Modern mathematics has frowned on the use of diagrams in proofs. However, times seem to be changing. The mathematician M. Harris, e.g., recently noted that diagrams are accepted in proofs in an arena of advanced mathematics. A relevant philosophical question, then, is: can diagrams be used to justify mathematical propositions? This question becomes more pressing in the light of the emergence of sophisticated image-generation software, making visualization a powerful tool in some fields of mathematics.
Norman provides a framework to address the issue. He assesses four basic philosophical explanations of the reasoning involved in Euclid’s I.32 (the internal angles of a triangle add up to two right angles). These four are the Inductive View and the views of Kant, Leibniz, and Mill. While each is found wanting, the Kantian view is favoured. A rough sketch of Norman’s assessment follows.
In the Inductive View, justification of a proposition consists of empirical generalization from the appearance of one or more diagrams. This does not capture the contribution of (nonvisual) deductive inference in conscious experience, a key reason for Norman’s negative assessment of this stance. (See pp. 59-60.)
For J. S. Mill, empirical generalizations justify the axioms and definitions of mathematics, while reasoning from these starting points is deductive. Norman argues empirical generalization does not seem to yield conscious certainty in the axioms because progressive refinement is required. In the case of “equals added to equals are equal,” a Common Notion used in I.32, one needs to assume, e.g., that counted objects do not reproduce or coalesce. The need for progressive refinement of axioms suggests that empirical evidence for the axioms is not as overwhelming as Mill supposed (Norman, p. 70). Regarding definition, Norman argues that the definition of a triangle in Euclid is a stipulation rather than an empirical generalization (p. 71).
In Leibniz’s view, mathematical reasoning is purely deductive. For this view to capture the experience of Euclid’s I.32, Norman argues, there would need to be a purely deductive proof of I.32 which is still faithful to Euclid ’s proof. This seems unlikely because some parts of Euclid’s thinking require diagrammatic inference (p. 86).
In Kant, Norman notes, justification of Euclid’s I.32 is independent of experience, including visual experience. However, the visual experience of the diagram can help to justify the proposition in this sense: seeing the diagram can prompt us to apply correct construction procedures (p. 103). In this way, our visual experience of a diagram can contribute to, e.g., justification which pertains to all triangles. Norman remarks:
The reasoner ‘constructs’ a concept of a triangle by drawing or visualizing a triangle. However, in doing so she is merely implementing the relevant construction procedures for triangles… Although the drawn diagram is determinate in respect of the size of its sides and angles, the reasoner does not take these properties of the diagram to restrict the class of triangles that she takes the diagram to represent. (p. 92)
It is construction procedures that determine what the diagram represents. They allow a diagram to represent a general situation, quite apart from the empirical appearance of the diagram.
[F]or Kant, the justification provided by Prop. I.32 does not derive from mere generalization of the visual awareness of a particular diagram or image. It is the knowledge of which figures can and cannot be constructed from a given sequence of construction procedures that constrains the generality and applicability of a given geometric property, and so provides the warrant for claims about such properties. (Norman, p. 99)
Thus, for Kant, the inferences involved in Euclid’s proof apply to all triangles, not just the depicted triangle.
Norman holds that Kant captures the experience of using the diagram in Euclid I.32. However, not fully satisfied with Kant’s general doctrine of intuition (of which the idea of construction procedures is a part), Norman develops a Neo-Kantian position. For this purpose he draws on recent philosophy of perception.
Kant’s doctrine of intuition and Norman’s alternative are both complicated, making for sometimes difficult reading. This book is a valuable contribution to the discussion about the use of diagrams in mathematics. The wide philosophical scope of Norman’s book is of particular value.
Harris, M. “‘Why mathematics?’ you might ask”. Online at http://www.math.jussieu.fr/~harris/ . To be published in the The Princeton Companion to Mathematics.
Dennis Lomas (email@example.com) has studied computer science (MSc), mathematics (half dozen, or so, graduate courses), and philosophy (PhD). He resides in Prince Edward Island (Canada).