Five Theories of Reasoning:

• Authors: Alison Pease , Andrew Aberdein
• Languages of publication: EN

Abstracts

EN

The last century has seen many disciplines place a greater prior- ity on understanding how people reason in a particular domain, and several illuminating theories of informal logic and argumentation have been devel- oped. Perhaps owing to their diverse backgrounds, there are several con- nections and overlapping ideas between the theories, which appear to have been overlooked. We focus on Peirce’s development of abductive reasoning [39], Toulmin’s argumentation layout [52], Lakatos’s theory of reasoning in mathematics [23], Pollock’s notions of counterexample [44], and argumen- tation schemes constructed by Walton et al. [54], and explore some connec- tions between, as well as within, the theories. For instance, we investigate Peirce’s abduction to deal with surprising situations in mathematics, rep- resent Pollock’s examples in terms of Toulmin’s layout, discuss connections between Toulmin’s layout and Walton’s argumentation schemes, and sug- gest new argumentation schemes to cover the sort of reasoning that Lakatos describes, in which arguments may be accepted as faulty, but revised, rather than being accepted or rejected. We also consider how such theories may apply to reasoning in mathematics: in particular, we aim to build on ideas such as Dove’s [13], which help to show ways in which the work of Lakatos fits into the informal reasoning community.

Keywords

EN

informal reasoning; mathematics; Lakatos; argumentation

• Publisher: Uniwersytet Mikołaja Kopernika (Nicolaus Copernicus University in Toruń)
• Journal: Logic and Logical Philosophy
• Year: 2011
• Volume: 20
• Issue: 1-2
• Pages: 7-57

Dates

published

2011-06-01

online

2013-07-02

Contributors

author

Alison Pease

• Centre for Intelligent Systems and their Applications Informatics Forum University of Edinburgh 8 Crichton Street Edinburgh, EH8 9AB

author

Andrew Aberdein

• Department of Humanities and Communication Florida Institute of Technology 150 West University Blvd Melbourne, Florida 32901-6975, U.S.A.

