Reason is not the already-accomplished apparatus of rationality, and the space of reasons can never be laid out in its totality under a single gaze. In short, it is not a “full body”, sufficient and all-comprehending.
Neither can a “rationalist project” be oriented by the thought of one day completing rationality, or take the measure of its accomplishments based on their perceived proximity to such a goal. Rationality is locally perfectible – there exist problems to which there are solutions, and even classes of problems to which there are general solutions – but not globally: there is no universal procedure which will render every circumstance as a problem to which there is a solution.
There is no end of phrases, and no end to the task of linking phrases together.
The obscure subject of a rationalist politics will be that which, in the name of a “full body” of accomplished (or to-be-accomplished) rationality, calls for everything to be restored to order. Society organised according to geometric principles! Its speech purged of fallacies, its politics free of antagonism…
We know that such an obscure subject must devote itself increasingly to destruction, culminating in a frenzy against the real. Only the destruction of error can restore the integrity of the full body; and there is no end of error.
The faithful subject of a rationalist politics will be that which proceeds by proofs, which is to say by logical invention. To trust in reason is to trust in a generic capacity, without any guarantee drawn from the particularity of this or that person or community. It is to trust in the next step of the proof, without the certain knowledge of the world’s approval (reason can still scandalise the world).
The obscure subject has no need of fidelity, since there is literally nothing left for it to prove. All that remains is the identification – and consignment to perdition – of the infidel.
It is by no means the rationalist orientation in politics that has generated the most ferociously indiscriminate instances of this subjective figure.
It is not immediately clear that the rationalism you describe is conscious of its historicity, of the way particular ‘rational’ subjects and knowledge practices are historically constituted — and therefore it appears to be unconscious of its limitations. The “rhetoric of transcendence” further suggests this inattention. Again, the question here is not one of intellectual imperialism, but about how, whether, and with what limitations you can know. Is a rationality that transcends the standpoints of particular rational subjects possible? Can it be practiced by these subjects? Are the tools being relied upon, whether maths or something else, capable or sufficient of producing knowledge that transcends given subjects’ limitations?
Yes, yes and yes, otherwise we might as well pack up and go home. But the trap’s in the word “transcendence”, which implies a magic trick that one could always remain unpersuaded had actually been performed. Actually, the notion that “the standpoints of particular…subjects” constrain rationality in such a way that it must somehow escape them in order to function as the rationality it thinks it is, is fatally question-begging. Such “transcendence” is in fact an everyday occurrence, banal and unmagical: it takes place every time you or I take up an argumentative form and commit ourselves to reasoning consequentially according to its rubric.
There is no fact of the matter about my standpoint that can have any bearing whatsoever on the validity of a mathematical proof: if I am able to know that the proof has been performed successfully, it is because I am able to follow it through. If I am able to detect an error, it is because in the process of following the proof through I have stumbled upon an invalidating condition (usually a contradiction of some kind). Formalisms enable us to get to places that are not prescribed or localised by our immediate situations as users-of-form. That is, in fact, precisely what we use them for.
Most of the things we know, we don’t know in quite that kind of way of course. Much of the time, what we know or think we know is stabilised as knowledge against a backdrop of assumptions and heuristics supplied by our “standpoints”, which is one of the reasons why we misunderstand each other so often and so deeply. So I’m not suggesting that we take proof-following reasoning as a model for reasoning in general, or that knowers in general are not situated and not simultaneously enabled and constrained by situational constraints and affordances. What I do want to argue is that our image of a knower’s “situation” as wholly-localised and wholly-limiting is false: we are in fact situated within view, and within reach, of a rich variety of navigational affordances which enable us to reason from context to context. Reason is not an instantaneous ascent into the empyrean heights, from whence the whole terrain is visible at once: it involves traversals, translations, the construction of linkages from context to context, and whereabouts you start on the map is often significant. The crux I think is that it’s significant but not wholly determining: we don’t have to have perfect, godlike freedom in order to have some degrees of freedom.
The label “rationalism” has already a somewhat anachronistic aura about it, as if it named something that had no proper place in the present. We have been (or, plausibly, “have never been“) rationalists; but who could be such a thing now? Both the rationalism of the past and the rationalism of the future have a phantasmal quality; it doesn’t seem unreasonable to many people to treat them purely as objects of fantasy, and to focus their critique, such as it is, at the level of libidinal investment. What do these strange people want from rationality? How do these wants relate to the usual generators of desire – anxiety about social position, for example? Why the embattled posture, the rhetoric of transcendence?
