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Intuition and Visualization in Mathematical Problem Solving

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Abstract

In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in mathematical practice. Then, I will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization.

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Notes

  1. For example, at the dawn of set theory, Cantor discusses how to access transfinite numbers. According to him, we get to Cantorian sets operating a double act of abstraction from sets of concrete things. The first act of abstraction brings us to the ‘ordinal number’ or enumeration; the second act of abstraction brings us to the cardinal number or power of the same set. The cardinal number of M, then, is the general concept that arises from the aggregate M by means of our active faculty of thought. It is thanks to this faculty that we can abstract, and, because of that, provide definitions. By abstracting, we obtain a whole Einheit (“Unity”) of undifferentiated Einsen (“Ones”): according to Cantor, these are ‘objects of our intuition’. See Cantor (1915).

  2. According to Hadamard, both the preparation and the illumination are mostly subconscious. Nevertheless, he does not deny that conscious thinking is necessary. In fact, once this unconscious illumination has occurred, it must be verified by means of conscious thinking. Intuition allows the mathematician to see the conclusion; then, it is only afterwards that this conclusion will be proved by traditional means. See Hadamard (1945).

  3. According to Gödel, in physics as well as in logic, we are able to describe, and in fact we do describe, the ultimate reality of things. This happens because we access this nature by means of some immediate capacity: by perception in the case of physics, and by intuition in the case of mathematics. It is mathematical intuition that provides mathematical content. The analogy between perception and intuition can be pushed further. Like perception, intuition is fallible: we can fail in our attempts to get to know the abstract world we are facing. This may mean that further and new intuitions are needed. Therefore, axioms are analogous to physical laws, since it is by means of them that we gain knowledge of the relationships among ‘things’, and we expect experiences to occur in accordance with what these laws prescribe. See Gödel (1986).

  4. Gödel (1986, p. 268).

  5. Barwise and Etchemendy (1996, p. 3).

  6. Ibid.

  7. Shin (2004).

  8. Shin (1994).

  9. Mancosu (2005, p. 23).

  10. Mancosu (2001).

  11. Rota (1997, p. 191).

  12. ‘Proofs without Words’ is the title of Nelsen (1997, 2001).

  13. Feferman (2000).

  14. Klein (2004, p. 202).

  15. Bråting and Pejlare (2008).

  16. Polya (1945).

  17. See Giardino and Piazza (2008), Ch. III.

  18. Netz (1999).

  19. Mancosu (2005, p. 26).

References

  • Barwise J, Etchemendy J (1996) Visual information and valid reasoning. In: Allwein G, Barwise J (eds) Logical reasoning with diagrams. Oxford University Press, Oxford, pp 3–25

    Google Scholar 

  • Bråting K, Pejlare J (2008) Visualizations in mathematics. Erkenntnis 68(3):345–358

    Article  Google Scholar 

  • Cantor G (1915) Contributions to the founding of the theory of transfinite numbers (transl: Jourdain PEB). Dover Publications, New York

  • Feferman S (2000) Mathematical intuition vs. mathematical monsters. Synthese 125:317–332

    Article  Google Scholar 

  • Fischbein E (1987) Intuition in science and mathematics. Kluwer, Dordrecht

    Google Scholar 

  • Giardino V, Piazza M (2008) Senza Parole. Ragionare con le Immagini. Milan, Bompiani

    Google Scholar 

  • Gödel K (1986) Collected works. Oxford University Press, Oxford

    Google Scholar 

  • Hadamard J (1945) An essay on the psychology of invention in the mathematical field. Princeton University Press, Princeton

    Google Scholar 

  • Klein F (2004) Elementary mathematics from an advanced standpoint (transl: Hedrick ER, Noble CAM). Courier Dover Publications. The first German edition is 1908

  • Mancosu P (2001) Mathematical explanation: problems and prospects. Topoi 20:97–117

    Article  Google Scholar 

  • Mancosu P (2005) Visualization in logic and mathematics. In Mancosu P, Jørgensen KF, Pedersen SA (eds) Visualization, explanation and reasoning styles in mathematics. Springer-Verlag, Berlin, pp 13–30

    Chapter  Google Scholar 

  • Nelsen R (1997) Proofs without words: exercises in visual thinking. The Mathematical Association of America, Washington, DC

    Google Scholar 

  • Nelsen R (2001) Proofs without words II: more exercises in visual thinking. The Mathematical Association of America, Washington, DC

    Google Scholar 

  • Netz R (1999) The shaping of deduction in Greek mathematics: a study in cognitive history. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Polya G (1945) How to solve it. Princeton University Press, Princeton

    Google Scholar 

  • Rota GC (1997) The phenomenology of mathematical proof. In: Kamamori A (ed) Proof and progress in mathematics. Kluwer, Dordrecht, pp 183–196

    Google Scholar 

  • Shin S-J (1994) The logical status of diagrams. Cambridge University Press, Cambridge

    Google Scholar 

  • Shin S-J (2004) Heterogenous reasoning and its logic. The Bullettin of Symbolic Logic 10:86–106

    Article  Google Scholar 

Download references

Acknowledgments

I thank Roberto Casati, Davide Crippa, Leon Horsten, John Mumma, and Mario Piazza for their useful suggestions on a first draft of the article. The research was supported by the European Community’s Seventh Framework Program ([FP7/2007-2013] under a Marie Curie Intra-European Fellowship for Career Development, contract number no. 220686—DBR (Diagram-based Reasoning).

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Correspondence to Valeria Giardino.

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Giardino, V. Intuition and Visualization in Mathematical Problem Solving. Topoi 29, 29–39 (2010). https://doi.org/10.1007/s11245-009-9064-5

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