Some Stretching, Squeezing, and Folding Caused by
John Paulos's Masterpiece: Once Upon A Number
By Doron Zeilberger
Written: Dec. 1, 1998
One of the many beautiful metaphors in Paulos's
``Once Upon a Number'' (henceforth #) is that the effect on us of reading
a book or article is like Smale's horseshoe map,
viz. iterative stretching, squeezing, and folding (p. 176).
I can't recall any other book, for a very long time,
that has stretched my mind so much.
Here, I wish to share with you some of the free
associations that # caused me. I am sure that
when YOU will read it, your mind will be stretched in completely
different, and probably more interesting ways!
Of course, in a hundred years, there would be
`the annotated #' by some late 21stcentury
analog of Martin Gardner. But until then, here is
(a tiny part of the) impact on my own peasize brain.

p. 17ff: The branching scam is a beautiful application
of the idea in the classical proof that for any
factorpreserving property of words (e.g. being cubefree)
the existence of ONE INFINITE word having that property
is equivalent to the
existence of INFINITELY many FINITE words having the property (see
Lothaire, Combinatorics on words, lemma 2.1.2 (p. 21)).
For a binary alphabet,
if there are infinitely many words that have the property,
then there must be infinitely many that start with 0 or
infinitely many that start with 1, let's denote that letter
(O or 1, as the case may be)
by w[1]. Now there are infinitely many words that start with
w[1]0 or infinitely many that start with w[1]1.
Let's call the successful prefix w[1]w[2].
Continue, ad infinitum, until you get an infinite word with
the property.

p.21: Regarding computergenerated fiction, some book series,
especially for children, SEEM like they were generated
by a (very bad) computerprogram. For example, the
American Girls series, that consists of the
cartesian product
{ Felicity, Josefina, Kirsten, Addy, Samantha, Molly}
x {Meet, Learns a Lesson, Surprise, Happy Birthday, Saves
the Day, Changes For...}.
A GOOD example of
cartesian product is given in some passages of Pirke Avot
(Ethics of the Fathers): e.g. V:13
{Mine is Mine, Mine is Yours} x{Yours is Mine, Yours is Yours}.
An excellent example of a branching novel is `Dictionary of
the Khazars'.
Speaking of `structured' stories, a paradigmatic
example of a structured proof is my
Proof of the Alternating Sign Matrix Conjecture, whose
treestructure enables this highly complex proof to be read
in any desired level of detail, and more importantly,
facilitates dividing the chore of checking formal correctness
between many referees, who each only has to check a small part.
 p. 28: I really liked Paulos's defense of stereotypes,
saying that they were unfairly stereotyped. This reminds
me of the currently fashionable practice among mathematicians and others
to denounce a subject because it is fashionable, implying that
if a subject (e.g. catastrophe theory in the seventies and
chaos and fractals in the eighties
or computer algebra in the nineties) is fashionable
or `chic', than it is just a superficial
fad. This fad of despising fads must be deplored!
If a subject is fashionable, it is because many people
are actively working on it, and find it stimulating!
 p. 30: John Paulos's prediction that the Unabomber is a mathematician
prompted me to look up Ted Kaczynski's papers. They are
paragons of precision! They are written in a nononsense
terse, yet complete, stepbystep style, that makes
most math papers look like longwinded sociology.
I strongly recommend that you look them up at
MathSciNet. The math is also beautiful! What a shame
that he quit math for bombing.
 p. 34: The mathematical analog of Very Very Short Stories
are math bites, that occasionally appear in Math Magazine.
Look up some of my own math bites in my website, e.g.:
A Very Very Short Rendition of Turing's halting theorem.
 p. 36: Paulos's youthful epiphany reminded me of
Dave Bayer's youthful insight, that the whole universe
is just a drop in some giant's cup of coffee, that was
one of many metaphors uttered in Dave Bayers's outstanding talk at
The MSRI SCGA workshop, Oct. 1216, 1998
(click on videos, then on Bayer's talk).

