Mathematics as public activity is evident in many ways. In an academic context, of course, the consummation of mathematical activity, as with every other academic discipline, is precisely publication. True, publication in learned societies, journals, and conferences is a relatively modern phenomena, beginning more or less in the
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I71'1 centuiy, picking up speed in the 19"’ century, and reaching its present scale only in the 20,h (for a review, see Barlie, 1995); still, communication of mathematical ideas in one form or another, widely or narrowly, openly or secretively, freely or selectively, has always been part of mathematical life. The qualifications just alluded to are not unimportant; nevertheless, what is essential is that the mere association between mathematical activity and communication places mathematical life within the range of community life and, to that extent, also public life. And although this is particularly true for the academy, it is also how we present mathematics to our pupils, even at an early age. Thus the NCTM Principles and Standards (2000) urges that communication as “...an essential part of mathematics and mathematics education” (p. 59) be made clear to all students, beginning with pre-kindergarten children!
Communication in mathematics and a mathematical community, not necessarily the academic community, are inseparably linked and mutually define each another:
emphasizing communication in mathematics and mathematics education is only a way of emphasizing that mathematics rests on collective activity, and, conversely, recognizing the existence of a mathematical community, even the restricted one of a mathematics classroom, is only a way of recognizing that in mathematics people speak to one another. There is no surprise here since communication and community themselves are inseparably linked. Thus, Aristotle says that it is speech, logos, that makes human beings political, creatures whose life is community life.3 But behind logos, behind dialogue, is a commerce of signs within a common shared system of signs, making speech and dialogue possible in the first place. Thus, a semiotic framework—though semiotics, in principle, takes in more than logos, even the phonai of non-human living beings, and not only their phonai (e.g., Sebeok, 1968)—-is, in fact, at the very heart of the link between community and communication and, simultaneously, their fullest expression. Indeed, this social orientation of semiotics is almost built into the very idea of semiotics. This is particularly pointed in Saussure’s (1974) approach to semiotics with its emphasis on language, which Saussure refers to as a social institution.
Whether mathematics should be thought of as itself a language, popular though that comparison may be, is, perhaps, unimportant. In his essay on the semiotics of mathematical discourse, Ard (1989) considers how symbols and words, for example, come together to produce mathematical texts: as lexis, mathematical discourse can be treated as belonging alternatively to language or to a recognizable class of concrete, visual, objects. In either case, mathematical texts are semiotic objects for which the insights and methods of Bakhtin are appropriate in the first case and those of Eisenstein in the second (Ard, 1989, p. 261).
The mention of the fiinimaker-semiolician Sergei Eisenstein is telling. Ard has in mind specifically Eisenstein Ts montage theories, but by placing mathematical texts in the context of cinema drives home the fact that a mathematical text, like a film in a theater, is an entity existing for an audience; its semiotic character—the signs that make it available for the understanding of the reader or viewer, regardless whether the signs are elements of a language or a montage of sensible images and symbols—puts it, as in a theater, in a public realm.
So, the public side of mathematics, starting with its institutional setting where it translates into the drive to publish, derives at a fundamental level from mathematics’
semiotic character. 1 wish to underline this point. Referring to the semiotic nature of mathematics does not mean speaking abstractly about its symbols, as if they were detached from the users of the symbols and the kinds of discourse engendered by them; it means seeing mathematics as visible, communal, public activity, or of the system of signs allowing such activity. As beings engaged in this kind of activity, our mathematical selves are, at this level, also public selves: from this perspective, we are what we are by how we are with and through others. For this reason, Radford (2006a) in working out this position (see also Radford, 2006b:
this volume), uses the illuminating, though slightly paradoxical, phrase, individu communauiaire or je coiwmmautaire [communal individual or communal “l"].
What has been said so far gives reason to believe that the mathematical self is a public self. But this line of thinking can be taken even further, namely, that a mathematical self does not just happen to be a public self, but that a non-public mathematical self is not even tenable. Among the most famous, and profound, arguments implying the latter conclusion is, surely, Wittgenstein’s argument against the possibility of a private language in his Philosophical Investigations (Wittgenstein, 1958).
Much can be said about this argument (and much has been said!), but let the following remark suffice. While the private language argument ostensibly refers to the problem of describing inner sensations, Kripke (1982) has pointed out that the crux of the argument lies in the nature of rules and is rooted in Wittgenstein’s paradox that no action can be determined by a rule, since some rule can be attached to any action (Wittgenstein, 1958, part I. §201). For us, the emphasis on rules is important, for it makes the application of the private language argument to an rule- driven activity such as mathematics incontrovertible. But besides bringing mathe
matics and other rule-driven activities, “language-games” to use Wittgenstein’s famous phrase,’’ squarely into the picture, seeing the private language argument in terms of rules highlights one of its main conclusions, namely, that rules only have meaning within a community: it is only within a community of-rule-users that thinking one was obeying a rule would be not be the same thing as obeying it.
Wittgenstein’s private language argument, then, suggests that private mathematical activity simply does not make sense, and with that, the idea of a private mathematical self ceases to be coherent.
To summarize, the argument for public character of the mathematical self begins with the centrality of communication in mathematical activity. This reflects the semiotic nature of mathematical activity. As such, it is activity directed towards a community, that is, an activity demanding the participation of others, and thus governed by a system of signs visible and open to others—in short, public activity'.
As creatures that engage in mathematics, then, we are public beings. Wittgenstein’s arguments further strengthen this line of thinking by suggesting that it is not even tenable to define the mathematical self as a private self.
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