## Abstract

This paper explores how mathematicians build meaning through communicative activity involving talk, gesture and diagram. In the course of describing mathematical concepts, mathematicians use these semiotic resources in ways that blur the distinction between the mathematical and physical world. We shall argue that mathematical meaning of eigenvectors depends strongly on both time and motion—hence, on physical interpretations of mathematical abstractions—which are dimensions of thinking that are typically deliberately absent from formal, written definition of the concept. We shall also show how gesture and talk contribute differently and uniquely to mathematical conceptualisation and further elaborate the claim that diagrams provide an essential mediating role between the two.

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## Notes

The distinction between

*ideational*and*conceptual mathematics*resembles Tall and Vinner’s (1981) distinction between concept image and concept definition, respectively.This observation, made by Mariolina Bartolini-Bussi (personal communication), can be seen in Archimedes’ and Cavalieri’s development of infinitesimals, as well as Desargues’s founding of projective geometry.

Indeed, based on her interviews with over 70 mathematicians, Burton (2004) identifies three styles of thinking of mathematicians, including one that she calls visual, and to which she adds (parenthetically), sometimes dynamic. Her analyses focus entirely on verbal utterances made by mathematicians, and unfortunately, she provides no examples of what she counts as a visual, dynamic style.

Furthermore, there are several examples of blind people doing mathematics who seem to draw on kinetic ways of thinking. For example, Helen Keller (1969), who was both deaf and blind, described straight lines as “I feel as if I were going forward in a straight line, bound to arrive somewhere, or go on forever without swerving to the right or to the left”. Also, Healy (2009) reports a blind student conceptualising a pyramid in terms of a gesture that involves wrapping his fingers around the base and tracing up the sides until his fingers meet at a point.

For example, Courant and Robbins (1978) write “The hyperbola approaches more and more nearly the two straight lines

*qx*±*py*= 0 as we go to farther and farther from the origin, but it never actually reaches these lines. They are called the asymptotes of the parabola” (p. 76). In using the active verbs in the following phrases “approaching more and more” and “go out farther and farther” and “reaches”, the authors seem to be describing moving lines and changing distances.We included commutativity (along with area, distance and symmetry) so as to have examples of concepts that do not point back in time to a spatiotemporal conception—this based on Harris’s (2008) critique of Núñez’s work which used limit and continuity, both arguably already imbued with temporal conceptions,

Ochs, Gonzales, and Jacobyet (1996)) note a similar phenomenon with physicists who, even in their spoken language, associated themselves with the physical phenomena under consideration, for example: “When I come down I’m in the domain state.”

He also sees the metaphor as another mediating link between the body and mathematics.

See also Sinclair (2010) for a discussion of the role of “covert” thinking in mathematics, which includes not only the surface intuitions mentioned by Châtelet, by also the more subconscious (irrational?) ways of knowing hinted at by Weil.

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## Acknowledgements

This research has been supported by the Social Sciences and Humanities Research Council of Canada. We would like to thank the mathematicians who participated in this study. We would also like to thank David Pimm for his feedback on previous versions. We extend our acknowledgements as well to three anonymous reviewers and to Norma Presmeg for her guidance in our revisions.

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Sinclair, N., Gol Tabaghi, S. Drawing space: mathematicians’ kinetic conceptions of eigenvectors.
*Educ Stud Math* **74**, 223–240 (2010). https://doi.org/10.1007/s10649-010-9235-8

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DOI: https://doi.org/10.1007/s10649-010-9235-8

### Keywords

- Language
- Metaphor
- Gesture
- Diagram
- Motion
- Time
- Conceptual mathematics
- Linear algebra