Eponyms in mathematics

Let S be the standard Smith class of normalized univalent Matcuzinski functions on the unit disc, and let B be the subclass of normalized Walquist functions. We establish a simple criterion for the non-Walquistness of a Matcuzinski function. With this technique it is easy to exhibit, using standard Hughes-Williams methods, a class of non-Walquist polynomials. This answers the Kopfschmerzhaus-type problem, posed by R. J. W. (“Wally”) Jones, concerning the smallest degree of a non-Walquist polynomial.

This fake abstract of a paper is from Merv Henwood and Ivan Rival, Eponymy in Mathematical Nomenclature: What's in a Name, and What Should Be? (PDF), from the Mathematical Intelligencer in 1980. It sounds to me like slight caricature -- but only slight. Henwood and Rival point out that such names are lazy. Names have at least two important functions -- to describe and to label -- and eponyms only label.

Perhaps such abstracts would be more common in areas which are small enough that all the major players talk to each other. I imagine that Smith, Matcuzinski, Walquist, etc. know each other.

Also of interest is David Rusin's list of eponyms occurring in the MSC classification. These names in general seem a bit less obscure than the names one would find in the abstract of a random paper, which isn't surprising as they're names of concepts big enough to get areas named after them.

(And can someone confirm or refute the story that Banach, in the paper in which he introduced Banach spaces, called them "spaces of type B" in an effort to get them named after himself? I've heard this one a few times but always unsourced.)

LaTeX equation labels

When writing a paper in LaTeX, you often want to put a numerical label on a displayed equation, say the number (1). So you write some code like
\begin{equation}\label{eq:basel-problem} \sum_{n=1}^\infty {1 \over n^2} = {\pi^2 \over 6} \end{equation}
which compiles to give something that looks like

Then later I can insert code like (\ref{eq:basel-problem}) and (1) appears in my docuemnt.

Now, as you may have noticed, I picked an equation that had a nice name, and I labeled it with that name. (The "eq:" in the label, of course, stands for "equation", a convention that I use to tell what sort of entity I'm referencing -- other things I use in that position are def:, thm:, prop:, cor:, lem:, and the like.)

But what do you do when the displayed equation doesn't have a nice "name" -- it's just an equation that occurs somewhere in the course of a calculation? For a while I tried to come up with a name, but I ended up with way too many generic names like "integral" and "sum" and "thing-with-binomial-coefficients". (Okay, so I'm exaggerating on the last one.) These names took time to think of but didn't make things easier on me later. So now I find myself using labels like \label{eq:feb-24-kappa} for the 10th labelled equation that I inserted on February 24. (Why do I use Greek letters? I tried using numbers, but it's too easy to get those confused with the actual numbers that are used to label equations.) But I'm wondering what sort of conventions people use for this; since it's the sort of thing that you can only see when you're looking at other people's LaTeX source, it's hard to know.

Somewhere, somebody is saying that I'm using LaTeX incorrectly. It might be you!

(Yes, I'm taking a break from rewriting a paper. How did you guess?)

On publishing your trash can

I'm rereading de Bruijn's book Asymptotic Methods in Analysis (which, sadly, appears to be out of print again!) -- one of the great mathematical expositions, of asymptotic methods in analysis as they stood at midcentury. It's one of the most readable math texts I know.

de Bruijn writes in the preface:

Many things in this book are not presented in the shortest possible form, as an attempt has been made to reveal, to a certain extent, the motives that lead to certain methods. Naturally one cannot go too far in this respect; a mathematician cannot possibly publish his waste-paper basket.

This seems worth remembering; terseness is not always a virtue.

Two questions on document preparation

1. Why are the default margins in LaTeX so wide? It's kind of useful, because it means that there's a lot of space to scribble in when editing, but it seems that by default they're wider than in just about any other program.

2. Why are dissertations usually double-spaced? I associate double-spaced with draft documents, because you can write things between the lines of text. But the dissertation isn't supposed to be a draft. It's supposed to be a final document!

