Suddenly, we're talking about Hamlet. This is supposed to help us understand the third synthesis, which produces a time that is "out of joint".
What does this mean: the empty form of time or third synthesis? The Northern Prince says 'time is out of joint'. Can it be that the Northern philosopher says the same thing: that he should be Hamletian because he is Oedipal? The joint, cardo, is what ensures the subordination of time to those properly cardinal points through which pass the periodic movements which it measures (time, number of the movement, for the soul as much as for the world). By contrast, time out of joint means demented time or time outside the curve which gave it a god, liberated from its overly simple circular figure, freed from the events which made up its content, its relation to movement overturned; in short, time presenting itself as an empty and pure form.
This was about as clear as mud to me when I first read it, and it turns out that part of the problem is that something is lost in translation. Twice. First, it's helpful to note that Shakespeare wrote in English, motherfucker. And Deleuze is quoting Hamlet in French. But then he goes on to translate the French term into Latin. If you look up "cardo in Latin" you'll find that it doesn't mean "joint" in the sense of an articulated joint of the body or a piece of furniture. Instead, it refers to the cardinal points of the compass the Romans used to orient the axis of the main street of a city. So "time is out of joint" would be better translated in this context as "time is off its axis". Or better still, since Deleuze wants to give a hint of madness to this time, as "time is unhinged". Unfortunately we're stuck with Shakespeare.
Shortly after putting this together I discovered that I had reinvented the wheel; here's a quote from James Williams' book Gilles Deleuze's Philosophy of Time.
The dramatic phrase uttered by Hamlet that Deleuze focuses on is 'the time is out of joint' (Hamlet: I, v, 189); in the French translation used by Deleuze, 'le temps est hors de ses gonds'. 'Gond' means hinge, as in door hinge, rather than simply joint; so some of the richness of the original is lost in this more precise word. According to Delueze, in line with a standard interpretation of Hamlet, time has become unhinged or disjointed (in the way the ball and socket joint of a shoulder might come separated).
The Latin root for 'gond' is 'cardo' the hinge or axle, the north-south axis in a city, or orienting direction and cardinal points for a circle, for instance north on a compass. The number of revolutions of a circle can then be measured thanks to cardinal points, for instance, in the number of times a clock passes midday or a horoscope passes a birthday. The passing of time, as much for the soul as for the world, he says, is measured thanks to such cardinal points. When these go missing time goes out of joint and becomes maddened and disordered ...
At first it might seem like a point moving around a circle is a great image for capturing the repetitive way that time keeps on happening without really changing form. The way it's always now ... now ... now. Every instant seems to be interchangeable with every other in terms of its form. But Deleuze is pointing out that the circle only works as a way of counting the passing of time if we keep track of the number of revolutions. We do this by picking out a special point on the circle, defined by where some line intersects it, and counting how many times we go past that same point. This is time with a "hinge" that joins the one revolution to the next. It's a time that's measured by the repetition of the same identity. We measure all of time this way, whether it's through the sun being in the same position on the solstice, the clock hand being in the same spot next to 12, or the caesium atom being at the same point in its frequency cycle. In short, we measure time by cycles of similarity.
The circle only captures the form of time if we pick out a special point, a bit of content. So, it doesn't capture "the pure and empty form of time" that we're after with the third synthesis. This is like the brute fact of time passing. The form of change, rather than the form of repetition. Ultimately, the question with time is why there's change at all, why there's constantly more difference. We can't get at this by counting the number of repetitions of the same. How could it be "the same" if time passed? This is precisely the problem we met back in the introduction; how can there be an exact repetition? Restating this question in terms of time brings it to its sharpest point. If time is real, how can any two things be exactly the same? And yet, we typically measure time by the fact that "exactly the same thing" is repeated. We measure the form of time by its contents. We drive a street through the heart of the circle to give it some coordinates.
By contrast, time out of joint means demented time or time outside the curve which gave it a god, liberated from its overly simple circular figure, freed from the events which made up its content, its relation to movement overturned; in short, time presenting itself as an empty and pure form. Time itself unfolds (that is, apparently ceases to be a circle) instead of things unfolding within it (following the overly simple circular figure). It ceases to be cardinal and becomes ordinal, a pure order of time. Holderlin said that it no longer 'rhymed', because it was distributed unequally on both sides of a 'caesura', as a result of which beginning and end no longer coincided. We may define the order of time as this purely formal distribution of the unequal in the function of a caesura.
