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.