use math, all or nearly all old, to solve problems usually from outside math. Apparently the OP is from the UK (United Kingdom). Here I attempt to provide a view and explanation of those two cultures from and for the US. Math got taken very seriously due heavily to WWII and The Bomb. There The Bomb was seen as heavily from Einstein's E = mc^2. So, suddenly the US (especially Congress) concluded that for US national security the US had to lead in science and math. Over the years after WWII, various events, e.g., Sputnik, reinforced this conclusion. One result was that the US NSF had funds for math research for culture (1) in the US research universities. The emphasis was on what was really new; that is, if there was to be another E = mc^2 result from math and science, then the US wanted to be the first to discover that result. But the NSF was not much interested in funding math in culture (2). Then there was some irony: For US national security, the NSF was funding math in culture (1) while also for US national security, especially the Cold War and the Space Race, the US DoD and NASA were heavily funding applications of math via culture (2). In those years, there were good culture (2) careers, especially near DC, for people with comparatively good backgrounds in math and computing. But in the profit seeking, practical, commercial US, away from the motivation of the Cold War and the Space Race, math from either culture was ignored, laughed at, or in rare but overwhelming cases terribly feared. The there was and is help for culture (2): The US research universities also commonly have schools of engineering, and there are journals eager to publish applications of math. E.g., the usual criteria for publication are "new, correct, and significant", and an application of some math that is a new and correct solution for a significant practical problem can be seen to satisfy these criteria and qualify for publication. So, net, currently, an application of math -- maybe all or nearly all old math -- that is powerful and valuable in practice -- i.e., some secret sauce -- can count on little or no competition.

dan-robertson 5y ago
You seem to imply that the two cultures are pure vs applied but that’s not really right. Something like Sturm-Liouville theory is generally very applied and useful but falls into culture 1. The sort of mathematics that Erdős did was pure but fell more into culture 2.

alberto_ol 5y ago

previous submissions https://hn.algolia.com/?dateRange=all&page=0&prefix=false&query=https%3A%2F%2Fwww.dpmms.cam.ac.uk%2F~wtg10%2F2cultures.pdf&sort=byPopularity&type=story https://hn.algolia.com/?dateRange=all&page=0&prefix=false&qu...

sdenton4 5y ago

Just to also a couple existing comments here, a problem solver is not necessarily an applied person or an engineer. It's someone who looks for interesting unsolved questions, and invents what needs inventing to get a solution. By contrast, an engineer uses well established techniques (with creativity!) to build bridges that they know can be built. Diving into unsolved mathematical problems - especially the ones that have picked up a reputation - is inherently a riskier endeavor. As the text says: 'the interesting problems tend to be open precisely because the established techniques cannot easily be applied.'

dan-robertson 5y ago

In case it is not familiar, the title of this alludes to https://en.m.wikipedia.org/wiki/The_Two_Cultures https://en.m.wikipedia.org/wiki/The_Two_Cultures

jonsen 5y ago
Thanks. I didn’t know this. So it’s a recursive phenomenon and as culture continues developing we can expect quite a mess of a fractal. That would explain the noticeable decline in ability of holistic judgement and the rise of kompetencelessness.

thomasahle 5y ago

Unfortunately the article quickly descends into a long defence of the merits of Combinatorics, rather than a description of the two cultures. Since combinatorics are so ubiquitious today in computer science, algorithms in particular, it would be more interesting for me to better understand the "algebraic number theory and differential geometry" world thst the author continously refers to. I guess I should just read the Michael Atiyah interview that this article sometimes feel like a direct reply to.

QuesnayJr 5y ago
It's a real phenomenon that you see at elite departments. I got into an argument here in the comment section with a mathematician on whether graph theory was a "core" area of mathematics. I don't even like graph theory or combinatorics.

josephg 5y ago

I wrote a blog post a few years ago[1] making a similar claim about computer science. I said there’s 3 main camps of programmers: - People who enjoy programming because it’s mathematically beautiful (eg Haskell programmers) - People who enjoy programming because they like reasoning about machines, and like mechanical sympathy (eg C programmers) - And people who like programming because it can solve real problems for their users. I got pushback here and elsewhere that lots of people fit into multiple camps - which makes sense. Maybe they’re better described as an ecology of values. But I also think there’s something real here. I worked in a consulting shop a few years ago and my boss came back from some front end development conference gushing about a talk he’d seen that I had to watch. The talk was a layman’s description of FP’s immutability approach applied to react. My boss had never heard of immutability before because it had never come up in front end web development circles. There’s lots of opportunities for our software to improve by cross contaminating all our best ideas. [1] https://josephg.com/blog/3-tribes/ https://josephg.com/blog/3-tribes/

