Modelling the way mathematics is actually done

Whereas formal mathematical theories are well studied, computers cannot yet adequately represent and reason about mathematical dialogues and other informal texts. To address this gap, we have developed a representation and reasoning strategy that draws on contemporary argumentation theory and classic AI techniques for representing and querying narratives and dialogues. In order to make the structures that these modelling tools produce accessible to computational reasoning, we encode representations in a higher-order nested semantic network. This system, for which we have developed a preliminary prototype in LISP, can represent both the content of what people say, and the dynamic reasoning steps that move from one step to the next.

[1]  Michael Jackman,et al.  Conceptual graphs , 1988 .

[2]  John F. Sowa,et al.  Chapter 5 Conceptual Graphs , 2008 .

[3]  Imre Lakatos,et al.  On the Uses of Rigorous Proof. (Book Reviews: Proofs and Refutations. The Logic of Mathematical Discovery) , 1977 .

[4]  Cezary Kaliszyk,et al.  Developing Corpus-Based Translation Methods between Informal and Formal Mathematics: Project Description , 2014, CICM.

[5]  H. Penny Nii,et al.  The Handbook of Artificial Intelligence , 1982 .

[6]  Ben Goertzel,et al.  Symbol Grounding via Chaining of Morphisms , 2017, ArXiv.

[7]  Susan E. Newman,et al.  Cognitive Apprenticeship: Teaching the Craft of Reading, Writing, and Mathematics. Technical Report No. 403. , 1987 .

[8]  Gerald J. Sussman.  Why programming is a good medium for expressing poorly understood and sloppily formulated ideas , 2005, OOPSLA '05.

[9]  Phil Blunsom,et al.  Teaching Machines to Read and Comprehend , 2015, NIPS.

[10]  Lauri Carlson.  Dialogue Games: An Approach to Discourse Analysis , 1982 .

[11]  D. Herrmann.  Proofs And Refutations The Logic Of Mathematical Discovery , 2016 .

[12]  Tom Addis,et al.  Drawing Programs: The Theory and Practice of Schematic Functional Programming , 2009 .

[13]  Allen Newell,et al.  Heuristic Problem Solving: The Next Advance in Operations Research , 1958 .

[14]  Mark O. Riedl,et al.  Using Stories to Teach Human Values to Artificial Agents , 2016, AAAI Workshop: AI, Ethics, and Society.

[15]  Michael Kohlhase,et al.  STEX: Semantic Markup in TEX/LTEX , 2010 .

[16]  Patrick Saint-Dizier,et al.  A Model for Processing Illocutionary Structures and Argumentation in Debates , 2014, LREC.

[17]  Steven L. Lytinen,et al.  CONCEPTUAL DEPENDENCY AND ITS DESCENDANTS , 1992 .

[18]  R. Carnap.  Logical Syntax of Language , 1937 .

[19]  Theodor Holm Nelson.  ZigZag (Tech briefing) , 2001, HYPERTEXT '01.

[20]  A. Turing.  Intelligent Machinery, A Heretical Theory* , 1996 .

[21]  Mohan Ganesalingam,et al.  The Language of Mathematics: A Linguistic and Philosophical Investigation , 2013 .

[22]  Paul Bernays,et al.  Natur und mathematisches Erkennen : Vorlesungen, gehalten 1919-1920 in Göttingen , 1992 .

[23]  Introduction to graph grammars with applications to semantic networks , 1992 .

[24]  M. Ganesalingam,et al.  A Fully Automatic Theorem Prover with Human-Style Output , 2016, Journal of Automated Reasoning.

[25]  Leslie Lamport,et al.  How to Write a Proof , 1995 .

[26]  Cezary Kaliszyk,et al.  HolStep: A Machine Learning Dataset for Higher-order Logic Theorem Proving , 2017, ICLR.

[27]  W. Mann.  Dialogue games: Conventions of human interaction , 1988 .

[28]  John F. Sowa,et al.  Handbook of Knowledge Representation Edited Conceptual Graphs 5.1 from Existential Graphs to Conceptual Graphs , 2022 .

[29]  Ben Goertzel,et al.  Engineering General Intelligence, Part 1: A Path to Advanced AGI via Embodied Learning and Cognitive Synergy , 2014 .

[30]  Roger C. Schank,et al.  Scripts, plans, goals and understanding: an inquiry into human knowledge structures , 1978 .

[31]  C. W. Tate.  Solve it. , 2005, Nursing standard (Royal College of Nursing (Great Britain) : 1987).