Answers to these questions are not difficult to produce – in a sense they’re encoded into the questions themselves – and so the desire-named-rationalism can without much effort be rendered transparent and intelligible. What the would-be rationalist really wants – we are immediately sure of it – is to recover a (fantasised-as-) lost position of mastery, no doubt imbricated with the self-image of the colonial slaveowner; they feel threatened by women and queers and people of colour, whose political demands they wish to subordinate to their own privileged sense of what would be “reasonable”; and so on. Inasmuch as all of this registers only at the level of unconscious fantasy, they are (for now) at least one step away from the out-and-out racists and sexists and reactionaries. If only they could be brought to acknowledge the unsavory unconscious content of all their high-minded talk, they might yet be saved.
Now, this hermeneutic has its own self-sufficient logic: it supplies to itself guarantees of its own correctness. It does not have to reckon with rationalism as a concrete position, taken in the here-and-now, because its founding gesture is one of incredulity that such a position could be held in earnest, that it might have any ramifications beyond the fugitive gratification it offers to a handful of hapless nerds. You cannot be serious. It will not, for example, distinguish between the doing of mathematics, an activity which has real ramifications inasmuch as one thing really does lead to another, and the performance of mathiness, the brandishing of the matheme as a totem of sophistication (or abstract fedora). In short, the source of its power (as a derailer of argument) lies in its capacity for inattention: since I already “know” that the object of your attention is a fantasy with no real purchase on the present, I am authorised to focus my attention on your attention, rather than upon the thing attended-to.
It’s in the specific polemical context in which proponents of rationalism encounter this hermeneutic – and while that is often a very narrow and specialised context indeed, it is nevertheless legitimately of concern to us – that we find ourselves both at bay, and empowered by concrete demonstrations of the viability of rationalism in the present. The terrain under dispute is not, or not immediately, that of the concrete conditions of everyday life. What we’re trying to do, ultimately, is strengthen the hand of a certain kind of argument, in the hope of bringing closer some of the goods that this kind of argument is – we believe – uniquely able to envisage. It’s all pretty meta. But we do think it’s important – or we wouldn’t bother – and I for one do find it galling when people whose reaction to the accelerationist manifesto was to describe its program as inextricably colonialist, then describe the accelerationists’ sense of being put somewhat on the back foot as histrionic.
A few words are in order about the use made of mathematics. I don’t believe, and don’t believe that anyone else believes, that a sound knowledge of category theory is necessary for salvation. We’re not trying to become Pythagorean sages here. What I think has become apparent during the course of the HKW summer school is that the current rationalist use of “higher” mathematics is partly revisionary and partly metaphorical: it’s about taking apart some old and creaky logico-mathematico-ideological constructions, which had trapped us in a false image of thought, and provoking new images of thought by giving a motivated and metaphorically suggestive account of the technical machinery used to do so. Some of the work involved in doing this is very technical, and requires those performing it to learn and practice some real and quite difficult mathematics. But the ultimate purpose is not to become surpassingly good at maths, but to get away from an inadequate sense of what “rationality” can mean, so that we are not presented with a bogus choice between (for example) first-order predicate logic on the one hand, and everything that isn’t first-order predicate logic on the other. Rationalism in the present moment means using whatever tools are available to reflect on rationality and extend our sense of what it is capable of. It turns out that fancy mathematics is quite indispensable to this endeavour, but we do not hold it to be synonymous with thinking itself. In fact, those of us who are good Badiousians will be well-accustomed to the vertiginous transit between mathematics and poetry:
Someone saw that very clearly, my colleague, the French analytic philosopher Jacques Bouveresse, from the Collège de France. In a recent book in which he paid me the honor of speaking of me, he compared me to a five-footed rabbit and says in substance: “This five-footed rabbit that Alain Badiou is runs at top speed in the direction of mathematic formalism, and then, all of a sudden, taking an incomprehensible turn, he goes back on his steps and runs at the same speed to throw himself into literature.” Well, yes, that’s how with a father and a mother so well distributed, one turns into a rabbit.
The good rationalist, I submit, will be a five-footed rabbit, composing a living present out of the energetic, irreconcilable distribution of antecedents.
Why am I so excited about the HKW Summer School? Because it represents an attempt to take some cultural initiative: this is “us” showing what we’ve got and what we can do with it, and showing-by-doing that what can be done in this way is actually worth doing.
