Elizabeth de Freitas and Nathalie Sinclair have written an admirable and provocative book. Ambitious, original, and theoretically accomplished, its purpose is to develop a new materialist approach, what they call ‘inclusive materialism’, to the learning of mathematics; one that includes and foregrounds the activity of the body, against the long-standing mentalist conception of mathematics as an activity of pure, abstract thought. Extending the current turn to materialism in philosophy and the humanities to mathematics, they reject Kantian-based epistemological schemes that understand knowledge as perception filtered through internal *a priori* intuitions and the conceptual categories, in favour of a more Humean, empiricist approach giving primacy to external sensation; ontologically they reject Platonic realism, the belief that mathematical objects – points, numbers, lines, and so on — are immaterial entities that exist in some Platonic heaven, ‘out there’, beyond time, space, matter, with mathematical activity consists of discovering truths about them analogous to scientists studying external reality. Despite numerous critiques — the chief of which asks: how can material beings make contact with things in a transcendent heaven — this metaphysical idealism remains the conventional belief, defended and widely embraced by mathematicians and others.

*Mathematics and the Body* is directed to mathematics educators and validates as well as explicates its ideas by critically examining a series of experimental classroom lessons, designed by the authors and by others, which focus on fundamental mathematical concepts such as number, parallelism, circles and diagrams. As they observe, the issue of embodied mathematics in education is topical. In the last decade or so a growing number of differently oriented initiatives – cognitive, phenomenological, enactive, communication-based approaches – have been devoted to examining the role played by students’ bodies; their gestures, hand, eye, and limb movements, their verbalizations, their drawings and diagrams, and their relation to the tokens, devices, physical objects and surfaces with which they interact. The book aims to explore the assumptions and consequences of this work. To do so, and go beyond it, they pose and confront the fundamental question: How are the “physical aspects of mathematical activity – be it that of students or mathematicians – transformed into the so-called abstractions and generalizations of formal mathematics?” Their answer involves formulating a new, extended notion of ‘body’ and correlatively a material understanding of mathematical concepts such a body engages with. The inclusion of mathematicians’ physical activity in their question indicates a possible parallel, between the creation of mathematics and its re-creation by students in the classroom; a link, that is, between the history of mathematics and the learning of it. Such is indeed the case as is evident in their opening sentence: “The idea for this book began as we read Gilles Chatelet’s (2000) stunning book on the history of mathematics, which challenges many long-standing as well as contemporary philosophies of mathematics.” Chatelet’s book,* Figuring Space*, opens up several key moments in the historical development of the subject*, *demonstrating how the interrelation of gesture – resulting from “disciplined movements of a body “ — and physical diagrams operate at the heart of mathematical invention. Sinclair and de Freitas embrace Chatelet’s linking of gestural bodies and formal abstractions and work to import it into the mathematics classroom.

But before they can accomplish this they need to establish the nature of embodiment. “When”, they ask, “does a body become a body?”. A survey of the mathematical embodiment literature finds them critical of approaches that fail to escape the “dualistic tradition of the mind/body split”; or “demote the body as the vessel or container of some higher act of cognition”; or “centre will or intention in the orchestrating of experience” assuming the human body to be “the principal administrator of its own participation”. Moreover, locating knowing and agency in the individual body does not adequately address the collective social body. Where, then, are the boundaries of a body? Against the common-sense view that “the body is an individual discrete entity and that cognition occurs within its borders”, the authors turn to posthumanist discourses of subjectivity and agency according to which subjects are dynamic assemblages of dispersed social networks, and that the “human body itself must be conceived in terms of malleable borders and distributed networks”; that is, a body understood as a “set of material relations that seems to structure the other material relations around it” In the classroom, as they illustrate in their analyses of students’ activities, such an assemblage-body will be composed of “humans, writing implements, writing surfaces, texts, desks, doors, chips, as well as disciplinary forces and habits of control and capitulation”. A consequence of conceiving the body in this way is that agency and thought becomes distributed across multiple sources in the student’s physical and psycho-social environment. Thus, analogous to Nietzsche’s insistence on ‘deeds without a doer’ one can have ‘thoughts without a thinker’ in the sense that the source of thought can come from material relations outside or beside the isolated thinking self; a phenomenon Gilles Deleuze, whose materialist ideas exert a profound effect on the authors’ project, calls the ’exteriority of thought’. In short, the power and efficacy of a body in relation to mathematics must be understood as distributed across an assemblage of heterogeneous relations, a posthumanist understanding not to be identified with the capacity that is “localized in a human body or in a collective produced (only) by human efforts”.

