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.