The Mathematical Logic of Stories

Do all stories have a mathematical logic? Is all mathematics storytelling? Is there a literary side to math and a mathematical side to literature? John Allen Paulos explores the surprising connections between the seemingly separate cultures of math and literature in his short, witty, sometimes quirky and sometimes profound book "Once Upon A Number: the hidden mathematical logic of stories". Indeed Paulos explicitly explores mathematics as a potentially unifying tool to address C. P. Snow's critique of The Two Cultures.

Once Upon A Number

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  • Sandy C.

    I just saw this video on the NY Times website. Alan Alda is talking about a program to teach story-telling and improvisation to scientists. Here's the link: http://www.nytimes.com/video/science/100000002723754/a-conversation-with-alan-alda.html?ref=afternoonupdate&nl=afternoonupdate&emc=edit_au_20140224

    1 · February 25, 2014

  • Lynn

    At this latest meetup, I mentioned game theory as a tool for the exegesis of stories, real or fictional. If/when this topic resurfaces, I am posting this here to remind myself and others to remind me to bring up an example of (a variation on) Prisoner's Dilemma that occurred to me on the way home, involving whether one should give change to a homeless person who might, or might not, spend the money unwisely on drugs or alcohol.

    1 · February 23, 2014

  • CJ F.

    Although John Allen Paulos' book "Once Upon A Number: the hidden mathematical logic of stories" is a short read, there isn't much time left to finish it by Saturday.

    If you want to get the gist of the book, this 19m TED Talk delivers: http://www.youtube.com/watch?v=XVMYTplQ158

    Here is my summary of the main ideas: The interplay between stories and statistics, narratives and numbers is a special case of C. P. Snow's "The Two Cultures". When we watch stories we suspend disbelief and so risk a false positive; when evaluating statistics scientifically, we suspend belief and so risk a false negative. In between these extremes lies journalism with context and statistics both held together by a story. Do great stories balance both?

    He also explores the different logics of story and mathematics: equals can be substituted for equals in math, but not necessarily so in stories!

    He suggests we can find balance and bridges between mathematics and story. I think he is onto something here!

    February 18, 2014

    • CJ F.

      On pp. 271-2 in Morris Kline's wonderful book "Mathematics: The Loss of Certainty", he wrote about Löwenheim–Skolem theory (LST): "one can find interpretations---models­---that are drastically different and yet satisfy the axioms."

      Does Löwenheim–Skolem theory (LST) imply that mathematics is extraordinarily rich in interpretations? Therefore, could mathematics engage and explore the imaginative space that LWT suggests is available to it? On the other hand, I wonder, is LWT a model for literary analysis? Does it explain, for instance, why there are so many different approaches to literary analysis? I wish Paulos' book had explored the nexus between story and mathematics implicit in Löwenheim–Skolem theory, but it doesn't.

      Does mathematics shirk its literary potential by excessively focusing on proof and technical details to the exclusion of the imaginative / interpretative possibilities. Ought the mathematics community embrace this literary potential?

      February 22, 2014

  • Greg G.

    Love that quote "The tyranny of the anecdote"

    1 · February 22, 2014

  • Rachel

    sorry, wanted to attend today but work has called...

    1 · February 22, 2014

  • Lynn

    Re: Chapter 4:

    "Although they differ ineradicably, both traditional religious and scientific approaches to a hoped-for theory of everything share the perhaps naive assumptions that such a theory can be found and that its complexity will be sufficiently limited to be understood by us. Why assume that?"

    I don't. Honestly I'm surprised this planet has even gotten as far as it has. But that doesn't mean all is hopeless:

    http://en.wikipedia.org/wiki/Postbiological_evolution

    February 15, 2014

    • CJ F.

      The passage you quote is on page 159 in my copy which is in the section "Our Complexity Horizons". In reviewing this passage, I am struck by the naivete of Paulos' idea that he might be "seven brain cells short" of understanding something. I find that even the most elementary theories require much effort and time to understand. I am "seven brain cells short" of understanding Ramsey Theory (pp. 163-4), Bayes' Theorem (pp. 49-51, 69-72), etc. What other mathematical ideas included in the book or videos are some of us "seven brain cells short" of understanding? If we discuss those ideas in the meetup tomorrow might some of us grow one more brain cell to be able to at least apprehend the idea? By a similar process of effort times discussion might a theory of everything or even God be understandable?

      I do not think our limitations are static in time and we have non-linear means of going beyond yesterday's limitations. Even without transhumanism! But I really could use more RAM :)

      1 · February 21, 2014

  • CJ F.