References

• [1] Aberdein, A., “The uses of argument in mathematics”, Argumentation 19 (2005): 287-301.
• [2] Aberdein, A., “Managing informal mathematical knowledge: Techniques from informal logic”, pages 208-221, in: J.M. Borwein and W.M. Farmer (eds.), MKM 2006, LNAI 4108, Springer-Verlag, Berlin, 2006.
• [3] Banegas, J.A., “L’argumentacio en matematiques”, pages 135-147 in: E.Casaban i Moya (ed.), XIIè Congrés Valencià de Filosofia, Valencia, 1998. http://my.fit.edu/~aberdein/Alcolea.pdf
• [4] Anderson, B., “Diary”, Spectator 313 (9481), May 15th 2010. http://www.spectator.co.uk/politics/all/5996573/part_2/diary.thtml
• [5] Bocheński, I.M., A History of Formal Logic, Chelsea Pub Co, New York, N.Y., 1970 [1956].
• [6] Boden, M.A., The Creative Mind: Myths and Mechanisms, Weidenfield and Nicholson, London, 1990.
• [7] Burley, W., On the Purity of the Art of Logic: The Shorter and LongerTreatises, Yale University Press, New Haven, CT, 2000. Translated by P.V. Spade.
• [8] Cauchy, A. L., Cours d’Analyse de l’École Royale Polytechnique, de Bure, Paris, 1821.
• [9] Colton, S., Automated Theory Formation in Pure Mathematics, Springer- Verlag, 2002.
• [10] Colton, S., A. Bundy, and T. Walsh, “On the notion of interestingness in automated mathematical discovery”, International Journal of HumanComputer Studies 53, 3 (2000): 351-375.
• [11] Conan Doyle, A., The Memoirs of Sherlock Holmes, Forgotten Books, 2008 [1894].
• [12] Dauben, J.W., “Peirce’s place in mathematics”, Historia Mathematica 9, 3 (1982): 311-325.
• [13] Dove, I. J., “On mathematical proofs and arguments: Johnson and Lakatos”, pages 346-351 in: F.H. Van Eemeren and B. Garssen (eds.), Proceedings of the Sixth Conference of the International Society for theStudy of Argumentation, volume 1, Sic Sat, Amsterdam, 2007.
• [14] Dunham,W., Euler: The Master of Us All, The Mathematical Association of America, Washington, DC, 1999.
• [15] Epstein, S. L., “Learning and discovery: One system’s search for mathe- matical knowledge”, Computational Intelligence 4, 1 (1988): 42-53.
• [16] Gasteren, A. J.M., On the shape of mathematical arguments, Lecture notes in computer science, volume 445, Springer, Berlin, 1990.
• [17] Gray, J., The Hilbert Challenge, Oxford University Press, Oxford, 2000.
• [18] Grosholz, E.R., Representation and Productive Ambiguity in Mathematicsand the Sciences, Oxford University Press, New York, NY, 2007.
• [19] Hairer, E., and G. Wanner, Analysis by Its History, Undergraduate Texts in Mathematics, Springer, New York, NY, 2008.
• [20] Hanson, N.R., “Is there a logic of scientific discovery?”, pages 20-35 in: H. Feigl and G. Maxwell (eds.), Current Issues in the Philosophy of Science, Symposia of scientists and philosophers. Proceedings of Section L of the American Association for the Advancement of Science, 1959, Holt, Rinehart and Winston, New York, NY, 1961.
• [21] Heath, T. L., The Thirteen Books of Euclid’s Elements, Cambridge Uni- versity Press, Cambridge, 2nd edition, 1925.
• [22] Hitchcock, D., “Toulmin’s warrants”, pages 69-82 in: F.H. van Eemeren, J. Blair, C. Willard, and A. F. Snoeck-Henkemans (eds.), Anyone WhoHas a View. Theoretical Contributions to the Study of Argumentation, Kluwer Academic Publishers, Dordrecht, 2003.
• [23] Lakatos, I., Proofs and Refutations, Cambridge University Press, Cam- bridge, 1976.
• [24] Lakatos, I., “Cauchy and the continuum: The significance of non-standard analysis for the history and philosophy of mathematics”, pages 43-60 in: J. Worral and C. Currie (eds.), Mathematics, science and epistemology, Philosophical Papers, volume 2, Cambridge University Press, Cambridge, 1978.
• [25] Lakatos, I., “Proofs and refutations (I)”, The British Journal for the Philosophyof Science 14 (1963): 1-25.
• [26] Lakatos, I., “Proofs and refutations (II)”, The British Journal for thePhilosophy of Science 14 (1963): 120-139.
• [27] Lakatos, I., “Proofs and refutations (III)”, The British Journal for thePhilosophy of Science 14 (1963): 221-245.
