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