the nondenumerable

The work of Graham Priest and Gilles Deleuze (and Félix Guattari) converge in significant ways on the concept of the nondenumerable.

Turning to Priest first, and to his Beyond the Limits of Thought especially, one finds in this book an interesting history of philosophy, and one with a particular narrative at work; namely, he uncovers numerous contradictions that are encountered as certain unthinkable limits to thought become the subject of thought itself (e.g., primary substance for Aristotle, God for Cusanus, the noumenon for Kant, among other examples). In the history of thought prior to Hegel, according to Priest, these contradictions were largely denied, primarily through a denial of the very limits that gave rise to them. But with Hegel there is an open recognition and affirmation of the contradictory nature of the limits of thought. It is for this reason that Priest claims that the ‘chapter on Hegel [in Beyond the Limits of Thought] is therefore the lynch-pin of the book.’ (7).

Priest cites Hegel’s own recognition of these contradictory limits in his Science of Logic where he argues that

great stress is laid on the limitations of thought, of reason, and so on, and it is asserted that the limitation cannot be transcended. To make such an assertion is to be unaware that the very fact that something is determined as a limitation implies that the limitation is already transcended. For determinateness, a limit, is determined as a limitation only in opposition to its other in general, that is, in opposition to that which is free from the limitation; the other of a limitation is precisely the being beyond it. (Logic, 134, cited by Priest, 108)

In writing of Hegel, for instance, Priest argues that Hegel’s understanding of the limits of thought, and the contradiction attendant upon this understanding, was disadvantaged because Hegel ‘had only a rudimentary understanding of the boundary-tearing mechanism which transcends limits,’ a mechanism such as that found with ‘diagonalisation’. (109).

Central to understanding the mechanism of diagonalization is the distinction between denumerable and nondenumerable sets. Put simply, if we take the set of natural numbers then any subset of natural numbers, whether finite or infinite, is countable if it can be paired one for one with a natural number. For example, if we take an infinite sequence of sets, (s1, s2, s3, s4,…), where each of these sets consists of an infinite sequence of 1s and 0s, these sets are nonetheless countable since they can be paired one for one with the natural numbers, and hence such sequences are said to be countably infinite. What Cantor was able to show was that one could construct a unique set, s0 let us say, that cannot be paired up with any of the natural numbers and hence is uncountable—or, it is nondenumerable. This set is nonetheless constructed from within the countable sets and hence the elements of the construction of the set are included within the infinite sequences but the set of these elements does not belong, or cannot be counted among one of the infinite sequence of sets. There is thus a paradox, Cantor’s paradox, where there is a set, s0, that both belongs to the set X (of natural numbers) in that it was constructed by drawing from the members of the sets that do belong to X, and at the same time this set does not belong to X since it cannot be countably related to the natural numbers. Hence the paradox: s0 X and s0 X. Such paradoxical sets are also referred to as inconsistent sets since it is not possible to construct a functional relationship that would map their elements one for one with the natural numbers, unlike consistent sets where this can be done.

For Priest this method of diagonalization assumes a prominent role in the arguments of the latter two thirds of his book, and he claims that there is a common paradox at the heart of much of ancient and modern philosophy, and in Nagarjuna as well, when it comes to attempts to think the limits of thought, a thought that entails a move beyond the limits of thought. Priest refers to this as the inclosure schema, or the inclosure paradox, which he states as follows:

(1) Ω = {y; φ(y)} exists, and ψ(Ω)

(2) if x is a subset of Ω such that ψ(x):            (a)            δ(x) x

(b)            δ(x) Ω

Stated in layperson’s terms, and tying in with the diagonalization argument from above, (1) corresponds to what Priest calls the Existence condition, by which he means there is a set of elements that exists and is definable. (2) brings us the paradox of Transcendence and Closure (which is a prominent and continuous theme in Priest’s book). By virtue of the diagonalization function, δ(x), we end up with a diagonalization of a subset that gives us a set that does not belong to that subset although it may belong to a larger, more inclusive set, denoted by the Ω. Where the inclosure paradox of limits comes in is when we consider the limit set itself, Ω, and apply the diagonalization function to it. What we end up with, Priest points out, is a case whereby we have a set that both belongs to Ω (or we have Closure) and yet does not belong to Ω (Transcendence). This is a case of a true contradiction, according to Priest, and it is such true contradictions that are central to what Priest refers to as dialetheism (see earlier post).

