John Allen Paulos, ONCE UPON A NUMBER: The Hidden Mathematical Logic of Stories (1998)

Here’s another John Allen Paulos book, one from 1998, and one that I had not previously read. When I noticed it on the shelf a few weeks ago I realized this was now of particular interest, since in recent years I’ve become fascinated by the nature of stories and they mold our understanding of the world.

What would the subtitle of this book mean? Something about how the idea number of acts in a play is 3, or 5, for whatever reasons? Perhaps the structure of the classic problem story – try three times before success? Is there any sense in which these ideas would apply to SF more than other kinds of stories?

Well, no, no. I finished thinking the subtitle is misleading, rather as that for the Baron-Cohen (see this post) book was.

The book is like many of Paulos’ others: some general themes illustrated by copious examples and anecdotes. And the actual theme, rather than “hidden mathematical logic,” is that some ideas are best understood with statistics, others with stories, and humans have a way of confusing the two. Which is to say, the book fits neatly with the many others I’ve read about human psychological biases to impose narratives on the world, to perceive causes where there are only coincidences, and so on. But otherwise, this is a hard book to summarize. I’ll just grab some highlights, almost at random.

  • Intro:
    • We mistake correlation for causation; we impose stories on random events; we are egocentric. Thus we have those who see the world as good guys vs bad guys, vs chance and number; literary vs scientific; conspiracy theorists and ‘nowhere men’. How do we account for both sides?
    • These concerns, author says, have in some way been a concern of all of his books.
  • Ch1, Between stories and statistics
    • Similar to Snow’s two cultures and other pairings. Notions of chance and probability weren’t formalized until the 17th century.
    • People often cite statistics without context (story); beware surveys conducted by those with ideological axes to grind. Examples of CPI and birth order. A rabbi who introduces only subjects for which he has parables.
    • Stories focus on individuals, but this can be deceptive and distort public policy. [[ comment: one sees on the news that no matter what the situation is, reporters can always find *someone* to illustrate whatever point they’d like to make; TV news especially is very anecdotal. ]]
    • Common sense leads to stereotypes, rules of thumb, which are useful for effective communication.
    • Statistical conclusions must be tested. This can be a drab business. People who hate making a Type I error (rejecting a truth) are perhaps inclined to like stories; vv for Type II and statistics. We accept sloppy reasoning from those closest to us, insist on exact protocols in public decision making.
  • Ch2, Between subjective viewpoint and impersonal probability
    • About likelihoods from minority viewpoints; Murphy’s law; Bayes’ theorem; regression to the mean and conspiracy theories.
    • How we rationalize coincidences and stories. Bible Code, sex scandals. People search for stories to explain things, especially those emotionally fraught.
    • In fiction coincidences are considered cheating, with some exceptions.
    • Card tricks. How probability and the law have an uneasy relationship; charts would help. Example of OJ Simpson, how the coincidence of factors greatly heightened the likelihood of his guilt, as author argued in op-ed at the time.
  • Ch3, Between informal discourse and logic
    • Formal facility with a subject, e.g. math, is not the same as intuitive understanding.
    • Informal logic, statistics: stories can make them meaningful. Examples of wild cards in poker. About “intensional” logic and combinatorial explosions.
    • About narrative common ground; parables about furious feminists, a village with philandering husbands.
    • Most statistical models, and most novels, are build with off-the-shelf components; it’s the telling details that make them unique. Things that are counter-intuitive. Almost identical plot lines can lead to very different novels.
    • Example of a sufi story in which two men hold up numbers of fingers, without speaking, and assume very different things about what the numbers of fingers mean in turn. P118ff. Similar to many partisan disagreements.
    • Appendix to Ch 3 – a talk the author gave about humor and computation. How could a machine be programmed for human? Compile standard human and joke schemas. Keep it short. Consider lessons of evolutionary psychology. And recall catastrophe theory.
  • Ch4, Between meaning and information
    • A story by Chekhov could be coded into a sequence of 0s and 1s. But one is a story and the other is information.
    • Information is bits, but also larger units that eventually analogous to narratives – like higher-order machine languages vs. machine code.
    • About cryptography and narratives; two-bit stories; Occam’s razor. Complexity horizons—how humans may simply be too stupid to understand the hypothetical “secret to the universe.” We all have personal complexity horizons.
    • For any sufficiently large set of elements, unlikely things will occur. E.g. guests at a dinner party who know each other; how many required? If 10,000 light bulbs flash randomly, stable cycles appear.
    • Stories too generate order and patterns, just as the news and talk shows report the same stories almost every day. A story read by a sophisticate, vs a newbie, will impart different information. Stories change us; yet they need context. But beware postmodernism
  • Ch5, Bridging the Gap
    • So how do we integrate stories and statistics? The literary and the scientific?
    • Beware category mistakes, and reporting in novelistic garb, the ‘new journalism’. Don’t elevate personal details into grandiose movements.
    • Lotteries and astrology can turn wishes into facts – as when people ‘fulfill’ those wishes.
    • About the two cultures; a doctor who can cure anyone with his bare hand, knowing that his skill will die with him—how proposals are quite different coming from literary folks vs scientific. Similar Princess Diana—soap opera, or just circumstances?
    • These come into play with the environment, and with religion. Religious texts are full of stories, while science tends to diminish the personal.
    • Ultimately the gap between stories and statistics is a facet of the mind-body problem. The two sides can exist if we avoid simple-minded attempts to reduce one to the other.


Comment: There’s a paradox here somewhere: most people, especially conservatives, want to see the world in black and white, and dismiss science because it’s always “changing its mind”… and yet science is the one expressed as math, ideally in specific, exact equations, whereas the ordinary world is heavily subjective. Hmm.

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