When mathematicians and philosophers talk about the beauty of « Cantor’s paradise » of higher and higher infinities or transfinite numbers, we should understand this in terms of Dante’s Paradiso as abstract poem. Dante’s ascent finishes on the poet’s vision of « the love that moves the Sun and the other stars ».

This would allow us to establish the link between Dante’s ascent to paradise and Plato’s ascension to the Ideas. It also highlights that Dante’s poem is incomplete, because after this ascent to infinite abstraction we need to descend to finite life again.

Otherwise we do not include the finite incarnated Dante, the poet who has visited these realms and who has returned to tell us of them. The poem needs to be completed by a fourth part: the RITORNO.

Badiou’s new book, THE IMMANENCE OF TRUTHS, is due be published in French in two weeks. Badiou starts his book with the analysis of the operators of finitude as the basis of the system of oppression. This is his INFERNO: the hell of living under the forced regime of finitude. The modern form of finitude is what he calls « covering ». It is based on the idea that the universe of sets is constructible.

The book then ascends, taking us on a voyage through the different types of infinity: inaccessible, resistant to partition, complete, and approaching the Absolute. In this ascent we are in Purgatory. The turning point comes when Badiou expounds a theorem allowing the « defeat » of covering. He then moves up the hierarchy of higher and higher infinities, mounting to the Absolute, « V », the class of all sets, which is not itself a set.

The analogy with Dante’s DIVINE COMEDY is striking, except that Badiou’s IMMANENCE OF TRUTHS provides the missing fourth part, as it accomplishes the re-descent, not only mathematically but also philosophically, into finitude « touched » by the infinite. Each of the four truth procedures (art, science, love, politics) are examined to determine their « index of absoluity ». This is the point in which, while still remaining within the regime of finitude, the « work » produced by each truth procedure can be said to touch the Absolute.

This idea of a missing fourth part has been with me since I read the DIVINE COMEDY in the light of Hubert Dreyfus and Sean Kelly’s notion in ALL THINGS SHINING that Dante’s poem is onto-theological precisely because it ascends to abstraction and remains there, unable to return from transcendence to concrete human life.

Another component of my response comes from my attending Deleuze’s classes on the cinema. In these seminars Deleuze emphasised the danger of a possible dualist dead-end in the move towards complete abstraction and he favoured the works of transformative re-descent, that comport what he called « re-injection », or the inclusion of abstraction into re-worked mundane forms.

Note: I am grateful to a discussion with Trent Knebel for helping me to clarify my ideas.

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JUNG/DELEUZE (3): re-naming, purging, and imaging

My hypothesis is that just as James Hillman tried to resolve Jung’s conceptual/theoretical insufficiencies (scientism, biologism, abstract universalism, dualist metaphysics, binary oppositions) by means of his « archetypal » turn Deleuze responded to those same inadequacies by a « structuralist » turn.

Deleuze’s « Lacanism » in DIFFERENCE AND REPETITION and LOGIC OF SENSE was an experiment in trying to preserve Jung’s insights by shedding their dogmatic shell. This solution proved unsatisfactory, and Deleuze quickly abandoned it.

Deleuze’s turn to Guattari was in essence a return to himself. Unfortunately, Guattari had been more Lacanised than Deleuze before rejecting Lacan’s theoretical apparatus. Hence the ambivalent pronouncement’s about Jung ( favourable and unfavourable) in the D&G collaborations.

Despite the ambivalence of their pronouncements about Jung, the use of the Jungian conceptual apparatus continues through the D&G collaborations, except that it has been renamed, purged of biologistic and metaphysical elements. Notably, « Anima » is called « becoming-woman », the Self is called « body without organs », and the Shadow is often referenced as such.

Beneath the level of explicit pronouncements (D&G’s ambivalence about Jung) and the conceptual apparatus (Jungian concepts re-named and purged), there is the imagistic level, which is pure Jungian-Hillmanian imaginative theorising.

This imaginal level becomes even more explicit and emancipated, to be explored for itself, in Deleuze’s cinema books, CINEMA I & II, where an image-ontology is proposed and implemented, with no relation to Freudian or Lacanian concepts and symbols.

