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 21st-century analog of Martin Gardner. But until then, here is (a tiny part of the) impact on my own pea-size brain.

  1. p. 17ff: The branching scam is a beautiful application of the idea in the classical proof that for any factor-preserving property of words (e.g. being cube-free) 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.
  2. p.21: Regarding computer-generated fiction, some book series, especially for children, SEEM like they were generated by a (very bad) computer-program. 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 tree-structure 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.

  3. 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!
  4. 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 no-nonsense terse, yet complete, step-by-step style, that makes most math papers look like long-winded 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.
  5. 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.
  6. 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. 12-16, 1998 (click on videos, then on Bayer's talk).
  7. 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.
  8. 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 hook-lengths formula (Art of Computer Programming, III, p. 63).
  9. p. 48: If you use the 25000-word built-in UNIX spelling dictionary list, let's call it words, and do `grep ^r words|wc -l' you would get 2700 words, that start with r, while if you do `grep ^..r words|wc -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.
  10. P. 52: A counterexample to the regression-of-the-means 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!
  11. 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 multi-colored Ramsey numbers, where one has to sweat so hard to construct graphs with `no-coincidences' (in this context, monochromatic cliques.)
  12. 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.
  13. 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.
  14. 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.
  15. p. 84: According to Thomas Kuhn and many others, so-called Natural rules are also man-made, 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).
  16. p. 86: Another beautiful non-transitive 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.
  17. 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.
  18. 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'.
  19. 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.
  20. 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 old-fashioned, modern mathematics. Third millenium postmodern (no relation to postmodern philosophy!) math, is going to be quasi-empirical, and semi-rigorous. See my prophecy.
  21. p. 124: This brilliant lecture offers a good example of meta-reference. # 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.
  22. p. 129: Paulos's suggestion for computer-generated humor also apply to computer-generated math. In particular, targeted proofs machine, like the WZ proof theory are thriving (Dave Bressoud).
  23. 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).
  24. p. 130: Sometimes we should be careful not to overdo Chaitin's Occamian obsession, as warned by Dave Bayer, see my Opinion 31.
  25. 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 n-4 parameters in the fit sequence.
  26. p. 142: There is a mathematical analog of George Carlin's parody: using 1-line notation for permutation. Instead of saying pi(1)=3,pi(2)=1,pi(3)=2, one just writes 312.
  27. p. 142: C++ seems better than Pascal, since it is object-oriented, and there is plenty of room to define classes for story-elements.
  28. 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.
  29. 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.
  30. 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 medical-doctors-bosses.
  31. 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.
  32. p. 146: Is the reference to `Edgar Allen Poe' in the index (p. 212) intentional?
  33. 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.
  34. 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 dumb-jock 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'.
  35. 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.
  36. p. 165: It seems to me that Kauffman's phenomenon can be explained via random permutations, and studying the distribution of cycle-lengths in a random permutation. See a recent article, by Richard Stong, in the Elec. J. Comb.
  37. 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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