How do you pronounce ≤ and ≥?

I'm taking a break from proofreading a paper. I'm reading it out loud, because I find this is the best way to catch mistakes; it forces me to look at every word.

There are inequalities in this paper, so the signs ≤ and ≥ come up a lot. How do you pronounce these? When I was in college I pronounced them "less than or equal to" and "greater than or equal to". But sometime around the first year of graduate school I seem to have shifted to "at most" and "at least", which have the obvious advantage of being shorter.

Edit (11:15 pm): It appears I've mentioned this before.

A couple of questions on the mechanics of writing mathematical papers

I'm writing a paper, and of course this requires the use of LaTeX. As many of you know, the way one creates cross-references inside a LaTeX document is a two-step process. First, you insert the command \label{big-important-theorem} where your Big Important Theorem is in the paper. Then when you want to refer to your theorem, you write something like "And as a consequence of Theorem \ref{big-important-theorem} we can prove Corollary \ref{million-dollar-problem}, and so I claim to be the winner of a million dollar prize."

No, I have not written that sentence. Nor do I plan to. In fact, I would add "claims to be the winner of one of the Clay prizes in the body of the paper" to John Baez's crackpot index. Baez couldn't have put that in his list, because it was written in 1998. He does mention the Nobel, though, which carries a similar monetary value.

But this leads to a question -- how do you label your theorems, equations, etc. in your own LaTeX code? I try to come up with names that reflect what the labeled object is about, but this isn't so easy, because sometimes there are lots of objects that are "about" the same sort of thing. I'm tempted to just find some source of extra-mathematical names. So I could name my theorems \label{market}, \label{chestnut}, \label{walnut}, \label{locust}, ... (Philadelphia streets), say. Of course, this sequence has the disadvantage that the streets come in order; a set of names with no natural order is probably better, because then I won't feel like I'm moving things out of order when I move them around.

On a related note, often one sees bibliographical references of the form
[Be74] Edward A. Bender. Asymptotic methods in enumeration. SIAM Review,
Vol. 16, No. 4, October 1974.
I prefer this form to the form where [Be74] is replaced by a number, because if something is cited more than once in the same paper, I only need to look at the reference once; the second time I see [Be74] in the paper I know it's the paper by somebody whose name starts with Be, written in 1974. (And if it's a paper I've heard of already, sometimes I don't have to look at the citation at all.)

But what's the convention for picking the letters to be used? First two letters of the name seems common for a single-authored paper, but by no means universal; I've definitely seen one-letter citations, the problem being that if you have a reasonably extensive bibliography you'll want to cite two different authors with the same initial. I'm currently using just the first letter of each name for multiple-author works -- [FS08] for Flajolet and Sedgewick's Analytic Combinatorics (coming out sometime in December, preorderable now! and readable online!), for example. I've tried to reverse-engineer whatever convention there is from other people's reference lists and I can't. Is there actually no convention?

Of course, there is no need for a convention, as long as each work cited has a unique identifier.

Organization of papers

I'm currently attempting to organize a paper out of a bunch of notes I've built up recently; a possibly useful suggestion I received is to write each theorem, definition, etc. on an index card, so that I can physically move them around to figure out how the paper should be organized.

Of course, definitions have to come before the theorems that use them, some theorems use other theorems in their proofs, and so on -- so to the extent that I'm remembering to do so, I'm indicating these sorts of dependencies on the index cards as well.

It occurs to me that what I am doing here is trying to extend a partial order (the ordering that comes from the dependency) to a total order. There are of course constraints on this order; certain results, although not logically related, are related in some philosophical sense and should perhaps be kept near each other. It's actually an interesting optimization problem.

Now if only I were writing a paper about extending partial orders to total orders...

(But my paper does talk quite a bit about permutations. And a total order will end up being a permutation of my index cards.)

LaTeX "errors"

Output from PDFLaTeX: "0 errors, 5 warnings, 33 badboxes".

