Saturday, September 21, 2013

"By the Usual Compactness Argument"

It's a sad truth, but the mathematics research literature is very tough going for beginners. By "beginners" I mean bright high-school students, or university students, or beginning graduate students, or even professional mathematicians who are trained in an area different from the article he/she is trying to read.

As a high-school student, I used to go to the mathematics library at the University of Pennsylvania to look up and try to read articles articles in number theory. Usually I couldn't understand them at a first reading, so I'd photocopy them and take them home to puzzle over. I remember being completely flummoxed by a paper on Bell numbers that used the "umbral calculus"; I just didn't understand that you were supposed to move the exponents down as indices. That is, in an equation like
B4 = (B + 1)3
you were supposed to expand the right hand side, getting
B3 + 3B2 + 3B1 + 1
and then magically change this to
B3 + 3B2 + 3B1 + 1 .

I had nobody to ask about stuff like that. Although my high-school teachers were great, they didn't know about the umbral calculus.

Things like this permeate the mathematical literature. Take compactness, for example. Compactness is a marvelous tool that lets you deduce -- usually in a non-constructive fashion -- the existence of objects (particularly infinite ones) from the existence of finite "approximations". Formally, compactness is the property that a collection of closed sets has a nonempty intersection if every finite subcollection has a nonempty intersection; alternatively, if every open cover has a finite subcover.

Now compactness is a topological property, so to use it, you really should say explicitly what the topological space is, and what the open and closed sets are. But mathematicians rarely, if ever, do that. In fact, they usually don't specify anything at all about the setting; they just say "by the usual compactness argument" and move on. That's great for experts, but not so great for beginners.

I really wonder who was the very first to take this particular lazy approach to mathematical exposition. So far, the earliest reference I found was in a 1953 article by John W. Green in the Pacific Journal of Mathematics 3 (2), 393-402. On page 400 he writes

By the usual compactness argument ([2, p.62]), there does exist a minimizing curve K.

Can anybody find an earlier occurrence of this exact phrase?


Harriet said...

Just a remark: often, the REFEREE makes you take out details and say things like "by the usual compactness argument".

A colleague and I are tempted to start a "Journal of Omitted Details".

James Cranch said...

I find myself hoping that journals and papers will be superseded in the near future as the standard way of disseminating mathematics.

My feeling is that the world would be a better place if maths was treated like open-source code, and farmed using some distributed version control tool like Git.

I can think of lots of reasons not to do this immediately, but they mostly seem stupid:

Ideas won't be attributable to individual mathematicians any more.
They aren't at the moment. A paper typically arises as a collaboration between several named mathematicians, following conversations with many unnamed mathematicians. Each paper may contain zero, one, or more ideas, not necessarily all of the same provenance. A system that allows more fine-grained contributions can only improve the situation.

Peer review will become impossible.
No, it could be exactly the same as it is at the moment. A mathematician looks at a version of a paper and offers their opinion anonymously.

Mathematicians will stop doing maths if they can't see published papers with their names on them.
I don't believe this for a minute: this isn't why people do maths. Mathematicians already do lots of things which don't result in published papers with equal (or greater) pride. Programmers didn't stop writing programs when this happened either, and they're just as proud.

Committees won't be able to evaluate mathematicians by counting papers. That's a good thing: we never wanted that to happen anyway.

g said...

Might be worth looking for earlier papers that cite Green's ref 2 (Eine Minimumaufgabe über Eilinien, Christiaan Huygens).

Gerry Myerson said...

I guess 1953 was a good year for the usual compactness argument; see F A Valentine, Minimal sets of visibility, Proc Amer Math Soc 4 (1953) 917-921. On page 918 we read,

Hence, by the usual compactness argument, we have $\prod_{x\in S}V(x)\ne0$, if $S\/$ is not convex.

Gerry Myerson said...

The earliest appearance in Math Reviews is in MR0155131 (27 #5071), the review by H R Pitt of William Feller, On the classical Tauberian theorems, Arch Math 14 (1963) 317-322.

It's in quotes in the review, which may mean the reviewer is quoting the author.

Robert Byers said...

I submit math be dropped as a subject except for the few who might paid for it.
Math is the modern latin. its useless to anything of discovery or invention and interferes with people applying themselves to other subjects that could progress the intelligence of mankind.
Math is just a language of reality and one needs not be bilingual to do cool things in science and humanity.
Down with math and up with intellectual insight and imagination .

Jeffrey Shallit said...

The depths of Byers' ignorance have still not be plumbed. Who knows what gems of inane stupidity we might find?

The fact that he's writing this on a computer system that probably uses error-correcting codes, principles of information transmission due to Shannon, cryptography based on number theory, and a variety of other mathematical techniques, is completely beyond him.

Robert Byers said...

Mr Shallit.
I understand that and many things have math as a important issue. i said its oksy for those few who get paid to use math.
Yet its unrelated to almost everything ever done in human intellectual progress and so science.
Its been a tool, like in making a house, but just a help.
Math is very overrated as relevant to scientific innovation or revolution.
I think it should only be studied by professionals who need it truly.
Dividing things up forever is mot needed for everyone else.

Miranda said...

Mr. Byers, up to what level of math should be taught to students in school?

Gerry Myerson said...

I posted Jeff's question to MathOverflow:

An answer by Benjamin Dickman cites three occurences in 1947, the earliest being W. Ambrose, Direct sum theorem for Haar measures, Transactions of the American Mathematical Society, 61(1) (1947) 122-127.

Moritz Firsching found it in German ("was aus der Kompaktheit von R in der üblichen Weise folgt") in a paper by Urysohn and Alexandroff, Ueber Räume mit verschwindender erster Brouwerscher Zahl, from 1928.

Robert Byers said...

Just math enough for life. Enough to avoid needs to upgrade for regular jobs.
Math is entirely a thing of memory save for the few who advance some discovery.
So computers should do the memory work.
Math is the modern LATIN in which a subject is taught thinking it helps somehow one to do unrelated subjects.
Its in the way as old Latin was in the previous centuries.
Its not a thinking mans subject.
Its just memorized operations and so mere attentiveness brings results.
Sharp minded people do math but its just a coincidence.
The great modern rule of thumb is IF a computer can do it then its not a thing of intellectual striving.
Its just memory application.

Jeffrey Shallit said...

Its just memorized operations and so mere attentiveness brings results.

You really have absolutely no idea what mathematicians do, do you?