[32]  Nicolas Bourbaki,et al.  Elements of mathematics , 2004 .

[33]  Willard Van Orman Quine,et al.  Word and Object , 1960 .

[34]  Aaron Sloman.  The well-designed young mathematician , 2008, Artif. Intell..

[35]  L. Lamport.  How to write a 21st century proof , 2012 .

[36]  Terry Winograd,et al.  Procedures As A Representation For Data In A Computer Program For Understanding Natural Language , 1971 .

[37]  I. Lakatos,et al.  Proofs and Refutations: Frontmatter , 1976 .

[38]  Ben Goertzel,et al.  Engineering General Intelligence, Part 1 , 2014, Atlantis Thinking Machines.

[39]  Ben Goertzel,et al.  Engineering General Intelligence, Part 2: The CogPrime Architecture for Integrative, Embodied AGI , 2014 .

[40]  Roger C. Schank,et al.  Conceptual dependency: A theory of natural language understanding , 1972 .

[41]  Cezary Kaliszyk,et al.  Learning-Assisted Automated Reasoning with Flyspeck , 2012, Journal of Automated Reasoning.

[42]  Barr and Feigenbaum Edward A. Avron,et al.  The Handbook of Artificial Intelligence , 1981 .

[43]  Silvia de Toffoli,et al.  'Chasing' the Diagram - the Use of Visualizations in Algebraic Reasoning , 2017, Rev. Symb. Log..

[44]  Chris Reed,et al.  Lakatos-style collaborative mathematics through dialectical, structured and abstract argumentation , 2017, Artif. Intell..

[45]  V. S. Ramachandran,et al.  Analogical Reasoning , 2012 .

[46]  Leslie Lamport,et al.  Specifying Concurrent Systems with TLA , 1999 .

[47]  Amy X. Zhang,et al.  Characterizing Online Discussion Using Coarse Discourse Sequences , 2017, Proceedings of the International AAAI Conference on Web and Social Media.

[48]  L. Corry.  The Origins of Eternal Truth in Modern Mathematics: Hilbert to Bourbaki and Beyond , 1997, Science in Context.

[49]  M. Richards,et al.  A city is not a tree , 2017 .

[50]  Alan Bundy,et al.  The interaction of representation and reasoning , 2013, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[51]  J. Barthélemy,et al.  Glossary Fifty Key Terms in the Works of Gilbert Simondon , 2014 .

[52]  William E. Byrd,et al.  The reasoned schemer , 2005 .

[53]  W. Thurston.  On Proof and Progress in Mathematics , 1994, math/9404236.

[54]  Michael Kohlhase,et al.  The Flexiformalist Manifesto , 2012, 2012 14th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing.

[55]  Tobias Nipkow,et al.  A FORMAL PROOF OF THE KEPLER CONJECTURE , 2015, Forum of Mathematics, Pi.

[56]  David Murray-Rust,et al.  Towards Mathematical AI via a Model of the Content and Process of Mathematical Question and Answer Dialogues , 2017, CICM.

[57]  Chris Reed,et al.  Lakatos Games for Mathematical Argument , 2014, COMMA.

[58]  Mateja Jamnik.  Mathematical Reasoning with Diagrams , 2001 .

[59]  Csr Young,et al.  How to Do Things With Words , 2009 .

[60]  John McCarthy.  The well-designed child , 2008, Artif. Intell..

[61]  David Moshman,et al.  : From inference to reasoning: The construction of rationality , 2004 .

[62]  Cezary Kaliszyk,et al.  Deep Network Guided Proof Search , 2017, LPAR.

[63]  Peter Koepke,et al.  Parsing and Disambiguation of Symbolic Mathematics in the Naproche System , 2011, Calculemus/MKM.

[64]  Avron Barr,et al.  The Handbook of Artificial Intelligence, Volume 1 , 1982 .

[65]  Roger C. Schank,et al.  Memory, meaning, and syntax , 1980 .

[66]  Marcos Cramer,et al.  Presupposition Projection and Accommodation in Mathematical Texts , 2010, KONVENS.

[67]  Simon Colton,et al.  Addressing the "Why?" in Computational Creativity: A Non-Anthropocentric, Minimal Model of Intentional Creative Agency , 2017, ICCC.

[68]  Brendan Larvor,et al.  Lakatos' mathematical Hegelianism , 1999 .

[69]  Ben Goertzel,et al.  Engineering General Intelligence, Part 2 , 2014, Atlantis Thinking Machines.