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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 defini

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 6y 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.
- graycat 6y ago
  You might be correct in part or in whole. We'd have to discuss back and forth! My (1) and (2) will have to be at best only rough, and to save space I omitted the real examples I have in mind to formulate those two. Writing clearly, briefly, and precisely about big subjects is not always easy! The time I encountered Strum-Louville theory was a course from the fairly well known Hildebrand, Advanced Calculus for Applications from MIT and by a professor with a recent MIT Ph.D. By the way, that book is available in PDF on the Internet for free. To me the topic looked like the math of vibrating strings, etc., but the course did not go deeply into Sturm-Louville theory or anything else. So, from that course I have regarded Sturm-Louville theory now as in (2); but if there is some continuing, deep work there, maybe it belongs in (1). Maybe I touched on that subject once more; talking to M. Athans at MIT, he explained that the optimal control problem I was considering was a "two point boundary value problem with mixed end conditions". So, maybe there are connections with control theory. For Erdős, I have just regarded him as all in (1), but he seemed to have gotten problems from anywhere, including from outside math so might be in (2). Sooo, for the problems outside math, I meant relatively practical problems, ones where solutions would a lot of value, financial or scientific, for fields outside of math. I've always regarded Erdős as working a lot on interesting, tricky, puzzle problems. But I don't know Erdős's work at all well. Maybe I can summarize: Back when I had one foot in each of (1) and (2), my definition of applied math was like a recipe for rabbit stew -- "First catch a rabbit." So, first get an application, i.e., from outside math, where an application solves a problem that in some sense is pressing and practical. Examples: At FedEx, how to schedule the fleet? How to project revenue for years in advance? How best to climb, cruise, and descend an airplane? How to plan an airplane tour under uncertainty. Back in military work, how to use the Navier-Stokes equations to design ship propellers? How to process the data to do beam forming from arrays of passive sonar? How to evaluate the survivability of the US SSBN submarines? How to target missiles? How best to search for a submarine? In a simulation, how to generate ocean waves with a given power spectral density? How to respond to ill-conditioned matrices in regression problems? How to do least squares spline fitting, including for multidimensional data? For each of these problems, there were people who cared a lot about getting the solutions -- these were applications. Puzzle problems seem to be applications of a different kind. Maybe now there are some good applications to be made analysis of DNA.
bigbillheck 6y ago
> But the NSF was not much interested in funding math in culture (2). I'm not sure that this is supported by the historical record. For example, from NSF's own page: https://www.nsf.gov/about/history/overview-50.jsp https://www.nsf.gov/about/history/overview-50.jsp > The first NSF grants are awarded to support computation centers and research in numerical analysis. Three years later, a separate budget is created for grants to enable academic institutions to acquire major computer equipment. That seems like it's very firmly in your (2). Going further, I found the distribution of NSF grants for 1952. Mathematics got one grant, to Lamberto Cesari, for work on "Asymtotic Behavior and Stability Problems". Cesari has a wiki page with links to some of his articles, for example : https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-56/issue-3/Area-and-representation-of-surfaces/bams/1183514764.full https://projecteuclid.org/journals/bulletin-of-the-american-... , and a bio at St Andrews: https://mathshistory.st-andrews.ac.uk/Biographies/Cesari/ https://mathshistory.st-andrews.ac.uk/Biographies/Cesari/ Going by those, I don't see how that would not be (2)? A little later we find UMN's "Institute for Mathematics and its Applications" (https://www.ima.umn.edu/about/history https://www.ima.umn.edu/about/history, sounds very (2)-ish) which was established with NSF funding in 1982. And a bit after that, in the 90s, I am fairly confident that some portion of my graduate work in applying mathematics to biological problems was NSF-supported.
- graycat 5y ago
  The Kelley, General Topology pure math book acknowledges NSF and Office of Naval Research support in years 1950-1952. Same financial supporters for Luenberger, Optimization by Vector Space Methods -- fun with the Hahn-Banach theorem. Good to hear that the NSF has supported some computing hardware -- my guess was that that was done mostly by the US Department of Energy. E.g., as I recall LINPACK was developed at Oak Ridge. At one time I was teaching computer science at Georgetown U., and some colleagues wanted to work in speech recognition, asked for NSF support, and were told that the NSF did not support "software development" or some such. Most of numerical analysis I saw was good definition, theorem, proof math. But recently I saw that the Formula 1 auto racing teams have been using CFD (computational fluid dynamics) for detailed design of the shape, wings, downforce, etc. of their cars. That's a lot of progress in fluid flow since I was working with the Navier-Stokes equations. So, apparently numerical methods for fluid flow have made a LOT of progress. As I recall, at times there was some such work at Courant Institute. I hope the NSF supported some of that progress. For NSF support of "applying mathematics to biological problems" very good to hear, but my guess would be that that came under NSF support of biology instead of applied math. Here is more what I had in mind: Daily the many US research-teaching hospitals take in sick people and make them well. There is good research, but the actual patient care is clinical, that is, serving people, and professional, e.g., with apprenticeship training, code of ethics, and liability and for professional practice. In the US I don't see anything similar for applied math or computer science, with or without NSF support. Generally US research universities have a huge fraction of their annual operating budgets from grants from the US Federal Government, especially NSF and NIH. Roughly professors apply for grants, and about 60% goes to overhead for the university and, thus, also supports the English department, the string quartet series, the drama company, the alumni magazine, etc. Since the research universities don't much like math applications outside of academics, that NSF/NIH support is not much for such math applications. E.g., university math departments have lots of talks by professors presenting their solutions without known non-academic problems but nearly no talks from non-academic people with problems looking for solutions. So, such academic math departments with NSF grants are not using that NSF funding to help non-academic people with problems find solutions. E.g., in grad school in an applied math department, I studied a lot in optimization and stochastic processes, but I was the only one there with any real non-academic problems that could use work in optimization or stochastic processes. When I was a B-school prof, I was shocked to discover that B-schools were fine with some quite pure math research but wanted nothing to do with education or practice as in the law school, medical school, school of pharmacy, or agricultural college.

