Applied Mathematics is Bad Mathematics

Yes, the the title is clickbait, but it is not our clickbait. It’s the title of an interesting and semi-provocative 1981 article by Paul Halmos. Halmos’s article came to mind after a brief conversation recently, about applied mathematics in Australia. As with pretty much everything Halmos wrote, it seemed worth sharing.*

Halmos begins his article by immediately walking back the title:

It isn’t really (applied mathematics, that is, isn’t really bad mathematics), but it’s different.

Halmos, however, spends a fair amount of time walking back his walking back, and there is a “teach the controversy” flavour to it all.

Early on, Halmos notes there are a number of this-or-that divisions in mathematics, and provides a couple simple, and admittedly artificial and imperfect, examples to illustrate:

Suppose you want to pave the floor of a room whose shape is a perfect square with tiles that are themselves squares so that no two tiles are exactly the same size. Can it be done? In other words, can one cover a square with a finite number of non-overlapping smaller squares all of which have different side-lengths? This is not an easy question to answer.

Here is another puzzle: if you have a perfect sphere, like a basketball, what’s the smallest number of points you can mark on it so that every point on the surface is within an inch of one of the marked ones? In other words, what’s the most economical way to distribute television relay stations on the surface of the globe?

Is the square example about sizes (numbers) or shapes (figures)? The answer seems to be that it’s about both, and so is the sphere example. In this respect the examples give a fair picture; mixed types are more likely to occur (and are always more interesting) than the ones at either extreme. The examples have different flavors, however. The square one is more nearly arithmetic, discrete, finite, pure, and the one about spheres leans toward being geometric, continuous, infinite, applied.

Halmos focusses upon the distinctive flavours and motivations of pure and applied mathematics:

Many pure mathematicians regard their specialty as an art, and one of their terms of highest praise for another’s work is “beautiful”. Applied mathematicians seem sometimes to regard their subject as a systematization of methods; a suitable word of praise for a piece of work might be “ingenious” or “powerful”.

Halmos also acknowledges the great overlap:

The differences between people are sometimes as hard to discern as the differences between subjects, and it can even happen that one and the same person is both a pure and an applied mathematician. Some applied mathematicians (especially the better ones) have a sound training in pure mathematics, and some pure mathematicians (especially the better ones) have a sound training in applicable techniques.

To confuse the issue still more, pure mathematics can be practically useful and applied mathematics can be artistically elegant. Pure mathematicians, trying to understand involved logical and geometrical interrelations, discovered the theory of convex sets and the algebraic and topological study of various classes of functions. Almost as if by luck, convexity has become the main tool in linear programming (an indispensable part of modem economic and industrial practice), and functional analysis has become the main tool in quantum theory and particle physics. The physicist regards the applicability of von Neumann algebras (a part of functional analysis) to elementary particles as the only justification of the former; the mathematician regards the connection as the only interesting aspect of the latter. De gustibus non disputandum est?

Just as pure mathematics can be useful, applied mathematics can be more beautifully useless than is sometimes recognized. Applied mathematics is not engineering; the applied mathematician does not design airplanes or atomic bombs. Applied mathematics is an intellectual discipline, not a part of industrial technology. The ultimate goal of applied mathematics is action, to be sure, but, before that, applied mathematics is a part of theoretical science concerned with the general principles behind what makes planes fly and bombs explode.

Halmos becomes more controversial again, when discussing the extent to which pure and applied mathematics need each other:

The deepest assertion about the relation between pure and applied mathematics that needs examination is that it is symbiotic, in the sense that neither can survive without the other. Not only, as is universally admitted, does the applied need the pure, but, in order to keep from becoming inbred, sterile, meaningless, and dead, the pure needs the revitalization and the contact with reality that only the applied can provide.

The first step in the proof of the symbiosis is historical: all of pure mathematics, it is said, comes from the real world, the way geometry, according to legend, comes from measuring the effect of the floods of the Nile … (If it’s true, the argument tends to prove only that applied mathematics cannot get along without pure, as an anteater cannot get along without ants, but not necessarily the reverse.)

