Math Funners:
I recently had some time to spend browsing my collection of American
Mathematical Monthlies going back to 2001 or so. I find the publication
immensely enjoyable to browse. Yet I felt a number of criticisms of the
magazine, which I wrote about to the Editor.
I'm interested in your feedback on these opinions, so I reproduce them here.
I go into excruciating detail about one article in particular in the last part of
this letter, and just a bit of detail about two other specific articles mentioned earlier. To best understand my reaction to these, needless to say, you may
want to get a hold of them from your own collection or a math library.
--Dan
-------------------------------------------------------------------------------------------
-------------------------------letter starts here-----------------------------------------
Recently I've had some time to go back and read a number
of articles in Monthlies from the last two or three years
that I hadn't read before. This has been very enjoyable.
At the same time, I've formed some thoughts about Monthly articles
in general, and I hope you won't mind my writing you about them.
Overall, I love reading Monthly articles. They are on the whole
written about very interesting topics and very rarely contain mathematical
errors or typos -- things I appreciate very much.
I also recognize that the Monthly probably relies heavily on the volunteer
of its editors, and the amount of time they have to donate is limited.
Still, there are a few things I'd ideally like to see in Monthly articles --
so I may as well mention them -- as follows:
1. I'd like to see standardized notation and terminology used for
common mathematical objects. Sure, the literature is filled with a
variety of notation and every mathematician needs to learn how to
read it. But I'd like to see the Monthly be a leader in this direction,
much as other august publications use a "stylebook" for
non-mathematical consistency.
2. In articles on applied math, I'd like to see a clear separation of the applied
and the math. Meaning this: Let the author describe the applied problem to
be solved, and the mathematical model which will be used to accomplish this.
But during any parts of the article where math is being done, non-mathematical
reasoning should be strictly forbidden.
Otherwise the math in the article becomes a mishmash of unverifiable steps
interlaced with solid math, and the end result becomes impossible to evaluate.
Two article that IMO flagrantly violate my desideratum are "The Mylar balloon
revisited" (December 2003, p.761) and "Quantum computation" (March 2003, p. 181).
The first article seems to contain a substantial amount of solid math, but it also
seems to contain a substantial amount of B.S. as well. It is patently absurd that
before they launch into long computation to determine the balloon's shape, they
neglect to state *at that point* that the problem they are solving is NOT the one
they appear to be solving (to find an isometric embedding of two flat 2-disks,
identified along their boundaries, into 3-space, such that certain mathematical
statements of physical conditions are satisfied). They never do state what
mathematical problem their computations purport to solve. Also, their mathematical "reasoning" as they go along often seems unjustified (such as the claim that the solution will of course be circularly symmetrical. Needless to say, there are zillions of math problems with circularly symmetric input but without circularly symmetric output).
The second article *pretends* to proceed formally by asserting a finite number of
"Postulates", but these are *not* well-defined postulates in terms of mathematics.
E.g., they assume the "flow" of time -- something not even physics knows how to
discuss in its own language. And this is purely a matter of personal taste, but as long as the article is supposed to be an introduction to quantum mechanics for
un-quantum-enlightened mathematicians, it drives me up the wall that the article
abandons mathematical notation for quantum notation. The material is esoteric
enough without throwing another great big roadblock in the reader's way. I fully
agree with the author's claim that to read further in the quantum literature, one needs to know standard quantum notation, but that does not justify using it in the boy of this article. The notational equivalence could have been relegated to an appendix.
3. I'd like to see tighter controls on the excellency of exposition.
Quite often the exposition of Monthly articles is unimpeachable. But
too often for my taste, the exposition is only fair. One egregious article
that comes to mind in this vein is "Life on the edge" (November 2002, p. 850).
Here are my thoughts on it (WARNING: They are lengthy, but possibly
worth reading since this article embraces almost every quality that I think
should be avoided in Monthly articles):
----------------------------------------------------------------------------------------------------
The first sentence is a little hard to swallow: I don't recall learning anything about
convergence on the boundary of a circle of convergence of a power series in C --
and *convergence* is clearly a central concept in this article. I had no idea
that -(sum_n=1^oo (z^n / n)) converges to log(1-z) for |z| = 1, z not equal to 1.
IF this is something to be proved later in the article, it should say so;
if not, it should be stated that we will assume this without proof.
Also shouldn't a word or so be said about which branch of log is being
referred to? (It can be easily guessed, but IMO it shouldn't have to be.)
On line 3, the last theta should be replaced by theta/2.
The first sentence of paragraph 2 is of totally unclear meaning to me. What does
"Be treated as if they were polynomials of [very] high degree provided ..." mean? Treated in what sense? I'm sure there's some reason he writes this, but I haven't
a clue what it is.
In the very next sentence he refers to *guessing* a certain mathematical
statement which I thought he had just proved. Why "guessing" ?
I recognize that he is being jocular where he says it is "scary stuff" --
but despite my having a well-developed sense of humor, I have no
idea what's funny about this, or even what is meant. Ditto for "Do not
try this at home."
The first sentence on page 851 is a grammatical monstrosity:
"I suggest that it is just [noun phrase] that, as theta -> 0, one has
[equation (2)]."
Perhaps I can prove (2) myself, but I do not know whether the author
just claimed to have proved it or not.
Eqn. (3) appears to be a standard complex analytic evaluation of an
improper integral -- so why wouldn't the author refer to this fact?
And the fact that I_theta is independent of theta has a trivial proof.
Why would the author not mention this?
Next paragraph -- starts out, once again, jocular but I cannot see either
the humor or the meaning of this first sentence.
Next sentence:
Utterly mystifiying! What can the author possibly mean by
saying that "the series 1 / (1-z) = sum_n=0^oo z^n is obviously well-behaved
on the boundary away from z = 1". The "boundary" presumably means
the circle |z| = 1, and on that circle there is *no* z for which the series
converges. Is the author being jocular again? I have no idea.
The sentence continues by claiming that "whence its integral
log(1-z) = -(sum_n=1^oo z^n / n) is also well-behaved on its boundary.
Aha -- that previous apparently false statement is being used to
prove this statement -- and Aha! This now tells us that the first
statement in the article was going to be proved later. But has it been?????
The paragraph finishes equally mysteriously: "Thus the argument just now
sketched " (WHAT @#$%^ argument?????) " proves that I = I_1 is
in fact pi/2, so (3) is well know to us." Huh?????
At this point I am so frustrated with the article I want to scream.
Next paragraph: "Mind you, I had better detail my main argument...."
The main argument for WHAT??????????????????????????????????
The end of this paragraph calls the "main argument" for god-knows-what
into question. So we have just maybe proved, and maybe not proved,
an unknown statement.
In the last paragraph, it appears that the author tries to put all preceding
statements -- or is it only one of them? -- on a firm mathematical footing
(i.e., prove them, or it? Or maybe not). But by this point I give up. The author has been unclear to an unprecedented degree. I am not going to allow him
even one more symbol to jerk my mind around with.
(Naturally, I check the cover again to see if this is the April issue. Nope.)
To borrow a phrase: With this article the author has filled a much-needed
gap in the literature.
--------------------------------------end of letter----------------------------------------
---------------------------------------------------------------------------------------------
