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My question is triggered by a quote from Manuel DeLanda which I find difficult to unpack as it is probably not only that they prefer different mathematical tools but that there is a profound difference in their world views.

I would like to start with the quote:

Badiou left me with a bad feeling after reading his book on Deleuze which is incredibly incompetent. He uses the word “the One” on just about every page when Deleuze never used it (other than when making remarks about the scholastic notion of the “univocity of being”). He is also a fanatic about set theory, whereas I tend towards the differential calculus as my mathematical base. (The idea that the latter was reduced to the former is yet another mistake we inherited from the nineteenth century).

In: New Materialism: Interviews & Cartographies by Rick Dolphijn & Iris van der Tuin, Page. 46, available here

I've done some research but I couldn't find a simple and clear support for DeLanda's statement that the differential calculus is reducible to set theory. I'm no mathematician but I would like to understand why DeLanda (surely based on Deleuze's work) stresses this distinction.

I've found here the question whether Deleuze's understanding of the infinitesimal calculus was primitive?. This helped clarity my concern. But it still left the question open how set theory is related to differential calculus and the philosophical implications.

My reasoning is that while doing some research about the difference of set theory and differential calculus I stumbled upon a very helpful beginner's explanation distinguishing between limits and infinitesimals, of which I take that

  • limits: are the modern and today's mathematically standard approach. However in order to be able to make proper calculations one needs to be content with cutting a certain part of the real, the full reality, away. This seems to be a rather reductionist view that if the difference is small enough it does not matter. Also using limits seems to be more related to objects than intensities. Thus would it also be more of a tool for set theorists?
  • infinitesimals: make use of a different "dimension", whose existence one has to accept. It seems to open up potentials as it is not clear which dimension you are using. It also allows one to "access" all of the real.

My questions are:

  • Is the differential calculus reducible to set theory? And how? And who came up with the needed theory?
  • Is set theory more related to limits while the differential calculus is more related to infinitesimals?
  • If Badiou is more into set theory, and probably monads, is this closer to limits? It seems for me that there are big implications in seeing the world as objects and not as differences or intensities.
  • How are set theory and the differential calculus related? Did Deleuze try to revive an alternative interpretation to make use of another branch of the scientific tree to open up room for different epistemological interpretations (a world of intensities in becoming)? Are these maybe better discussed in the form of rates of change (differential calculus) rather than sets (definition of borders and separation of objects).

I would appreciate any pointers.

