Mathematical practice, crowdsourcing, and social machines

The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. Mathematical practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice - for example the community question answering system {\it mathoverflow} contains around 40,000 mathematical conversations, and {\it polymath} collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of "soft" aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal. Crowdsourced mathematical activity is an example of a "social machine", a new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.Comment: To appear, Springer LNCS, Proceedings of Conferences on Intelligent Computer Mathematics, CICM 2013, July 2013 Bath, U

Improving the Efficiency of Reasoning Through Structure-Based Reformulation

Abstract. We investigate the possibility of improving the efficiency of reasoning through structure-based partitioning of logical theories, combined with partitionbased logical reasoning strategies. To this end, we provide algorithms for reasoning with partitions of axioms in first-order and propositional logic. We analyze the computational benefit of our algorithms and detect those parameters of a partitioning that influence the efficiency of computation. These parameters are the number of symbols shared by a pair of partitions, the size of each partition, and the topology of the partitioning. Finally, we provide a greedy algorithm that automatically reformulates a given theory into partitions, exploiting the parameters that influence the efficiency of computation.

The algebra of lexical semantics

Abstract. The current generative theory of the lexicon relies primar-ily on tools from formal language theory and mathematical logic. Here we describe how a different formal apparatus, taken from algebra and automata theory, resolves many of the known problems with the gener-ative lexicon. We develop a finite state theory of word meaning based on machines in the sense of Eilenberg [11], a formalism capable of de-scribing discrepancies between syntactic type (lexical category) and se-mantic type (number of arguments). This mechanism is compared both to the standard linguistic approaches and to the formalisms developed in AI/KR. 1 Problem Statement In developing a formal theory of lexicography our starting point will be the informal practice of lexicography, rather than the more immediately related for-mal theories of Artificial Intelligence (AI) and Knowledge Representation (KR). Lexicography is a relatively mature field, with centuries of work experience an

Challenges Using Extrapolated Family-level Macroinvertebrate Metrics in Moderately Disturbed Tropical Streams: a Case-study From Belize

Family-level biotic metrics were originally designed to rapidly assess gross organic pollution effects, but came to be regarded as general measures of stream degradation. Improvements in water quality in developed countries have reignited debate about the limitations of family-level taxonomy to detect subtle change, and is resulting in a shift back towards generic and species-level analysis to assess smaller effects. Although the scale of pollution characterizing past condition of streams in developed countries persists in many developing regions, some areas are still considered to be only moderately disturbed. We sampled streams in Belize to investigate the ability of family-level macroinvertebrate metrics to detect change in stream catchments where less than 30% of forest had been cleared. Where disturbance did not co-vary with natural gradients of change, and in areas characterized by low intensity activities, none of the metrics tested detected significant change, despite evidence of environmental impacts. We highlight the need for further research to clarify the response of metrics to disturbance over a broader study area that allows replication for confounding sources of natural variation. We also recommend research to develop more detailed understanding of the taxonomy and ecology of Neotropical macroinvertebrates to improve the robustness of metric use