A Historically and Philosophically Informed Approach to Mathematical Metaphors

Abstract

This article discusses the concept of mathematical metaphor as a tool for analyzing the formation of mathematical knowledge. It reflects on the work of Lakoff and Núñez as a reference point against which to rearticulate a richer notion of mathematical metaphor that can account for actual mathematical evolution. To reach its goal this article analyzes historical case studies, draws on cognitive research, and applies lessons from the history of metaphors in philosophy as analyzed by Derrida and de Man.

Acknowledgements

I would like to thank Eitan Grossman, Lin Haluzin-Dovrat, and Rachel Giora for their valuable comments and discussion. I thank two referees of this journal for their criticism.

Notes

[1] For an earlier articulation of mathematical metaphors within the same theoretical field, see Sfard (1995).

[2] The analysis refers to the specific case studies presented, and does not necessarily reflect the overall approach of the relevant mathematicians. I shuffled the chronological order of the examples in favour of the logical build-up of the analysis. Wherever I used anachronistic notations, I used them in a way that will not tamper with my analysis here.

[3] There's an issue here of whether such transfer respects the concrete-to-abstract directionality of Lakoff and Núñez's metaphors, but we will set this concern aside for now.

[4] Moreover, this blend does not fall under Lakoff and Núñez's ‘fundamental metonymy of algebra’ (Lakoff and Núñez 2000, 74–75), where x is a variable that represents any possible value of a variable. Here, a determined sign (a line of a given length) represents a specific unknown line, just as x represents a specific unknown value.

[5] See also Grosholz (2007) for an example of a non-fixed blend in the work of Leibniz and a general discussion of ‘useful ambiguities’; see Fisch (1999) for another example in the work of Peacock. Practically everything I have ever written on mathematics is about ambiguous correspondences.

[6] Arithmetic/algebra designates arithmetic expanded by the explicit use of an unknown, as in the examples above. This is more than arithmetic, but less than a variable-based algebra of the kind assumed in Lakoff and Núñez's basic metonymy of algebra quoted above.

[7] Some vague evidence for this comes from classical descriptions of the Pythagorean tradition. Explicit evidence appear in the work of Hero and Ptolemy—late evidence indeed, but, according to Netz (2004), representing the earliest surviving written forms of a much older tradition.

[8] One may suspect that Mowatt and Davis, whose model is not quantified enough to be discussed in such terms, fell victim to a trend of referring to too many complex phenomena as ‘scale free’. People with strong mathematical background are referred to Clauset, Shalizi, and Newman (2009) for a statistical methodological critique of scale freeness; a more accessible analysis of scale freeness is provided by Keller (2005).

[9] Cf. Mason's (2010) note on the role of cards in embodying a notion of permutations and Klein's (2003) notion of ‘paper tools’.

[10] Pointwise convergence means that the distance between any point of the limit line and the point immediately above it on the curve converges to zero as the curves proceed in their sequence. Uniform convergence means that the maximal such distance for a given curve converges to zero as the sequence of curves proceeds.

[11] In C¹ convergence, both the curves and their derivatives converge uniformly to the limit curve and its derivative respectively.

[12] For the gender aspect of controlling metaphors, consider the following comments on Locke by de Man:

It is clear that rhetoric is something one can decorously indulge in as long as one knows where it belongs. Like a woman, which it resembles (‘like the fair sex’), it is a fine thing as long as it is kept in its proper place. Out of place, among the serious affairs of men (‘if we would speak of things as they are’), it is a disruptive scandal—like the appearance of a real woman in a gentlemen's club where it would only be tolerated as a picture, preferably naked (like the image of Truth), framed and hung on the wall. There is little epistemological risk in a flowery, witty passage about wit like this one, except perhaps that it may be taken too seriously by dull-witted subsequent readers. But when, on the next page, Locke speaks of language as a ‘conduit’ that may ‘corrupt the fountains of knowledge which are in things themselves’ and, even worse, ‘break or stop the pipes whereby it is distributed to public use’, then this language, not of poetic ‘pipes and timbrels’ but of a plumber's handyman, raises, by its all too graphic concreteness, questions of propriety. Such far-reaching assumptions are then made about the structure of the mind that one may wonder whether the metaphors illustrate a cognition or if the cognition is not perhaps shaped by the metaphors. (de Man 1978, 15–16).

The conduit metaphor is indeed one of those most favoured by Lakoff (1992, 204).

[13] Sometimes there is even discussion of un/natural vs. other usage, e.g. the ‘naturally continuous line’, as opposed to discretized versions (Lakoff and Núñez 2000, 289) or natural (finite) vs. metaphorical (transfinite) numbers (Núñez 2005).

[14] Another aspect of controlling metaphors is to minimize the set of metaphors required to explain mathematics. Lakoff and Núñez's main ‘universal’ metaphor is the basic metaphor of infinity, which I discuss in Wagner (2012). Ernest insists that ‘there is not a single, uniquely defined semiotic system of number, but rather a family of overlapping, intertransforming representations constituting the semiotic systems of number’ (Ernest 2006, 94). The same, I argue, goes for mathematical infinities.

[15] This may be explained as follows. At one point in his article, Derrida (1974, 26) questions the very possibility of properly mathematical metaphors. Arkady Plotnitzky, in an unpublished discussion with Barry Mazur on mathematical metaphors, rightly interpreted this claim as arguing that it is impossible to set a properly mathematical grounds for metaphor—namely that the metaphors of mathematics can always be pursued down to other metaphors in and outside mathematics. This open-endedness of mathematical metaphors prevents protecting a supposed cognitive ‘outside’ from their intervention.

[16] The main references range from de Saussure's Course in General Linguistics to Lévi-Strauss's Structural Anthropology. A special place should be reserved for Lacan's analysis in ‘The Instance of the Letter in the Unconscious’ (Lacan 2005, 412–444). A good starting point is Barthes (1971).