Deleuze on Number
Deleuze’s Philosophy of Number
This is the first of a two part post on Deleuze's philosophy of number. I'm still working on part II, but would still love any feedback, criticism, or requests for clarification in the meantime. I've tried to write this in a way that requires as little prior mathematical knowledge as possible.
§1: Extensive and Intensive Number
Here I want to explicate Deleuze’s philosophy of number. More specifically, I want to explicate his thesis about the genesis of extensive numbers. First, what are extensive numbers? Extensive numbers are simply numbers as we know them, for example: 1, 2, 3, etc. These are usually called the “natural numbers”, and are one class (type) of numbers among others (there are also the irrationals, the reals, the imaginary numbers and so on). What Deleuze calls ‘extensive numbers’ in fact covers all these kinds of numbers, but we’ll take our starting point from the naturals because they are simple and familiar. It’s useful to think about extensive numbers in terms of “extension”, or ‘size’. On a number line, for instance, “1” occupies one unit of space, while “2” occupies two units of space and so on.
The most important aspect of extensive numbers is that they are commensurable: they ‘fit into’ each other without remainder: there are two “1”s in “2”. Commensurability is important because it allows for precise calculation*.* One way to think about calculation is as the comparison and manipulation of like-elements for specific purposes. Extensive numbers are perfect for this.
Now, Deleuze’s basic claim about numbers is effectively this: that extensive numbers are the end result of a process which generates them. Specifically, they are generated from what Deleuze calls intensive numbers. In other words, there is a process by which intensive numbers are ‘explicated’ into extensive numbers. But what is an intensive number? Well, if extensive numbers are commensurable, intensive numbers are strange in that they are, for the most part, incommensurable. They are numbers which do not fit nicely among each other. But how is this to be understood? One basic way is to think about it is with the distinction between ‘ordinal’ and ‘cardinal’ numbers. Cardinal numbers are, again, numbers as we typically think about them: they are counting numbers. One rabbit, two rabbits, etc. Ordinal numbers, on the other hand, define position (or rather order, hence ordi-nal): first, second, third.
The first thing to notice is that ordinal numbers, unlike cardinal numbers, are thoroughly relational*.* “First” only exists in relation to “second” and vice versa. These numbers only exist in relation to each other. Unlike cardinal numbers, ordinal numbers do not represent units. They do not have size. As such, the distances between ordinal numbers (first, second, third) cannot be divided in equal measure: they have no unambiguous numerical identity which can neatly ‘fit’ into each other. Instead, they can only be spoken about in terms of proximity and distance. These distances however, cannot be ‘fixed’ in any way. In fact, as markers of order, ordinal numbers are markers of difference.
Examples that Deleuze gives are those of frequency, pressure, and potentials: measurements of frequency are always measurements of difference, such that you cannot have a frequency that is, as it were, in itself: “there are no reports of null frequencies, no effectively null potentials, no absolutely null pressure” (DR 234). Frequency, pressure, and potentials are intensive to the extent that, if they were not measures of difference, they would be measures of nothing at all. As Deleuze says: “The expression 'difference of intensity' is a tautology. Intensity is the form of difference”. What this also means is that intensive numbers are differential ‘all the way down’ – even the smallest possible measure of frequency would be a measure of difference: “intensity affirms even the lowest; it makes the lowest an object of affirmation”.
What is interesting here is that, with respect to the relation between extensive and intensive numbers, extensive numbers “cancel” the differences that inhere in intensive numbers. They turn incommensurable into commensurables, order and position into units and size: “Every number is originally intensive and vectorial in so far as it implies a difference of quantity which cannot properly be cancelled, but extensive and scalar in so far as it cancels this difference on another plane that it creates and on which it is explicated” (DR 232). We’ll see this operation of ‘cancelling’ at work more down below. For now, to sum what we’ve said in a metaphor: extensive numbers, like 1, 2, 3, are like external shells. They are hardened calcifications of intensities of number which otherwise rumble beneath them. Or to paraphrase Deleuze: extensive numbers ‘dance upon a volcano’.