As far as the interaction between pure and applied mathematics is concerned, the truth seems to be that it exists, in both directions, but it is much stronger in one direction than in the other. For pure mathematics the applications are a great part of the origin of the subject and continue to be an occasional source of inspiration-they are, however, not indispensable. For applied mathematics, the pure concepts and deductions are a tool, an organizational scheme, and frequently a powerful hint to truths about the world-an indispensable part of the applied organism. It’s the ant and the anteater again: arguably, possibly, the anteater is of some ecological value to the ant, but, certainly, indisputably, the ant is necessary for the anteater’s continued existence and success.

Halmos ends on a not-very-conciliatory conciliatory note:

So, after all that has been said, what’s the conclusion? Perhaps it is in the single word “taste”.

A portrait by Picasso is regarded as beautiful by some, and a police photograph of a wanted criminal can be useful, but the chances are that the Picasso is not a good likeness and the police photograph is not very inspiring to look at. Is it completely unfair to say that the portrait is a bad copy of nature and the photograph is bad art?

Much of applied mathematics has great value. If an intellectual technique teaches us something about how blood is pumped, how waves propagate, and how galaxies expand, then it gives us science, knowledge, in the meaning of the word that deserves the greatest respect. It is no insult to the depth and precision and social contribution of great drafters of legislative prose (with their rigidly traditional diction and style) to say that the laws they write are bad literature. In the same way it is no insult to the insight, technique, and scientific contribution of great applied mathematicians to say of their discoveries about blood, and waves, and galaxies that those discoveries are first-rate applied mathematics; but, usually, applied mathematics is bad mathematics just the same.

The reader can decide whether Halmos’s last line is sincere or just a tease.

We’ll also add a tease of our own, with an unsubstantiated claim:

Australian applied mathematics is bad mathematics.



*) Halmos’s article appears in a collection titled Mathematics Tomorrow, edited by Lynn Arthur Steen. We haven’t been able to locate a (legitimate) online copy of the article.

41 Replies to “Applied Mathematics is Bad Mathematics”

  1. I put the difference down to my motive.

    If I am trying to forecast the incidence of cancer in a particular region, then I am doing applied mathematics.

    If I am trying to find the best constant in some inequality, then I am doing pure mathematics.

    PS: In addition to pure and applied mathematics, there is a third area – mixed mathematics. An ancient term.

  2. In my experience what one region calls applied and pure is not what another would call it. I’ve been in both departments despite producing similar research throughout my career.

    I’m somehow wondering if this ever appeared in print. It seems so unusual. Halmos is Halmos, of course. I don’t agree in general though. Applied mathematics is not bad mathematics.

      1. I suppose “in print” was a weird way of asking what I meant, which was if it appeared in a peer-reviewed journal with an editorial board. Not that these two things ensure any kind of quality. I was curious if Halmos’ essay had a larger implied stamp of approval than just himself and the editor of the collection. Now that I’ve looked at the front matter to the collection I can see that the editor explicitly distances some of the AMS bodies from the “opinions” in the collection. I just find it interesting, that’s all.

        Australian applied mathematics is not bad mathematics. Some Australian mathematics, just like math anywhere, is certainly bad mathematics. But I don’t think the level of “bad” is enough to justify the blanket statement that Australian applied mathematics is bad mathematics.

        I don’t think it is very helpful to work out how much of whatever kind of mathematics is “bad” either. Instead I think it is more helpful to explain what practices constitute “bad” mathematics, or, in many cases, not mathematics at all. Or poor scientific practice. For instance, plotting on a computer the set of solutions to a PDE whose solution set is empty.

          1. Yes. I’m not sure any mathematics education is good mathematics education (I think you had a post to this effect earlier).

            1. So, it’s not that you think apriori that Australian applied mathematics can’t (sufficiently and sufficiently commonly) be bad mathematics.

              1. Yes. I’ve seen lots of what I consider to be good applied Australian mathematics. Much more good stuff than bad. I think being “applied in the EU/USA, pure in Aus” had given me decent exposure to Australian applied math, but I do not e.g. attend ANZIAM. I do see quite a bit of stuff though and had quite a few good conversations with applied mathematicians in Australia.

