Levi Bryant writes a mixed piece on irreduction as a concept. At first he regrets previously adopting the concept in is 'Ontic Principle' it but then he goes on to defend irreduction anyway.
It's shocking how often people misunderstand this (to my mind) quite simple principle. As I read it, Latour's principle of irreduction actually has four parts - well, three plus a clarification. People tend to fixate on the first part and miss the rest. Here it is in full:
"1.1.1 Nothing is, by itself, either reducible or irreducible to anything else.1. Nothing is reducible to anything else.
• I will call this the "principle of irreducibility", but it is a prince that does not govern since that would be a self-contradiction"
To exist is to differ. If A is entirely reducible to B then A and B cannot meaningfully be said to be separate entities, they are one and the same. If any thing were to be reduced to any other thing then only the other thing would exist, by definition. That's the ontology. Epistemologically this also means that no matter how much you study a thing that thing is always, necessarily beyond you. Reality exceeds knowledge not just in fact ('alas we are merely flawed humans') but in principle. Indeed, this principle of 'excess' applies to any form of relation, not just knowledge.
2. Nothing is irreducible to anything else.
Nothing is beyond relation, there are no hermetically sealed spheres separating things that cannot possibly touch. Anything can be brought to bear on anything else. No two things are sufficiently enemies that they cannot become allied. Reality is promiscuous. There are no dualities, no lines that cannot be crossed; there are only pluralities, many lines that can be crossed if you can summon strong enough allies. Furthermore, to ally, to join, to relate is to reduce. For X to form an alliance with Y each must reduce the other to some degree; they must simplify each other. But they must do this because they are irreducible to each other! It is because that one is irreducible to the other that, in order to relate, they must translate and simplify one another. The irreducible/reducible dyad is not so much a contradiction as a self-reinforcing circle. For things to relate and exist they must be both reducible and irreducible.
3. Nothing is reducible or irreducible by itself.
And things must be brought to bear. Nothing can persist by itself; it's only through complex tangles of (vicarious) relations that anything can happen. Every thing is a swarm, every event is a cascade. Reality is a chaotic, raucous, poly-dimensional game of dominoes.
4. This is a prince(iple) that does not govern.
This is principle 1.1.1 of Latour's thesis. It kick-starts the discourse but the axioms that follow from it do not 'follow from it' in the sense that they can be deduced from it or that they already exist in it 'in potentia'. This contrasts to classical metaphysics, e.g. Spinoza, Descartes, where everything is supposed to be deducible from the first principle. Furthermore, however, I think this also means that this thesis doesn't reproduce, replicate or represent the 'heart of reality'; this axiom isn't the source code from which the universe is pieced together. That would be a contradiction since it would reduce reality to words. No, what this axiom does is put something new out into the world, a new semantic creature that can form new alliances, perform new assemblages, etc. Thus it demonstrates its metaphysical truth by performative alliance building, not by summing up in words something that exists 'out there' behind appearances.
That's my interpretation anyway. Later theses in Irreductions clarify and add to the principle but most of it is there in some form.
Latour is never saying that we must stop 'reducing' things altogether. That's the criticism that's usually levelled at this principle and it's nonsensical. Ray Brassier's 'critique' of Irreductions in the Speculative Turn is based almost entirely upon this elementary misunderstanding.
Reduction is more or less a synonym for relation - but you can only relate two things that are irreducible to each other (since otherwise there would not be two things at all).
The argument that science can only work by reduction fits with Latour's axiom perfectly well. To reduce is to form a network. There's no issue. But while science may 'reduce to explain' it doesn't 'reduce away' as in 'explain away' - there's always a remainder.
What we must be wary of is the claim that science captures what it studies completely, that knowledge of a thing is equivalent to the thing itself. What we must resist is the claim that you can reduce a thing completely without destroying it. Essentially it's all an argument against the ideal of pure mastery - it's all Nietzschean. Yes, we must resist the idea that things can be reduced without remainder but to resist reduction as such is an impossibility. In fact, to cease to reduce is to submit to unrelation: death.
Latour today distances himself from Irreductions, calling it a 'naive' book that he has 'fondness' for (off the top of my head, I think he uses these words). Certainly, he has moved on from much of it but the principle of irreduction is evident throughout his work, right up to the present. It's one of the few unbroken threads running throughout his work.
When Latour talks about things being 'irreduced and set free' he's releasing them from reduction*ism*, but not from reduction per se. Each flap of the seagull's wings is a simplification and a reduction of the air around it, but it is irreduced insofar as it is set free from the notion that it is 'nothing more' than the diagram in the textbook or the DNA sequence digitised by the computer or the phenomenon apperceived by the thinking subject.
In short, you were right to defend the principle of irreduction, Levi! Irreduction isn't the antonym of reduction, it's a broader concept that contains it.
You can't spell irreduction without reduction!