§2: Generating Types of Numbers
Here is where things get really cool. If extensive numbers are generated from intensive numbers, how does this happen? Well, if extensive numbers ‘cancel’ the differences which inhere in intensive numbers, one way to see this reflected is in the historical development of numbers. We spoke earlier of the natural numbers, 1, 2, 3, and so on. The naturals themselves belong to a class of numbers known as rationals, which include fractions which can be neatly expressed in whole numbers: ½, ¾, and so on. What are not included in the class of rational numbers are irrational numbers: numbers cannot be expressed as fractions, and thus do not neatly resolve into wholes, such as √2 ( =1.4142…). Famously, the irrationals were said to have been discovered by a disciple of Pythagoras, who, as the apocryphal story goes, promptly drowned the student for having done so – irrationals breaking the neat holism provided by the ‘rationals’. Here is how Deleuze recapitulates this history:
“In the history of number, we see that every systematic type is constructed on the basis of an essential inequality, and retains that inequality in relation to the next-lowest type: thus, fractions involve the impossibility of reducing the relation between two quantities to a whole number; irrational numbers in turn express the impossibility of determining a common aliquot part for two quantities, and thus the impossibility of reducing their relation to even a fractional number, and so on”. (DR 232)
In other words, every type of number (from natural to rational, from rational to irrational, and so on) is a creation that in some way, aims at taming some element of ‘inequality’ (read: incommensurability) that cannot be tamed at the ‘lower level’. We invent fractions because what we want to express by them cannot be expressed by the natural numbers in any commensurable manner; In turn, we invent the irrationals because what we want to express by them cannot be expressed by fractions. Each ‘higher-order’ type of number “is constructed on the basis of an essential inequality”. These inequalities, for Deleuze, express nothing other than the inherence of intensity ‘beneath’ each explicated or extensive series of numbers. While every type of number aims to tame these inequalities, they cannot do so in a way which eliminates those incommensurabilities entirely.
Let’s examine one particular instance of this in a bit of detail. In his lectures on Spinoza, Deleuze goes over the discovery of irrational numbers, which we briefly mentioned above. What is of interest to Deleuze is the way in which the irrationals had to be created in response to a problem: specifically, you need an irrational number to measure the hypotenuse (diagonal) of a right-angled triangle. That is, if you have a right-angled isosceles triangle whose length and breadth = 1, its hypotenuse would = √2 (see image below). However, √2 is incommensurable with any whole numbers. You can’t make a whole-number fraction that would = √2. As Deleuze puts it, “the irrational numbers… differ in kind from the terms of the series of rational numbers”. Irrationals are, as it were, a different species of number from the rationals. Moreover, their existence responds to a problem that cannot be ‘resolved’ at the level of the rationals: the measurement of a certain kind of length.
For Deleuze, this attests to the ancient Greek preference for what he calls the ‘primacy of magnitude over number’. This itself should be a surprising idea: that ‘magnitude’ is not the same as ‘number’. What it implies however, is that irrational numbers themselves were a kind of solution to a problem posed in a domain which is not, in itself, numerical. In other words, it’s not just that there are mathematical solutions to mathematical problems, i.e. 1 +1 = ? Math itself, as such, is a solution to a problem! In Deleuze’s words, “number has always grown in order to respond to problems posed to it”. But these problems are not necessarily internal to math, but imposed upon math from an extra-mathematical ‘outside’.
In terms of the vocabulary of commensurability and incommensurability we have been using, one can say that the development of number has always been as a matter of rendering commensurate what is, in itself, incommensurate. Extensity – the development of new types of number - ‘cancels’ the difference involved in intensity – but only ever provisionally, only ever “locally”, in response to specific problems. We can say one last thing then, about Deleuze’s general orientation with respect to the status of numbers in general: unlike certain ‘Platonic’ conceptions of numbers, in which numbers inhere in some Platonic heaven of Number (this is a caricature, but a useful one for pedagogic purposes), for Deleuze, numbers are only ever ‘local’ solutions: “numbers have no value in themselves … there is no independence of the number system”, and instead, “numbers are always local numbers”.