Turning to Deleuze now, and in particular to Deleuze and Guattari’s A Thousand Plateaus, we find that here too the nondenumerable assumes an important position. As Deleuze and Guattari put it in A Thousand Plateaus,

What characterizes the nondenumerable is neither the set nor its elements; rather it is the connection, the “and” produced between elements, between sets, and which belongs to neither, which eludes them and constitutes a line of flight. (470).

First off, there’s a clear similarity between the Deleuzo-Guattarian notion of a line of flight and the function of diagonalization. In both cases the result is something that eludes belonging to countable sets. In the case of DG, however, they place the emphasis neither on sets nor on the elements of sets, but rather on the ‘“and” produced between elements, between sets.’ This emphasis upon the “and” connects with Deleuze’s own concern with empiricism, which he believes is best characterized by the “and” (I’ll turn to Hume below). But as we have seen, the diagonalization method was a method of producing a nondemumerable set by taking an element from one set, “and” another, “and” another, such that the result is a nondenumerable set (though Deleuze will prefer the term multiplicity). DG are thus not offering a heterodox reading of the nondenumerable. Moreover, we gain an important insight concerning ‘line of flight’ by comparing it to the diagonalization function. A line of flight, contrary to Peter Hallward’s reading of Deleuze (in Out of this World) is not an attempt to get out of this world, to escape this world, but it is rather an effort to construct and compose a nondenumerable multiplicity while remaining fully within the world, much as Cantor was able to construct a nondenumerable set by drawing solely from countable sets.

We can also better understand DG’s critique of Badiou in What is Philosophy? For Badiou what is the process whereby an inconsistent set becomes consistent and definable. For Badiou Cohen’s notion of forcing plays a key role in this understanding. What this entails, however, is a subjective intervention that imposes consistency upon the inconsistent. The problem with this perspective, and one that echoes Priest’s own arguments, is that whatever consistency is produced by way of a subject’s faithfulness and forcing can itself, by way of the diagonalization function, become yet another inconsistent set. Rather than a subject that needs to intervene from outside a given, countable situation in order to “force” a consistency that heretofore did not exist, for Deleuze and Guattari the revolutionary project entails creating the connections, the between of elements and sets, the “and,” that gives rise to the nondenumerable that problematizes any given situation and prompts its possible transformation (this is all too brief, but for more on my critique of Badiou see my Deleuze’s Hume). With this in mind we can turn to DG’s analysis of Badiou in What is Philosophy?.

After describing in their own terms much of what we have detailed above about Badiou and nondenumerable sets, DG refer to Badiou’s invocation of the void as an effort to render such sets consistent, an effort which DG claim reintroduces ‘the transcendent’. We can see why they would interpret Badiou in this way. Rather than working immanently from within situations and states of affairs to compose a nondenumerable that leads to the efforts to create a consistency, Badiou introduces the void of a situation, that which cannot be related to any situation, and calls upon a faithful subject to ‘naturalize’ such events so that they can acquire the consistency that enables the events to become placed within the continuity of historical processes. Another important difference, as DG read Badiou, is that whereas Badiou begins and ends with functional relationships between states of affairs, situations, and sets (e.g., functional relationships between countable sets) that are characteristic of science, while DG begin with problematic, nondenumerable multiplicities that are inseparable from countable sets and states of affairs (recall that δ(x) Ω). In doing the former, DG believe that Badiou ultimately invokes the transcendent, despite his claims to the contrary (which is also probably why Badiou could not see himself in DG’s characterization), whereas the latter approach tracks the path of immanence. To state the difference as this is developed throughout What is Philosophy?, DG are interested in developing philosophical concepts while Badiou is interested in formulating scientific concepts.

I’ll close with one more example – Hume. On my reading of Deleuze’s Hume, or the Deleuzian Hume that accounts for the profound sense in which Deleuze was a Humean throughout his career, the impressions and ideas are not to be understood as a countable set but are rather a nondenumerable multiplicity that becomes, when actualized, the bifurcations of conceptual thought. In Priest’s terms, the multiplicity Hume thinks in his Treatise is a limit to thought itself and gives rise to contradictions when it is thought. For example, there is the well-known contradiction in Hume’s thought, and one Hume himself despaired of in the appendix to the Treatise, regarding the self. On the one hand the self is understood to be nothing but a bundle of impressions and yet, through much of the latter half of the Treatise and in his essays, the self is the assumed and unquestioned condition for many of his analyses of the passions, justice, politics, etc. If the multiplicity of impressions and ideas is understood as a nondenumerable condition inseparable from thought, then as this condition is thought it gives rise to the bifurcations and contradictions that attend conceptual identifications (or intellectual mitosis as this was discussed in an earlier post), as is evidenced in this case with Hume’s attempts to delimit the identity of the self – he too encountered the contradictory limit.