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    I'm not qualified to address most of your questions, but (almost?) any undergraduate real analysis textbook will give a rigorous definition of the derivative of a function, starting only with sets, relations and functions (so objects from set theory). In this sense the differential calculus is reduced to set theory. I believe the question of who exactly should get how much credit for developing this theory has been much debated in the history of mathematics. – Simon L Rydin Myerson Jul 11 '18 at 18:25
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    Under the standard meanings of terms the answers to the bulleted questions are 1) Yes, Weierstrass and Cantor; 2) No, infinitesimals are an alternative to limits approach to calculus (currently standard), but both are reducible to set theory; 3) No, "monad" is Leibniz's term used in modern versions of infinitesimal analysis; 4) See 2). What DeLanda is talking about is hard to tell from his angry passing remark, but he likely refers to some early version of calculus with infinitesimals as primitives (I am not sure he is aware of contemporaneous Newtonian/kinematic versions without them). – Conifold Jul 11 '18 at 18:46
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    When I get more free time I can venture a full response, but a quick one, just to clarify a major point: the question of the relation between set theory & diff calculus for Deleuze relates back to the theory of multiplicities ALWAYS. Set theory is inadequate to multiplicities because it is not the limited thing that sets a limit to the infinite but the limit that makes possible a limited thing. So the concern at this extremely abstract level philosophically is not a sphere of reference (which is not primary object of philosophy), it's the sphere of immanence from which sets abstract. – ClearMountainWay Jul 11 '18 at 20:43
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    Anyone who so casually dismisses set theoretic foundations as a mistake can be just as casually dismissed as a crank, at least where any math or history of math is concerned. – Tim kinsella Jul 11 '18 at 23:18
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    DWSmith, 2003, Mathematics and the Theory of Multiplicities: Badiou and Deleuze Revisited, Southern J of Phphy, v. 41, p. 411 philarchive.org/archive/SMIMAT-6 (later perhaps smth like an answer) – sand1 Jul 12 '18 at 17:44
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From the perspective of these two philosophers there's not a debate about whether set theory or differential calculus are better, more important, more appropriate or anything in general terms. The discussion framing the book mentioned by DeLanda revolves around metaphysics and the specific claim made by Badiou that Set Theory was sufficient to found what Deleuze would call "any multiplicity whatever" covering all classes of multiplicity. If the two mathematical references are posed in any sense against each other, it is due primarily to the reception of a particular framing set up by Badiou of his opposition to Deleuze in which Deleuze is posed by Badiou simultaneously as a philosopher of "the One" that only appears to promote a philosophy of difference and also as his biggest enemy because despite being a philosopher of "the One" (his term), he fails to recognize Set Theory as the basis for philosophical ontology, (which ultimately shapes the differences in their conceptions of what philosophy is/does). In the framing of this opposition by Badiou the universality of the Set is perhaps framed against the crowned anarchy of difference for which differential calculus played a role in the enunciation of the case presented in Difference & Repetition, but this perception of some opposition is the result of the dust blown up by the polemic, or from the brief mention in Clamor of Being where Badiou states that upon their meeting Deleuze was occupied with this set of mathematical problems and he with that. But there is no actual opposition of these mathetical references as suggested by DeLanda'a quote. It is key to note that both shared similar views on the role of mathematics in the formalization of problematics independently of the other, and perhaps that is what initially attracted Badiou to Deleuze's work despite not being of his camp. The differences, which seem larger than they really are (such as if you were to compare either to two other names in philosophical history) are primarily political --this having a number of consequences which people have to understand in order to properly contextualize the absolutely crazy history of Badiou's historical antagonism of Deleuze; and secondarily their different approaches to and perspectives on ontology.

And before moving on, I would say that DeLanda's characterization of the book as "incompetent" misrepresents what Badiou is doing. Anyone who reads Badiou can see an extremely sharp mind at work, it's not an issue of competence (and his book on Wittgenstein can be read for comparative purposes on this point). His book contains eggregious falsifications and misrepresentations (from the non-Badiouan perspective), not because of incompetence but because his politics demands it. The product of his work remains incomprehensible until you understand his politics IMO. Please see Badiou's "Politics as Truth Procedure". (I'd be happy to expound on this and it's relevance to the question, but I didn't want to go into it here for fear this would be extremely long)

Direct answers to your questions:

Is the differential calculus reducible to set theory? And how? And who came up with the needed theory?

In just about any discussion of these two the answer is clearly yes. But one should be clear about what the question behind your question is. To philosophy the question seems to be more about the specificity of the tools appropriate for a particular Idea or problem (allow me to make clear that the 'or' here isn't synonymizing). If Deleuze in his hat of historian of philosophy is retracing the steps of Leibniz it would make no sense to discuss set theory in lieu of calculus. He would simply say that both play different roles in the formulating of "problems" for philosophy. In Badiou, where set theory plays the central role in formal thinking of entities, the question takes on importance and the question of reducibility is key, but in his framework of thinking alongside Deleuze this is not ever an issue of contention, and that's because the discussions (around the particular problems being confronted philosophically) are different. He reads Deleuze's univocity thesis in the same terms as he develops his own Platonic theory of multiplicity. A theory of mulitplicities and one of differentials are directed towards two different planes/problems.

Is set theory more related to limits while the differential calculus is more related to infinitesimals?

With regards to set theory it's not simply a concern about axioms and limits but of the meaning of "being" in the most basic terms or of the most general and abstract formal and structural conditions for anything to exist at all. For Badiou, his development of a set-theoretical ontology was to establish general formal conditions for the consistent presentation of any existing thing, "counted-as-one" and as a coherent unit. Whereas being in itself is simply "pure inconsistent multiplicity" or multiple-being without any organizing structure.

