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The Two Cultures of Mathematics (2000) [pdf]

dpmms.cam.ac.uk

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Re: The Two Cultures of Mathematics (2000) [pdf]

#61
post #4

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 o…

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".

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), 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).

Re: The Two Cultures of Mathematics (2000) [pdf]

#62
post #11

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…

"I got pushback here and elsewhere that lots of people fit into multiple camps - which makes sense."

People do fit into multiple camps... however, the camps definitely exist.

One of the several reasons I didn't go into academia, and one of the bigger ones overall, is that I could tell I wanted to split the difference between the "practicals" and the "mathematicians". And while this may not have been impossible, it was certainly at the very least a "high risk" move, because you end up with the support of neither camp, and there aren't enough people in the middle to make up for it.

Re: The Two Cultures of Mathematics (2000) [pdf]

#63
post #39

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 nota…

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.

Re: The Two Cultures of Mathematics (2000) [pdf]

#64

Earlier quoted context omitted.

Perhaps you are a detail-oriented person? I've met quite a lot of people who - like you it seems (although perhaps you disagree) - find things that are low-level and procedural to be clearest. Such people often like languages like C and Go where this style is encouraged. On the other hand, I tend to think in terms of abstractions. And thus find things much clearer when they are based on high level abstractions (thing…

i think it may be that. When I see some code like this in C++: y = f(x); I'm extremely terrified to what is going on. It is like looking into an abyss. Is this code calling a function named "f" with argument "x"? If so, there may be several different functions "f" depending on the type of "x". Maybe none of them has the same type as "x", but there are some conversions that may happen and lead to that. Or maybe "f" is…

> i think it may be that. When I see some code like this in C++

Indeed and I think this is the root of all the internet shouting on abstraction-heavy languages vs procedural/"simple" languages. A lot of folks that think in abstraction find it tedious and unsafe to deal with detail-oriented languages, while folks who are more detail-oriented find the abstraction disorienting and unsafe in their own way. Given that one group can't do away with another group, I hope we can all learn to work together.

Re: The Two Cultures of Mathematics (2000) [pdf]

#65
post #4

Earlier quoted context omitted.

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".

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 ac…

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

Re: The Two Cultures of Mathematics (2000) [pdf]

#66
post #4

Earlier quoted context omitted.

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".

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 ac…

I gave the prof's remark of "academic suicide" as an example of a significant attitude in computer science. I didn't claim that the remark was correct, and your examples show that the remark was not correct.

But the more likely explanation for the professor's reaction was not your (1) -- (3) but just that prof's strong belief that doing anything in programming language design would be "academic suicide". Again, I didn't claim that the prof's attitude was correct, but it WAS his attitude.

The date of his remark was about 1974, and I was suggesting improvements in PL/I (or similar languages) that I'd been using for about 4 years, for US Navy sonar work, scheduling the fleet at FedEx, etc.

I would guess that the prof's 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 and more show that the prof was wrong: Long after 1974 others in academic computer science 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 differences of opinion are common, i.e., make horse races.

But that encounter in 1974 was just one example. In the 1990s I had another one from a prof in a top level computer science department. Gee, at the time the President of that prof's university was one of my Ph.D. dissertation advisors, and I could have warned the prof that he was trying to paddle upstream against the relatively practical orientation of his university president!

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 several papers in computer science conferences.

Moreover, beyond just language design, I was our lead on our joint work with GM Research, the person writing our paper, and 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 worthy of publication in academics. Indeed, MIT published a book with papers from the conference, and our paper was one of the ones published.

So, with my work, now both of us have shown 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 science, no doubt the work -- both research and teaching -- of academic computer science departments has been affected. In particular, students who are eager to get high proficiency in using programming languages, and, maybe, writing compilers, can find, sadly, that their department is not much interested.

I have a third example of academic computer science demeaning programming languages: When I was prof in a Big 10 B-school and MBA program, I'd quickly upset the apple cart of the campus CIO (soon I served on a committee to pick another CIO) and been named the Chair of the Computer Committee of the B-school. And I'd given a grad course in computer selection and management. We were considering what our B-school might do in research and teaching in computing and had a site visit of computer science profs from some other Big 10 universities.

At one of our meetings with the site visit committee, one of the visitors asked me what programming I'd done and in what languages. So, I mentioned some of my 15 years or so of work with a variety of applications with a variety of common languages. Right away his reaction was that from that experience with programming my "brain was ruined for computer science". Again, I'm not saying he was correct! So here is my third example of some academic computer science hostility to programming languages!

This hostility seems to be a special case of a larger, common pattern in parts of academics: Some graduate students and non-tenured faculty live under continual yellow rain from leaks from higher up latrines.

The situation is harmful for all productive purposes for all concerned.

E.g., as a grad student, I got the best in our class on four of the five Ph.D. qualifying exams and in the fifth topic already, from independent work, had a first manuscript of my Ph.D. dissertation, but the yellow rain continued. It wasn't just me: Junior faculty were falling like soldiers at Gettysburg.

By accident I found a solution, a water tight hazmat suit and a stainless steel umbrella: A course in optimization had gotten fairly deep into the Kuhn-Tucker conditions. I saw a question, saw no answers in the likely literature, and asked for a "reading course" to address the question. I had a rough idea how to proceed, and the course was approved. I looked around for existing tools for a solution, found none, so just pursued what I had in mind. Two weeks later I had a nice solution, written up, submitted, and accepted. So, I was done with the "reading course" in two weeks. Part of my solution was a surprising, curious theorem -- with the set of real numbers R, a positive integer n, R^n with the usual topology, and a set C a subset of R^n, set C is closed if and only if it is the level set of an infinitely differentiable function f: R^n --> R. My work in constraint qualifications also answered a question stated but not answered in the famous paper in mathematical economics by Arrow, Hurwicz, and Uzawa. Yes, I found it easy to publish the paper.