Karrot_Kream 5y ago
Fantastic! Thanks for this. The FP vs non-FP split has always just felt like a less rigorous version of the "analysis vs algebra" or "problem-solving vs theoretical" discussion and I'm glad I'm not alone in thinking this. This is also what rubs me the wrong way about the simplistic level with which FP languages (like Haskell or Scala+monads) are either cargo culted or hated. The truth is, the two perspectives really go hand-in-hand, and we're all the worse for not realizing this.
- Kalium 5y ago
  In my opinion, the FP/IP split is fueled in no small part by a wildly divergent set of vocabularies. A dozen different <greek letter> transforms are inscrutable to anyone who has not spent much time studying Lambda calculus, and the concept of a monad is famously difficult to explain. You wind up with two camps that cannot communicate. It's often made worse by the functional programmers - who are mostly also imperative programmers - doing a poor job of enabling translation.
- ukj 5y ago

dang 5y ago

Some past threads: The two cultures of mathematics and biology - https://news.ycombinator.com/item?id=8819811 https://news.ycombinator.com/item?id=8819811 - Dec 2014 (69 comments) The Two Cultures of Mathematics (2000) [pdf] - https://news.ycombinator.com/item?id=7970284 https://news.ycombinator.com/item?id=7970284 - July 2014 (28 comments) "Problem Solvers" vs "Theory Developers": The Two Cultures of Mathematics. [pdf] - https://news.ycombinator.com/item?id=682913 https://news.ycombinator.com/item?id=682913 - July 2009 (5 comments)

alberto_ol 5y ago
the first thred "The two cultures of mathematics and biology" is not the same essay
- dang 5y ago
  Ah yes, sorry - I noticed that and meant to take it out but evidently just imagined that I did!

jorgenveisdal 5y ago

Sylvia Nasar's take on John Forbes Nash, Jr.'s type: "He was a mathematician who viewed mathematics not as a grand scheme, but as a collection of challenging problems. In the taxonomy of mathematicians, there are problem solvers and theoreticians, and, by temperament, Nash belonged to the first group. He was not a game theorist, analyst, algebraist, geometer, topologist, or mathematical physicist. But he zeroed in on areas in these fields where essentially nobody had achieved anything. The thing was to find an interesting question that he could say something about."

276 5y ago
let me try an example... ^^ '12 x 23 = 276' Most often listed is the number '2'. The highest numbers together make '7' and if you defer 'the multiplication one position to the right', you have '2 x 3' -making '6'. 276 (-;

Double_Cast 5y ago

To me, the distinction maps onto the Effectual vs Causal distinction. For some path "A to B", some people prefer to consider their present tools/resources (point A) and work forward opportunistically. While others prefer to consider the end-goal (point B) and work backwards recursively. If the analogy is unclear: a theory is a tool; a specific problem is an end-goal.

hatmatrix 5y ago

It sounds like the distinction is between pure and applied mathematics, though the writer says that the battle is within pure mathematics. I've also read about two other categorizations of mathematicians: active and passive. The active are out trying to prove new theorems while passive mathematicians are trying to collect and generalize past theorems. Finding generalizations also requires new theorems and could be argued to be an active task, but this sounded like a distinction between researcher and educator, and has stuck with me.

isolli 5y ago

Only tangentially related, but a friend of mine had to interrupt his doctoral thesis in mathematics because the set he was studying turned out to be empty. This gave a lot of ammunition to our Russian colleagues, who routinely make fun of French mathematicians for being far too ensconced in theory.

dTal 5y ago
That still sounds like an interesting and publishable result, if unfortunate if you're trying to spin an entire PhD out of it.

piadista 5y ago

Actually the late Prof Leo Breiman expanded this metaphor into the statistical modelling world. The two cultures being the explanatory and predictive modelling folk. The explanatory modelling culture (that he calls the data modelling culture) being those that first come up with a guess of how the data is generated and then try to test that hypothesis using goodness of fit measures, and the predictive modelling culture (he calls algorithmic modelling culture) being the modern machine learning researchers that are purely interested in predictive power. The full essay is here: https://projecteuclid.org/journals/statistical-science/volume-16/issue-3/Statistical-Modeling--The-Two-Cultures-with-comments-and-a/10.1214/ss/1009213726.full https://projecteuclid.org/journals/statistical-science/volum...

ukj 5y ago

This paper makes a ton of brilliant points all of which strongly resonate with me: * The results that will last are the ones that can be organized coherently and explained economically to future generations (yes! effective compression!) * How effectively a result can be communicated to another mathematician (and perhaps even s/mathematician/person/) 90% of my time spend 'studying mathematics' is spent lexing the notation. What do those symbols even mean? I can't even copy-paste this into Google to get any meaningful results!!!!!!!!!!!!@#!$@#$@#$@#$#@!$@#$ Have you noticed how we don't have this problem in Computer Science? Because the source code gives you the context in which to interpret the meaning of the grammar! Homoiconicity is the panacea of formal languages.

theflyinghorse 5y ago
I remember in our intro to AI course our prof would sometime run out of English letters for symbols, then start introducing German ones, and eventually start using triangles and squares for stuff
frakt0x90 5y ago
I do think there needs to be a better search system for latex/math symbols. That would be amazing. As far as using the notation, I forget where I read this but I remember seeing that one excuse for the use of abstract symbols is keep the ideas abstract so as to not narrow your mind into just what you're working on. So many areas of math cross over so keeping things abstract could aid in that recognition.

yummypaint 5y ago

There is a similar kind of division of labor in physics between theorists and experimentalists. The cultural difference is closer to that between mathemeticians and engineers, but of course both groups are entirely dependent on each other and most successful collaborations have both. The days of solo publishing are mostly over.