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I was given a photocopy of an article: "In Defense of Alain Badiou" by Robert Michael Ruehl, published in Philosophy Now. The article is behind a paywall, but here's the idea that caught my attention:

Against more traditional evaluations of it, Badiou argues that ontology is not an area of philosophical specialization: he argues that mathematicians specialize in ontological thought without knowing so, and philosophers are left to explain the radical implications of the mathematicians' thinking.

The article goes on to describe arguments that Badiou makes about what set theory shows about the world (e.g. "there is no transcendent unity to the world.")

I didn't know that this was... allowed, to extract information about the world from a formal system in this way.

Further reading (Wikipedia) reveals that Badiou has been criticized by some mathematicians and others who say he misuses set theory.

If I'm going to read "Being and Event", where Badiou first (?) makes his set theory arguments, I would be interested to know about other philosophers who have made similar kinds of arguments, or any other contextual information that might be relevant to evaluating this kind of argument in general.

[EDITED to get rid of a paywalled link and to add words]

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    The article is behind a paywall, making it difficult to get enough context. However, it is worth nothing that one must assume reality is perfectly describable using a particular model (such as set theory) before making claims about reality based on the model. It might be worthwhile to look at Tarski's nondefiniton theorem. It shows there are strong limits to the power of a formal language which tries to describe its own semantics. – Cort Ammon Sep 18 '15 at 19:50
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    As Cort points out, the linked article is for subscribers only. I read Badiou's book "Number and Numbers" last year and I must confess that I found it to be more poetic than anything mathematical. He seemed to misunderstand (or at least re-define) a lot of the terms and concepts used my mathematicians. – Nick R Sep 18 '15 at 20:42
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I didn't know that this was... allowed, to extract information about the world from a formal system in this way.

There's a tradition in Philosophy that uses mathematics to characterise physical ontology; starting with the Pythagoreans and continuing with Platos Timaeus; in the contemporary modern world, Physics - what was once Natural Philosophy - describes an aspect of the world - it's physical ontology; and it most definitely uses mathematics - though it's not reducible to it.

Consider too the first statement in Wittgensteins Tractatus, which relies on philosophical thinking on propositions (facts).

1.1 The world is the totality of facts not of things

Badiou is returning to this tradition but it relies on a tradition of philosophical thinking of mathematics that is distinct from the Anglo-American tradition.

For example Lautman theorised a dialectic of ideas in a mathematical Platonism; ie a unity of opposites. (In Kuhnian terms it's dynamic is a paradigm shift). This is renamed or rather rethought as inconsistency in Badioun thought.

In a way, he's returning to Socrates: in his thinking Philosophy does not produce truths (think of the inconclusively of the Platonic dialogues); but seizes them from other domains - there are four - Politics, Art, Love, Science; and they have their own truth-procedures of producing truths; it is also the condition of their compossibility - that they don't deny the claims of each other - a distinctive feature of Post-Modernity, the disavowal of 'grand narratives'.

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