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Philosophia Mathematica
, Volume 26 (1) – Feb 1, 2018

6 pages

/lp/ou_press/roi-wagner-making-and-breaking-mathematical-sense-histories-and-7qBPzMEGgT

- Publisher
- Oxford University Press
- Copyright
- © The The Authors [2018]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com
- ISSN
- 0031-8019
- eISSN
- 1744-6406
- D.O.I.
- 10.1093/philmat/nkx038
- Publisher site
- See Article on Publisher Site

Is mathematics a reflection of some already-given realm? It would not matter whether we are talking about the empirical world in a Millian way, or the domain of a priori truths in Leibnizian or maybe Kantian style, or some world of analytical truths à la Carnap. Or perhaps — could mathematics be something more, or something less, than such a reflection? Might it be human, perhaps, dependent on our kind, on our forms of life? Could it just be a set of techniques and conceptions that we develop and employ to establish order and measure in the world around us, in our activities and our environment? Consider any basic mathematical notion that you may wish: e.g., number, function, or space. There is a long history to be told of changing conceptions of number, from the rigid idea codified by Euclid to the proliferation of number systems (real, complex, transfinite, hypercomplex, surreal) that defies any crisp characterization of the word ‘number’. Similarly for conceptions of function, from the initial glimpse by Leibniz all the way to so-called ‘arbitrary’ functions around 1850, back to definable functions, and then on to notions of morphism, etc. in a process that remains open, to be continued. And the same goes for the idea of space, once restricted to a 3-dimensional world that was assumed to be God-given, but which now includes $$n$$-dimensional and infinite-dimensional spaces, Hilbert spaces, function spaces, and so on. We will encounter something similar if we pay attention to a given mathematical theory (say, basic real analysis) and try to understand some of the crucial changes that have reshaped mathematicians’ presentations of the theory. Basic analysis was presented (and understood) very differently by the promoters of the infinitesimal calculus, and new alternative ways of understanding it keep being proposed. In fact one must admit that relevant mathematical theories keep being reshaped and reconceived in an open-ended process. This is why Cavaillès liked to say, almost a century ago, that mathematics is a becoming: “la mathématique est un devenir”. This phenomenon should be well-known to anyone seriously interested in mathematics, and the call for serious study of mathematical practice implies the need to engage with that central dynamic aspect of mathematical work. The more standard viewpoints, the classical approaches to the philosophy of mathematics, most of the philosophical elaborations around mathematics published in the English language, are perfectly orthogonal to such a concern. This is the contrast between philosophy of mathematics focused on ‘what it is’ (statically understood) and philosophy of mathematical practice focused on ‘how it works’. Yet even for those of us who favor the study of mathematical practice, there is a danger of falling into staticism: e.g., if we concentrate on the most central structures of mathematics and the long-lasting questions and problems (as I have done myself [Ferreirós, 2016]) a lot of mathematical activity may escape and disappear from our focus. As is obvious from the title itself, Roi Wagner’s Making and Breaking Mathematical Sense sets out to explore and emphasize such dynamic processes. In his view, mathematics is centrally configured as an exploration of sign systems and their shifting interpretations. Thus, the central topic of this work is signs and their changing meanings as they emerge through contingent semiotic and interpretive processes, above all the mathematical processes of sense making. Wagner’s approach will be perceived as radical by many readers, but it is not thoughtless: the semiotic processes are not pictured as living their own lives, but rather seen to work in the context of many constraints. Mathematical practice, according to Wagner, is under many different kinds of constraints — internal, natural, cognitive, and social; and all of them condition the ‘contingent’ semiotic and interpretive dynamics of the discipline. Chapter 1 is a brilliant introductory commentary on the canonical philosophies of mathematics, presented in terms of metaphilosophical debates behind the more obvious ones: tension between the idea that mathematics reflects an assumed natural order, and conceptual freedom; whether mathematics has an intermediary constitutive epistemological position or not; different ways of handling the emergence of ‘monsters’;1 and questions of authority (including political and religious) versus creative freedom. Wagner is inclined to see all of those as productive tensions, and he does not side with any of the classical viewpoints — hence his preference for a philosophy which accepts “all of the above” (p. 38). But the author’s constraints-based philosophical position is delineated in Chapter 3; we come back to this in a moment. The author combines good mathematical training with philosophical sensibility and a serious engagement with historical developments. The cases he discusses range from ancient mathematics to contemporary work, and he moves with agility from case