With the notion of the nondenumerable, therefore, there is indeed an interesting and important convergence between Priest and Deleuze

24 thoughts on “the nondenumerable

  1. Forgive me if my question seens naïve, but I haven’t understood the passage from the mathematical results (say, Cantor’s paradox) to more substantive results, be they epistemological or metaphysical. For example, you claim that we should consider Hume’s impressions and ideas as “a nondenumerable multiplicity”. But this assumes, at minimum, that Hume’s impressions and ideas are isomorphic to sets, or even that they are a model for set-theory, but I don’t see (at least not immediately) an argument to that effect. Unless you meant that the notion of a true contradiction should provide the middle term for comparison, but even that seems very thin. Could you expand a bit on this? (I’m not familiar with Priest’s work, incidentally, so that may account for the problem)

    • Hi Daniel,
      The basic idea for Priest is that the philosophical tradition has repeatedly encountered a limit to thought, a limit that it is taken to be something that cannot be thought, but, Priest goes on, in making this very claim this limit is itself thought. That was the insight Hegel had as was quoted by Priest and is in my post. Cantor’s paradox is related to this. I didn’t go into this for lack of time, but Priest shows how Cantor made distinctions between the finite, the transfinite, and the absolutely infinite. The absolutely infinite is again another limit, a limit that in thinking it entails going beyond it, which is precisely what occurs through the diagonalization method. Now tying this in to the metaphysical and epistemological concerns, and Deleuze in particular, the arguments concerning the nondenumerable within mathematics are most commonly associated with showing that the set of real numbers is greater than the set of natural numbers, and yet the real numbers are uncountable since to be countable entails a functional relation to the natural numbers. In Deleuze’s hands, the significance of the nondenumerable is to capture the sense that there is a reality (what he most frequently refers to as the reality of the virtual) that eludes the reality of the actual. Deleuze’s reading of Hume, or at least my reading of Deleuze here, understands the multiplicity of impressions and ideas in this sense, and hence as irreducible to the actually identified ideas and impressions. Hope this helps and thanks for your comment.

  2. Great connections here. It’s also probably worth reading the connections that Deleuze draws to Russell’s paradox, etc., in The Logic of Sense with Priest in mind.

    A couple of small quibbles, though, if I may: it’s a bit misleading to say that we can “construct” a nondenumerable set by diagonalization, since diagonalization is a paradigm of a “nonconstructive” procedure. Also, Cantor at any rate uses “inconsistent multiplicity” to refer to sets like the Russell set that can’t exist without contradiction; nondenumerable sets (such as those with the power of the continuum) are perfectly “consistent” and don’t seem to raise exactly Priest’s problem, at least not all by themselves.

    • True true. Thanks for this Paul. This actually brings out another similarity since lines of flight are not, on my reading, constructive procedures either. And your point about the consistency of the power of the continuum is another point well taken. I’ve not worked my way through Cohen’s arguments related to the continuum, but from what I know you are exactly right.

  3. I think that there are potential problems here.

    First, multiplicities (in the Deleuzian-Gauttarian sense) are not simply nondenumerable, they are non-numerical in a Bergsonian manner: that is, what distinguishes their members from one another is quality and not quantity. Nondenumerable sets are entirely numerical. Priest’s work is not on non-numerical multiplicities but on inconsistent sets.

    Second, not all non-numerical multiplicities are uncountable. A bag of apples is a multiplicity, because it is (i) a unity which is already many, synthesized conjunctively (and…and…) and (ii) because the apples are not merely distinguished quantitatively (this is apple 3 of 5) but qualitatively (THIS apple is a haecceity, qualitatively different than the other apples).