If only that meant that there are zero mathematical errors in my paper.

Telescopic text

Telescopic Text, by Joe Davis.

The web page starts out with the words "I made tea. by Joe" and various words can be clicked on; when you click on them they expand, so "Joe" becomes "Joe Davis", for example, when you click on it. Clicking on "I" reveals the word "Yawning" preceding it; "tea" becomes "a cup of tea", and so on. As you expand the text, some biscuits that weren't there before, comments on how to make tea, and so on materialize.

This reminds me of something that's been thrown around the mathematical blogosphere as a possible way to write papers that might be well-adapted to our present computer technology; start with a very high-level sketch of a proof, and make each step clickable. Upon clicking on a word, the proof is expanded to remind you what that word means, how exactly one uses that particular technique here, etc.

This would require more work than writing a paper in the usual way, though; it's not clear whether it's worth the trouble. And there's always the issue that some people like to read papers away from the computer, they eventually end up in journals which are printed on paper, and so on; what level of detail should be published there?

Mathematical infallibility

In proving the fundamental theorem of arithmetic to my students, I was establishing the fact that any number has a factorization into primes. The proof goes as follows:

By way of contradiction, say there are positive integers without prime factorizations. Then there is a smallest such integer; call it N. N is not prime, because then it would have a prime factorization. So N has some divisor a such that 1 < a < N, and we can write N = ab for some integers a, b greater than 1. By assumption, N was the smallest integer without a prime factorization, so a and b have prime factorizations and we can concatenate these to get a prime factorization of N.

I want to bring your attention to the bolded "we can write", which I definitely said while presenting the proof. (The other language might not be exactly what I used.)

Sure, we can write that. But we can also write "2 + 2 = 5". Or we could have just written "N = ab" at the beginning When a mathematician says "we can write X" for some statement X, they mean something like "X is true, for suitable values of some variables which might be contained in X that we haven't mentioned yet, and which we'll talk about now."

In short, mathematicians are only capable of writing true things, or so we'd want people to think from our writing. If only it were so easy!

Gowers on examples

Tim Gowers wrote My favorite pedagogical principle: Examples first a few days ago. (The comments are worth reading too.) As an example, he gives an axiomatic definition of a field, and then compares this with defining a field by saying "look, you know about the rational numbers, the real numbers, and the complex numbers; fields are `like these', and now here's what we formally mean by that.) People have said most of what I'd want to say about this topic, but there's an interesting question here. It's been pointed out that it's often a good idea to read mathematics "out of order". But then why doesn't the writer write things out of order, since most people's natural impulse is to read things in the order they're written? Presumably the writer understands the material better than the reader, and therefore has a better idea of what order things should be presented in.

The fundamental theorem of enumeration, and the Princeton Companion to Mathematics

From Doron Zeilberger's chapter on "Enumerative and Algebraic Combinatorics, to be included in the currently-in-preparation Princeton Companion to Mathematics

"The fundamental theorem of enumeration, independently discovered by several anonymous cave dwellers, states that
|A| = Σ_a∈A 1.
In words: the number of elements of A is the sum over all elements of A of the constant function 1."

Sounds kind of silly, but it's true. The whole chapter is a nice fourteen-page answer to "what is enumerative combinatorics?", mentioning most of the classic problems and most common methods of solution, which appears to be its raison d'être; I know most of this stuff but I can imagine how useful similar blurbs on subjects I'm not so familiar with would be, and indeed most of the book is intended to be at about the first-year undergraduate level; that's low enough that I should be able to read it without stopping for breath. (The guidelines for contributors say that the articles about various subjects should be something like the beginning of a very good colloquium talk, the sort where you really get the feeling that you know something about how some other area of mathematics works.) The PCM has a semi-official blog, which is Tim Gowers' blog. Several dozen of the component articles are available online, on a password-protected site; the password is in the linked-to post by Gowers. I suspect I'll have more to say about the PCM in the future.