The Lacanian « codes » of the imaginary and the symbolic are avoided as overly reductive « slowing down » of Jungian concepts. The plane of immanence of the image (the imaginal plane) is far vaster than a supposedly inherently binary « imaginary ». Jung’s « symbolic » life pluralises Lacan’s notion of the symbolic. These deconstructions are already implicit in the concepts of becoming-animal, becoming-woman, becoming-molecule (Alchemy!) in A THOUSAND PLATEAUS.

Translations that are not sensitive to this Jungian resonance misrepresent the text, e.g. in English we have:

« This is the central chamber, which one need no longer fear is empty since one fills it with oneself » (Deleuze, FOUCAULT, page 123).

In French the last part reads: « puisqu’on y met le· soi »: « since one fills it with the self ». In the translation the concept of « le soi » « the self » has been omitted in favour of the less Jungianly connoted « oneself ».

In the FOUCAULT book the re-naming continues, e.g. Jung’s « process of individuation » (realisation of the Self) becomes the process of subjectivation. The link between the two concepts (individuation/subjectivation) can be seen in a note:

« The question would be: is there a Self or a process of subjectivation in Oriental techniques » (FOUCAULT, 148).

Deleuze’s FOUCAULT is the great book on the Self, there are multiple references to it on almost every page in the second half of the book (the self, techniques of the self, relation to self.

This omnipresence of the concept of « self » is a little obscured in the English translation, which translates « rapport à soi » as « relation to oneself ». This rendition is linguistically correct, but conceptually unfortunate. A whole continent of Deleuze/Jung convergences is partially submerged and needs to raised again.



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JUNG/DELEUZE (2): Shadow, Anima, and Self as conceptual personae

This is a sequel to JUNG/DELEUZE (1): schizophrenia, individuation, and alchemy.

Anyone who has read Jung even a little can see that Deleuze’s writing up to DIFFERENCE AND REPETITION is pervaded by Jungian terms, concepts, and symbolic approach: the Shadow, the Anima, and the Self are all present by name.

Deleuze’s early work on Nietzsche, including his book NIETZSCHE AND PHILOSOPHY and the smaller NIETZSCHE, draws on Jung, for example explicitly calling Ariadne the Anima. Some quotes:

« Ariane (et Thésée). — C’est l’Anima… quand Dionysos-Taureau approche, elle apprend ce qu’est la véritable affirmation, la vraie légèreté. Elle devient l’Anima affirmative, qui dit Oui à Dionysos » (NIETZSCHE, page 44).

« Ariane (and Theseus). – This is the Anima…when Dionysus-Bull approaches, she learns what true affirmation is, true lightness. She becomes the affirmative Anima, who says « Yes » to Dionysus » (my translation).

We can see here the early stages of Deleuze’s concept of « conceptual personae »

« Ariadne is Nietzsche’s first secret, the first feminine power, the anima, the inseparable fiancée of Dionysian affirmation » (Deleuze, NIETZSCHE AND PHILOSOPHY, 20).

It is important to note that in Deleuze’s French use of the term the « Anima » begins with a capital letter, making it more explicitly a reference t Jung. The passage continues:

« But the infernal feminine power is altogether different; negative and moralising, the terrible mother, the mother of good and evil, she who depreciates and denies life ».

This analysis explicitly situates Nietzsche’s thought and life in a Jungian matrix.

According to Deleuze Nietzsche projects elements of his personal inner (or « intensive », see below) drama onto the characters in his immediate environment, transmuting them into conceptual personae.

Cosima is the Anima, Wagner is the Shadow,  Nietzsche’s mother is the « Terrible Mother », his sister is the negative Anima, and Nietzsche himself is Dionysus, both as fragmented unconscious and as individuated « affirmative » Self.

« Ariadne, abandoned by Theseus, senses the coming of a transmutation which is specific to her : the feminine power emancipated , become beneficient and affirmative , the Anima » (NIETZSCHE AND PHILOSOPHY, 187).

Deleuze here also uses the Jungian/alchemical term of « transmutation ».