                Halmos is in the US mathematiscape, where a massive amount of amazing things are being done by applied mathematicians. In 2011 I spoke at, attended and organised a mini session at the annual SIAM PDE meeting. The “AM” in SIAM is Applied Mathematics. The math at that meeting was remarkable, one of the best conferences I’ve ever been to. Maybe Halmos whacked a straw man.

                  1. I wouldn’t say “lots”. I’ve certainly seen some. (I’ve also seen some bad Australian pure mathematics.) I would guess that there exists bad math in every country in the world.

  3. My background is more on the applied mathematics side, but when I was doing research I always felt that what I was doing was real mathematical research. As such, I used to wonder why some pure mathematicians were adamant about their work not having any application whatsoever. It seemed to me to be a sort of weird snobbery, “my thinking is more intellectual pure than yours”.

    But then I realized what it actually meant, and we see it in Halmos’ reference to “bombs” quoted above by Marty. To a certain type of politically left-wing, mid-20th century pure mathematician, applied mathematics means applying mathematics to military problems, to waging war in developing countries, to killing people. It’s no surprise that they would view applied mathematics as essentially immoral.

    Now I don’t know anything about the Australian school mathematics curriculum. But like Glen, I should note that what mathematicians think of as applied mathematics is not really the same thing that educationalists are thinking about when they talk of applied mathematics. Applied mathematics isn’t “STEM”, in the sense that you only do M if you have an explicit (possibly contrived) S, T or E justification for it.

    1. Thanks, Marc. I think your last paragraph is key, and not just for understanding maths ed and the STEM fetish. Note Halmos’s paragraph on the various things applied mathematics is not.

      As to pure mathematicians concerned to not be seen as applied, your thoughts about the militarising of applied mathematics rings true. In general, though Halmos thinks the concerns come more from the applied side. Here’s another part of the article:

      The only other curiosity along these lines that I’ll mention is that you can usually (but not always) tell an applied mathematician from a pure one just by observing the temperature of his attitude toward the same-different debate. If he feels strongly and maintains that pure and applied are and must be the same, that they are both mathematics and the distinction is meaningless, then he is probably an applied mathematician. About this particular subject most pure mathematicians feel less heat and speak less polemically: they don’t really think pure and applied are the same, but they don’t care all that much. I think what I have just described is a fact, but I confess I can’t help wondering why it’s so.

  4. From my own, relatively short and confined, experience in schools there are actually two types of Mathematics teachers.

    There are some who feel the need to show an “application” of every set of skills and assess “application” or “modelling” frequently (VCAA examiners, particularly Paper 2 Methods and Specialist seem to fall into this category); they tend to enjoy writing SACs that meet VCAA’s requirements, and, to a certain extent, they are able to produce open-ended tasks.

    There are others who believe that Mathematics should be taught for its own sake and (seem to) be stricter with use of Mathematical language.

    The second group (or set?) I would imagine are much less common. They do (those that I have met) seem to meet the criteria above though in that they (small sample size I know, so…) seem to think that there are two distinct types of Mathematics teachers but don’t always seem to care.

        1. Yes, but the point is these “application” SACs are not, in the proper sense of the word, applied mathematics. These idiots, including the idiots at ACARA and VCAA, know as little about applied mathematics as they do about pure mathematics.

              1. No; but it may reduce his credibility. Of course, if he had done some applied mathematics, then to say all applied mathematics is bad mathematics would amount to being critical of his own work.

                I wonder, would he regard the work of Newton and Einstein as bad mathematics? In Australia I’d say that the work of Terry Speed is outstanding applied mathematics. Perhaps Halmos was exaggerating to get our attention.

                BTW, I admire the work of Halmos a great deal. In fact, he signed my copy of “Finite dimensional vector spaces”.

            1. Halmos was John von Neumann’s assistant and either wrote or at least assisted in the writing of von Neumann’s “Mathematical Foundations of Quantum Mechanics”. Whether that counts as pure or applied is a different question. I also think Halmos’s last sentence was indeed a tease — fits right in with his personality. On the other hand, there were negotiations here at U. Mass. (Amherst) for Halmos to be hired. The negotiations became contentious when Halmos objected to the departmental intention to hire more applied mathematicians. The upshot was that he didn’t join the faculty and the department hired several applied mathematicians. I was a grad student at Michigan and took several courses from Halmos. He was a great teacher and gave great parties.