to case, making the discussion lively and engaging. Perhaps most noteworthy is the amount and quality of material devoted to a discussion of changing conceptions of algebra and its relationship to geometry. Indeed this constitutes an excellent field of study for Wagner’s semiotic approach. Chapter 2 features a very interesting discussion of the early emergence of algebraic signs and ideas: particularly striking is the argument that the sign of the unknown did not have to be invented, but was appropriated from economic practices. In Chapter 2 we also see mathematicians of former times handling monsters like the negative numbers, developing flexible algebra systems, applying changing and often inconsistent criteria to them, interrelating arithmetic and geometry. The discussion of algebra is continued in the first part of Chapter 6, focused on the complex interplay between algebra and geometry, with a solid discussion of authors as diverse as Khayyam, Descartes, and Bombelli. The general idea is that mathematics has a semi-autonomous and constitutive position, even though reacting to all sorts of circumstances — most importantly, the constellation of constraints that shape mathematics. The human capacity to obtain consensus by establishing and following rules becomes the basis for practices and norms, including mathematical rules (p. 66). In so far as this is a matter of dismotivated rules,2 the enterprise may seem analytic; but since it arises in reaction to constraints and forms of life, it is a posteriori: “Mathematics as a whole is much too dependent on empirically grounded semiotic activity to be confined to the realm of the a priori” (p. 67). In the case of contemporary mathematics, Wagner accepts that consensus about validity reaches an incredible degree, compared not only to humanities but also the sciences, thanks in particular to the standard of ‘syntactic validity arbitration’ (formalization). One of the usual traditional ideas is that mathematics is a fully autonomous discipline. Those who disagree have tended to emphasize the role of astronomy and physics as the context for many key ideas and results. But it is unusual to emphasize economic life as context, even though it is rather obvious that mathematics has been a crucial part of many human techniques or technologies, including the handling of economic interactions. This becomes quite central to Wagner’s message, since he conceives mathematics to have a formative position, more precisely, an intermediate constitutive epistemological role (an idea that Kant pursued in his own way). Mathematics does not merely ‘reflect’ an independent reality (be it platonistic or empirical), it makes a crucial contribution to shaping reality in human ways of doing, and through them in the world. But, of course, such a capacity is not unconditioned; it works under constraints as we have already mentioned. What are those constraints? Mathematics is learned and developed by humans in relation to their concrete forms of life; so there are obvious cognitive constraints, but also inter-communal constraints, and empirical constraints in all kinds of practical applications. Wagner insists on the obvious reality of mathematical institutions and practices (the sociological side of things) while acknowledging a continuum of natural-to-social constraints which strongly condition mathematical statements and objects. These many constraints, he argues, cannot be neatly disentangled into purely internal versus external (social). And he writes: Instead of trying to pull the conceptual blanket of existence and truth across too many domains, the philosophy of mathematics might want to study the specifics of the contingent pools of constraints that make mathematical objects and statements non-arbitrary and sometimes even uniquely determined. This approach would allow us to compare mathematical truth and objectivity constraints with those imposed on other, not necessarily scientific, branches of knowledge. (p. 90) Finally we have what may well be the central notion in Wagner’s constraints-based semiotic philosophy of mathematics: interpretation. This is central to the whole treatment: mathematics is interesting and useful because it is interpreted; in practice we always work with some inherited interpretations, but mathematical interpretations are in flux (p. 77). Mathematicians often work at once with formal syntactic approaches and with several semantic interpretations, in an ambiguous and sometimes even contradictory way (for an example, see the discussion of variables in generating functions, Chapter 4). Tensions between interpretations are the source of shifts of meaning and mathematical innovations. Interpretation always comes to an end in our concrete practices, but it is always provisional; the process is open-ended, and Wagner argues that this is not a marginal aspect, but rather the core of mathematical activity (p. 81). He elaborates his philosophical ideas about this central issue in dialogue with philosophers such as Peirce and Wittgenstein, but also Derrida and Deleuze. In fact, his theoretical preference is for a post-structuralist approach incorporating ideas such as the “iterable sign” in a limitless open-ended process (Derrida) and the notion of “haptic vision” (Deleuze). Nevertheless, references to such ideas have been intentionally reduced to a minimum in this book, which can be read independently of questions about post-structuralism. Judged from this point