    And third, and perhaps most importantly, those multiplicities which are non-numerical in a stricter Bergsonian sense — for instance, the multiplicity which is constituted by the successive states of a sensation — are not of the cardinality of nondenumerable sets. Deleuze is clear in Difference and Repetition that he opposes the arithmetization of the Continuum in the 19th Century, that continuity is in some sense deeper than uncountably dense topologies. But, by Cantor’s Continuum Hypothesis, nondenumerable sets are precisely the cardinality of the Real Continuum. Non-numerical multiplicities like the successive states of some sensation are not of the cardinality of the Real Continuum, but are successive without distinction; they are qualitative multiplicities, whose members bleed into each other. This is why Deleuze highlights the ‘and’ when writing about nondenumerability — it is about conjoining the members and occupying the space in between them so as to produce a non-numerical multiplicity. It is not the nondenumerable that is a multiplicity (at least not as mathematicians or as Priest conceive of it) but the space between the members of the nondenumerable, the conjunctive space which produces real continuity and not merely a mathematized model thereof.

    Here I think Deleuze is to be differentiated from Priest — Priest’s work is precisely on the nondenumerable (as it is understood commonly in set theory), while Deleuze’s seems to be on the non-numerical. Lines of flight and diagonalization are related by the latter’s ability to discover the spaces in between the members of countable sets. This is related to Priest’s work on inconsistency. But isn’t it important not to confuse nondenumerable sets with non-numerical multiplicities?

    • Thanks for this Tony. You are right to differentiate between non-numerical multiplicities and nondenumerable sets. As I say in the paradox of expressibility post, there are important differences between Deleuze and Priest – most notably and most obviously, Deleuze’s understanding of paradox is incorporated into his ontology and metaphysics, which follows a Bergsonian, Whiteheadian line (as you point out) whereas Priest is attempting to expand logic such that it can incorporate true contradictions (what he calls dialetheism). This post did not highlight those differences, so you were correct to bring up that fact. In this post I was trying to think through the connection DG make in A Thousand Plateaus between nondenumerable and lines of flight, where I did see some connections with Priest. Thanks again for your helpful comments.

  4. The mathematical theory of Cantor and others is precise, beautiful and totally non-mysterious. It is a natural and correct completion of the mathematics that is adequate to everyday life, the life in which we deal in finite numbers of objects. It may be a little counter-intuitive at first, but if we dive into the theory, our intuitions soon catch up. It has nothing to do with limits to our thought.

    So I still do not see how it can be invoked in any clear thinking on the kinds of broad philosophical problem that are considered here, whether by Priest or by Deleuze. It is certainly not a source of easy answers, or easy analogies that are any use.

    When considering whether there are any useful links between the work of people like Deleuze and the work of truly great thinkers like Cantor, it is helpful to bear in mind the work of Alan Sokal.

    • I have no basis for dialogue with you since you think Priest and Deleuze are lacking in intellectual substance, as your reference to Sokal implies. A dialogue can therefore not get off the ground even though I agree with you about the beauty of Cantor’s thought. Wishing you well.

  5. Hello Jeffrey, you may well be right that we cannot sensibly have a dialogue on this specific topic, given our very different starting-points. It might even be difficult to have a dialogue on the extent to which Deleuze does have intellectual substance.

    I would however like to add one point. I deliberately did not mention Graham Priest in my original post. I would certainly not accuse him of any lack of intellectual substance.

    I wish you well too. It is Deleuze’s birthday tomorrow, and we can celebrate in our different ways.

      • Why not extend your accusation to Priest, too? You claim:

        “The mathematical theory of Cantor and others is precise, beautiful and totally non-mysterious. It is a natural and correct completion of the mathematics that is adequate to everyday life, the life in which we deal in finite numbers of objects. It may be a little counter-intuitive at first, but if we dive into the theory, our intuitions soon catch up. It has nothing to do with limits to our thought.”

        Well, Priest is the one here who is claiming that Cantor’s argument has everything to do with “the limits to our thought” and its obvious that Bell’s post assumes this work as a backbone to his own (perhaps even to the point of incurring into some distortions of Deleuze’s own thought, as Tony has remarked above). While, admittedly, I haven’t checked this particular work by Priest, the few reviews that I read were enough to see that it is constructed around the idea of drawing the analogies you apparently think are forbidden. So, why only one of them classifies as a dishonest, obscurantist thinker, when both of them are practicing the same procedures around the same concepts?

  6. I have not looked at Priest on this point. But I have read some of his other work, and he regularly states important positions very clearly.

