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Deleuze and the History of Mathematics: In Defence of the 'New'
Gilles Deleuze's engagements with mathematics, replete in his work, rely upon the construction of alternative lineages in the history of mathematics, which challenge some of the self imposed limits that regulate the canonical concepts of the discipline. For Deleuze, these challenges are an opportunity to reconfigure particular philosophical problems - for example, the problem of individuation - and to develop new concepts in response to them. The highly original research presented in this book explores the mathematical construction of Deleuze's philosophy, as well as addressing the undervalued and often neglected question of the mathematical thinkers who influenced his work.In the wake of Alain Badiou's recent and seemingly devastating attack on the way the relation between mathematics and philosophy is configured in Deleuze's work, Simon Duffy offers a robust defence of the structure of Deleuze's philosophy and, in particular, the adequacy of the mathematical problems used in its construction. By reconciling Badiou and Deleuze's seeming incompatible engagements with mathematics, Duffy succeeds in presenting a solid foundation for Deleuze's philosophy, rebuffing the recent challenges against it.
- GenresPhilosophyMathematics
208 pages, Kindle Edition
First published January 1, 2013
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December 3, 2022
Simon Duffy is surely the premier interpreter of Deleuze from a mathematical point of view. While this may seem idiosyncratic for a philosopher who’s best known references are philosophical figures above all (Nietzsche, Bergson, Kant, etc), it is simply the case that Deleuze’s texts are abound with mathematical talk. From the centrality of the differential calculus all the way to the well known discussions of multiplicity and manifolds, math is everywhere in Deleuze. How lucky we are then, that there is someone like a Duffy out there, who, having done the hard work of tracing those references back through time, has made accessible those vast vistas of Deleuzian thought that might have otherwise remained only ever latent, extra-philosophical curiosities to be noted but not pursued. Well, Duffy has pursued them, and thanks to him, our reception of Deleuze has been enriched … dare I say - immeasurably.
But more than just a patient explication of these shining points in the Deleuzeoverse, Duffy’s book also functions - as it says on the tin - as a ‘defence of the New’. In the background here is the challenge, as laid down by Deleuze’s emergent philosophical Other, Alain Badiou, that Deleuze’s use of math remains merely metaphorical, and as a result, not particularly well grounded. Against this, Duffy ‘defence’ in fact, turns into something of an offence. Showing not only that Deleuze’s use of math is perfectly consistent and philosophically well-grounded, but also, and as its own result, in fact superior to Badiou’s particular brand of mathematically inflected philosophy. I should be clear that this thematics of one-upmanship is largely implicit with the exception of the last chapter that deals with Badiou directly, and for the most part Badiou’s name is absent other than as a lurking shadow in the background. But lurking it remains throughout.
Regardless of motivation however, every chapter in here stands alone as an instrument of illumination in its own right. Together there are five, each structured in roughly the same way: Duffy begins by bringing out a key Deleuzian philosophical reference (Leibniz, Maimon, Riemann, Lautman, and Badiou), walks through the relevant mathematics, before linking them back to the use Deleuze makes of both. As far as pulling it off, it’s a success. While Duffy isn’t the most exciting writer in the world, what one loses in the pulse-rending prose of a Deleuze, one gains in the exactitude and clarity of someone who knows work - the relevant history of mathematics - inside out. Which isn’t to say that this is easy reading either. While I’m not convinced that I could tell you what a Riemann surface or Galois group is on their own terms, I know enough now about their properties to say why they were important to Deleuze, and how they fit into his wider philosophical project. Which, short of studying the mathematics on their own, is a pretty good result I reckon.
Having said all this then, a word about the exact place that mathematics occupies in the philosophical vision advanced by Deleuze, according to Duffy. Essentially, mathematics functions a model; it isn’t, as with the Platonists (like a certain Badiou, say), a foundational science from which all else follows. But neither is it a mere metaphor, an inert store from which to draw certain analogies. Rather, the impasses of math, along with the intra-mathematical innovations developed each time to overcome them, teach us a certain how: a manner of approach (or approaches) that can be similarly redeployed at the level of philosophical thought itself. This implies a certain reciprocity too: that problems in philosophy itself can themselves inspire approaches in math. The exact nature of this ‘how’ - which relates to what Deleuze/Duffy call a problem-solution complex - exceeds this review*, and I will say only that this is exemplary of the idea of “univocity” - the idea that “being is said in the same sense of all of which it is said” - operative in math, philosophy, and elsewhere too.
