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Main problems of diagrammatic reasoning. I: The generalization problem. (English) Zbl 1178.03011

Summary: The paper attempts to analyze in some detail the main problems encountered in reasoning using diagrams, which may cause errors in reasoning, produce doubts concerning the reliability of diagrams, and impressions that diagrammatic reasoning lacks the rigour necessary for mathematical reasoning. The paper first argues that such impressions come from long neglect which led to a lack of well-developed, properly tested and reliable reasoning methods, as contrasted with the amount of work generations of mathematicians expended on refining the methods of reasoning with formulae and predicate calculus. Next, two main groups of problems occurring in diagrammatic reasoning are introduced. The second group, called diagram imprecision, is then briefly summarized, its detailed analysis being postponed to another paper. The first group, called collectively the generalization problem, is analyzed in detail in the rest of the paper. The nature and causes of the problems from this group are explained, methods of detecting the potentially harmful occurrences of these problems are discussed, and remedies for possible errors they may cause are proposed. Some of the methods are adapted from similar methods used in reasoning with formulae, several other problems constitute new, specifically diagrammatic ways of reliable reasoning.

MSC:

03A05 Philosophical and critical aspects of logic and foundations
00A30 Philosophy of mathematics

Software:

ActiveMath; Mizar
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Full Text: DOI

References:

[1] Arnheim R. (1969) Visual thinking. University of California Press, Berkeley, CA
[2] Bundy A., Richardson J. (1999) Proofs about lists using ellipsis. In: Ganzinger H., McAllester D., Voronkov V. (eds) Logic for programming and automated reasoning. LNAI 1705. Springer-Verlag, Berlin, pp 1–12 · Zbl 0938.03017
[3] DIAGRAMS. (1992). Reasoning with diagrammatic representations (1992 AAAI spring symposium). Menlo Park: CAAAAI Press.
[4] Dove, I. (2002). Can pictures prove? Logique & Analyse, 179–180, 309–340. · Zbl 1075.00003
[5] Dubnov, Y. S. (1955). Oshibky v geometricheskykh dokazatyelstvakh (in Russian). Moscow: GITTL. [English translation: Dubnov, Y. S. (1963). Mistakes in geometric proofs. Boston, MA: Heath].
[6] Foo, N. Y., Pagnucco, M., & Nayak, A. C. (1999). Diagrammatic proofs. In Proceedings of 16th International Joint Conference on Artificial Intelligence (IJCAI-99) (pp. 378–383), Stockholm, August 1999. Morgan Kaufmann.
[7] Frege, G. (1879). Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. S: Louis Nebert [English edition: Frege, G. (1879). Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought. Reprinted in: van Heijenoort, J. (Ed.), (1967). From Frege to Godel: A source book in mathematical logic (pp. 1879–1931). Cambridge, MA: Harvard University Press].
[8] Giaquinto M. (2007) Visual thinking in mathematics: An epistemological study. Oxford University Press, Oxford · Zbl 1175.00023
[9] Greaves M. (2001) The philosophical status of diagrams. CSLI Publications, Stanford, CA · Zbl 1025.00001
[10] Jamnik M. (2001) Mathematical reasoning with diagrams: From intuition to automation. CSLI Publications, Stanford, CA · Zbl 1040.68111
[11] King, J. R., & Schattschneider, D. (Eds.), (1997). Geometry turned on!: Dynamic software in learning, teaching, and research. Washington, DC: Mathematical Association of America. · Zbl 0876.68108
[12] Kulpa Z. (1983) Are impossible figures possible? Signal Processing 5(3): 201–220
[13] Kulpa Z. (1987) Putting order in the impossible. Perception 16: 201–214 · doi:10.1068/p160201
[14] Kulpa Z. (2003a) Self-consistency, imprecision, and impossible cases in diagrammatic representations. Machine GRAPHICS & VISION 12(1): 147–160
[15] Kulpa, Z. (2003b). From picture processing to interval diagrams. IFTR Reports 4/2003, Warsaw, 313 pp. See http://www.ippt.gov.pl/zkulpa/diagrams/fpptid.html . · Zbl 1173.65314
[16] Kulpa, Z. (2009a). Main problems of diagrammatic reasoning: Part II: The imprecision problem (in preparation). · Zbl 1178.03011
[17] Kulpa, Z. (2009b). Representing sets in diagrams: Spaces of all triangles (in preparation).
[18] Le T.L., Kulpa Z. (2003) Diagrammatic spreadsheet. Machine GRAPHICS & VISION 12(1): 133–146
[19] Le T.L., Kulpa Z. (2004) Diagrammatic spreadsheet: An overview. In: Blackwell A., Marriott K., Shimojima A. (eds) Diagrammatic representation and inference. LNAI 2980. Springer-Verlag, Berlin, pp 420–423 · Zbl 1091.68647
[20] Lemon O., Pratt I. (1997) Spatial logic and the complexity of diagrammatic reasoning. Machine GRAPHICS & VISION 6(1): 77–88
[21] Lindsay R.K. (2000) Playing with diagrams. In: Anderson M., Cheng P., Haarslev V. (eds) Theory and application of diagrams. LNAI 1889. Springer-Verlag, Berlin, pp 300–313 · Zbl 0973.68546
[22] Luengo, I. (1995). Diagrams in geometry. Ph.D. Thesis, Indiana University, Bloomington, IN.
[23] MacInnis, R., McKinna, J., Parsons, J., & Dyckhoff, R. (2004). A mechanised environment for Frege’s Begriffsschrift notation. In Proceedings of Mathematical User-Interfaces Workshop 2004, Bialowieza, Poland, September 18, 2004. See http://www.activemath.org/\(\sim\)paul/MathUI04/proceedings/FregesBegriffsSchrift.html .
[24] Maxwell E.A. (1959) Fallacies in mathematics. Cambridge University Press, Cambridge · Zbl 0083.24103
[25] Miller, N. (2001). A diagrammatic formal system for Euclidean geometry. Ph.D. Thesis, Cornell University, Ithaca, NY.
[26] Muzalewski, M. (1993). An outline of PC Mizar. Brussels: Foundation Philippe le Hodey. See http://mizar.org/project/bibliography.html .
[27] Needham T. (1997) Visual complex analysis. Clarendon Press, Oxford · Zbl 0893.30001
[28] Nelsen, R. B. (1993). Proofs without words: Exercises in visual thinking. Washington, DC: The Mathematical Association of America. · Zbl 1160.00303
[29] Otte M. (2006) Proof-analysis and continuity. Foundations of Science 11: 121–155 · Zbl 1151.00010 · doi:10.1007/s10699-004-5915-0
[30] Pollet M., Sorge V., Kerber M. (2004) Intuitive and formal representations: The case of matrices. In: Asperti A., Bancerek G., Trybulec A. (eds) Mathematical knowledge management. LNCS 3119. Springer-Verlag, Berlin, pp 317–331 · Zbl 1108.68600
[31] Shin S.-J. (1994) The logical status of diagrams. Cambridge University Press, Cambridge, MA · Zbl 0829.03002
[32] Winterstein, D. (2004). Using diagrammatic reasoning for theorem proving in a continuous domain (263 pp.). Ph.D. Thesis, The University of Edinburgh, Edinburgh.
[33] Winterstein D., Bundy A., Jamnik M. (2000) A proposal for automating diagrammatic reasoning in continuous domains. In: Anderson M., Cheng P., Haarslev V. (eds) Theory and application of diagrams. LNAI 1889. Springer-Verlag, Berlin, pp 286–299 · Zbl 0973.68544
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