I don’t expect everyone to be convinced by such a demonstration – in fact, I expect quite a few people to be dismayed about it, to feel that this is an upstart, renegade movement with distinctly not-for-People-Like-Us values and practices (maths! logics! don’t we know Lawvere* was a worse fascist than Heidegger?). It’s likely that not a few leftish PLUs will be rocking up any moment now to tell us all to curb our enthusiasm. But a glance over the history of Marxist thought will show that there have been plenty of times and places in which the initiative has indeed been held by rationalists – albeit often by warring rationalists, who disagreed ferociously with each other about how a rational politics was to be construed and practised. It’s not at all clear that the present moment, which places such overriding importance on affective tone, is not in fact the anomaly. That’s not to say that we should ditch everything that has declared itself over the past decade – on the contrary, it represents a vast, complex, necessary and unfinished project to which we should aim to contribute meaningfully. But we can only do so by approaching that project from a perspective which it does not encompass, and is hugely unwilling – and perhaps unable – to recognise as valid. To do so requires confidence, of a kind that those who are already confident in their moral standing will find unwarranted and overweening. We are going to be talked down to a lot; we are going to be called names; we are going to have to develop strong memetic defenses against the leftish words-of-power that grant the wielder an instant power of veto over unwelcome ideas. We have a lot to prove. Calculemus!
What itinerary would a gentle introduction to sheaves have to take? I would suggest the following:
A basic tour of the category Set, introducing objects, arrows, arrow composition, unique arrows and limits. (OK, that’s actually quite a lot to start with).
Introduction to bundles and sections, with a nicely-motivated example.
Enough topology to know what open sets are and what a continuous map is, topologically speaking.
Now we can talk about product spaces and fiber bundles.
Now we can talk about the sheaf of sections on a fiber bundle.
Now we back up and talk about order structures – posets, lattices, Heyting algebras, and their relationship to the lattice of open sets in a topological space. We note in passing that a poset can be seen as a kind of category.
Functors, covariant and contravariant.
That’s probably enough to get us to a categorial description of first presheaves (contravariant functor from a poset category to e.g. Set) and sheaves (presheaves plus gluing axiom, etc). Show how this captures the fundamental characteristics of the sheaf of sections we met earlier.
Then to applications outside of the sheaf of sections; sheaf homomorphisms and sheaf categories; applications in logic and so on. This is actually where my understanding of the topic falls of the edge of the cliff, but I think that rehearsing all of the material up to this point might help to make some of it more accessible.
Anything really essential that I’ve missed? Anything I’ve included that’s actually not that important?
If, as is sometimes said, software is eating the world, absorbing all of the contents of our lives in a new digital enframing, then it is important to know what the logic of the software-digested world might be – particularly if we wish to contest that enframing, to try to wriggle our way out of the belly of the whale. Is it perhaps object-oriented? The short answer is “no”, and the longer answer is that the ontology of software, while it certainly contains and produces units and “unit operations” (to borrow a phrase of Ian Bogost’s), has a far more complex topology than the “object” metaphor suggests. One important thing that practised software developers mostly understand in a way that non-developers mostly don’t is the importance of scope; and a scope is not an object so much as a focalisation.
The logic of scope is succinctly captured by the untyped lambda calculus, which is one of the ways in which people who really think about computation think about computation. Here’s a simple example. Suppose, to begin with, we have a function that takes a value x, and returns x. We write this as a term in the lambda calculus as follows:
The symbol means: “bind your input to the variable named on the left-hand side of the dot, and return the value of the term on the right-hand side of the dot”. So the above expression binds its input to the variable named x, and returns the value of the term “”. As it happens, the value of the term “” is simply the value bound to the variable named x in the context in which the term is being evaluated. So, the above “lambda expression” creates a context in which the variable named x is bound to its input, and evaluates “x” in that context.
We can “apply” this function – that is, give it an input it can consume – just by placing that input to the right of it, like so:
This, unsurprisingly, evaluates to 5.
Now let’s try a more complex function, one which adds two numbers together:
There are two lambda expressions here, which we’ll call the “outer” and “inner” expressions. The outer expression means: bind your input to the variable named x, and return the value of the term “ ”, which is the inner expression. The inner expression then means: bind your input to the variable named y, and return the value of the term “”.