of view, a key difference between contemporary mathematics and more classical forms of mathematics lies precisely in the increment of semantic complexity. While classical mathematics stressed some basic structures (conceived to be univalent or rigid, like $${\mathbb R}$$) and the univocality of classical logic, contemporary mathematics has opted for a complex plurality of structures and categories, and associated logics. Both options make sense as different phases in the open-ended process of interpretation that is at the core of mathematics, but Wagner adds the following explanation for the contemporary setting: the stronger the authority of formal-syntactic arbitration, the more its semantic fluidity. A sizable part of this book is devoted to a discussion on cognitive science ranging from studies of the so-called number sense (Dehaene, Walsh) to alternative integrated analyses of brain dynamics (Freeman), but with a special emphasis on the theory of conceptual metaphors. In fact, Chapter 6 on mathematics and cognition is the longest in the whole book. The detailed discussion here goes beyond the better-known ideas, to reveal problems of formulation and interpretation that haunt the empirical discussion on neural cognition. For example, the work of W.J. Freeman III (a brain scientist expert in olfactory systems) rejects the basic assumptions of representationalist cognitive neuroscience, and instead of looking for microsystems encoding perceptual features, considers patterns of action in bigger systems (complex loops of interconnection and not just one-directional processing, semi-autonomous activity, preafference). Wagner elaborates a parallelism — which may be problematic — between such ideas and aspects of mathematics. Nevertheless, the question he deals with in greatest detail is conceptual metaphor theory (Lakoff & Núñez) and its deficiencies as an analysis of metaphors in the practice of mathematics. This is just natural, given Wagner’s interest in the web of semiosis, in mathematical signs, their meanings and their changing interpretations. His approach leads naturally to an emphasis on semantic changes, reinterpretation and metaphor, but it deals with these issues in a way that conflicts with the well known work of Lakoff and Núñez [2000]. In a detailed and very critical discussion, Wagner argues that their “concept of metaphor is far too thin and rigid to account for mathematical practice”. Three main criticisms against it are given, challenging first the notion of ‘conceptual domain’, second the directionality of the theory (concrete $$\rightarrow$$ abstract), and third its dependence on an abstract formal reconstruction of mathematics. While acknowledging the importance of metaphors (and semiosis) and the relevance of “transfer of ideas between domains” in mathematics (p. 150), the author suggests that we should break with Lakoff and Núñez’s rigid and too strongly formal-mathematical conception of metaphors as “a grounded, inference-preserving cross-domain mapping”, and opt instead for regarding metaphors as a “relative articulation across mathematical contexts” which is based not only on embodied experience but on practice with signs and tools (p. 155). Articulating one context in terms of another is a flexible process which implies many things beyond inferences, and involves much more than just neural networks. In sum, this is an interesting book that should help refine and improve current ways of analyzing mathematical practices. Wagner’s detailed work is rich in historical material, sophisticated in point of methodology, and will be found challenging by most philosophers of mathematics. He encourages us to avoid preconceptions that narrow our perspective on mathematics and its role (not only in science, but in social life) — and to consider that philosophy’s role is not merely to offer unimpeachable analyses or justifications but to reconsider established practices, and in the process, perhaps, to open up new avenues for human life. I have not discussed the case studies from combinatorics (in Chapter 4), but I should mention that there the author underlines the need for scientists to “assume responsibility for their language” and the ethical and political connotations of their work. This is not merely a matter of humanist concerns: it is a natural outcome of the view that mathematics does not merely reflect a ‘given reality’, but rather, through its constitutive role in science and life, mathematics has the effect of shaping our reality. Thus, a more reflective mathematics also has the potential for changing reality. Footnotes 1 Living with monsters such as irrationals and negatives (as practical mathematicians did), barring them (as the scholarly elite tended to do), or taming them. 2 Wagner follows Wittgenstein in this notion of dismotivation: first something is observed as empirically correct, next it becomes a rule that no experience can refute — a reference against which to measure experience. REFERENCES Ferreirós, José [ 2016]: Mathematical Knowledge and the Interplay of Practices . Princeton University Press. Google Scholar CrossRef Search ADS Lakoff, G., and Núñez R.E. [ 2000]: Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being . New York: Basic Books. © The The Authors [2018]. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oup.com

Philosophia Mathematica – Oxford University Press

**Published: ** Feb 1, 2018

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