But how does this material body-assemblage become entangled with mathematical concepts? In what sense can we consider concepts, mathematical or otherwise, to be related to *matter*? The question goes to the theoretical heart of *The Body and* *Mathematics. *Their aim is to show how “mathematical concepts partake of the material in operative agential ways”. In order to accomplish this they need to go outside a humanist conception of matter and ‘materiality’ as well as construct a new approach to the nature of concepts. They derive this from contemporary feminist philosophers, principally Karen Barad, but also Jane Bennett, and Diana Coole and Samantha Frost, whose common aim is to re-orient how we think ‘matter’ and the material world. From Barad’s theory of ‘agential realism’, derived from Niels Bohr’s explication of quantum phenomena, they take her understanding of a concept not as immaterial mental object but “a material arrangement of things” and of relations preceding, in some sense constituting, that which they relate, so that things are always “intra-related” rather than interrelated. Jane Bennett’s concept of “vibrant matter”, a (non-animistic) understanding that credits matter with agency, and the ‘new materialisms’ surveyed by Coole and Frost provide the wherewithal for constructing a body-concept nexus. This, along with the anthropological work of Lambros Malfouris and Bruno Latour, allows them to rethink the concept of ‘mere’ matter. Rejecting the Cartesian split between the active cognizing human mind and inert ‘dead’ matter — the contemporary orthodoxy underpinning the physical sciences’ engagement with matter — these various thinkers urge materialisms in which the freedom and agency Descartes restricted to the embodied human mind is opened up and dispersed across human and non-human agents. The ontology of mathematics the authors weave from these diverse materialisms, with their insistence on the extra-human and material dimensions of thought, complements the authors’ construction of the assemblage-body. With this theoretical meshing in place Sinclair and de Freitas are ready to enlarge on how the two, bodies and formal mathematical concepts, might in practice become entangled. They accomplish this through the essay of Chatelet that inspired them to pursue their ambitious body-mathematics project.

Chatelet’s interest is how mathematics comes into being, its genesis, the becoming rather than the ‘being’ of mathematics, and his essay is a series of analyses of specific mathematical inventions such as Grassman’s creation of algebras over vector fields, Cauchy’s method of integrating complex functions – to reveal the physico-conceptual movements that can be seen as constituting them.

His starting point is actual physical movement. According to Chatelet the “amplifying abstractions” of mathematics, whatever their ultimate immaterial representation as formal constructs, have bodily beginnings. They originate in gestures, “disciplined distributions of mobility” that are not signs or representations of anything prior to or outside themselves, but material events that, through their action, by the fact of their occurrence, bring new mathematical meanings into being. They are not, Chatelet insists, describable by formal languages, cannot be determined by algorithms, are not expressions of an intention, though they can be retrospectively seen as such, and not in fact consciously produced: “One is”, he says, “ infused with the gesture before knowing it”. And they do not work through reference or signification, but by pointing, through allusions that, in interaction with diagrams (themselves responses to problems), give rise to “dynasties of problems” and correlatively families of ever more precise allusions. A diagram, for Chatelet, is a frozen gesture, a gesture caught midflight in its path toward a formal abstraction: it can “transfix a gesture, bring it to rest, long before it curls up into a sign.” Diagrams are intermediaries between bodies and mathematical objects and operations. They are, like gestures, material events. Contrary to the customary view of them, they are not depictions, illustrations or visual icons of mathematical objects or concepts (though they can be), but pivotal devices in the creation of mathematical meaning. “kinematic capturing mechanisms”, as the authors neatly describe them, “for direct sampling that cut up space and allude to new dimensions and new structures.”

