Seventy four minutes of mathematics

An analysis of the third mini-polymath project

Alison Pease (Lead / Corresponding author), Ursula Martin

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    10 Citations (Scopus)

    Abstract

    Alan Turing proposed to consider the question, “Can machines think?” in his famous article [38]. We consider the question, “Can machines do mathematics, and how?” Turing suggested that intelligence be tested by comparing computer behaviour to human behaviour in an online discussion. We hold that this approach could be useful for assessing computational logic systems which, despite having produced formal proofs of the Four Colour Theorem, the Robbins Conjecture and the Kepler Conjecture, have not achieved widespread take up by mathematicians. It has been suggested that this is because computer proofs are perceived as ungainly, brute-force searches which lack elegance, beauty or mathematical insight. One response to this is to build such systems which perform in a more human-like manner, which raises the question of what a “human-like manner” may be.
    Timothy Gowers recently initiated Polymath [4], a series of experiments in online collaborative mathematics, in which problems are posted online, and an open invitation issued for people to try to solve them collaboratively, documenting every step of the ensuing discussion. The resulting record provides an unusual example of fully documented mathematical activity leading to a proof, in contrast to typical research papers which record proofs, but not how they were obtained.
    We consider the third Mini-Polymath project [3], started by Terence Tao and published online on July 19, 2011. We examine the resulting discussion from the perspective: what would it take for a machine to contribute, in a human-like manner, to this online discussion? We present an account of the mathematical reasoning behind the online collaboration, which involved about 150 informal mathematical comments and led to a proof of the result. We distinguish four types of comment, which focus on mathematical concepts, examples, conjectures and proof strategies, and further categorise ways in which each aspect developed. Where relevant, we relate the discussion to theories of mathematical practice, such as that described by Polya [34] and Lakatos [22], and consider how their theories stand up in the light of this documented record of informal mathematical collaboration.
    Original languageEnglish
    Title of host publicationAISB/IACAP World Congress 2012
    Subtitle of host publicationSymposium on Mathematical Practice and Cognition II
    EditorsA. Pease, B. Larvos
    PublisherSociety for the Study of Artificial Intelligence and Simulation of Behaviour
    Pages19-29
    Number of pages11
    ISBN (Print)978-1-908187-10-9
    Publication statusPublished - 2012
    EventSymposium on Mathematical Practice and Cognition II - University of Birmingham, Birmingham, UK, Birmingham, United Kingdom
    Duration: 2 Jul 20124 Jul 2012
    http://homepages.inf.ed.ac.uk/apease/aisb12/home.html

    Conference

    ConferenceSymposium on Mathematical Practice and Cognition II
    CountryUnited Kingdom
    CityBirmingham
    Period2/07/124/07/12
    Internet address

    Fingerprint

    Turing
    Four colour theorem
    Mathematical reasoning
    Formal Proof
    Kepler
    Human Behavior
    Logic
    Series
    Experiment
    Human
    Collaboration
    Strategy
    Concepts
    Intelligence

    Cite this

    Pease, A., & Martin, U. (2012). Seventy four minutes of mathematics: An analysis of the third mini-polymath project. In A. Pease, & B. Larvos (Eds.), AISB/IACAP World Congress 2012: Symposium on Mathematical Practice and Cognition II (pp. 19-29). Society for the Study of Artificial Intelligence and Simulation of Behaviour.
    Pease, Alison ; Martin, Ursula. / Seventy four minutes of mathematics : An analysis of the third mini-polymath project. AISB/IACAP World Congress 2012: Symposium on Mathematical Practice and Cognition II. editor / A. Pease ; B. Larvos. Society for the Study of Artificial Intelligence and Simulation of Behaviour, 2012. pp. 19-29
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    title = "Seventy four minutes of mathematics: An analysis of the third mini-polymath project",
    abstract = "Alan Turing proposed to consider the question, “Can machines think?” in his famous article [38]. We consider the question, “Can machines do mathematics, and how?” Turing suggested that intelligence be tested by comparing computer behaviour to human behaviour in an online discussion. We hold that this approach could be useful for assessing computational logic systems which, despite having produced formal proofs of the Four Colour Theorem, the Robbins Conjecture and the Kepler Conjecture, have not achieved widespread take up by mathematicians. It has been suggested that this is because computer proofs are perceived as ungainly, brute-force searches which lack elegance, beauty or mathematical insight. One response to this is to build such systems which perform in a more human-like manner, which raises the question of what a “human-like manner” may be. Timothy Gowers recently initiated Polymath [4], a series of experiments in online collaborative mathematics, in which problems are posted online, and an open invitation issued for people to try to solve them collaboratively, documenting every step of the ensuing discussion. The resulting record provides an unusual example of fully documented mathematical activity leading to a proof, in contrast to typical research papers which record proofs, but not how they were obtained.We consider the third Mini-Polymath project [3], started by Terence Tao and published online on July 19, 2011. We examine the resulting discussion from the perspective: what would it take for a machine to contribute, in a human-like manner, to this online discussion? We present an account of the mathematical reasoning behind the online collaboration, which involved about 150 informal mathematical comments and led to a proof of the result. We distinguish four types of comment, which focus on mathematical concepts, examples, conjectures and proof strategies, and further categorise ways in which each aspect developed. Where relevant, we relate the discussion to theories of mathematical practice, such as that described by Polya [34] and Lakatos [22], and consider how their theories stand up in the light of this documented record of informal mathematical collaboration.",
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    Pease, A & Martin, U 2012, Seventy four minutes of mathematics: An analysis of the third mini-polymath project. in A Pease & B Larvos (eds), AISB/IACAP World Congress 2012: Symposium on Mathematical Practice and Cognition II. Society for the Study of Artificial Intelligence and Simulation of Behaviour, pp. 19-29, Symposium on Mathematical Practice and Cognition II, Birmingham, United Kingdom, 2/07/12.

    Seventy four minutes of mathematics : An analysis of the third mini-polymath project. / Pease, Alison (Lead / Corresponding author); Martin, Ursula.

    AISB/IACAP World Congress 2012: Symposium on Mathematical Practice and Cognition II. ed. / A. Pease; B. Larvos. Society for the Study of Artificial Intelligence and Simulation of Behaviour, 2012. p. 19-29.

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

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    Pease A, Martin U. Seventy four minutes of mathematics: An analysis of the third mini-polymath project. In Pease A, Larvos B, editors, AISB/IACAP World Congress 2012: Symposium on Mathematical Practice and Cognition II. Society for the Study of Artificial Intelligence and Simulation of Behaviour. 2012. p. 19-29