§3: Inventing Infinity
Here, we’re going to deal with the largest metaphysical stakes of this conception of number. Given what we’ve said above – that all numbers are ‘local’ responses to problems, and there is no Platonic Sky-Realm of Number – does this mean that numbers are simply arbitrary? That is, are they just human inventions employed at our whim and fancy? Or, to put the question contrariwise – without a Platonic heaven of Number, is there any sense of necessity to the way in which numbers are as they are? Deleuze’s answer is yes. Precisely to the extent that numbers are solutions to problems, their necessity derives from the fact of being specific responses to specific problems. To see what this means, we’re going to examine another aspect of Deleuze’s philosophy of number: his conception of infinity.
What is infinity? Preliminarily, if all (types of) number are a response to a problem, then infinity is a response to a problem. The question is – which problem? A: the problem of irrational numbers. And what is the problem that irrational numbers pose? It has to do with the problem of populating the number line. As we’ll come to see, without irrational numbers, we can’t populate the line. Let’s see what this means. Imagine a finite line, two units long. We’ll label the left-most point “0”, and the right-most point “2”. Now, we’re going to populate the line with numbers. We’re going to fill it right up, so it is completely dense, without gaps. Dead in the centre will be “1”. A quarter of the length along will be ½, and three-quarters of the length along will be 1½. Here’s the challenge: using only the natural numbers (1,2,3 …) and the rational numbers (½, ¾ …), can you populate the line so as to make it a pure continuum of numbers? The answer is no.
In
other words, without irrational numbers, one will be left with ‘gaps’
in the line. There will be places on the line at which, if you were to
point a tiny, tiny finger, there would be no corresponding number. The
reason this is the case has to do, once again, with the
incommensurability of the irrational numbers with the rational numbers.
As we saw above, we know that no ratio of rational numbers can ever
equate to an irrational number. Irrational numbers are sui generis, a different species of number. We had to invent
the irrationals precisely in order to overcome the problem of measuring
the hypotenuse of a right-angled isosceles triangle, which cannot be
done with rationals alone. This means, however, that if we laid out the
length of the hypotenuse on a number line, no number would correspond to
it on the line!
Or, to put it as Bertrand Russell did: “No fraction will express exactly the length of the diagonal of a square whose side is one inch … where √2 ought to be, there is nothing”. As such, in order to render Number adequate to populating a line such that it will ‘fill it’, it is necessary to introduce the concept of the infinite. In Deleuze’s own words: “only the irrational number founds the necessity of an infinite series”. And moreover, “infinite series can exactly only be said to exist when the development cannot occur in another form”. In other words, the invention of infinity responds to the problem of densely populating the distance between any two points.
In order to properly understand this sense of ‘necessity’, it is useful to contrast this against what Deleuze calls the mere ‘possibility’ of formulating a concept of infinity. For most people, the concept of infinity is simply something ‘very large’, or that ‘goes on forever’. If we add, for instance, 1 + 2 + 3 … one might think this would be enough to furnish us with a concept of infinity. However, Deleuze, following Leibniz, distinguishes between what is merely an ‘indefinite sequence’ from an ‘infinite sequence’. And from this he notes that while there is always the possibility of forming an infinite sequence with rational numbers, it is only the introduction of irrationals that makes infinity something that “cannot be developed otherwise”. Infinity is a necessary response to the problem of the irrationals. The infinity that corresponds to the mere possibility of introducing it, is, on the other hand, a ‘false infinite’, or ‘bad infinite’, to use Hegel’s famous term.
§3.1: Unconcluding Non-Scientific Mid-Script
This is only part I of a two-part post, but, by way of anticipation, I'll simply note the the next part of this will deal with infinitesimals, infinity (again, but this time we'll solve Zeno's paradoxes!), as well as zero and negative numbers.
For now, we can stop half-way by way by asking how all of the above might deal with the age-old question: is math invented or discovered? Well, first, this isn't about Deleuze's philosophy of math per se: this is about his philosophy of number. And math, for Deleuze, is broader than number. There's more to say about this, but not here. Nonetheless, the philosophy of number offers a window into the philosophy of math. And, if Deleuze's understanding of number is correct, this question - invention or discovery? - is misplaced. Or rather, this question papers over another, more subject-appropriate one: is math necessary or arbitrary? For Deleuze, for answer lies with necessity: but a necessity that calls for creation, invention, in a way that cannot be otherwise. Number, like everything else in Deleuze's philosophy, responds to the "claws of absolute necessity" (DR 139), which result from the encounters to which it is subject.
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