• [28] Lakatos, I., “Proofs and refutations (IV)”, The British Journal for thePhilosophy of Science 14 (1964): 296-342.
• [29] Larvor, B., Lakatos: An Introduction, Routledge, London, 1998.
• [30] Lenat, D., AM: An Artificial Intelligence Approach to Discovery in Mathematics, PhD thesis, Stanford University, 1976.
• [31] Leng, M., “What’s there to know? A fictionalist account of mathematical knowledge”, pages 84-108 in: M. Leng, A. Paseau, and M. Potter (eds.), Mathematical Knowledge, Oxford University Press, Oxford, 2007.
• [32] Macedo L., and A. Cardoso, “Creativity and surprise”, in: G. Wiggins (ed.), Proceedings of the AISB’01 Symposium on Artificial Intelligenceand Creativity in Arts and Science, 2001.
• [33] Mancosu, P., “Mathematical explanation: Problems and prospects”, Topoi 20, 1 (2001): 97-117.
• [34] Marquis, J.-P., “Abstract mathematical tools and machines for mathe- matics”, Philosophia Mathematica 5 (1997): 250-272.
• [35] Muntersbjorn, M., “Construction, articulation, and explanation: Phases in the growth of mathematics”, pages 19-42 in: B. Van Kerkhove, J.-P. van Bendegem, and J. De Vuyst (eds.), Philosophical Perspectives on MathematicalPractice, College Publications, London, 2010.
• [36] Pease, A., A Computational Model of Lakatos-style Reasoning, PhD thesis, School of Informatics, University of Edinburgh, 2007. http://hdl.handle.net/1842/2113
• [37] Pease, A., M. Guhe, and A. Smaill, “Analogy formulation and modifica- tion in geometry”, pages 358-364 in: Proceedings of the Second InternationalConference on Analogy, 2009.
• [38] Pease, A., A. Smaill, S. Colton, and J. Lee, “Bridging the gap between argumentation theory and the philosophy of mathematics”, Foundationsof Science 14, 1-2 (2009): 111-135.
• [39] Peirce, C. S., Collected Papers of Charles Sanders Peirce, Eight Volumes, Harvard University Press, Cambridge, Mass, 1931-58.
• [40] Peirce, C. S., The New Elements of Mathematics, Mouton, The Hague, 1976.
• [41] Plato, The Republic, OUP, Oxford, 1993.
• [42] Pollock, J., “The structure of epistemic justification”, American PhilosophicalQuarterly, monograph series 4 (1970): 62-78.
• [43] Pollock, J., Contemporary Theories of Knowledge, Rowman and Little- field, Totowa, NJ, 1986.
• [44] Pollock, J., Cognitive Carpentry, The MIT Press, Cambridge, MA., 1995.
• [45] Polya, G., Mathematics and plausible reasoning, volume 1, Induction and analogy in mathematics, Princeton University Press, 1954.
• [46] Quine, W.V.O., “Two dogmas of empiricism”, From a Logical Point ofView, Harvard University Press, 1953. First version of the paper, without any reference to Duhem, in The Philosophical Review 60 (1951): 20-53.
• [47] Reed, C., and G. Rowe, “Translating Toulmin diagrams: Theory neutral- ity in argument representation”, Argumentation 19, 3 (2005): 267-286.
• [48] Robinson, A., Non-standard Analysis, Princeton University Press, Prince- ton, New Jersey, 1996. Revised Edition.
• [49] Robinson, R., “Analysis in Greek geometry”, Mind 45 (1936): 464-73.
• [50] Sandifer, C.E., The early mathematics of Leonhard Euler, The Mathe- matical Association of America, 2007.
• [51] Seidel, P. L., “Note uber eine Eigenschaft der Reihen, welche Discontinuir- liche Functionen Darstellen”, Abhandlungen der Mathmatisch-PhysikalischenKlasse der Königlich Bayerischen Akademie der Wissenschaften 5 (1847): 381-93.
• [52] Toulmin, S., The uses of argument, CUP, Cambridge, 1958.
• [53] Toulmin, S., R. Rieke, and A. Janik, An Introduction to Reasoning, Macmillan, London, 1979.
• [54] Walton, D., C. Reed, and F. Macagno, Argumentation Schemes, Cam- bridge University Press, Cambridge, 2008.
• [55] Walton, D.N., Argument Schemes for Presumptive Reasoning, Lawrence Erlbaum Associates, Mahwah, NJ, 1996.
• [56] Walton D.N., and C.A. Reed, “Diagramming, argumentation schemes and critical questions”, pages 195-211 in: F.H. van Eemeren, J. Anthony Blair, Ch.A. Willard, and A. Francisca Snoeck Henkemans (eds.), AnyoneWho Has a View: Theoretical Contributions to the Study of Argumentation, Kluwer Academic Publishers, Dordrecht, 2003.

Identifiers

DOI

10.12775/LLP.2011.002