              1. Thanks very much, JH. Definitely there’s a teasing element to Halmos’s article, and definitely he’s not just teasing.

                I stupidly missed out on Halmos teaching a term at Stanford. Didn’t know what I was missing.

            1. I taught a course at the summer school in 2022. I think it really depends on who is proposing courses as to what gets up, which may depend a lot on the organisers. Usually a friendly tap can prompt people to action if something is lacking.

              I’m not sure what you’re aiming at (I guess you don’t like one or more of these offerings, I wish you’d just be explicit about it, maybe I can cross my fingers for a post about it). For me, it looks alright. Home universities are expected to have the standard things covered, and hopefully two of those subjects at least will appeal to every attending student. I’m open to being convinced that something is wrong with it, but right now I don’t see it.

              Total enrolment was pretty low in 2022. Unfortunately it isn’t very similar to those summer schools in years gone by.

  5. Once in Melbourne, I saw a machine that was labelled “applied street sweeping”. I wondered what “pure street sweeping” would be.

  6. I’m on your side but for the wrong reason! 🙂

    I actually think teaching the math straight is pedagogically simpler. This is because word problems are hard. So, initial learning should be at the “x pushing around” stage, not the “imagine a rod with one end heated and one end cooled” stage. Once the student has learned the x pushing around, some practice with word problems can be done (after he has drilled and mastered the x pushing stage) since it just gives a bit more drill in the topic and since it will be a little more interesting than drill in something he’s already run several of.

    This does give some window into possible applications (like just knowing what stuff is good for and starting to learn the applied subjects) and helps groove the way for problems in engineering and physics, if the student continues in that direction. It’s actually not vital though. What’s vital is mastering the x pushing. The chemistry/physics/engineering teachers can help the students when they get to those lessons with word problems in those topics (learning the content, the units, and getting practice in setting up problems). But those teachers want the product from math class to have a base of the x pushing done. [After all most word problems involve going from description to equaions, manipulating equations, getting a result, and translating back in the units/quantity requested.]

    This has nothing to do with the “beauty of math”. That’s just a side benefit. It’s just that x pushing is actually easier than “how much Mg is in the mixed Mg/Zn oxide sample”. Realio trulio it is. Cognitive load theory. Yes…it is real. We are not silicon. Are not rules based structures. Takes work and training to perform quantitative problems. And even “simple” things like a word problem versus equation are extra cognitive load.

    For the kids “it will be on the test” is the major motivation. If they ask more, just say, you’ll need it in physics and chemistry (and even if you’re a doctor, you have to take some). Don’t feel obliged to prove it to them. Just make a flat assertion and then move back to teaching the lesson.

  7. Functional analysis is NOT the main tool of quantum mechanics. That’s the sort of silly thing math people say about technical fields, they’ve never taken and just based off some random high end papers or the like. I wonder if Halmos ever solved the SE for hydrogen? Or even a particle in a box. The main tool of quantum mechanics is differential equations. For complicated systems (and things get complex fast), you can maybe choose perturbation theory as a key technique (there are others).

    I’m not a super physicist, but I’ve had courses in QM at undergrad and grad level. And I never took functional analysis. Barely know what it is.

    1. Hi Anon. You have a point.

      I’d note however that functional analysis is only slightly more specific than the immensely broad “analysis”. So in a sense that kind of statement is trivially true. Solutions to DEs are functions and so are members of function spaces, the prime object of study in functional analysis. The sentiment is fairly common, as a lot of the theory of modern DEs is enabled by advances made in the study of Sobolev and Banach spaces, specifically related to weak derivatives and generalised notions of functions. There’s more to it, of course. Perhaps another thing worth mentioning is that the Journal of Functional Analysis publishes quite a lot of papers dealing with differential equations. I’m not sure if they have many papers that study quantum mechanics, but I’d not be shocked to learn that it is the case.

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