This ‘openness’ as to the place of math - neither foundational, nor metaphorical, but with lessons to teach nonetheless - also informs Duffy’s sense of the future of the relation between mathematics and philosophy. Indeed, in the development of category theory - a ‘new’ field of math unbroached by Deleuze himself - Duffy sees potential avenues for the extension of Deleuzian philosophy - one that, intriguingly, exceeds Badiou’s own recent attempt to incorporate it into his own philosophical edifice. So while this book is indeed about Deleuze and ‘the history of mathematics’, so too is it really about the future of both Deleuze and mathematics too, with all the excitement - such as one might have - that that would entail. A note to close: Duffy does not in fact speak of univocity, which is shame, because I think it would have worked very nicely to bring together the relation between philosophy and mathematics that he tries so hard to establish; and if I mention it here it’s only because Duffy himself wrote the book on this topic - his previous The Logic of Expression, which fills the Spinoza-shaped hole that otherwise punctures Deleuze and the History of Mathematics.
*For more on the problem-solution complex, check out Duffy's reply to James Williams' review of this book, found here.
But more than just a patient explication of these shining points in the Deleuzeoverse, Duffy’s book also functions - as it says on the tin - as a ‘defence of the New’. In the background here is the challenge, as laid down by Deleuze’s emergent philosophical Other, Alain Badiou, that Deleuze’s use of math remains merely metaphorical, and as a result, not particularly well grounded. Against this, Duffy ‘defence’ in fact, turns into something of an offence. Showing not only that Deleuze’s use of math is perfectly consistent and philosophically well-grounded, but also, and as its own result, in fact superior to Badiou’s particular brand of mathematically inflected philosophy. I should be clear that this thematics of one-upmanship is largely implicit with the exception of the last chapter that deals with Badiou directly, and for the most part Badiou’s name is absent other than as a lurking shadow in the background. But lurking it remains throughout.
Regardless of motivation however, every chapter in here stands alone as an instrument of illumination in its own right. Together there are five, each structured in roughly the same way: Duffy begins by bringing out a key Deleuzian philosophical reference (Leibniz, Maimon, Riemann, Lautman, and Badiou), walks through the relevant mathematics, before linking them back to the use Deleuze makes of both. As far as pulling it off, it’s a success. While Duffy isn’t the most exciting writer in the world, what one loses in the pulse-rending prose of a Deleuze, one gains in the exactitude and clarity of someone who knows work - the relevant history of mathematics - inside out. Which isn’t to say that this is easy reading either. While I’m not convinced that I could tell you what a Riemann surface or Galois group is on their own terms, I know enough now about their properties to say why they were important to Deleuze, and how they fit into his wider philosophical project. Which, short of studying the mathematics on their own, is a pretty good result I reckon.
Having said all this then, a word about the exact place that mathematics occupies in the philosophical vision advanced by Deleuze, according to Duffy. Essentially, mathematics functions a model; it isn’t, as with the Platonists (like a certain Badiou, say), a foundational science from which all else follows. But neither is it a mere metaphor, an inert store from which to draw certain analogies. Rather, the impasses of math, along with the intra-mathematical innovations developed each time to overcome them, teach us a certain how: a manner of approach (or approaches) that can be similarly redeployed at the level of philosophical thought itself. This implies a certain reciprocity too: that problems in philosophy itself can themselves inspire approaches in math. The exact nature of this ‘how’ - which relates to what Deleuze/Duffy call a problem-solution complex - exceeds this review*, and I will say only that this is exemplary of the idea of “univocity” - the idea that “being is said in the same sense of all of which it is said” - operative in math, philosophy, and elsewhere too.
This ‘openness’ as to the place of math - neither foundational, nor metaphorical, but with lessons to teach nonetheless - also informs Duffy’s sense of the future of the relation between mathematics and philosophy. Indeed, in the development of category theory - a ‘new’ field of math unbroached by Deleuze himself - Duffy sees potential avenues for the extension of Deleuzian philosophy - one that, intriguingly, exceeds Badiou’s own recent attempt to incorporate it into his own philosophical edifice. So while this book is indeed about Deleuze and ‘the history of mathematics’, so too is it really about the future of both Deleuze and mathematics too, with all the excitement - such as one might have - that that would entail. A note to close: Duffy does not in fact speak of univocity, which is shame, because I think it would have worked very nicely to bring together the relation between philosophy and mathematics that he tries so hard to establish; and if I mention it here it’s only because Duffy himself wrote the book on this topic - his previous The Logic of Expression, which fills the Spinoza-shaped hole that otherwise punctures Deleuze and the History of Mathematics.
*For more on the problem-solution complex, check out Duffy's reply to James Williams' review of this book, found here.
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