    By contrast, take a sample of Deleuze’s writing, and try to extract some propositions that are clear, significant and plausible. They need not be incontrovertible – that would be far too tough a test. But they do need to be clear, significant and plausible to someone who is not already committed to the Deleuzian way of thinking.

    Take, for example, the quotation above: it is fairly typical, although it is far from the worst that could be found: “What characterizes the nondenumerable is neither the set nor its elements; rather it is the connection, the “and” produced between elements, between sets, and which belongs to neither, which eludes them and constitutes a line of flight.”

    This appears to say that the operation of set union, and/or the fact that several elements constitute a set, is or are something that we can call a line of flight. That is entirely vacuous. The context might help a bit – I do not have this text to hand. But my experience of Deleuze is that the context does not usually help much at all. In any case, sentences out of context should generally mean something, even if the context is needed to get their full meaning.

    I hope this makes clear why I think Deleuze lacks intellectual substance. I know that many commentators here will disagree, and I do not expect to convert anyone.

    • I’m afraid you’ve entirely missed the point of Cantor’s diagonalization argument. For Cantor the nondenumerable is not, as Paul was correct to remind me in an earlier comment, constructed, and hence diagonalization is not ‘the operation of set union’ nor does it come down to claiming that ‘several elements constitute a set’ as you say. Key to the diagonalization argument is that it does not involve ‘the operation of set union’ but rather entails showing how there is a set that cannot be constructed of sets or even the union of all sets but is diagonal to these sets, or it is ‘the “and” produced between elements, between sets’ to cite Deleuze again, and it is for that reason nondenumerable since it does not belong to to any of the infinitely countable sets. I greatly simplify Cantor here, but the point of my post and of the quote you find vacuous is that lines of flight as Deleuze understands them are much like Cantor’s understanding of diagonalization, although Deleuze will push this in an ontological direction whereas Cantor does not (but that is a point I make in another post). I understand that Deleuze is quite challenging to read. His conceptual inventions can be quite unforgiving. But being difficult to read or understand does not necessarily mean it is without intellectual substance. Cantor and Cohen are difficult to read, and there’s a fair amount of terminology within the analytic tradition as well. I’ve recently had a quite favorable review of Deleuze’s Hume in Hume Studies by Martin Bell (no relation), who is a noted Hume scholar. In this review he admits that there is an immense conceptual apparatus that comes with Deleuze, but he nonetheless finds that Deleuze, and Hume obviously, have much to say that is intellectually substantial and important. In sum, there are arguments that are vacuous because they are devoid of argumentation, or argumentation that makes no sense, and there are arguments that one finds vacuous because they fail to understand them and take this failure to be a failure of the arguments themselves. Although you would like to claim that the first option is the one that applies to Deleuze, I would say it is the second one that applies to you. I’d love to have a dialogue with you about Deleuze, but if you stubbornly refuse to grant that there might even be something there in Deleuze’s work from which we might begin a conversation then again I don’t see where this can go. But then my question is, why contribute to this post if you are not interested in participating in the conversation with those such as myself who do take Deleuze to have something to say?

  7. I am aware the diagonalization is not a matter of set union. I was simply trying to make some sense of the DG quotation. I am grateful for your interpretation, Jeffrey, which does help to make sense of the quotation, although I cannot see that your interpretation gives a proper role to the word “and”. (I am assuming that nothing has been lost or gained in translation from the French.) “Line of flight” still sounds like a bit of hand-waving, but no matter.

    If, however, DG meant something along the lines of your interpretation, they should have said it. There simply is no excuse for their level of obscurity. Books can be justifiably challenging by virtue of their subject matter: both Kant and physics texts are like that. Maybe DG (and D on his own) are justifiably challenging because of the nature of the subject matter. But there remains no doubt in my mind that they have added a thick layer of wilful obscurity, which makes it impossible to tell whether their work should be challenging. In taking this view, I am of course assuming that their aim is to state some truths.

    I am grateful for your offer of a dialogue, but I could not participate unless I first had a translation of the relevant works, not into English but into plain language which made the meaning transparent. That would take too long.

    As to why I contributed at all, I found the post via another blog, and got momentarily excited by the contrast between beautiful mathematics and something rather different. I shall now withdraw.