    I love the way Paulos summarizes the book near the end "Metaphor and analogy stretch the narrow literalness of mathematical and scientific understanding, and mathematical calculations and constraints ground literary imagination. ... The gap between stories and statistics must be filled somehow by us."

    I wonder: why isn't more mathematics presented as compelling story (rather than tedious proof or arcane equations) and why isn't more literary analysis built around mathematical models?

    I look forward to discussing the topic & the book tomorrow.

    BTW, I received an e-mail from John Allen Paulos himself. He plans to make "an appearance" during the meetup tomorrow. It is unclear when he will show and how long he can stay (his other obligation may prevent his appearance). So I recommend we start the meeting promptly at 11 AM and run for at least the full two hours (until 1 PM). If you arrive late or leave early, you could miss the opportunity to meet the author!

    February 21, 2014

  • Lynn

    The intensional / extensional distinction mentioned is equivalent to „Sinn und Bedeutung” ("sense and reference"):

    http://en.wikipedia.org/wiki/Sense_and_reference

    The most succinct way of capturing that idea is to compare the following: the meaningful sentence "The morning star is the evening star." vs. the apparently redundant sentence "Venus is Venus."

    February 14, 2014

    • CJ F.

      I have now read a logic book on my shelf and most of the Wikipedia entry on Sense and Reference https://en.wikipedia.o...­

      This is a subtle concept. So far I'm finding Paulos' treatment is understandable but it is diluted and I'm not sure I understand all the subtleties.

      Muhammad, I'm glad you will be at the meetup as we will probably need your help working through this subtle concept.

      BTW, did you study formal semantics in logic or in linguistics. Wikipedia thinks there is a big difference: http://en.wikipedia.o...­

      February 19, 2014

    • Lynn

      The formal semantics course I took applied first order logic and extensions thereof to natural language. I thought it was a kind of brittle approach but there it is.

      February 20, 2014

  • CJ F.

    Here is a nice 52 minute video of John Allen Paulos talking about his book "Once Upon A Number: the hidden mathematical logic of stories": http://www.c-span.org/video/?121900-1/book-discussion-upon-number

    Although math is indubitable, the applications of math are subject to critical discussion. We may question the appropriateness of the mathematical model, the assumptions used, and how things are measured. 1 cup of water + 1 cup of popped popcorn = 1.5 cups of soggy popcorn.

    Statistics tells us a few things about a lot of people (so it can seem simple-minded), whereas stories tell us a lot about a few people (so they seem rich but can also be full of bias).

    In statistics multiple correlation can give many spurious associations but in stories this is what we do and where the meaning lives. But there is the risk in stories of the "tyranny of the anecdote".

    "Storytelling always trumps number crunching."

    I love the way he illuminates the role of math in story and vice versa.

    1 · February 20, 2014

  • CJ F.

    In the book "Once Upon a Number" John Allen Paulos establishes a dramatic tension between stories and mathematics via dichotomies. Here are a few that I pulled out of the first chapter:
    The Humanities --- The Sciences
    Narratives --- Numbers
    Stories --- Statistics
    Psychology --- Logic
    personal --- objective
    particular --- general
    subjective --- universal
    intuition --- proof
    drama --- the timeless
    first person --- third person
    special --- standard
    particular viewpoint --- agentless impersonal overview
    human perspective --- overweening abstraction
    richness of detail --- abstraction and generality
    entertaining & beguiling --- bored & certain

    Do these dichotomies characterize the gulf between mathematics and stories? Between the sciences and the humanities?

    Does Paulos' convince you that the gap between these dichotomies is bridgeable? Can mathematics help bridge this gap? Is there mathematics in all stories and story in all math?

    February 17, 2014

  • Lynn

    John Allen Paulos makes the unfortunate mistake of describing the parameter α in null hypothesis statistical testing as the cutoff point at which the results can be considered "due to chance". (Probably not out of his own ignorance but in an attempt to vulgarize the concept.) There are a lot of mistakes in this arena so I don't think it's terribly pedantic to point out how you can go wrong with that mindset.

    For more info on how (not) to interpret p-values, see here, esp. Misconception #1 in this context: http://www.perfendo.org/docs/BayesProbability/twelvePvaluemisconceptions.pdf

    And for more info on why NHST might not be a good idea in the frst place, see here:

    http://www.ics.uci.edu/~sternh/courses/210/cohen94_pval.pdf

    In the medical and/or behavioral science literature, you may notice there is increasing concern for "effect sizes", "practical" or "clinical significance" and the like.