« This is why the Dionysian universe , the eternal cycle, is a wedding ring, a wedding mirror which awaits the soul (anima ) capable of admiring itself there, but also of reflecting it in admiring itself » (NP, 187).

The « wedding » terms are a reference to the Jungian/alchemical concept of the alchemical union, the conunctio.

Even more overtly Jungian:

« The labyrinth is a frequent image in Nietzsche. It designates firstly the unconscious, the self; only the Anima is capable of reconciling us with the unconscious , of giving us a guiding thread for its exploration » (NP, 188).

Note the explicit use of « the self » as one of the names of the unconscious.

For the use of the Shadow:

« The shadow has lost its goal, not because it has not reached it but because the goal which it has reached is itself a lost goal » (NP, 170).

On the Shadow and the transforming power of the light of consciousness:

« The shadow is the activity of man, but it needs light as a higher instance; without light it vanishes; with light it is transformed and disappears in another way, changing in nature when it is midday » (NP, 170).

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DELEUZE/BADIOU/INFINITY (2): Cantor and the Absolute

The popular stereotype about Cantor is that he invented transfinite arithmetic, but that he was also mentally deranged and theologically obsessed, in a naive autodidactic sort of way. However, Cantor was quite philosophically literate, and I find it interesting that he had a well-worked out philosophy of the infinite, and that he read Spinoza and Leibniz very closely. His discoveries led him to distinguish different « sizes » of infinity, allowing him to re-read the classical philosophers from a new point of view.

« Cantor’s inquisitive « how infinite » was an impossible question. To minds like Spinoza and Leibniz, the infinite in this absolute sense was incomprehensible, as was God, and therefore any attempt to assign a basis for determining magnitudes other than merely potential ones was predestined to fail » (GEORG CANTOR His Mathematics and Philosophy of the Infinite, Joseph Warren Dauben, 123).

Unlike Badiou, Cantor does not identify the Absolute with « V », the class of all sets. On the contrary, he distinguishes them for theological reasons, that I reject, and so subsumes the transfinite to the regime of the One.

However, I think that there is still good reason to distinguish the Absolute from V, and that Badiou does so implicitly in that he carefully distinguishes philosophical concepts as metaphorisations or poetisations of mathematics from mathematics itself.

(This is in reference to Badiou’s thesis that philosophy is a « poetisation » of mathematics).

One could argue from a Badiousian perspective that Deleuze’s concepts of infinity, as occurring for example in WHAT IS PHILOSOPHY? (where the words « infinite », « infinity », « absolute » – and their synonym, the « outside » and the composite terms « absolute horizon » and « absolute velocity » occur on nearly every page) are too qualitative, too vague and imprecise, remaining too intuitive and insufficiently theorised, too close to the « poetic » end of the spectrum.

In contrast, from a Deleuzian perspective, Badiou’s concepts, which are based on the mathematical hierarchy of infinite cardinals, are insufficiently philosophical. While still « poetic » they are much closer to the mathematical end of the spectrum, and so represent a slowing down of the plane of consistency. However, even this critical term of « slowing down » is itself intuitive and poetical.

See also:

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I think Badiou explains the sense and interest of his definition of attribute quite well in the conclusion (5), where he invokes the perception of an unprecedented intensity in the finite situation as a perception of the infinite, by the human (inevitably finite) mode.

His mathematical explanation in (4) of what it means to be an « aspect » of God/Nature/ the Absolute/Substance is an amazing example of translating from one type of discourse into another.

Deleuze’s and Badiou’s systems, though different, can be seen as both « right » (i.e. useful, stimulating, and insightful) and they are often convergent especially as Badiou’s evolving system brings him ever closer to Deleuze (although they will never coincide).

Badiou’s « Immanence of Truths » project begins and ends in finitude, after ascending to the Absolute in the middle via the mathematics of higher infinites. The Spinoza lecture That I have just summarised represents a fragment that has its place in this middle part.

Badiou’s new book « Immanence of Truths » will be published in French at the end of this month, but its general outline has been covered in his last seminars (2012 – 2017) and in a few other videos dating from that period.