In a sense diagrams are works in progress, never complete in themselves: “When a diagram immobilizes a gesture in order to set down an operation it does so by sketching a gesture that then cuts out another.” Diagrams and gestures interact, mutually presupposing each other, participating in what the authors call each other’s “provisional ontology”. Overall, the gesture-diagram nexus operates as a “dynamic process of excavation that conjures the sensible in sensible matter.” The authors relate this conjuring to Barad’s realist understanding of concepts as material arrangements. “The concept itself”, they observe, “is entailed in the hands that gesture, the mouth that speaks, and the circulating affect across an interaction.” They concretize this entailment through a variety of examples from discussion of how “the point at infinity” is cognized in projective geometry to describing at length the results of an experiment with a class of undergraduates asked to draw diagrams in response to a simple film of moving circles.

The gesture-diagram apparatus of allusions to mathematical meanings is one half of what the authors find valuable in Chatelet’s approach; the other is his deployment of the notion of the *virtual*. He takes this from Deleuze’s materialist and immanentist philosophy according to which the physical world of matter constantly comes into being – becomes — by making actual that which is virtual: “The virtual must be defined as strictly a part of the real object – as though the object had one part of itself in the virtual into which it is plunged as though into an objective dimension.” The virtual is what is latent in matter, the source of all that it could become, which the authors interpret as its “mobility, vibration, potentiality and indeterminacy” and it is the link Chatelet provides between the mathematical and physical worlds.

Following Gottfried Leibniz in conceiving space as “a flexible, folding and animated substance”, Chatelet observes that the supposedly immovable objects of mathematics divorced from “sensible matter” are on the contrary always in a state of potential movement and change; a geometrical point (line, circle) cannot be confined to a designated entity, the representation of a position within a fixed absolute space. As he observes in the case of Cauchy’s treatment of a singular point in the complex plane, the virtuality of a point, probed by mathematicians within “thought-experiments” becomes the source of radically new concepts. A point is the simplest example of a diagram, but the effect is quite general. As the authors observe for any diagram its “virtuality consists of all the gestures and future alterations that are in some sense ‘contained’ in it.” Mathematical entities, then, are material objects with virtual and actual dimensions. The virtual is not so much a bridge, an interrelation between mathematics and the physical worlds as if they were initially separate and then joined, but rather an intra-relation in the sense of Barad, a mutual fabrication or co-constitution wherein the two are thoroughly entangled. This means mathematical concepts engage in a process of becoming which binds them to the actions of mathematicians, leading to the authors’ striking conclusion that “The body comes into being through actualizing the virtual – through gestures, diagrams and digital networks *we become mathematics*; we incorporate and are incorporated by mathematics.” (original emphasis)

Summarized in this way and taken in isolation, the concept of ‘becoming’ mathematics will doubtless strike many potential readers of *The Body and Mathematics* as a strange and counter-intuitive characterization of ‘we’ and of mathematics; but hopefully this will not to be their experience. Throughout Sinclair and de Freitas seem fully aware of the unfamiliarity of the ideas they mobilize and the conceptual demands of their thesis; they go to considerable lengths to present matters as accessibly as possible. Not only does their book carefully develop the idea of the body-as-assemblage and its dynamic relation with abstract concepts that is the basis of how we become mathematics, but it contains a wealth of material and a rich texture of connections that elaborate and contextualize their thesis. Thus, beside constantly rooting their ideas in the concrete classroom observations and experiments which feature throughout, they step back and offer a series of illuminating and provocative chapter-length discussions of key aspects of their field, ranging from the “sensory politics of the body mathematical” and “mapping the mathematical aesthetic” to the “materiality of language” and the material dimension of “inventiveness in the mathematics classroom”.

In a final reflection on what becoming mathematics might mean, generally and in the context of the classroom, they invoke Deleuze’s concept of a ‘minor science’, a ‘minor literature’, and indeed a ‘minor mathematics’ – forms of thought and creation which escape the constrictions of the dominant ‘state’ or orthodox version. They describe “a mathematics that is not the state-sanctioned discourse of school mathematics, but might be full of surprises, non-sense and paradox.” and which, though at odds with institutional demands and the domination of a fixed curriculum “is likely to engage students-teachers in more expansive ways, and our hope is that it would engage *more* students in mathematics.” Whether or not it does remains of course to be seen, but in any event the minor mathematics Nathalie Sinclair and Elizabeth de Freitas usher onto the mathematics education scene constitutes a major theoretical intervention in their field. *Mathematics and the Body* is a valuable, radical and challenging work.