SH now asks, “what are the characteristics we find in intensity? We first note important differences between intensity and extensity. When you combine two extensive magnitudes, like two spatial distances, you get one whose value is the numerical total of the two. That is not how intensive magnitudes work, however. For example, if you combine fluids of different temperatures, the new temperature is not the addition but rather the average of the two. “intensity understood prior to its location in extensity will have different characteristics to extensity itself” (174). So for Deleuze, the “nature of | the extensum and the spatium as a whole are radically different,” but he also focuses on the differences between intensive and extensive elements and also the differences in the ways each kind of element relates to others of its own type (174-175). Deleuze’s account discusses cardinal and ordinal numbers, while all the while in the background is Bergson’s account of two multiplicities. [SH discusses Bergson’s multiplicities in his Hegel, Deleuze… book, Pt2.Ch3.Sb4]. For Deleuze, intensity has three characteristics: 1) it includes the unequal in itself, 2) it affirms differences, and 3) it is an implicated, enveloped, or embryonized quantity (SH 175).
1) Intensity includes the unequal in itself.
SH will follow Delanda’s analysis in his Intensive Science and Virtual Philosophy. Extensive quantities, we noted, maintain their nature when added, [and perhaps intensive quantities do not. If intensive quantities change their nature when added, I am not entirely sure how the prior example of combining fluids with different temperature applies, since I would think they keep the same nature even if they average. For, they would form a fluid with its own temperature that could be added to yet another, repeating the same process.] Extensive quantities “can be measured numerically, and […] these measurements are comparable (or commensurate) with one another” (175). [The next ideas are slightly complicated, but they are straightforward. It seems we need to establish different orders of ordinal numbers, since within each order certain values are suggested or desired, but that order cannot express them. Another order is then posited which can express those other values. So we begin with natural numbers, which do not have decimal values between them. So among the natural numbers, we have 2, 7, and 8, for example. We can divide 8 by 2 and get 4, which is also a natural number. But if we divide 7 by 2, we get something greater than 3 but less than 4. So we need to turn to another order of numbers, fractions, which would give us seven halves, in this example. But fractions do not allow all quantities to be expressed, since for example there is the value of pi, for which there is no exact ratio of integers that could express it. So we need another order of numbers, the real numbers (which will include both rational numbers, that is, those that can be expressed in fractional form, as well as irrational numbers, which cannot be expressed fractionally). So again, within each order, there is an incommensurability that cannot be expressed within that order and which thus calls for another order to express it.]
As Deleuze notes, this difference reflects one of the key features of extensive magnitudes: that they can be measured numerically, and that these measurements are comparable (or commensurate) with one another. Now, if we look just at the natural numbers (0, 1, 2, 3 . . .), we find that frequently we come across magnitudes that cannot be expressed in these terms. For instance, provided we remain with the natural numbers, we cannot divide 7 by 2, as the result is not itself a natural number. The obvious solution to this difficulty is to introduce another order of numbers that does allow us to relate these two quantities to each other, in this case, fractions. Similarly, we will discover that fractions do not allow all quantities to be related to one another, leading to the instigation of a new order of numbers: real numbers (such as √2 or π). In each case, we have an incommensurability between quantities that cannot be cancelled within the order of numbers themselves, but only by instigating a new order of numbers.
(SH 175)
[So we note that fractions proceeded from natural numbers, and reals from fractions.] Now we ask, from what order of numbers do natural numbers proceed ? (175). We turn now to the difference between ordinal and cardinal numbers. Cardinal numbers can have identities in the sense of forming equivalents and compositions. So the difference in value from one to three is the same as from two to four (175). Ordinal numbers, however, “just give us a sequence without requiring that the difference between the elements is the same in each case (thus, the difference between first and third does not have to be the same as the difference between second and fourth)” (SH 175). [Cardinal numbers can be said to have a metrical distance. For example, the “distance” between one and three is the same “distance” as between two and four. (That of course sounds right, but I wonder if numerical differences obtained through subtraction imply a geometrical sort of distance like on a number line. I guess the idea is that so long as the differences between units are standardized, that this creates a sort of metric.)] Ordinal numbers can also be thought of as having distances, but it is not metrically consistent or measurable like with cardinal numbers [so first is more distant from third as it is from second, but that does not mean we can say there is the same distance from first to second as there is from second to third.]
Since ordinal numbers do not operate on the basis of metrical units spanning between their values, “the kinds of operations we can perform with cardinal, natural numbers cannot be performed, meaning that we cannot produce equalities within this domain. Rather, it is only by the addition of a common measure between numbers (and thus the conversion of ordinal numbers to cardinal numbers) that we can begin to talk about equalising quantities” (176); [we then turn to a Deleuze quote. It seems we first need to understand the spatium as having distances enveloped in it, which might be like how the syntheses of space somehow explicate extensive space from an intensive spatium. What is happening here is a bit vague in my mind. The next thing is that when ordinals are converted to cardinals, it involves such an explication of the spatium.] “ ‘In fact, ordinal number becomes cardinal only by extension, to the extent that the distances enveloped in the spatium are explicated, or developed and equalised in an extensity established by natural number’ (DR 233/292)” (SH 176). SH continues:
We can note that, in these cases, we have a model that parallels the account of intensity we have seen so far. An uncancellable difference (intensity) gives rise to a new domain (extensity) within which that difference is cancelled. ‘Here, however, we rediscover only the duality between explication and the implicit, between extensity and the intensive: for if a type of number cancels its difference, it does so only by explicating it within the extension that it installs. Nevertheless, it maintains this difference in itself in the implicated order by which it is grounded’ (DR 232/292).