I think that the pluralism of attributes accessible to humans that Badiou expounds is an interesting step, that is close in spirit to Deleuze and Guattari’s pluralism of attributes in A THOUSAND PLATEAUS, where attributes are equated with types or genuses of bodies without organs:

« After all, is not Spinoza’s Ethics the great book of the BwO? The attributes are types or genuses of BwO’s, substances, powers, zero intensities as matrices of production. The modes are everything that comes to pass: waves and vibrations, migrations, thresholds and gradients, intensities produced in a given type of substance starting from a given matrix » (page 178).

Deleuze, like Badiou, wants to get round the dualist limitation to two attributes, and push Spinoza towards a full blown pluralism. At the same time they want to fuel this pluralism with the mathematical conception of different sized infinities, which was not available to Spinoza.

Cantor, as discoverer of different sized infinities, expresses an intermediate position as he was very drawn to Spinoza’s system but felt that it did not provide sufficient room for freedom. Interestingly, Cantor talked philosophically of the « Absolute » but, unlike Badiou, did not identify it with V the class of all sets. Cantor himself regarded V as something like the intellect of God, or part thereof. Unfortunately, too little of Cantor’s philosophical and theological reflections have been published in English.

It is true that we can find the concept of an infinity (Substance) of infinities (attributes) in Spinoza, but it remains a « qualitative » concept, and we come across the same dilemma that we find in comparing Deleuze and Badiou.

Deleuze’s concept of infinity is qualitatively differentiated, and I have shown, in the light of Badiou’s set theoretical explications, that it can be specified into at least four types of infinity (inaccessible, resisting, affirmative power, horizon) but it can be seen as lacking articulation compared to Badiou’s more complete differentiation.

On the other hand, Badiou’s own articulation can be seen as a scientistic reduction that requires the constant metaphorical commentary to « re-philosophise » it. I think that each option clarifies the other, and I feel no necessity to choose between them but am willing to maintain them in « divergent » dialogue.

Strictly speaking Badiou proposes four modes of access to the Absolute: love, art, science, and politics. That’s enough to allow individuals and groups to find their own access or for teachers of « true life » to find their own style of pedagogy. The problem comes when Badiou singles out one of those modes of access, science, and proceeds to further specify it as mathematics, as synonymous with « ontology ».

I do not think that Badiou has demonstrated the necessity of this last step, as step that Deleuze is unwilling to take. However, I do think that he has demonstrated the heuristic and pedagogical power of mathematics, not for an élite (of Platonic « guardians ») but for those who are in sufficient sympathy with that mode of approach (mathematics) or who are capable of developing their sympathy further.

This pedagogical outcome of acquiring sympathy for (some) mathematics is interesting, as one of Badiou’s major themes is empowerment: discovering that we are capable of something we did not know we were capable, and could not foresee.

Note: I am indebted to a conversation with Frank Dixon for helping me to clarify my ideas.

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BADIOU ON SPINOZA (5) : what is the interest?

Continuing with the summary of the video, from 54m to  1h08m.

According to Badiou this investigation is of much mathematical interest because it is an exploration from the inside of the characteristics of the Absolute.

This Absolute cannot be « injected » into the theory because it is the place of the theory, it is not itself a set. We can work on interior attributes that though they too are not sets are more approachable because they are defined by a particular property (e.g. « is an ordinal number », or « is a cardinal number », etc.)  that is very general but specifies something in the Absolute.

The Absolute can thus be said to express itself inside itself, immanently, not only at the level of quantitative immanence (the class is inside it, and in that sense « smaller »), but at the level of the truths that it deploys, i.e. in the effective definition of the properties of a multiplicity. The Absolute is thus repeated in the attribute but also displaced, because there will be cases when j (x) ≠ x. So the attribute represents the Absolute from the inside while differing at the level of truths.

This means that the theory of the general forms of the multiple can admit correlations or internal movements which slightly displace a particular characteristic of the truths from the point of view of their application.