    • Richard,

      You say: “If, however, DG meant something along the lines of your interpretation, they should have said it. There simply is no excuse for their level of obscurity. Books can be justifiably challenging by virtue of their subject matter: both Kant and physics texts are like that. Maybe DG (and D on his own) are justifiably challenging because of the nature of the subject matter. But there remains no doubt in my mind that they have added a thick layer of wilful obscurity, which makes it impossible to tell whether their work should be challenging. In taking this view, I am of course assuming that their aim is to state some truths.”

      I don’t find Deleuze’s works particularly obscure, though it’s obvious that he (and Guattari) coined an unusual style. My question to you is this: could there not be a philosophical point behind this stylistic choice? Instead of assuming that they adopted such a style in order to gain fame, money, or other personal benefits that presumably would follow from their greater profits, why not assume that they did it because they were philosophically compelled to do so? Notice that I’m not saying that their position was the correct one. Maybe it wasn’t. Maybe this wrong position justifies you cautious distance and unwillingness to read or consider their works. Even so, isn’t more charitable, however, to assume that D&G honestly thought that their stylistic choice was guided by philosophical considerations, insetad of assuming from the outset that they were cretins looking for profit?

  8. I was going to withdraw, but so long as my contrary views attract attention, I should continue to respond.

    I do not know why DG chose their style. It had not occurred to me that they might have done it either for fame, or for money. It is not how I would pursue either objective. So yes, they may have thought that their philosophy required their style. If so, then it seems to me very likely that they were mistaken, and that someone, if only the editors at their publishers, should have told them so. What proportion of really worthwhile philosophers from the past have been driven to such a level of obscurity? And how likely is it that the thought of DG is so revolutionary that reasonably straightforward language would be unequal to the task of expressing it? (The adjective “straightforward” is not meant to exclude the use of technical terms. But they should have stable and clearly-defined meanings.)

    • Richard,

      At least you agree that D&G were not charlatans, as Sokal and Bricmont, not to mention Bouveresse, seem to suggest. If this is so, I think you should drop references to Sokal in contexts like this one, as you apparently do not share his diagnosis of continental philosophy. Anyway, D&G are certainly not alone in thinking that certain subject matters require a different stylistic treatment. Notable predecessors include Nietzsche, Hegel, Kierkegaard, and Wittgenstein. This does not mean that they necessarily thought that such a subject matter was “revolutionary” in any sense. Why would they? You seem to assume that there is a correlation between “being revolutionary” and “being difficult to express”, but that need not be the case. Maybe there are things which are ordinary, but precisely in being ordinary require a different stylistic treatment than most philosophers are used to (that seems to be the position of Wittgenstein). But that’s irrelevant. What is relevant is simply that they probably thought that their philosophical position, revolutionary or not, required that stylistic treatment (Incidentally, I mentioned only the cases where the style is more readily apparent. Most authors of the classical period, including Hume and Rousseau, were worried with rhetorical questions and their texts show a very conscious choice of text structure and vocabulary, even at the level of syntax. I think this lack of care for rhetorical questions is a contemporary phenomenon).

      As I said, it’s possible that they were mistaken, though it’s hard to judge this a priori. It’s also possible that D&G’s position in relation to their style make it difficult for you to engage with them. That’s fine. D&G were pretty clear that they would not appeal to everyone. Should this be a motive to denigrate those who engage with them?

  9. I do not know whether or not D and G were charlatans, in the sense of peddling doctrines that they knew to be worthless. But I shall continue to regard Sokal and Bricmont as having done the intellectual community a great service. They drew our attention to how much nonsense is talked, and how much science is misused, by several philosophers. Whether those philosophers do so deliberately or accidentally, is beside the point. And their comments on D and G’s words, leaving aside any motive that D and G may have had, are, in my view, entirely apposite.

    I mentioned revolutionary thought in the charitable search for an excuse for obscurity of language. The most revolutionary thought might not find the language to express it ready to hand. I did not say that it was the only reason for writing in a distinctive style. I agree that Nietzsche, Kierkegaard and Wittengstein chose special styles, and did so for good reasons. They did not choose especially obscure styles. Each sentence makes sense. It is a challenge to assemble a single philosophical system to attribute to each thinker, but it is quite clear that, certainly in the case of Nietzsche, that is not what you are supposed to do. Hegel is different. He, like D and G, used an unjustifiably obscure style.

    Incidentally, Daniel, it is all too easy accidentally to attribute to people, positions which they have not stated. You suggested in your previous post that I assumed that D and G were cretins looking for profit. In your latest post, you suggest that I denigrate those who engage with them.

