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.