Badiou has a deep problem, carried over from Althusser’s structuralism, in regards to the relation between stasis and change. (I have commented on this problem at a more general level, without mentioning Badiou, in terms of the relation between diachronic and synchronic ontology, in my article IS ONTOLOGY MAKING US STUPID? However, I do not think that Althusser is in a position to give Badiou lessons in avoiding the idealist tendencies implicit in structuralism.
Much post-Althusserian thought was devoted to finding a principle of de-stabilisation of structures tied to subjectivity: Derrida’s writing, Kristeva’s subject-in-process, Lyotard’s drift, Deleuze and Guattari’s group-in-fusion and later their process of subjectivation, and Foucault’s care of self. Badiou’s project is clearly inscribed within this field of responses to the perceived excessive determination of the subject by the structure and to its link with the problem of change.
Another problem that comes with inheriting from Althusser’s philosophy is his structuralist scientism. Badiou attempts to mitigate this excessively static and monlolithic privilege given to science in several ways:
1) by characterising science as as a site of truth production, rather than simply of knowledge production. One must remember that for Badiou truth is linked to change, whereas knowledge is linked to stasis.
2) by asserting the contemporary existence of four figures of truth production (science, love, politics, art).
3) by incorporating the subject within the scientific process of truth production, and more generally within all of the processes of truth production.
4) by emphasising that science as a truth procedure exists only in diverse “crossings” with other truth procedures, some of which are legitimate (this acceptance of the legitimacy of crossings distinguishes Badiou’s position from Althusser’s policing of frontiers): this theorisation of crossings can be found in Badiou’s latest seminar, class of October 19th 2015.
On the queston of the priority given to axiomatics by Badiou, this is an ambiguous move in terms of the stasis/change (or synchronic/diachronic) distinction. “Axiomatics” refers to a formal structure and so is a force of stabilisation, but also to the (subjective) decision to adopt certain axioms. This priority of axiomatics comes under fire from François Laruelle, who proposes “quantum” thinking in its place. Laruelle’s quantum thinking is, at least in intention, more process-oriented, than structure-oriented axiomatics. Yet this critique ignores Badiou’s more recent work-in-progress on the “immanence of truths”, an enquiry into how truths (and not knowledge) modify and transform worlds from within.
“Quantum” thinking as used by Laruelle explicitly involves a principle of “indetermination in the pre-first instance” (this is Laruelle’s quantum reformulation of “determination in the last instance”. It belongs, despite Laruelle’s denials, to the same conceptual field as the other principles of de-stabilisation cited above: writing, subject-in-process, group-in-fusion, process of subjectivation, care of self.
My running argument with the Laruelleans has been:
(1) it is necessary to give an important place to this concept, as Laruelle’s pre-quantum non-philosophy is too structuralist and synchronic (Laruelleans strenuously avoid doing so, because they are remarkable uninformed about Anglophone philosophy of science, and
(2) quantum thinking is not really developped by Laruelle but functions as a gesture towards the need to supplement his thought with diachrony in order to be able to account for change.