The important thing to understand here is that the inner expression is evaluated in the context created by the outer expression, a context in which x is bound, and that the right-hand side of the inner expression is evaluated in a context created within this first context – a new context-in-a-context, in which x was already bound, and now y is also bound. Variable bindings that occur in “outer” contexts, are said to be visible in “inner” contexts. See what happens if we apply the whole expression to an input:
We get back a new lambda expression, with 5 substituted for x. This expression will add 5 to any number supplied to it. So what if we want to supply both inputs, and get ?
Some simplification rules in the lambda calculus notation allow us to do away with both the nested parentheses and the nested lambda expressions, so that the above can be more simply written as:
There is not much more to the (untyped) lambda calculus than this. It is Turing-complete, which means that any computable function can be written as a term in it. It contains no objects, no structured data-types, no operations that change the state of anything, and hence no implicit model of the world as made up of discrete pieces that respond as encapsulated blobs of state and behaviour. But it captures something significant about the character of computation, which is that binding is a fundamental operation. A context is a focus of computation in which names and values are bound together; and contexts beget contexts, closer and richer focalisations.
So far we have considered only the hierarchical nesting of contexts, which doesn’t really make for a very exciting or interesting topology. Another fundamental operation, however, is the treatment of an expression bound in one context as a value to be used in another. Contexts migrate. Consider this lambda expression:
The term on the right-hand side is an application, which means that the value bound to f must itself be a lambda expression. Let’s apply it to a suitable expression:
We “pass” a function that multiplies a number by itself, to a function that applies the function given to it to the number 4, and get 16. Now let’s make the input to our first function be a function constructed by another function, that binds one of its variables and leaves the other “free” – a “closure” that “closes over” its context, whilst remaining partially open to new input:
If you can follow that, you already understand lexical scoping and closures better than some Java programmers.
My point here is not that the untyped lambda calculus expresses the One True Ontology of computation – it is equivalent to Turing’s machine-model, but not in any sense more fundamental than it. “Functional” programming, a style which favours closures and pure functions over objects and mutable state, is currently enjoying a resurgence, and even Java programmers have “lambdas” in their language nowadays; but that’s not entirely the point either. The point I want to make is that even the most object-y Object-Oriented Programming involves a lot of binding (of constructor arguments to private fields, for example), and a lot of shunting of values in and out of different scopes. Often the major (and most tedious) effort involved in making a change to a complex system is in “plumbing” values that are known to one scope through to another scope, passing them up and down the call stack until they reach the place where they’re needed. Complex pieces of software infrastructure exist whose entire purpose is to enable things operating in different contexts to share information with each other without having to become tangled up together into the same context. One of the most important questions a programmer has to know how to find the answer to when looking at any part of a program is, “what can I see from here?” (and: “what can see me?”).
Any purported ontology of computation that doesn’t treat as fundamental the fact that objects (or data of any kind) don’t just float around in a big flat undifferentiated space, but are always placed in a complex landscape of interleaving, interpenetrating scopes, is missing an entire dimension of structure that is, I would argue, at least as important as the structure expressed through classes or APIs. There is a perspective from which an object is just a big heavy bundle of closures, a monad (in the Leibnitzian rather than category-theoretical sense) formed out of folds; and from within that perspective you can see that there exist other things which are not objects at all, or not at all in the same sense. (I know there are languages which model closures as “function objects”, and shame on you).
It doesn’t suit the narrative of a certain attempted politicisation of software, which crudely maps “objects” onto the abstract units specified by the commodity form, to consider how the pattern-thinking of software developers actually works, because that thinking departs very quickly from the “type-of-thing X in the business domain maps to class X in my class hierarchy” model as soon as a system becomes anything other than a glorified inventory. Perhaps my real point is that capitalism isn’t that simple either. If you want a sense of where both capitalism and software are going, you would perhaps do better to start by studying the LMAX Disruptor, or the OCaml of Jane Street Capital.
Reading the opening chapter of Katerina Kolozova’s Cut of the Real, I’m struck with a kind of delighted awe, the kind you might feel when observing a dazzling card-trick or reaching the conclusion of a felicitous mathematical proof. I want to record this sensation at the outset, as I believe it points to something about the book: an effect of suddenness, of striking or leaping or arcing. A short-circuit. And this is in spite of the book’s style of plain and patient argumentation, without overt rhetorical showiness or show-stopping flourishes. Its sublimity is argumentative rather than oratorical. It does something unexpected and, in doing it, shows that it can be done.