Some thoughts on coordinate systems

I'm teaching multivariate calculus this term. The course, as it's taught here, begins with a unit on geometry in three dimensions; here we introduce cylindrical and spherical coordinates. (I'm not entirely convinced that they should be introduced at the point they are, because basically all we can ask the students to do is to convert between the various coordinate systems, but my hands are tied. It almost seems to make more sense to wait to teach the coordinates until we get to the point where we're doing integrals over regions that are best expressed in cylindrical or spherical coordinates.)

Anyway, a student e-mailed me a question today. One problem on the homework was to find equations in cylindrical and spherical coordinates for x² + y² + z² + 2z = 0. In cylindrical coordinates, (where x² + y² = r² this becomes r² + z² + 2z = 0. (My aesthetic sense is that this is perhaps better written as r² + (z+1)² = 1, because then it's immediately obvious that there are in fact solutions to the equation.) She asked if it was acceptable to leave it in one of these forms, or if it needed to be converted to, say, something with z in terms of r and θ (which would involve square roots).

I responded as follows:

In rectangular coordinates it might be preferable to solve for z in terms of x and y if it's possible to do so without making things too ugly, since we have a tendency to think of surfaces given in Cartesian coordinates as a "graph" of a function of x and y. The same thing is sort of true in spherical coordinates, in that ρ is often seen as a function of θ and φ (This is like polar coordinates, where r is usually a function of θ.) In cylindrical coordinates, though, none of the three coordinates really seem to be "dependent" or "independent".

Of course, all these rules can be violated; for example, one would never write the equation of a unit sphere centered at the origin as

z = ± (1-x²-y²)^1/2

unless it were to graph it on a system that can't handle implicitly defined surfaces. The guiding principle should be that you want the simplest equation possible, i. e. the one which takes the least writing.

"Takes the least writing" is admittedly a bit sloppy here; how does one define it? The amount of "writing" any mathematical expression takes depends on one's notation. For example, subscripts or superscripts take up a lot of keystrokes in HTML, not so many in TeX (which is of course optimized for mathematical writing), and if one is writing by hand writing a subscript is just one character. (Indeed, one could even argue that subscripts and superscripts require less writing than full-size characters in handwriting -- they take less ink! But I shudder at the thought of taking this to its logical conclusion.)

Another principle I might invoke is "symmetry" -- another reason the equation about for a sphere looks wrong is because it fails to capture the fact that x, y, and z all behave "the same way". The form x² + y² + z², on the other hand, doesn't have this problem. But I think "symmetry" might not be so useful a concept when trying to teach students; the ones who will understand this rather vague idea are the same ones who would probably just give the equation in a "more symmetric" form even if I didn't tell them to.

And why do cylindrical coordinates not seem to have a "dependent" and an "independent" variable, whereas three-dimensional Cartesian and spherical coordinates do? (I admit that this might just be my intuition, and I don't know if this agrees with anyone else's intuition.)

how maps are like mathematics, and maps of mathematics

Yesterday I wrote about different ways of drawing the Interstate highway system, and introduced this map by Nat Case that clearly shows it as a grid. Case had written about the map at Map Head, and says there that people accused him of copying this map by Chris Yates.


Apparently some people aren't happy about Case's map, because they're claiming that Case just ripped off Yates' idea. I don't know anything about intellectual property law, but it seems to me that there are a lot of maps of the same area that look very similar and convey almost the same information -- maps that are much more similar than those of Case and Yates. For example, two ordinary road maps of the same area should look very similar, because they'll depict the same roads! The impression I have is that the information (where the roads are) is freely available but the particular way in which it's drawn (if this isn't just a simple transcription of the information) is not. To quote one of the commenters here:

Copyright law expressly does NOT apply to "ideas", only to the execution of ideas. If this dude had copied the colors, font, or any other specific element from Chris' map, then it would be a different story (and "straight lines" isn't quite a specific element). If he had made his own map look like a subway map, even then it might have been a fuzzy area.

