Linked bibliography for the SEP article "The Epistemology of Visual Thinking in Mathematics" by Marcus Giaquinto

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If everything goes well, this page should display the bibliography of the aforementioned article as it appears in the Stanford Encyclopedia of Philosophy, but with links added to PhilPapers records and Google Scholar for your convenience. Some bibliographies are not going to be represented correctly or fully up to date. In general, bibliographies of recent works are going to be much better linked than bibliographies of primary literature and older works. Entries with PhilPapers records have links on their titles. A green link indicates that the item is available online at least partially.

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Adams, C., 2001, The Knot Book, Providence, Rhode Island: American Mathematical Society. (Scholar)
Azzouni, J., 2013, “That we see that some diagrammatic proofs are perfectly rigorous”, Philosophia Mathematica, 21: 323–338. (Scholar)
Brown, J., 1999, Philosophy of Mathematics: an introduction to the world of proofs and pictures, London: Routledge. (Scholar)
Barwise, J. and J. Etchemendy, 1996, “Visual information and valid reasoning”, in Logical Reasoning with Diagrams, G. Allwein and J. Barwise (eds) Oxford: Oxford University Press. (Scholar)
Bolzano, B., 1817, “Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation”, in Ewald 1996: vol. 1, 225–248. (Scholar)
Brooks, N., Barner, D., Frank, M. and Goldin-Meadow, S., 2018, The role of gesture in supporting mental representations: The case of mental abacus arithmetic. Cognitive Science, 45(2), 554–575. (Scholar)
Carter, J., 2010, “Diagrams and Proofs in Analysis”, International Studies in the Philosophy of Science, 24: 1–14. (Scholar)
Cauchy, A., 1813, “Recherche sur les polyèdres—premier mémoire”, Journal de l’Ecole Polytechnique, 9: 66–86. (Scholar)
Chen, F., Hu, Z., Zhao, X., Wang, R. and Tang, X., 2006, Neural correlates of serial abacus mental calculation in children: a functional MRI study. Neuroscience Letters, 403(1–2), 46&dash;51. (Scholar)
Chvátal, V., 1975, “A Combinatorial Theorem in Plane Geometry”, Journal of Combinatorial Theory, series B,18: 39–41, 1975. (Scholar)
Dedekind, R., 1872, “Continuity and the Irrational Numbers”, in Essays on the Theory of Numbers, W. Beman (trans.) New York: Dover Publications. (Scholar)
De Toffoli, S. and V. Giardino, 2014, “Forms and Roles of Diagrams in Knot Theory”, Erkenntnis, 79: 829–842. (Scholar)
Eddy, R., 1985, “Behold! The Arithmetic-Geometric Mean Inequality”, College Mathematics Journal, 16: 208. Reprinted in Nelsen 1993: 51. (Scholar)
Euclid, Elements, Published as Euclid’s Elements: all thirteen books complete in one volume, T. Heath (trans.), D. Densmore (ed.). Santa Fe: Green Lion Press 2002. (Scholar)
Ewald, W. (ed.), 1996, From Kant to Hilbert. A Source Book in the Foundations of Mathematics, Volumes 1 and 2. Oxford: Clarendon Press. (Scholar)
Fisk, S., 1978, “A Short Proof of Chvátal’s Watchman Theorem”, Journal of Combinatorial Theory, series B, 24: 374. (Scholar)
Fomenko, A., 1994, Visual Geometry and Topology, M. Tsaplina (trans.) New York: Springer. (Scholar)
Frank, M. and Barner, D., 2012, Representing exact number visually using mental abacus. Journal of Experimental Psychology: General, 141(1), 134–149. (Scholar)
Giaquinto, M., 1993b, “Visualizing in Arithmetic”, Philosophy and Phenomenological Research, 53: 385–396. (Scholar)
–––, 1994, “Epistemology of visual thinking in elementary real analysis”, British Journal for the Philosophy of Science, 45: 789–813. (Scholar)
–––, 2007, Visual Thinking in Mathematics, Oxford: Oxford University Press. (Scholar)
–––, 2011, “Crossing curves: a limit to the use of diagrams in proofs”, Philosophia Mathematica, 19: 281–307. (Scholar)
Gromov, M., 1993, “Asymptotic invariants of infinite groups”, in Geometric Group Theory, A. Niblo and M. Roller (eds.), LMS Lecture Note Series, Vol. 182, Cambridge: Cambridge University Press, (vol. 2). (Scholar)
Hahn, H., 1933, “The crisis in intuition”, Translated in Hans Hahn. Empiricism, Logic and Mathematics: Philosophical Papers, B. McGuiness (ed.) Dordrecht: D. Reidel 1980. First published in Krise und Neuaufbau in den exakten Wissenschaften, Fünf Wiener Vorträge, Leipzig and Vienna 1933. (Scholar)
Hatano, G., Miyake, Y. and Binks, M., 1977, Performance of expert abacus operators. Cognition, 5, 47–55. (Scholar)
Hilbert, D., 1894, “Die Grundlagen der Geometrie”, Ch. 2, in David Hilbert’s Lectures on the Foundations of Geometry (1891–1902), M. Hallett and U. Majer (eds) Berlin: Springer 2004. (Scholar)
Hishitani, S., 1990, Imagery experts: How do expert abacus operators process imagery? Applied Cognitive Psychology, 4(1), 33–46. (Scholar)