What Kolozova does in her first chapter is to show that the canonical “postmodern”, and especially queer or feminist, theory of the subject rests on a disavowal of the unitary – of any sense that the subject’s protean capacity for being woven out of “multiple discursive positions” might be borne by a single self, something that is itself even as it is this multiplicity of self-positings in discourse. This disavowal has become routine to the point of being “axiomatic”, and is guarded by an ethical apparatus that trains us to associate the unitary with the totalising, one-ness with soliloquy, identity with domination. (“Identities” are permitted, on precisely the condition that they are multiple – that they are identifications-with discursive positions, rather than the identity-of a self prior to its co-responsibilities). What we might call the “mystic” or “spiritual” sense of one-ness (wince all you like) is always taken to be a mystification of the coercive forming-into-one of imposed fixity and manufactured consensus. As sexual beings, for example, we are encouraged ethically to focus on the “…which is not one” part of Irigaray’s formula “this sex which is not one” – rather than the prior “this sex”, an identity which proceeds from and as itself into not-one-ness. Kolozova shows clearly that the indicative gesture which picks out this sex, this “I”, this instance of the real in its this-ness, is frequently the disavowed precursor of the common gesture of dispersal, the gesture towards multiplicity. In the work of Judith Butler, for example, there is an unavoidable recourse to, a repeated looping-back towards, the “I” to whom and in whom the psychic life of power occurs. We can treat this as an unfortunate but unavoidable lapse into a theoretically-unsophisticated manner of speaking; but we can also treat it as a symptomatic intrusion, a resurfacing of what’s really always there, a return of the repressed.
If we can disarm or circumvent the ethical apparatus which enforces these associations between identity and domination, then (and perhaps only then) we can get to the point of treating the axiomatic decision it secures as an option amongst other options. Perhaps it is a good option, but we ought nevertheless to be able to suspend it, to treat it as material for thought rather than the indispensible precondition of thinking at all. This is of course the “non-philosophical” move par excellence, and Kolozova makes the best use I have yet seen of Laruelle’s strategies of “unilateralisation” and “dialysis” to render the seemingly-unmovable bedrock of (a certain) gender studies amenible to questioning and transformation. (My view of Laruelle remains that what he proposes to do is extremely interesting, but that he himself does it quite badly. He’s rather like Lacan in that respect: all the best of Lacan is in the work of his disciples, whether faithful or rebellious). To put it simply, Laruelle turns out to be useful for a feminist writer such as Kolozova insofar as he outlines strategies for disloyalty and disaffiliation (or non-loyalty and non-affiliation), for separating oneself from the suffocating self-sufficiency of closed systems of thought. Once you learn to detect gestures of auto-position, you can also learn how to avoid being positioned by them as inexorably subject to the law they propose.
This is a surprising and very promising start, and as I continue through the book I’ll report back here on where Kolozova’s speculative journey takes her. For the moment, though, here’s a passage from Douglas Oliver’s An Island That Is All The World that her first chapter sent me back to:
My companion set off with a strong sidestroke and I liked watching her progress before plunging in and striking up a crawl designed to catch her up. But she was 12 years younger and the cigars had affected my blood. In the lake’s centre I watched her climbing out on the far side; and discovered I was completely out of stamina. For 20 seconds I flailed about wildly or tried to float, which only made me lose precious breath, and I thought myself sure to drown. She was too far away to help. (We found police notices afterwards warning against swimming there.)
It came to me that the mind must have some hidden rescue of its own. There stabilized within me a steady, confident self, which I imagine to be the self I had often speculated about, the unconscious unity of everything we have experienced and incorporated throughout our length of days, an entity that persists, minutely changing, very minutely, as our conscious self goes through its wilder swings of mood. Much modern linguistic philosophy argues this large entity out of all real existence, but I simply don’t believe it. A larger self instructed me to let my limbs do the work while it lay back, almost entirely uninvolved. After great calm – the panic holding off on the periphery – I realised I had ground under my feet, staggered up the shore, and collapsed, as everyday conscious awareness flooded back.
There may be much more to say about the relationship between Douglas Oliver’s conception of harmlessness, connected to the “almost entirely uninvolved” passivity of this “larger self”, and Kolozova’s sense of the real of the subject as a kind of animal-corporeal selfhood; but I’ll need to read more before I can come back to this.