Chapter 12: Appreciating the Embodied Social Nature of Mathematical Cognition

In the introductory quote to this chapter, Eric Livingston, who in his doctoral dissertation has studied a pair of mathematicians doing Gödel’s proof (Livingston, 1986), notes the distraction that occurs as part of the disattention to the “ordinary circumstance and aim of provers’ activities.” That is, the way in which mathematics is depicted, often in the transcendental properties of its objects that are said to exist independent of the material properties of the proof accounts, we do not see the embodied work that mathematics consists of (but see Núñez’ chapter 11). In fact, the study of the mathematical objects, such as a proof as it appears on paper or whiteboard, cannot ever lead us to the process that brings it about (Henry, 2000). This process or work is invisible in the description of mathematical cognition. That is, there are actually two parts to mathematical proof. On the one hand, there is a proof account, which exists in and through the communicative devices that make the proof available to others (e.g., in a journal). On the other hand, there is the lived work of proving, without which no proof exists. This embodied work is required not only of the person who does some proof for the first time, but also on the part of the reader, in participative understanding, who reproduces and transforms the lived work of proving. This lived work not only is embodied, it is social as well, because directed toward the (anonymous) other who, in his or her own lived work, actually reifies the independence of the proof from the material particulars in which it is presented. But a proof is a proof only if another person, using the materials at hand, reproduces the proof account in and through his or her own lived work, which, as such, becomes an expression of mathematical actions that another recognizes as his or her own. Mathematics is both embodied and social simultaneously, and the two aspects, the embodied work and the proof account produced, constitute a Lebenswelt (life-world) pair (Lynch, 1997). Mathematics, therefore, becomes transcendental precisely when its objects, the proofs, are detached from the actual details of doing proving. In contrast to the contention of some who continues to consider mathematics as a body of transcendental phenomena, the position I have been taking throughout this book to organize the chapters is one in which mathematics simultaneously is embodied and social.