So there's a number of things he attemped to derive from this framework, but the two most central aspects are:

1. a priori conditions of any possible ontology
2. a theory of pure multiples (including a typology of multiples, and implications for what this means for truth, Events and singularity)

Whether he failed or succeeded or not in Being & Event, a proper way to assess the issue of his formalization was as a measure to rigorously define a space of action and creativity as opposed to simply be controling/limiting operations.

With respect to differentials, they can't be reduced to infinitesimals. Insofar as infinitesimals were invoked in Deleuze it was to demonstrate how Carnot and Leibniz demonstrated how problems resist being absorbed by solutions.

If Badiou is more into set theory, and probably monads, is this closer to limits? It seems for me that there are big implications in seeing the world as objects and not as differences or intensities.

With respect to the differences between objects and intensities, there were major limitations to the framework Badiou developed in Being & Event, but by Logiques des Mondes Badiou adopts Deleuze's understanding of intensive difference pretty much to the letter. Also Badiou's work expands a lot beyond his philosophical formalization of set theory, so if we were to do him justice here we would also have to include all of the developments he made around category theory.

How are set theory and the differential calculus related? Did Deleuze try to revive an alternative interpretation to make use of another branch of the scientific tree to open up room for different epistemological interpretations (a world of intensities in becoming)? Are these maybe better discussed in the form of rates of change (differential calculus) rather than sets (definition of borders and separation of objects).

"What matters to us is less the determination of this or that break [coupure] in the history of mathematics (analytic geometry, differential calculus, group theory ... ) than the manner in which, at each moment of that history, dialectical problems, their mathematical expression and the simultaneous origin of their fields of solvability are interrelated." - D in Difference & Repetition p. 180

Others can explain the relations between set theory & differential calculus in mathematical terms more concisely than I, so let me focus on your second & third questions. If we approach these passages as attempt to found alternative epistemological interpretations it obfuscates the DIRECT objects of these works. Deleuze had a great interest in mathematics and "hard" sciences, but equally in arts, music, linguistics, anthropology etc. Generally if something specific in a field is being addressed it's at the service of some larger target (e.g. making clear semiotic operations of deterritorialization from reterritorialization, to show what role axiomatization plays vis a vis a-signifiying ruptures etc, which as semiotic concerns, are not restricted to any particular field or discipline, thus you'll see an example as related to science here, another related to music there, another related to economics over there). When Deleuze revisits the issue of the development of the thinking behind calculus in Leibniz (both in Difference & Repetition and in his stand alone book The Fold: Leibniz & The Baroque) it's first to elucidate the complex web of concepts surrounding Representation, Faculties, Ideas, Problems, Solutions, what it means to think, and the nature of differentials as unsubsumable to solutions; the second use is to trace the development of an Idea (specifically differential relations) via the problematizations of thought actually ran through cases of philosophical preocupations (regarding perception, extrinsic physical causality, affective qualities) to the "convenient or well-founded fictions" (as Leibniz described his calculus to Varignon) and extensive reality. In both cases, while Badiou is correct in saying that Deleuze was extremely interested in mathematics, it's not because he is concerned with what mathematics does with these problem, the development he makes of them is within philosophy. Alternative examples can be seen in something Badiou would equally criticize him for, writing books invoking hundreds of cinematic cases that are of no use to cinematographers or film theorists, but rather to make statements about movement, sensory-motor schemas and time, a book about a painter that is really about the relationship between thought and sensation. Badiou has a different agenda that of course priviledges the mathematical that really is of no interest to Delezue. But the point here is that there are no epistemological direct motives here, there may be epistemological consequences, but they would be secondary to the things concepts and worlds that Deleuze was actually focused on creating.