    • Richard,

      I apologize if I’m misreading you. I was misled by your mention of Sokal, who does seem to imply that the intellectuals he has examined, among them D&G, are close to charlatans (look at the title of his book with Bricmont!). And your accusation that Deleuze lacks “intellectual substance” at least appears to imply that those who are engaging with his thought are pursuing chimeras. But, if those claims do not represent your position, as I said, I would be happy to retract them.

      That being said, I wonder what is your measure of “obscure” here. I ask that because I don’t think that D&G are more obscure than Wittgenstein, or that Hegel is more obscure than Kant. That’s why I find such charges of “obsucrantism” baffling. To use another example, Crispin Wright has recently accused John McDowell of being wilfully obscure, and even suggested that he be excluded from the analytical philosopher’s club (a suggestion that McDowell, rightly, found risible). McDowell replied that he wrote as clearly as he could, that he thought his book was very straightforward, even if the style was a bit unusual. How are we to judge?

    • That Hegel ‘s style is not straight forward is beyond doubt. That it is unjustifiable is debatable, especially if you consider the political climate he wrote in..

  10. Different people may judge obscurity in different ways. I use two criteria. First, does each sentence make sense? Second, could I state in my own words what the author had said, listing some specific and worthwhile propositions, and be confident that I had not just invented what I thought he should have said?

    McDowell is an interesting test case. I place him on the right side of the line. He is like Kant: his work is challenging because the subject matter is difficult.

    Those who engage with Deleuze may well obtain the useful result of worthwhile thoughts of their own, which can then be published and which may deepen our understanding. Interaction with others often have that effect. They may also contribute to our understanding of intellectual history. But if they conclude that Deleuze himself said certain things and that those things were worthwhile, I would ask them to reflect on whether they have really found those things clearly in the text, or whether they have had to read their own thoughts into the text in order to extract some definite meaning. In particular, could they just have easily have extracted some other, contradictory, meaning? And could they tell when two pieces of writing in the Deleuzian style did contradict each other?

    Now I really am going to withdraw from this discussion. Please feel free to have the last word. But please also re-read Sokal and Bricmont with the same open mind as that with which you might like me to approach Deleuze.

  11. I’m an amateur at this as I’ve been forced by a fixating learning disability to route around and compensate for my broken short term memory with high information density contexts. It allows me to connect things that refused to click the straightforward way.

    The result has been that I can read Deleuze without trouble while other supposedly simpler material makes me rant and rave about articulate morons and their ability to draw people toward the vacuum between their ears.

    The key part is that high detail isn’t the same as high information context.

    Let me try to describe the virtual in a different way and see if I can help. It’s burning the candle at both ends, tying knots with both hands moving in concert rather than using one as a pivot or reference point. It’s the sort of approach that tries to keep conflicting ideas dancing long enough for the audience to see that the dance is the whole point. And also that the conflicting ideas are at worst patsies in the game of illustrating the dance and at best they are the infinitesimal endpoints of the tight rope walkers baton, only there to define the tensions necessary to represent a much larger picture.

    This is the ingenuity of beginning with the problematic rather than the infinite series of adjustments required when using axioms as a foundation. I dabble a bit in trying to reintroduce mechanics back into quantum mechanics and the problematic approach has led me to believe that a complete picture of QM requires not one but 2 scales of quantization, namely the quantization of measurement and then the quantization necessary so that the first quantization is possible to do in the first place. It’s easy enough to say I have 5 items, but to have a fruitful discussion I have to say I have 5 apples and immediately we need a scale of quantization that allows us to differentiate between apples and oranges.

  12. Teasing out the relation between Deleuze and Priest would be a similar feat to accounting for the former’s relation to Hegel. I do not envy you this task.

    Here is one passage from The Logic of Sense which I find interesting and which is relevant here:

    “[W]e cannot invoke the contradictory character of the insinuated entities [i.e. paradoxes], nor can we say that the barber cannot belong to the regiment. The force of paradoxes is that they are not contradictory; they rather allow us to be present at the genesis of the contradiction. The principle of contradiction is applicable to the real and the possible, but not to the impossible from which it derives, that is, to paradoxes or rather to what paradoxes represent.” (p. 74-5)

    Perhaps Priest’s noneism is more relevant here than his dialetheism?



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