But, what seems to have happened was that he looked at Chris' map, then started from scratch creating his own expression of a similar idea. While it may have been courteous for him to credit Chris for the inspiration more obviously than he did, he certainly isn't legally required to do so. In fact, this very process of re-interpreting others' work is basically how art progresses in society in general. To use your analogy, I think this guy looked at a Picasso, said, "Hmm, cubism is cool," and painted his own cubist painting of the same model that Picasso used.

To this I would add that mathematics works the same way. Most proofs of the same mathematical fact, most expositions of the same concept, etc. will look the same in outline, simply because they have to respect how the ideas stand in logical relation to each other. But the choice of notation, of words to go between the equations explaining what's going on, and so on is unique to each author, and probably has a lot to do with what else is going on in a particular book or article. One might choose to emphasize certain parts of a proof -- doing some simple algebra in full instead of just saying "the reader can verify this" -- because the same calculation will come in handy later. Notation seems analogous to graphic design elements like color; a good map uses colors which are clearly different for things which are different, and good notation does the same. (I've had professors who used a and α, u and μ, or v and ν in the same problem, and had bad handwriting. No! I'd even go so far as to say that using letters which look similar in typed work is bad practice, because the conscientious reader will be copying your notation by hand in order to check things.) I doubt there are two proofs out there of, say, the Fundamental Theorem of Calculus (to take a result that's been reproduced in a zillion textbooks) that are word-for-word and symbol-for-symbol the same, just like there probably aren't two maps of Manhattan that are pixel-for-pixel the same.

The point here is that there's good exposition and bad exposition, which coincide roughly with good maps and bad maps. A good map, or a good writer, will help you navigate a tricky area; a bad map will just confused you.

And what would a map of mathematics itself look like? Dave Rusin has given it a shot, but each area of mathematics is just a circle to him, and it's not clear to me how they're connected to each other, or why his circles are arranged the way they are. He says at his A Gentle Introduction to the Mathematics Subject Classification Scheme that "[t]he welcome page for this site shows an image of the areas of mathematics which shows the relative numbers of recent papers in each area (arranged so as to illustrate the affinities among related areas)." It's a decent job -- nothing seems too far from where it should be, and I imagine that projecting the whole thing down into two dimensions makes it quite difficult! -- and it's not just based on Rusin's prejudices. It seems to come from correlations between classifications in that scheme -- two classifications are "close together" if there are papers which are classified in both of them. This somehow gives rise to a 61-dimensional space (the 61 dimensions presumably corresponding to the 61 two-digit classes in that scheme) of which this map is a two-dimensional projection. The vertical direction seems to work out with "discrete" mathematics at the top and "continuous" mathematics at the bottom; I'm not sure what the horizontal direction represents. (I think one could make an argument for "pure" on the left versus "applied" on the right, but it's a weak one.)


The same can be done with individual papers, or with mathematicians; see, for example, the papers of eight Fields medalists.

I sense that more could be done. Which areas of mathematics do you need to learn before learning certain others? Which areas historically grew out of each other? Within an area, how are the important results related to each other? And how could this be illustrated pictorially? I'm not sure what good this would be, though.



(At least two people seem to have done something similar for motorways in England); not surprisingly they both did their maps in the style of the London Underground map. It probably won't surprise you to learn that Great Britian's road numbering scheme isn't a grid, but rather divides England and Wales into zones emanating from London, and Scotland into zones emanating from Edinburgh.)

On mathematical communication

A three-part blog post on how a theoretical physics paper gets made: inspiration, calculation, culmination. This tells the story through the example of a particular paper on cosmological inflation. (From Cocktail Party Physics.) The comments are probably worth reading, too.

Somewhat relatedly, although more about the mechanics of writing, Terence Tao on rapid prototyping of papers -- basically, sketch out the outline of the paper first, making the statements of the key intermediate results, and then fill in the gaps, rather than writing from beginning to end. I can't vouch for this working on the level of writing research papers for the simple reason that I have written none. (I hope this changes soon.) But it seems to work reasonably well for writing, say, solutions to rather involved homework problems that can take a few pages, and have three or four major intermediate results.

