The social machine of mathematics Ursula Martin, University of Oxford
RAs: Alison Pease (Dundee), Gabriela Nesin (Brighton), Chris Hollings (Oxford) Joe Corneli (Edinburgh), Lorenzo Lane (Oxford), Fenner Tanswell (Oxford) The social machine of mathematics Ursula Martin, University of Oxford
RAs: Alison Pease (Dundee), Gabriela Nesin (Brighton), Chris Hollings (Oxford) Joe Corneli (Edinburgh), Lorenzo Lane (Oxford), Fenner Tanswell (Oxford) Large machine proofs … talked about on the ground …and from the mountain tops Lurie: I would like to see a computer proof verification system with an improved user interface, something that doesn’t require 100 times as much time as to write down the proof. Can we expect, say in 25 years, widespread adoption of computer verified proofs?
Tao: I hope [we will Simon Donaldson, Maxim Kontsevich, Jacob eventually be able to Lurie, Terence Tao, Richard Taylor: award of $3 million Breakthrough Prizes, 2014 verify every new paper by machine.]. Perhaps at some point we will write our papers… directly in some formal mathematics system.
Model process – social machines EPSRC SOCIAM project: Southampton, Edinburgh, Oxford
Social machines - purposeful human interaction on the web • Designing social computations • Accessing data and information • Accountability, provenance, trust, incentivisation • Governance, finance and management
A social machine - OEIS http://oeis.org/ Social machines - OEIS Polymath collaborative projects
Polymath - deep interdisciplinary collaborations October 2009 Nature 461,15 Unyurl.com/ jt75z2n
Analysing mini-polymath Martin and Pease, 2012
U Martin, Computational logic and the social, Journal of Logic and Computation, 26, 467-477 (2016) U Martin, A Pease, Stumbling around in the dark: lessons from everyday mathematics, Springer LNCS, 9195, 29 - 51 (2015) P is equivalent to P’
P is not equivalent to P’. Here is a counterexample.
P is equivalent to P’’
I surrender - it is not the case that P is equivalent to P’ Lightweight Social Calculus
Describes user input, communication, (personal) data flows Completely decentralised Specification is descriptive and executable Records history of the interaction
Robertson, D. (2005). A Lightweight Coordination Calculus for Agent Systems DALT2005
Mathematical Context
• Want to look at exchange and discussion of mathematical entities
• Introduce identifiers for bits of maths
• Combine using ad hoc predicates:
• any_permutation = "Show that for any permutation s in Sn, the sum a_s(1)+a_s(2)…+a_s(j) is not in M for any j= • equivalent( main_problem, any_permutation ) • ql = property( quite_large(s_n), any_permutation ) • suggest_strategy( any_permutation, pigenhole, using(ql) ) Predicates Statement Content The following reformula0on of the problem may assert( equivalent( main_problem,any_permuta be useful: tion )) Show that for any permuta0on s in Sn, the sum a_s(1)+a_s(2)…+a_s(j) is not in M for any j= Now, we may use the fact that Sn is "quite large" large_sn=property( quite_large(s_n), and prove the existence of such permuta0on with any_permutation ) some kind of a pigeonhole-ish principle. sn_large = assert( has_property( Sn, quite_large ) ) suggest( strategy( any_permutation, pigeonhole, using( sn_large ) ) Following (2 3); assert( sub_conjecture( problem, For any x in M, there are two possibili0es: x_in_M ) ) 1. x can’t be represented as a sum of (dis0nct) ai’s. 2. x=a_j1+a_j2…+a_jk. In this case, we may assign assert( case_split( x_in_M, x_not_sum, x the set {j1, j2…jk} x_sum ) ) So M can actually be regarded as a subset of P({1, assert( equivalent( x_sum, set_j1 ) ) 2…n}) assert( m_powerset ) Statement Content Addressing Haim(2 5): challenge( equivalent( problem, any_permuta0on )) assert( equivalent( problem, exist_permuta0on ) ) That’s preYy strong; all you need is that there exists a {weaker( any_permuta0on, exists_permuta0on) } permuta0on where that is true. And it doesn’t work; there are numbers $a_1,a_2,\ldots,a_n challenge( any_permuta0on) $ and sets $M$ of $n-1$ points such that, for instance, $a_1 \in M$. Then any permuta0on star0ng with $a_1$ would counter(any_permuta0on, ... ) not sa0sfy your conjecture for $j=1$. But, just looking for *one* permuta0on that sa0sfies suggest( strategy( exist_permuta0on, $a_s(1)+a_s(2)…+a_s(j) \not \in M$ for any $j \leq n$ (which is basically the statement of the theorem), could induct_on_n, using( subset_solu0on ))) lend itself well to induc0on. In other words, use the fact that for every subset $M’ \subset M$ of size $j$ not containing $a_s(1)+a_s(2)…+a_s(j)$, there is a way to permute those $j$ numbers to avoid $M’$. Sorry, indeed I meant: “Show that for *one* permuta0on…” retract( equivalent( problem( exist_permuta0on ) ) agree( equivalent( problem, exist_permuta0on ) ) What might a protocol look like? a( reformulator(X) ) :: equivalent( X, Y ) => all <- reformulate(X,Y) then ( assert( Facts ) => all suggest_strategy( Y, S, Facts ) => all ) <- Facts ^ strategy( Y, S, Facts ) or null a( counterexample ) :: X <= collaborator then ( not(X) => collaborator counterexample(X, Cx ) => collaborator ) <- find_counterexample( X, Cx ) or agree( X ) => collaborator A visual representation of the structure of interaction Modelling polymath with argumentation theory / LSC A Pease, J Corneli et al, Lakatos-style Collaborative Mathematics through Dialectical, Structured and Abstract Argumentation, Artificial Intelligence online first (2017) Used polymath as evidence base for “exp-phil” testing hypotheses about explanation A Aberdein, U Martin, A Pease, An empirical investigation into explanation in mathematical conversations, submitted, (2017) Ethnographic study of collaboration Newton Institute: varieties of collaboration; importance of “habitus” Lorenzo Lane, The Bridge Between Worlds: Relating Position and Disposition in the Mathematical Field, submitted, (2017) Virtue theory and mathematical collaboration Fenner Tanswell, A Problem with the Dependence of Informal Proofs on Formal Proofs, Philos Math, 23, 295-310 (2015) How does mathematics have impact? Impacts Instrumental, Conceptual, Capacity- building, Attitude/Culture, Enduring Connectivity 2014 REF 209 case studies + 52 environment Conclusions • Variety of impact types: conceptual important • Long-term relationships + interdisciplinarity • Culture for impact • Knowledge intermediaries • Limitations of linear narratives U Martin and L Meagher, Slightly dirty maths: the richly textured mechanisms of impact, Research Evaluation, online first (2017)