dan-robertson 6y 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.

graycat 6y ago
You might be correct in part or in whole. We'd have to discuss back and forth! My (1) and (2) will have to be at best only rough, and to save space I omitted the real examples I have in mind to formulate those two. Writing clearly, briefly, and precisely about big subjects is not always easy! The time I encountered Strum-Louville theory was a course from the fairly well known Hildebrand, Advanced Calculus for Applications from MIT and by a professor with a recent MIT Ph.D. By the way, that book is available in PDF on the Internet for free. To me the topic looked like the math of vibrating strings, etc., but the course did not go deeply into Sturm-Louville theory or anything else. So, from that course I have regarded Sturm-Louville theory now as in (2); but if there is some continuing, deep work there, maybe it belongs in (1). Maybe I touched on that subject once more; talking to M. Athans at MIT, he explained that the optimal control problem I was considering was a "two point boundary value problem with mixed end conditions". So, maybe there are connections with control theory. For Erdős, I have just regarded him as all in (1), but he seemed to have gotten problems from anywhere, including from outside math so might be in (2). Sooo, for the problems outside math, I meant relatively practical problems, ones where solutions would a lot of value, financial or scientific, for fields outside of math. I've always regarded Erdős as working a lot on interesting, tricky, puzzle problems. But I don't know Erdős's work at all well. Maybe I can summarize: Back when I had one foot in each of (1) and (2), my definition of applied math was like a recipe for rabbit stew -- "First catch a rabbit." So, first get an application, i.e., from outside math, where an application solves a problem that in some sense is pressing and practical. Examples: At FedEx, how to schedule the fleet? How to project revenue for years in advance? How best to climb, cruise, and descend an airplane? How to plan an airplane tour under uncertainty. Back in military work, how to use the Navier-Stokes equations to design ship propellers? How to process the data to do beam forming from arrays of passive sonar? How to evaluate the survivability of the US SSBN submarines? How to target missiles? How best to search for a submarine? In a simulation, how to generate ocean waves with a given power spectral density? How to respond to ill-conditioned matrices in regression problems? How to do least squares spline fitting, including for multidimensional data? For each of these problems, there were people who cared a lot about getting the solutions -- these were applications. Puzzle problems seem to be applications of a different kind. Maybe now there are some good applications to be made analysis of DNA.