Also, Can Scientists be Great Communicators?, from The Accidental Scientist. I would say that regardless of whether or not we are (and I'm including mathematicians in this "we"), we have to be. This is true both in terms of communication among scientists (which is tremendously useful for driving along the whole scientific enterprise, because otherwise we'd all be reinventing the wheel) and in communication with the non-scientific public, which I think is quite important. For one thing, ultimately a lot of the money that funds science comes from taxes; if these people are in the end paying our salaries, don't we owe them some explanation what we're doing? But also, communicating complicated ideas in non-technical terms forces us to actually understand them. Feynman, when he was preparing his famous freshman physics lectures at Caltech, said that if he couldn't reduce something to the level where he could explain it to freshmen, it meant that he didn't really understand it. When you can't fall back on technical terms and convoluted equations you have to understand what you're doing. So communicating with the hypothetical "educated layman" perhaps pays dividends within science as well. I just wish that people didn't automatically glaze over when they heard I'm a mathematician, though...

Communicating with this person is becoming more and more feasible thanks to the web 2.0-ification of science. Write something. Google will find it. You'd be surprised to see how many hits I get from what looks like people trying to buy used furniture, for example. And although I offer no advice there on how much used furniture should cost, I feel like I'm doing something by just exposing them to the idea that perhaps mathematics can be used to figure out such things. It's a subtle propaganda campaign.

Another subtle propaganda campaign might be the sculptures at Bathsheba Sculpture (by Bathsheba Grossman), which are for the most part models of various mathematical objects, done via 3D printing in metal; she has both mathematical and artistic training. What other sorts of training might be useful for mathematicians?

checks for nothing, and why English is useful

Karl Fogel attempts to pay a bill for $0, because of course you have to pay bills for zero, because otherwise the companies that issue them keep sending them.

In this case, the bill was for the purchase of a book from the Mathematical Association of America, so he wrote a check for e^iπ+1 dollars. They didn't deposit it, because "check needs to be wrote out in U. S. dollars", as they put it.

I can see a more legitimate reason for rejecting the check. Usually, when writing a check, one puts, say, "3.14" in the little box on the right, and "Three and 14/100" on the line. The reason for writing out the value of the check in both figured and words is for redundancy. (Although then why don't we write "three dollars and fourteen cents" on the line? I suppose redundancy doesn't matter quite as much when we're talking about sub-dollar amounts.)

So you might say he should have written "e to the i π plus one dollars" on the line. But even that seems a bit suspect, because e, i, and π are themselves bits of mathematical notation. It seems that he really should have written something like

"The base of the exponential function, raised to the product of the imaginary unit and the ratio of a circle's circumference to its diameter, plus one"

for the number of dollars he wanted. Of course, this is the sort of thing that makes it obvious why having a compact mathematical notation is a good idea. I am not enough of a mathematical historian to have looked at the way things used to be written, but from what I understand this is the sort of thing they would have written five centuries ago, and I can't imagine working like that.

Unfortunately, the fact that we have such a good mathematical notation creates another problem -- people think that they can just put a bunch of symbols on a page and not explain what they mean by them, and that's "mathematics". Terry Tao, at his blog, has lots of writing advice; of particular interest in this discussion is his advice to take advantage of the English language. Here he gives a couple dozen ways to say that two statements are true, which are logically equivalent but have a wide variety of connotations. To take two examples of his examples at random, "P(x) is true. Unfortunately, Q(y) is also true." and "P is satisfied by x. Similarly, Q is satisfied by y." might be logically equivalent but are philosophically (psychologically, emotionally, morally -- what's the right word here?) quite distinct. I think that mathematicians as a whole are not sensitive enough to the connotations of their words; this is useful when doing formal mathematics but not so useful when trying to express the results of it. Perhaps we kneel too much at the altar of Bourbaki.