Hoffman, D., 1987, “The Computer-Aided Discovery of New Embedded Minimal Surfaces”, Mathematical Intelligencer, 9: 8–21. (Scholar)
Hu, Y., Geng, F., Tao, L., Hu, N., Du, F., Fu, K. and Chen, F., 2011, Enhanced white matter tracts integrity in children with abacus training. Human Brain Mapping. 32, 10–21. (Scholar)
Jamnik, M., 2001, Mathematical Reasoning with Diagrams: From Intuition to Automation, Stanford, California: CSLI Publications. (Scholar)
Joyal, A., R. Street, and D. Verity, 1996, “Traced monoidal categories”, Mathematical Proceedings of the Cambridge Philosophical Society, 119(3) 447–468. (Scholar)
Kant, I., 1781/9, Kritik der reinen Vernunft, P. Guyer and A. Wood (trans. & eds), Cambridge: Cambridge University Press, 1998. (Scholar)
Klein, F., 1893, “Sixth Evanston Colloquium lecture”, in The Evanston Colloquium Lectures on Mathematics, New York: Macmillan 1911. Partially reprinted in Ewald 1996: vol. 2: 958-65. (Scholar)
Kojima, T., 1954, The Japanese abacus: Its use and theory. Tokyo, Japan. Tuttle. (Scholar)
Landau, E., 1934, Differential and Integral Calculus, Hausner and Davis (trans.), New York: Chelsea 1950. (Scholar)
Leinster, T., 2004, “Operads in Higher-dimensional Category theory”, Theory and Applications of Categories, 12(3): 73–194. (Scholar)
Littlewood, J., 1953, “Postscript on Pictures”, in Littlewood’ Miscellany, Cambridge: Cambridge University Press 1986. (Scholar)
Lyusternik, L., 1963, Convex Figures and Polyhedra, T. Smith (trans.), New York: Dover Publications. (Scholar)
Mancosu, P., 2005, “Visualization in Logic and Mathematics”, in P. Mancosu, K. Jørgensen and S. Pedersen (eds), Visualization, Explanation and Reasoning Styles in Mathematics, Dordrecht: Springer. (Scholar)
–––, 2011, “Explanation in Mathematics”, The Stanford Encyclopedia of Philosophy (Summer 2011 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/sum2011/entries/mathematics-explanation/>. (Scholar)
Maxwell, E., 1959, Fallacies in Mathematics, Cambridge University Press. (Scholar)
Montuchi, P. and W. Page, 1988, “Behold! Two extremum problems (and the arithmetic-geometric mean inequality)”, College Mathematics Journal, 19: 347. Reprinted in Nelsen 1993: 52. (Scholar)
Miller, N., 2001, A Diagrammatic Formal System for Euclidean Geometry, Ph. D. Thesis, Cornell University. (Scholar)
Mumma, J. and M. Panza, 2012, “Diagrams in Mathematics: History and Philosophy”, Synthese, 186: Issue 1. (Scholar)
Needham, T., 1997, Visual Complex Analysis, Oxford: Clarendon Press. (Scholar)
Nelsen, R., 1993, Proofs Without Words: Exercises in Visual Thinking, Washington DC: The Mathematical Association of America. (Scholar)
Palais, R., 1999, “The visualization of mathematics: towards a mathematical exploratorium”, Notices of the American Mathematical Society, 46: 647–658. (Scholar)
Pasch, M., 1882, Vorlesungen über neuere Geometrie, Berlin: Springer 1926, 1976 (with introduction by Max Dehn). (Scholar)
Rouse Ball, W., 1939, Mathematical Recreations and Essays, Revised by H. Coxeter, 11^th edition. (First published in 1892). New York: Macmillan. (Scholar)
Russell, B., 1901, “Recent Work on the Principles of Mathematics”, International Monthly, 4: 83–101. Reprinted as “Mathematics and the Metaphysicians” in Mysticism and Logic, London: George Allen and Unwin 1918. (Scholar)
Shin, Sun-Joo, Oliver Lemon, and John Mumma, 2013, “Diagrams”, The Stanford Encyclopedia of Philosophy (Fall 2013 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/fall2013/entries/diagrams/>. (Scholar)
Starikova, I., 2012, “From Practice to New Concepts: Geometric Properties of Groups”, Philosophia Scientiae, 16(1): 129–151. (Scholar)
Stigler, J., 1984, “Mental Abacus”: The effect of abacus training on Chinese children's mental calculation. Cognitive Psychology, 16, 145–176. (Scholar)
Tappenden, J., 2005, “Proof style and understanding in mathematics I: visualization, unification and axiom choice”, in Mancosu, P., Jørgensen, K. and Pedersen, S. (eds) Visualization, Explanation and Reasoning Styles in Mathematics, Dordrecht: Springer. (Scholar)
Tennant, N., 1986, “The Withering Away of Formal Semantics?” Mind and Language, 1(4): 382–318. (Scholar)
Van den Dries, L., 1998, Tame Topology and O-minimal Structures, LMS Lecture Note Series 248, Cambridge University Press. (Scholar)
Weyl, H., 1995 [1932], “Topology and abstract algebra as two roads of mathematical comprehension”, American Mathematical Monthly, 435–460 and 646–651. Translated by A. Shenitzer from an article of 1932, Gesammelte Abhandlungen, 3: 348–358. (Scholar)
Zimmermann W. and S. Cunningham (eds), 1991,Visualization in Teaching and Learning Mathematics, Washington, DC: Mathematical Association of America. (Scholar)

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