For example in Spinoza we can affirm that extension and thought are the same, because they have the same order-structure, but this does not explain how they are also different. It gives us a structural identity, without explaining how a thing and an idea are different. But in Badiou’s schema we have an explanation. It explains how the truths concerning a form of the multiple can manifest themselves in an attribute, in integrating a principle of differentiation at the same time as a principle of identity. One may say that this schema is a dialectical version of Spinoza’s attribute.

This takes care of Substance, attributes, and modes (elements x and j (x) are modes), but what about infinite modes? Here there is an absolutely magnificent theorem. We cannot demonstrate with the axioms of set theory that attributes of the Absolute Place exist. So perhaps there is only V the Absolute Place, perhaps it never expresses itself in different attributes. However, as we have seen this creates a problem for distinguishing thought and extension.

The theorem states that if there exists a non-trivial elementary embedding, then there exists a new previously unknown type of infinite cardinal, larger than all those we have defined up to now, called a measurable cardinal, i.e. there exists what mathematicians call a « witness » to the existence of attributes. Inversely, it can be demonstrated that if such a cardinal exists, then there exists a non-trivial elementary embedding.

So it has been demonstrated mathematically, which in Spinoza remained at the level of a brilliant but vague idea, that there is a constitutive link between something infinite, and the existence of attributes.

This theorem is one of the great masterpieces of modern set theory. It shows that from a structural hypothesis (the existence of a correlation internal to the Absolute Place under the form of an attribute), if it is valid, one can deduce an existential witness, i.e. the effective existence of a form of the multiple previously unknown.

This is an admirable political theorem. If you manage to think or to edify, inside or on the surface of the world as it is, something that is the possibility of expressing it, but within a difference, because j (x) is not always equal to x, if you manage to introduce into the world something that does not destroy it but which preserves it with a difference, and this difference can be very great, if you really do something without destroying, which is neither in the figure of destruction nor in that of simple preservation, something which is a differentiated internal expression, then a masterpiece will bear witness to it, i.e. a new existential infinite, a creation, something that did not previously exist.

So it establishes the political (in the widest sense) link between the structural actions of politics, i.e. its effective deployment in the possible dimension of an attribute of what is, and something which is a witness of this effort, something which is an existence of an unprecedented intensity. The reverse is also true, for the mathematician. If you are in the feeling of an unprecedented intensity then there is probably somewhere a structural effect that is both expressive and differentiated.

All this, Spinoza together with modern mathematics, teaches us the extreme value of the notion of attribute. That is to say, one must not allow oneself to be confined within the idea that there is only the alternative between conservation and destruction.

The attribute is an expression of the absoluteness of Substance totally differentiated from every other. Spinoza insists on this, he affirms that two attributes have nothing in common. Even if we are more prudent, and we maintain that an elementary embedding produces more localised differentiations, nonetheless it is a matter of difference. If you have the testimony of the existence of a form of life of an unprecedented intensity, K the giant cardinal, then it is the case that this structural modification has occurred.

There is a link of witnessing, and so a subjective link, between that which attempts to express absoluteness inside an uncontrolled situation and the creation or appearing of an unprecedented form of intensity.







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BADIOU ON SPINOZA (4): mathematical definition of attribute

Continuing with the summary of the video, from 31m to 54m.

Badiou has just identified the Absolute with V, i.e. the class of all sets (possible forms of the multiple), which is not itself a set (as then it would have to be a member of itself).

Badiou remarks on the peculiar aptness of certain mathematical symbols, such as V, as if chosen with an unconscious philosophical intelligence. The choice of the symbol « V » for the class of all sets opens on to the acknowledgement that the only veritable Absolute is the void, « the place of all places ».

The Place posited as the place of all the forms of the multiple corresponds to substance in Spinoza’s system, the sets (the possible forms of the multiple) would correspond to the modes (particular things), and one is entitled to think that certain classes correspond to the attributes.

So we would have

1) The Absolute, Substance – the general Place that allows us to explore set theory as the theory of the forms of the multiple

2) Modes -the particular forms of the multiple that we manage to know and to explore, and which are also the forms of the multiple of empirical objects

We may ask ourselves how mathematics can apply to empirical objects, one has only to see that mathematics is the general theory of all the possible forms of multiplicity. So it is only to be expected that it will come up in the study of particular multiplicities.