bigbillheck 6y ago

> But the NSF was not much interested in funding math in culture (2). I'm not sure that this is supported by the historical record. For example, from NSF's own page: https://www.nsf.gov/about/history/overview-50.jsp https://www.nsf.gov/about/history/overview-50.jsp > The first NSF grants are awarded to support computation centers and research in numerical analysis. Three years later, a separate budget is created for grants to enable academic institutions to acquire major computer equipment. That seems like it's very firmly in your (2). Going further, I found the distribution of NSF grants for 1952. Mathematics got one grant, to Lamberto Cesari, for work on "Asymtotic Behavior and Stability Problems". Cesari has a wiki page with links to some of his articles, for example : https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-56/issue-3/Area-and-representation-of-surfaces/bams/1183514764.full https://projecteuclid.org/journals/bulletin-of-the-american-... , and a bio at St Andrews: https://mathshistory.st-andrews.ac.uk/Biographies/Cesari/ https://mathshistory.st-andrews.ac.uk/Biographies/Cesari/ Going by those, I don't see how that would not be (2)? A little later we find UMN's "Institute for Mathematics and its Applications" (https://www.ima.umn.edu/about/history https://www.ima.umn.edu/about/history, sounds very (2)-ish) which was established with NSF funding in 1982. And a bit after that, in the 90s, I am fairly confident that some portion of my graduate work in applying mathematics to biological problems was NSF-supported.

graycat 5y ago
The Kelley, General Topology pure math book acknowledges NSF and Office of Naval Research support in years 1950-1952. Same financial supporters for Luenberger, Optimization by Vector Space Methods -- fun with the Hahn-Banach theorem. Good to hear that the NSF has supported some computing hardware -- my guess was that that was done mostly by the US Department of Energy. E.g., as I recall LINPACK was developed at Oak Ridge. At one time I was teaching computer science at Georgetown U., and some colleagues wanted to work in speech recognition, asked for NSF support, and were told that the NSF did not support "software development" or some such. Most of numerical analysis I saw was good definition, theorem, proof math. But recently I saw that the Formula 1 auto racing teams have been using CFD (computational fluid dynamics) for detailed design of the shape, wings, downforce, etc. of their cars. That's a lot of progress in fluid flow since I was working with the Navier-Stokes equations. So, apparently numerical methods for fluid flow have made a LOT of progress. As I recall, at times there was some such work at Courant Institute. I hope the NSF supported some of that progress. For NSF support of "applying mathematics to biological problems" very good to hear, but my guess would be that that came under NSF support of biology instead of applied math. Here is more what I had in mind: Daily the many US research-teaching hospitals take in sick people and make them well. There is good research, but the actual patient care is clinical, that is, serving people, and professional, e.g., with apprenticeship training, code of ethics, and liability and for professional practice. In the US I don't see anything similar for applied math or computer science, with or without NSF support. Generally US research universities have a huge fraction of their annual operating budgets from grants from the US Federal Government, especially NSF and NIH. Roughly professors apply for grants, and about 60% goes to overhead for the university and, thus, also supports the English department, the string quartet series, the drama company, the alumni magazine, etc. Since the research universities don't much like math applications outside of academics, that NSF/NIH support is not much for such math applications. E.g., university math departments have lots of talks by professors presenting their solutions without known non-academic problems but nearly no talks from non-academic people with problems looking for solutions. So, such academic math departments with NSF grants are not using that NSF funding to help non-academic people with problems find solutions. E.g., in grad school in an applied math department, I studied a lot in optimization and stochastic processes, but I was the only one there with any real non-academic problems that could use work in optimization or stochastic processes. When I was a B-school prof, I was shocked to discover that B-schools were fine with some quite pure math research but wanted nothing to do with education or practice as in the law school, medical school, school of pharmacy, or agricultural college.