When I submitted my work at the end of the two weeks, word spread quickly in the department. The prof I regarded as the best gave me congrats in the hall. Result -- the hazmat suit and umbrella. For the rest of my time through my Ph.D., no more yellow rain.

My Ph.D. was in applied math and not computer science.

It appears that it there is a wide range of topics in computer science and for most of the topics there are profs who regard the topic as good and profs who regard the topic as junk.

The critical stuff of the yellow rain can cause stress, clinical depression, and suicide -- that's what happened to my wife, sweet old-fashioned girl, won prizes in cooking, sewing, raising chickens, Valedictorian, Summa Cum Laude, PBK, Woodrow Wilson and NSF Fellowships, and Ph.D.

I do not now, nor have ever had, any interest in being a college prof or publishing papers. I regard such a career as financially irresponsible. My interests are in business, the money-making kind, and there I regard math, existing and/or original, and computing as the main tools. I went to grad school to learn more about the math tools. I was a college prof for a while as part of taking care of my suffering wife.

Re: The Two Cultures of Mathematics (2000) [pdf]

#67
post #3

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…

> 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 > 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…

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.

Re: The Two Cultures of Mathematics (2000) [pdf]

#68
post #67

Earlier quoted context omitted.

> 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 > 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…

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 R…

> 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.

So what if they did? Your original claim was "NSF funds (1), and does not fund (2)". Symbolically, we can represent this as "A and (not B)", where A is "NSF funds (1)" and B is "NSF funds (2)". To refute that, we need to show the negation, i.e. "(not A) or B" (This is, of course, just a De Morgan's law)

If I were going for "not A", then Kelley would destroy that line of argument, but in fact I was going for "B", and neither Kelley nor Luenberger bear on that.

> 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.

"In 1985, NSF began funding the creation of five new supercomputing centers" https://en.wikipedia.org/wiki/National_Science_Foundation_Ne...

> 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.

Sounds to me like they wrote the grant for the wrong thing, or spun it incorrectly. NSF turns down grants all the time (more than they fund, in fact), and if they didn't make it clear that there was actual research being conducted, then of course they'd get turned down.

In fact, when it comes to speech recognition, NSF says they funded a lot of it back when: https://www.nsf.gov/od/lpa/nsf50/nsfoutreach/htm/n50_z2/page...

"""Much of the initial research, performed with NSF funding, was conducted in the 1980s. This research led to further product development from Dragon, AT&T, IBM and other companies. """

(And presumably your colleagues would have had a reasonable expectation of having had a shot at NSF money, otherwise they wouldn't have wasted their time)

> 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.

See, this claim, to me, is bizarre, as a sizable portion of HPC use has historically been doing CFD and of course the NSF was heavily involved.

For example : https://nsf.gov/pubs/1996/nsf9646/nsf9646.pdf and https://www.nsf.gov/od/lpa/news/03/tip031113_topsupercompute...

> 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.

I went and checked, and the grant was via DMS, which stands for "Division of Mathematical Sciences" and not, as one might have thought, "DNA, Metabolism, and Snakes".

> 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.

This is an entirely different kettle of fish and also irrelevant.

>Since the research universities don't much like math applications outside of academics

.... What? https://www.pacm.princeton.edu/about ??? https://amath.washington.edu/history ???

> 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.

Non-academics usually use back channels and personal communication instead of formal presentations. Maybe in a better world it'd be different.

> 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.

I used to teach math to business school students, and in the interest of decorum I will say no further on the matter.

Re: The Two Cultures of Mathematics (2000) [pdf]

#69

Earlier quoted context omitted.

There's no real difference, if your sole interest is a paycheck, any programming is boring.

Yep. Does it contradict to what I wrote though?

Your post is formulated as if there's difference.

Re: The Two Cultures of Mathematics (2000) [pdf]

#70
post #67

Earlier quoted context omitted.

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 R…

> 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. So what if they did? Your original claim was "NSF funds (1), and does not fund (2)". Symbolically, we can represent this as "A and (not B)", where A is "NSF funds (1)" and B is "NSF fu…

Uh, let's see: When I was buying math and physics books, it seemed that a lot of them credited the NSF with funding. I gave two examples, Kelley and Luenberger, but there were more.

Soooo, there was some NSF pure math funding in the 1950s, to respond to a small point. The funding seemed to be generous since the joke was "While you are up get me a grant.".

And from my non-representative sample way back there, it looked like the NSF loved "The analytic-algebraic topology of the locally Euclidean metrization of infinitely differentiable Riemannian manifolds."

I should have been more clear -- LINPACK is software and IIRC was written at Oak Ridge. So, by then some Federal organization was funding software. So, the Georgetown pair, in about 1973, that wanted funding for speech recognition and was told by the NSF that they didn't fund software was, shall we say, early and later the NSF changed their mind.

1973 was a long time ago. So, with your impressive data, the NSF has evolved!

Oh, by the way, the grad and ugrad B-school students I taught did okay: They liked that I got them into matrix theory easily, used duality to prove the two person game theory saddle point result, and made min cost capacitated network flows easy to understand.

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