Badiou presents a simple picture of mathematics, where it follows what might be called a scientific method, with models being the grist for experimenting on formal systems that those models instantiate. No doubt this picture is influenced by Bourbaki. This description of mathematical practice is then used to argue that positivist model-oriented philosophy of science more-or-less puts the cart before the horse by reversing the terms of the relationship. This is tied into a more general Marxist observation that model-first approaches might be inclined to prioritize representation over asking questions that would involve modelling interventions in the model system. He suggests a modelling approach that prioritizes interventions equal to representations would be better for scientific practice and would enable science to go beyond idealist positivist conceptions of science.

I buy that at least parts of mathematics have the described form. I think that there is an independent case in this text that models should focus on interventions as much as the task of bare representation. I am not sure I buy the idea of using set theoretic models as a metaphor for scientific models, but perhaps that was more of a salient point against the background of logical positivism, which Badiou addresses himself to often. Some of his philosophical interpretations of mathematics are a bit off (e.g. that the independence of choice from ZF calms all fears about using it), however they're mostly forgivable.

Good book, well worth a read for an introduction to a more grounded approach to philosophy of mathematics, chase with Fernando Zalamea if you can handle algebraic topology and/or modern algebra.

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