    February 14, 2014

    • Lynn

      The problem with the "due to chance" phrasing is that it omits the conditional nature of the test. One is led to think that α gives the cutoff for P(data) rather than P(data | H_0). Because, as Cohen points out in the second paper, null hypotheses are frequently implausible in the first place (i.e. the prior probability P(H_0) is rather low), I would consider this a substantial oversight.

      February 15, 2014

    • Lynn

      Although it is easy to see how Paulos would be forced to make an unfortunate compromise like this one, given he wanted to have a broad audience.

      February 15, 2014

  • CJ F.

    I'm really enjoying my re-reading of John Allen Paulos' "Once Upon A Number: the hidden mathematical logic of stories". I just finished re-reading the first chapter and wrote two pages of notes.

    I like the way the book grapples with the interrelationship between story and mathematics. It really engages my wondering faculty. What are your impressions of the book?

    Is statistics the abstraction and generalization of patterns from illions of stories? Is all mathematical abstraction simply the encapsulation of illions of stories, scenarios, and events? Does a particular statistic or mathematical fact suggest each of the illions of stories that could illustrate it (that it could have been abstracted from)? Is mathematics really this overwhelming rich in stories? Could mathematics teachers engage our facility with stories to enliven and enrich the subject?

    I love the way Paulos explores the gap between the sciences and the humanities: C.P. Snow's "Two Cultures". Can math bridge this gap?

    February 13, 2014

  • Lynn

    Thought I'd share this:

    http://www.amazon.com/Unimaginable-Mathematics-Borges-Library-Babel/dp/0195334574

    IIRC, the stories referenced in the above are to be found largely in "Labyrinths". "The Library of Babel" most certainly is.

    February 13, 2014

    • CJ F.

      That book is on my TODO list. If it is as good as it sounds, we will discuss it at a future meetup.

      1 · February 13, 2014

  • Lynn

    At the previous meetup, I suggested the title "Discovering Artificial Economics", which you can find online for free (and not even through a piracy site). I think it's good overall, but I'd like to stress that it shouldn't be taken for gospel. In particular, I remember coming across the claim that the right hemisphere of the brain is the "intuitive" half. This is a false but unfortunately popular belief:

    http://www.livescience.com/39373-left-brain-right-brain-myth.html

    It's a bit ironic that a book dedicated to challenging economic conventional wisdom manages to perpetuate pop psychology. Caveat lector!

    1 · January 29, 2014

    • CJ F.

      "Discovering Artificial Economics" looks interesting except it seems to be out of print with only 3 new copies available from Amazon.

      The thesis that the right hemisphere is "intuitive" is more than pop psychology. I believe Jeanette Norden also suggests this idea in her lecture "Memory and the Brain" which is available at http://www.thegreatco...­

      But I did not take notes and may be wrong about that.

      I agree that there is evidence suggesting the left/right dichotomy is at least way over-simplified.

      January 29, 2014

    • Lynn

      It is out of print, but as mentioned, you can find it online, specifically here:

      http://www.econ.iasta...­

      I suppose if you prefer print (I'm indifferent) that's a problem though.

      January 29, 2014

  • Greg G.

    I went to the library to pick up the book. The database said it was in the shells but it was missing. It's hard to believe that somebody would take a five finger discount on that particular book you just never know

    January 29, 2014

  • Greg G.

    Sounds really interesting but I don't know if I'll be able to attend since I'm having my knee replaced on Feb 6th. I will definitely take a look at the book thanks for the opportunity

    January 25, 2014

    • CJ F.

      Greg, If your recovery hasn't reached the point where you can travel, you can still post comments, thoughts, and questions about the book in the comment section here. I'd love your feedback on the book!

      January 26, 2014

    • Lynn

      Best of luck with your surgery!

      January 29, 2014

  • Susan

    There are numerous books recent reads for me are Stumbling on Happiness, The Drunkard's Walk, Is G-d a Mathematican and A Mathematician Reader the New York Times. The latter apparently is frowned upon by peers.

    1 · January 27, 2014

    • CJ F.

      The Drunkard's Walk is exquisite! I have been thinking about suggesting it for a book discussion for Math Counts. Thanks for the suggestions.

      1 · January 27, 2014

  • Lynn

    Another title in the vein of the Paulos book mentioned here is "Game Theory and the Humanities" by Steven Brams. It is highly rated but very terse and assumes fairly sophisticated knowledge of game theory, such as the so-called "theory of moves". The book reads more like a collection of academic papers than something by one author. I will not strongly recommend it for that reason but if you're really interested you could check it out. Here is a more elementary game theoretic approach to literature originally due to von Neumann and Morgenstern themselves:

    http://books.google.com/books?id=75DSyyqiG34C&pg=PA62

    1 · January 26, 2014

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