(SH 176)
We saw how heat was cancelled in its own domain of energy and temperature [but this is not a case of intensities in the intensive spatium, but rather of intensities explicated into the extensive physical world of energy transfers.] Yet, intensity is not equalized in its own domain of intensity. It can only be equalized after being explicated into the constituted realm of extensities. “Whereas in the thermodynamic model, difference is cancelled within its own domain, leading to the idea of the heat death of the universe, for Deleuze difference can only be equalised in a constituted realm, leaving it unequalised in its original domain” (176). Cardinal numbers, as we have gathered, presuppose extensive space. [I am not sure I get the next point about time and ordinal numbers. The Bergson quote seems to be saying that a succession of increasing numbers needs a spatial structure, since without it, each successive number would displace the prior and we would never have more than just one thing at a time. Then it seems SH is saying that Deleuze follows Bergson’s claim that a) cardinal number requires space, and also that b) there is a temporality to ordinal numbers, perhaps since they are understood only in terms of their succession, but I am not sure I follow correctly, and c) this temporal succession requires space. The final point that quotes from the Deleuze lecture, in the SH quotation below, I do not grasp so well. Time arises in an ordinal series. But it arises secondarily as a spatialized time. Perhaps the idea is that is that the pure form of time is somehow intensive, and in some way it is ordinal succession, and out of it somehow arises an extensive time of cardinal succession. I probably have that wrong, and even so, I do now grasp how that works. Let me quote:]
As cardinal numbers are constituted from elements that are absolutely identical with one another, they presuppose an extensive space. Bergson makes the point as follows [the following up to citation is Bergson quotation, and bracketed text within it and within the next Deleuze quotation is SH’s]:
And yet [numbers] must be somehow distinct from one another, since otherwise they would merge into a single unit. Let us assume that all the sheep in the flock are identical; they differ at least by the position which they occupy in space, otherwise they would not form a flock. But now let us even set aside the fifty sheep themselves and retain only the idea of them. Either we include them all in the same image, and it follows as a necessary consequence that we place them side by side in an ideal space, or else we repeat fifty times in succession the image of a single one, and in that case it does seem, indeed, that the series lies in duration rather than in space. But we shall soon find out that it cannot be so. For if we picture to ourselves each of the sheep in the flock in succession and separately, we shall never have to do with more than a single sheep. In order | that the number should go on increasing in proportion as we advance, we must retain the successive images and set them alongside each of the new units which we picture to ourselves: now, it is in space that such a juxtaposition takes place and not in pure duration. (Bergson 1910: 77)
Thus, Deleuze follows what he takes to be Bergson’s claim that ‘space [is] a condition of number, even if only an ideal space, the time that arises in the ordinal series arising only secondarily, and as spatialized time, that is to say as space of succession’ (L 00/00/70).
(SH 176-177)
[All this material was to explain how intensity includes the unequal in itself, but I am not sure how to sum up how that is so. I suppose that it contains the unequal within itself, because equalization is only possible when intensity is explicated and thus when it is not within itself. Therefore, it seems, within intensity is only unequality somehow. The ordinals for example do not have a standarized metric, so in a sense their differences are unequal, perhaps.]
2) Intensity affirms difference.
The first claim “aimed to show that intensity couldn’t be understood in terms of extensity” (177). The second claim is that intensity “also cannot be understood as a quality” (177). Deleuze supports this claim with another one: qualities are understood in terms of negation, but intensity, when characterized by difference, is not understood in terms of negation. Recall how for Aristotle’s notion of definition, the difference between species involves a negation. [see again for example section 1.6, and before that section 1.2. As we noted, in Aristotle’s system of division, things are differentiated on the basis of clear defining limits that determine what is special and proper to each thing. These limits serve to define what makes one thing what it is and what makes something else not that thing but rather something different entirely, and thus Aristotle’s system makes use of negation.]
As we saw in Chapter 1, within the Aristotelian notion of definition, a difference presupposes negation. That is, when we wanted to talk about the essence of man, we did so by attributing a property to him called a difference. This difference allowed us to divide the genus into two opposed classes: the rational and the non-rational. Negation was thus fundamental to the process of definition, and to the specification of properties. We can sum up this characterisation of difference with the claim that if x differs from y, x is not y.