p. 42: I gave up a long time ago worrying whether
my socks are matching or not, so I don't have this
problem. However, I still have a positive chance
of having matching socks, since I draw them at random.
But, Richard Garfield, of Magic The Gathering fame,
ALWAYS, wears unmatching socks! That's why he is rich,
and I am not.
 p.42: Sometimes Pr(A and B and C and ..)=Pr(A)Pr(B)Pr(C) ...
holds even when the events are NOT independent.
My favorite example is Knuth's (of course intentionally) fallacious
`proof' of the hooklengths formula (Art of Computer Programming,
III, p. 63).

p. 48: If you use the 25000word
builtin UNIX spelling dictionary list, let's call it
words, and do `grep ^r wordswc l' you would get 2700 words, that
start with r, while if you do `grep ^..r wordswc l' you would get
less than one third of that, 855. On the other hand, the
analogous tests for k yield only slightly less: 155 for the
former, and 128 for the latter.

P. 52: A counterexample to the regressionofthemeans phenomenon
is # itself. It looked like after `A Mathematician Reads a
Newspaper', Paulos's next book would not be quite as good.
But, as great as MRN was, # is MUCH MUCH BETTER!

p. 63: Concerning the difficulty of avoiding coincidences,
constructing good Ramsey graphs, a very difficult task,
is one of the many fortes of my brilliant student
Aaron Robertson , look up his paper on lower bounds of
multicolored Ramsey numbers, where one has to sweat so hard
to construct graphs with `nocoincidences' (in this context,
monochromatic cliques.)

p. 73: Raising the probability from 0 to .0001 is indeed
much worse than raising it from .0001 to .0002, since the
relevant scale here should be logarithmic.

p. 73, line 2 from bottom: I assume that between
`will' and `be' there was intended to be `not', but
then again, it makes perfect sense the way it is now.

p. 80: Sometimes it is advantageous to forget about structures,
and to do things `formally' . For example, the theory of
formal power series is much more rigorous and elegant
than the theory of convergent power series.

p. 84: According to Thomas Kuhn and many others, socalled
Natural rules are also manmade, and the `unreasonable
effectiveness of math in nature' raved about by
Einstein and Wigner are just artifacts of the way we think.
The only parts of Nature that we are capable of understanding
are mathematical. (Compare the quotation from Wittgenstein
on p. 173).
 p. 86: Another beautiful nontransitive game is
Penney Ante, described in Graham, Knuth, and Patashnik's
classic `Concrete Math'. A Maple implementation can be
found in John Noonan's and Doron Zeilberger's Maple package
David_Ian.

p. 88: Context is also sometimes important in math proper.
There are many proofs that 0=1, by proving that
for some A, 0=A and A=1. The fallacy is that
0=A in some context, and A=1 in another, mutually
exclusive context.

p. 100: the discussion of the copula `is' reminds me
of Derrida's warning of the futility of defining
deconstruction, and of Clinton's lame excuse that
he did not lie, since, `is' does not necessarily mean
`is and never has been'.

p. 101: I first heard of Kripke when someone
showed me, ca. 1977, a New York Times Magazine
article heralding him as the new genius. Then
I was told that Rebecca Goldstein's `Mind Body
Problem' was by his (ex(?))wife.

p. 112: `The most casual observer knows that mathematical
statements are either posited (axioms) or proved (theorems),
not tested or confirmed in the way that scientific laws
and hypotheses are'. This is only true for oldfashioned,
modern mathematics. Third millenium postmodern (no relation
to postmodern philosophy!) math, is going to be quasiempirical,
and semirigorous. See my
prophecy.

p. 124: This brilliant lecture offers a good example of
metareference. # quoting the lecture, the lecture quoting
the 1980 book, the 1980 book quoting Russel quoting
Godel. It would have been even better if Godel would have
quoted #, making a cycle.

p. 129: Paulos's suggestion for computergenerated humor also
apply to computergenerated math. In particular, targeted
proofs machine, like the WZ proof theory are thriving (Dave Bressoud).