  • My sense is that part of the problem is with Deleuze trying to map methodology onto content. The "problematic/axiomatic" is similar to Peirce's distinction between deductive (mechanical) and abductive (creative) sides of mathematics, but slapping them on calculus and set theory is just off-key. Here and elsewhere Deleuze ends up with concepts labeled by intended "paradigmatic examples" that do not really instantiate them. Together with loose use of the resulting vocabulary it makes for a lot of cacophony and confusion. The same with attaching his multiplicities (and "problematic") to Riemann. – Conifold Jul 20 '18 at 23:06
  • for clarity I would say that the "methodology" (or modes of approaching particular subject matters, of which works of Archimedes, Poncelet or Monge become examples) becomes precisely the "content" of particular distinctions in activity he makes in his most commonly cited work. The axiomatic/problematic distinction however never poses set theory vs calculus. Problematization and axiomatization are from the perspective of history moments in the development of all scientific and mathematical fields or any formalized systematization. But I'm not sure I misunderstood your point above or not. – ClearMountainWay Jul 24 '18 at 1:05
  • His interest in this distinction is not ultimately mathematical, its more elementary, relating to the issue of thought, which is why it's always immediately related to so many other distinct fields. But agree that there is lots of lose and confusing vocab and "unique" readings of figures, all over, from mathematics, to philosophers, to painting, to literature etc. I would say that the two hold the latter trait in common – ClearMountainWay Jul 24 '18 at 1:09
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Hopefully, the paper already referred to (DWSmith 2003) should answer most of this extended question and its last 10 pages of notes contain enough references.

As I read it, DeLanda says about analysis being reduced to set theory "is yet another mistake we inherited from the nineteenth century", so a discussion about reduction (ontological or methodological, etc) could be skipped. Since Robinson exposed his non-standard approach to analysis, set theory might be seen mostly as an unnecessary detour. Smith mentions that Robinson called 'monads', the non-standard parts surrounding usual numbers. Badiou is well aware of this development and has written at some length about the surreal numbers in his Number and Numbers .

Actually the polemic Badiou / Deleuze is perhaps better grasped in Smith's terms of axiomatization vs. problematization. Ultimately it is about platonism vs antiplatonism and most anything that connects with either horn of the alternative. 'Overcoming platonism" was Deleuze's slogan and and the opposition is clearly when the discussion comes to 'events'. Subject and predicate is the aristotelian and settheoretical logic while Deleuze was intersted in predication as an effect exhibited by propositional logic. The originality of his postion is clearly seen in his Logic of Sense.

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    "Since Robinson exposed his non-standard approach to analysis, set theory might be seen mostly as an unnecessary detour"??? Robinson's approach is based on set theory and the ultrafilter construction in particular. Non-standard analysis features hyperreal numbers, not surreal ones. I also do not follow "subject and predicate is the aristotelian and settheoretical logic". What is "settheoretical logic" (predicate calculus with quantifiers?), what does it have to do with Aristotle, and how is predication restricted to propositional logic? – Conifold Jul 12 '18 at 21:50
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    "... axiomatization vs. problematization." -- This reminds me of Tim Gowers's idea that mathematicians are either theory builders or problem solvers. dpmms.cam.ac.uk/~wtg10/2cultures.pdf – user4894 Jul 13 '18 at 1:31
  • @ Cornifold. Admittedly, surreals are a more general case than hyperreals and Badiou, being a phpher, chose to mention them; he writes (disapprovingly?) about Robinson. Set theory has been touted as an (indispensable) foundation for calculus but nonstandard analysis shows this to be an overstatement. Lastly, Deleuze was interested in propositional logic, not in "some men are wise". Anyway, the authors and papers discussed here deserve more interest than my "answer". – sand1 Jul 15 '18 at 8:46
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    @sand1 "Set theory has been touted as an (indispensable) foundation for calculus but nonstandard analysis shows this to be an overstatement." What? That's complete nonsense. Nonstandard analysis is itself founded in set theory - how precisely do you show that a hyperreal field exists in the first place? (Or, do you not consider ultrapowers to be set-theoretic?) The facts that the birth of set theory was historically intertwined with skepticism of infinitesimals and that there is a consistent approach to infinitesimal calculus in no way imply tension between set theory and nonstandard analysis. – Noah Schweber Jul 15 '18 at 18:58
  • There is serious research into non-set-theoretic foundations of mathematics, but there is no connection whatsoever with nonstandard analysis. – Noah Schweber Jul 15 '18 at 19:02

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