In studying a particular multiplicity one begins by fixing the possible form of multiplicity and so convoking the Absolute in the thought of a singular, modal, figure of existence.

The problem arises of what corresponds to the attributes. An attribute cannot be a set, for then it would be a mode, a singular thing. It cannot be the Place, the Absolute, or the class of all sets, for this corresponds to Substance. An attribute must be a class that is not a set. If we try to think the attribute as something that expresses Substance without being identical to it, then we can clarify what a class is.

3) Attributes – an attribute is a class belonging to V that without being a set and without being V has properties very similar to V’s properties, and so « expresses » V, i.e. a class in V that expresses the most important immanent possibilities of V.

Take the expression x V.

This is strictly an incorrectly formed expression, as can be affirmed only between sets, and V is not a set, but a class. It can be read philosophically as

« x is a possible form of the multiple, identifiable as a set, and from this point of view figures in V, the total collection of all the possible forms of the multiple »

If we want a Class C similar to V it will be composed of sets, and will itself be a part, or subclass of V: C V.

Given the definition the question is can there exist something like C, a class contained in V, composed of sets of V, that would be similar to V. What could be the conditions of such a resemblance? The resemblance cannot be quantitative, because C is a piece of V

A set x will have certain specific properties that distinguish it from other sets (finite, infinite, well-ordered or not, etc.)

Let φ (x) mean x has the property φ.

C can be said to « resemble » V if whenever a set x in C has a property φ in V it has a corresponding property in C. In this case we may say that C expresses V. The property delegated by the Absolute can be found in the attribute, and the attribute will in that sense « express » the Absolute, in Spinoza’s sense.

Between C and V there must exist a certain relation. It is not enough to say that C is in V. Similarly in Spinoza to distinguish two attributes it is not enough to say that both are « in » Substance. The infinite multiplicity of the attributes requires that the representation of the Absolute by the attribute be also a transfer of the characteristics of the object under discussion from Substance to the attribute, internal to Substance itself.

For C to resemble V, between V and a subclass C there must exist a correlation that is stronger than the subclass relation, which is too weak a relation to differentiate between attributes. Let us call this correlation j (called in mathematics an « embedding » of V in C, an embedding of the Absolute in the attribute). This embedding is « elementary » because it correlates elements of V with elements of C.

We can now specify our correlation: if we have φ (x) in V then we have φ [j (x)] in C. Thus the correlation of C and V will be not only quantitative (C V) but also truth-preserving as what is true in V, φ (x), is projected as true in C, φ [j (x)].

This definition of attribute is interesting only if the correlation j is non-trivial, i.e. if it does not correspond to a simple relation of identity, where we would have j (x) = x for all x element of V. We need that at least for one point a we have j (a) ≠ a.

We will then be able to differentiate attributes from each other by means of the system of their differences from the absolute, given that the truths will be preserved.

The attribute amounts to the introduction of a principle of novelty and differentiation, as the attribute introduces novelty immanent to Substance at the same time as it expresses Substance. This is the dialectic of difference as supreme form of identity. It is because the attribute is different that it is particularly expressive of the absolute identity of the thing.

To sum up, we have the formal table of the attribute:

V – the Absolute, general form accessible to mathematical thought of the class of configurations of the multiple

C V – C a class internal to V

x V ↔ φ (x) – x an element of V defined by the property φ

j – the correlating relation, or embedding

j (x) C – the j-correlate of x in C, such that φ [j (x)]

j (x) ≠ x, for some x – j is non-trivial (it is not equivalent to the relation of equality)

C is an attribute if all these conditions are satisfied, i.e.

C is interior to Substance, it is immanent to Substance yet quantitatively different from it (it does not cover all of it), and it is in expressive truth-preserving correlation without it preserving truth by repetition but truth preserved in difference by a systemic operation of correlation of the two levels.

In conclusion, we have reached the following definition of an attribute:

an attribute of the Absolute is a class of this Absolute such that between the Absolute and this class there exists a non-trivial elementary embedding.

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