p. 129: A simpler example of Chomskian ambiguity is
a Gym at the Univ. of Illinois called `Old Men's Gym',
which could either mean (Old Men)'s Gym or Old (Men's Gym).

p. 130: Sometimes we should be careful not to overdo
Chaitin's Occamian obsession, as warned by Dave Bayer,
see my
Opinion 31.

p. 131: the problem with continuing 2,4,6, ... by 38 is that
we should have at least one slack term. Salvy and Zimmermann's
Maple package gfun guesses the `rule' that the sequence
obeys, if it exists, as does my own Maple package
SCHUTZENBERGER, that guesses the rule that sequences
that belong to either the algebraic paradigm or the holonomic
paradigm obey. BUT, there are always a few slack terms, i.e.
when we fit n terms successfully, we use at most
n4 parameters in the fit sequence.

p. 142: There is a mathematical analog of George Carlin's
parody: using 1line notation for permutation. Instead of
saying pi(1)=3,pi(2)=1,pi(3)=2, one just writes 312.

p. 142: C++ seems better than Pascal, since it is objectoriented,
and there is plenty of room to define classes for storyelements.

p. 143: The Talmud should also be studied for deep combinatorics.
For example, the diagrams implicit in tractate Yebamot, about
incest laws, are reminiscent of boolean circuits, with the extra
complexity that there are male and female nodes.

p. 144: Fuzzy logic, invented by Zadeh, may also be relevant
to the issues here. An excellent new book about Fuzzy Logic
(and neural networks and Genetic Algorithms), by Arturo
Sangalli, just came out from Princeton University Press.

p. 144: about abusing statistical packages, I know of at least
two cases of mathematical statisticians who worked in
medical research, and who were fired because they did not
go along with the `conclusions' claimed by their
medicaldoctorsbosses.

p. 144: Not only is the D+ student a faker, but most of us
are occasionally, when we talk of things other than our
specialty. Quite a few times I have made a fool of myself,
when I tried to fake it in topology.

p. 146: Is the reference to `Edgar Allen Poe' in the index
(p. 212) intentional?

p. 158: Regarding the complexity horizon, I was reminded of one
of my favorite quotes, from Pierre Cartier
(quoted by Ruelle, in his `Chance and Chaos'), that says
that the axioms of set theory are actually inconsistent,
but the proof of inconsistency is too long to be carried in
our physical universe.

p. 160: It is amazing that in our society, where journalists
are a few notches above scientists in the status ladder, someone
like James Horgan can be taken seriously. His book, `The End of
Science' was so annoying not so much because of its central message
(parts of which I even agree with), but because his
dumbjock disdain to us scientists as a bunch of nerdy
eccentrics. By the Way, the token `cool' scientist, that passes
the Horgan test, is Ron Graham (see his profile by Horgan
in Sci. Amer. about a year or two ago). It is clear that Horgan respects
Graham much more than Popper and Witten, since the latter are
nerdy, but Ron Graham is `cool'.

p. 164: R(5,5) is now known, thanks to McKay and Radziszowski
to be less than 50. Of more direct relevance to debunking
Bible Codes is Van der Waerden's theorem about the
inevitability of ESLs consisting of the same letter.

p. 165: It seems to me that Kauffman's phenomenon can
be explained via random permutations, and studying
the distribution of cyclelengths in a random permutation.
See a recent article, by Richard Stong, in the Elec. J. Comb.

p. 174: If you want to find out the possible ways of making
change, look up my Maple program (that looks like a poem!)
All The Ways of Making Change.
One last remark. Not only did Paulos manage to bridge
the Snowian gap between the literati and the techies,
but he went a long way toward bridging a much wider gap
in our contemporary society: that between the bimbo culture of
the mass media, and the nerd culture of science, computers,
and literature. If nothing else, even more than
Ron Graham, John Paulos is the coolest mathematician
around!
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