If we can't understand the true form time as a circle, how should we conceive of it? Deleuze introduces two new images here. The pure order of time. And the caesura.
The idea of the order of time references the mathematical distinction between cardinal and ordinal numbers. Cardinal numbers measure the size of a set (1, 2, 3 ... elements) while ordinal numbers describe its position in a series (first, second, third ...). One way to think about this would be to say that ordinal numbers are qualitatively distinct whereas cardinal numbers are quantitatively distinct. We only know that one ordinal number is "bigger" than another because it comes "after" another in the series. Third comes after second. But we don't know how much bigger third is than second, and we also don't know whether fourth is that same "amount" bigger than third. In other words, the space ordinal numbers doesn't have a metric, a measuring system that lets us compare size or distance. They only deal with what comes before and after, first and second.
[As an aside, it turns out that one can define some arithmetic operations on ordinals. Addition, multiplication, and exponentiation all makes sense, though they are a bit weird (non-commutative). Subtraction, however, is pretty much completely busted.]
The cardinals that measure the size of a set do have a familiar metric, the one we use for arithmetic. This makes them perfect for counting the revolutions of a circle if we just click another off every time we go past the point on it marked 'North'. You can't do this with the ordinal, because while you may say conceptually that this is the first, second, etc ... time you've been around the circle, you can only determine that by referring to the cardinal point. Thinking about this also illustrates that ordinal time is irreversible -- you can't go back from the third revolution to the first -- whereas it makes perfect sense to go around the circle in the opposite direction and decrement the cardinal numbers as you pass the point; cardinal time is reversible.
Ordinal time, the synthesis of an order of time, is exactly what we need to think about the idea that all thought happens within time. The most basic form of time is that it has a before and an after. Thinking may involve a long time or a short time, but the idea that it takes time, that it is within time or inherently requires time as an a priori condition, is perfectly captured by the pure order of time. "Being in time" requires a first part, a second part, and a third part. It is precisely the fact that thinking has parts, that it is composite, that it unfolds as a series, that makes Descartes' instantaneous thinking a myth. The process of thinking needs time not because it moves through or takes up a specified block of pre-existent time, but because it actually synthesizes time within itself insofar as it has an order. Though I think Deleuze would probably reverse these terms and say that the third synthesis of time is thought. Or how we define thinking.
... the future and the past here are not empirical and dynamic determinations of time: they are formal and fixed characteristics which follow a priori from the order of time, as though they comprised a static synthesis of time. The synthesis is necessarily static, since time is no longer subordinated to movement; time is the most radical form of change, but the form of change does not change.
By now the attentive reader has heard me mention three parts to the order of time, but only offer two -- before and after. This begs the question: before and after what? Before and after the caesura. This turns out to be another Latin term for a break or pause introduced into a line of poetry. So it's a fracture or gap or cut that occurs between the before and after. Basically, the during, the moment of creation. However, to avoid falling back into the problems we've discussed, this has got to be a strange and paradoxical during. It cannot function as a special point defined in advance that acts as a hinge jointing past with future. Instead, it's a gap. It's a missing piece or cut that prevents the circle from closing. It's not a point with a solid identity in and of itself, but is in-itself just the distinction of past from future. The order of time and the caesura are two sides of the same coin meant to enable us to stop thinking about time as the repetition of some identity, and start thinking of it as the repetition of difference itself, as the way difference keeps on differentiating itself.
The caesura is pure paradox. It marks a time that has come un-hinged and a circle that never repeats. As we'll see more next time, it's a monumental moment that defines the entirety of time as split irrevocably between before and after, yet at the same time, it's nothing in itself but the distinction between these two. The definition is completely circular -- before and after are defined relative to a caesura, and the caesura is defined as the transition between the before and after. You might think of it as whatever non-number there is between any two ordinals, the nothing between first and second place. Though my favorite image is the one Deleuze is alluding to in Cinema 2 -- an irrational Dedekind cut is a very pure illustration of the caesura.
But cinema and mathematics are the same here: sometimes the cut, so-called rational, forms part of one of the two sets which it separates (end of one or beginning of the other). This is the case with 'classical' cinema. Sometimes, as in modern cinema, the cut has become the interstice, it is irrational and does not form part of either set, one of which has no more an end than the other has a beginning: false continuity is such an irrational cut.
The caesure is a tiny gap, equal to all of history, that splits it into before and after.
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