Explanatory power of visual proofs

Joachim Frans

Vrije Universiteit Brussel

Mathematics seems to have a special status when compared to other areas of human knowledge. This special status is linked with the role of proof, that is a formal argument allowing a unique level of certainty and leaving no room for unclarity. Nonetheless, mathematicians use other types of argumentation as well. Indeed, the ubiquity of computerized, intuitive, visual, analogical or metaphorical arguments in mathematical practice has been well documented. These types of argumentation are more informal, do not share the same level of certainty, and their use for justificatory means is controversial. One might therefore investigate what specific role these various non-formal methods play in mathematical practice. A possible answer to the above question could be that (some of) the non-formal methods referred to (sometimes) exhibit explanatory power. There is a growing consensus that explanation in mathematics exists. The main idea is that mathematical activity is not merely driven by justificatory aims, such as the collection of mathematical truths. For example, in many cases mathematicians will search for alternative proofs of known results in order to find a (better) explanation of the theorem. Only a few authors have tried to explicate the nature of explanatory proofs. The most well-known are Steiner (1978), introducing the notion of characterizing property, and Kitcher (1989), arguing that his unificationist account of explanation captures both scientific and mathematical explanation. As mentioned above, one of the ways mathematicians deviate from formal reasoning is by using visualizations or diagrams. Certain visualizations are specifically intended to represent a mathematical theorem in a purely visual manner, with limited or no textual guidance. Philosophical discussions have focussed on the question whether such visual representations are merely adjuncts or parts of proofs, or can be an independent method of justification. One of the most profound defenders of the latter view is Brown (1999), who argues that visualizations and traditional arguments both equally prove a theorem. Rather than further discussing the notion of proof, we want to analyze the potential explanatory value of these visualizations. We start from one of the dominant views on explanation in general, namely that it increases our understanding of a phenomenon by answering an explanation-seeking question. Proofs or other mathematical arguments can consequently only be parts of answers, they do not answer to such a question on their own. They can however provide information that is useful in order to answer such an explanation seeking question. Further analyzing mathematical explanation entails, in this view, clarifying what information we can get from mathematical proofs and arguments, which questions are explanation-seeking and how we identify satisfactory answers. In the case of visual proofs, given the absence of a written proof, a person analyzes the visualization in order to see what mathematical idea it conveys and how one could begin to prove this. We are interested in the heuristics of this specific activity. More precisely, we want to see whether it also entails formulating and answering certain explanation-seeking questions.