Abstract
The more extensive use of diagrammatic representations as a tool for managing complexity and communication problems of mathematical knowledge is advocated in the paper. The specifics of this representation tool are introduced, including the problems with using diagrams in mathematics, issues of proper design of diagrams, specification of main usage types of mathematical diagrams and ways of their implementation. The discussion is illustrated by a number of diagrams, mostly taken from the diagrammatic notation for interval algebra recently developed by the author. These and other issues of diagrammatic representation and reasoning are investigated by the recently emerging discipline of diagrammatics.
The paper was supported in part by the grant No. 5 T07F 002 25 (for years 2003–2006) from the KBN (State Committee for Scientific Research). The author is also indebted to anonymous referees of the paper for their comments and suggestions.
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References
Anderson, M., Cheng, P., Haarslev, V. (eds.): Diagrams 2000. LNCS (LNAI), vol. 1889. Springer, Heidelberg (2000)
Anderson, M., Meyer, B., Olivier, P. (eds.): Diagrammatic Representation and Reasoning. Springer, Berlin (2002)
Arnheim, R.: Visual Thinking. University of California Press, Berkeley (1969)
Barker-Plummer, D., Bailin, S.C.: The role of diagrams in mathematical proofs. Machine Graphics & Vision 6, 25–56 (1997)
Barwise, J., Etchemendy, J.: Visual information and valid reasoning. In: Allwein, G., Barwise, J. (eds.) Logical Reasoning with Diagrams, vol. 3–25. Oxford University Press, Oxford (1996)
Barwise, J., Etchemendy, J.: Heterogeneous logic. In: ibidem, pp. 209–232
Blackwell, A.F., Marriott, K., Shimojima, A. (eds.): Diagrams 2004. LNCS (LNAI), vol. 2980. Springer, Heidelberg (2004)
Buchberger, B.: Logicographic symbols: A new feature in Theorema. In: Tazawa, Y. (ed.) Symbolic Computation – New Horizons, pp. 23–30. Tokyo Denki Univ. Press (2001)
Goodstein, D.L., Goodstein, J.R.: Feynman’s Lost Lecture: The Motion of Planets Around the Sun. W.W. Norton & Co. (1996)
Hadamard, J.: The Psychology of Invention in the Mathematical Field. Princeton University Press, Princeton (1945)
Hammer, E.: Logic and Visual Information. Cambridge Univ. Press, Cambridge (1996)
Hegarty, M., Meyer, B., Narayanan, N.H. (eds.): Diagrams 2002. LNCS (LNAI), vol. 2317. Springer, Heidelberg (2002)
Jamnik, M., Bundy, A., Green, I.: On automating diagrammatic proofs of arithmetic arguments. Journal of Logic, Language and Information 8, 297–321 (1999)
Kulpa, Z.: Diagrammatic representation for a space of intervals. Machine Graphics & Vision 6, 5–24 (1997)
Kulpa, Z.: Diagrammatic representation for interval arithmetic. Linear Algebra Appl. 324, 55–80 (2001)
Kulpa, Z.: Diagrammatic analysis of interval linear equations. Part I: Basic notions and the one-dimensional case. Reliable Computing 9, 1–20 (2003)
Kulpa, Z.: Self-consistency, imprecision, and impossible cases in diagrammatic representations. Machine Graphics & Vision 12, 147–160 (2003)
Kulpa, Z.: From Picture Processing to Interval Diagrams. IFTR Reports 4/2003, Warsaw (2003), See: http://www.ippt.gov.pl/~zkulpa/diagrams/fpptid.html
Kulpa, Z., Markov, S.: On the inclusion properties of interval multiplication: A diagrammatic study. BIT Numerical Mathematics 43, 791–810 (2003)
Kulpa, Z.: Designing diagrammatic notation for interval analysis. Information Design Journal + Document Design 12, 52–62 (2004)
Le, T.L., Kulpa, Z.: Diagrammatic spreadsheet. Machine Graphics & Vision 12, 133–146 (2003)
Miller, N.: A Diagrammatic Formal System for Euclidean Geometry. Ph.D. Thesis. Cornell University, Ithaca, NY (2001)
Needham, T.: Visual Complex Analysis. Clarendon Press, Oxford (1997)
Nelsen, R.B.: Proofs Without Words: Exercises in Visual Thinking. The Mathematical Association of America, Washington, DC (1993)
Shimojima, A.: The graphic-linguistic distinction: Exploring alternatives. In: Blackwell, A. (ed.) Thinking with Diagrams, pp. 5–27. Kluwer, Dordrecht (2001)
Tufte, E.R.: The Visual Display of Quantitative Information. Graphics Press, Cheshire (1983)
Winterstein, D., Bundy, A., Gurr, C., Jamnik, M.: Using animation in diagrammatic theorem proving. In: [12], pp. 46–60
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Kulpa, Z. (2004). On Diagrammatic Representation of Mathematical Knowledge. In: Asperti, A., Bancerek, G., Trybulec, A. (eds) Mathematical Knowledge Management. MKM 2004. Lecture Notes in Computer Science, vol 3119. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27818-4_14
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DOI: https://doi.org/10.1007/978-3-540-27818-4_14
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