(177)
[For the next point, recall from section 1.4 how for Scotus, there is a difference in quantitative intensity between God’s infinite perfection and the finite perfection of our world. So also, our finite (im)perfection is not to be understood as a negation of God’s infinite perfection. Rather, we are different degrees on the same scale of the intensive variation of being. We also saw, again from section 1.6, how the negation-based models involve a notion of extensive space, since if one thing is not another, they cannot occupy the same place, whether that by physical space or a sort of “conceptual space” you might say. But intensive differences never involve one being a negation of another. I am not entirely sure I understand how to conceive intensities, unlike extensities, which I can picture in my mind more readily. I wonder if the idea is something like this: say you have two intensive magnitudes of heat, with one being greater than the other. What makes one have its own value is understood only by its relation to the other, or to some third value, perhaps zero (or becoming-zero or infinitesimally-away-from zero or something like that). Since each of the intensive values is such only in relation to the other, what defines each of them is not that one is the negation of the other, (that it is “not the other”), but instead that together they differentially relate, and it is that relation itself which defines them jointly. Perhaps one problem with this heat example I used is that differences in heat are not intensive in the way we mean it here, since they have been explicated in such a way that they may be assigned numerical values which themselves are extensive. At any rate, the next idea I think comes from a possible objection, which is that an intensity can diminish to zero, thus intensities can be negated. But Deleuze notes that when it comes to physical intensities, there is no zero value. I am not sure, but perhaps the idea is that intensities can get smaller and smaller to the infinitely small, but never to zero, and when we say ‘zero’ for some intensive value, we really mean infinitely little. I am not entirely sure why this is. Maybe it is because intensities are always the difference between values, but total zero has no value and thus it cannot enter into a differential relation with other intensities.]
As we saw in Chapter 1, this constraint on the concept of difference was not inherent to difference itself, but only difference thought in terms of extensity. Scotus’ intensive conception of difference avoided the need to define it in terms of negation. We can, in effect, note that the introduction of negation into difference rests on the need to see contrary properties as not inhering in the same object, or occupying the same ‘space’. Since intensity is prior to the emergence of both objects and (extensive) space, these restrictions do not apply to it. As Deleuze points out, even if we do look at intensity as it occurs within extensive space, we do not find the strict absence of intensity: ‘It is said that in general there are no reports of null frequencies, no effectively null potentials, no absolutely null pressure, as though on a line with logarithmic graduations where zero lies at the end of an infinite series of smaller and smaller fractions’ (DR 234/294).
(SH 177)
[The next idea is that negation can be applied to properties, but not to intensities. From the Plato example, one instance of a property seems to be ‘equal to’ or at least just physical size. But I am not sure what would be other properties. They would not for example I guess be the ones mentioned above, like frequency and pressure. They would be extensive sorts of things I suppose like spatial dimensions, and perhaps physical distributions of component parts (or maybe not, if density is an intensity). But I wonder if these properties includes color, which can be understood quantitatively (and perhaps intensively) in terms of light wave frequency or qualitatively, with each basic color being different in kind from the others (blue is not more or less of something than red is, in how it appears). At any rate, for the next idea, recall how for Plato, contrary properties, like objects that under certain conditions appear equal, and under others appear unequal, shock us to think more about the Ideas we use. Deleuze also thinks that contrary qualities have this power. Yet, Plato thinks that the shock turns our mind toward timeless Ideas. Deleuze, however, thinks that the contrary properties arise because their source is (somehow) intensive difference. Thus, these shocking moments are an invitation to think more directly about the intensive basis of thought.]
Thus, whereas negation can be applied to properties, we never actually discover the negation of an intensity, but only its difference from other intensities. Deleuze here supports Plato’s insight that the fact that objects | can possess contrary properties presents a shock capable of leading to thinking. Rather than seeing these contrary properties as leading us to contemplation of a timeless realm of Ideas, Deleuze argues that they refer us to a field of intensive difference responsible for the change in qualities we find in the world around us. What makes the qualities in becoming contradictory is that they actualise an underlying intensive difference, and it is this difference that provides the real opening to thought.
(177-178)
3) Intensity is an implicated, enveloped, or embryonized quantity.
SH explains that this third characteristic of intensity is derived from the first two. [We now make a distinction between cardinal numbers and qualities. Cardinal numbers are divisible, but qualities are not. The reason we can divide cardinal numbers is because both their unity and their division are matters of mental actions. They are unified insofar as we mentally treat the value as a whole unit, but they are divided insofar as we regard them as a collection of smaller values. There is also discussion in the quote about continuity and discontinuity. I do not follow it readily, so I will just quote it and return to it if it factors in later. The next idea is that intensity can be divided, but not like extensities can be, since intensity is not metrically homogenous. But I do not understand how it can be divided. What we know is that if you divide an intensity, it changes the nature of what was divided, with the example being Bergson’s notion of consciousness or duration. I am not exactly sure how you divide consciousness in the first place, and then how the result of the divisions would be different in kind. Perhaps we might check section Pt2.Ch3.Sb4 from his Hegel, Deleuze book. But it gets a bit clearer when we look at the Bergson example (I am just not sure how it works for intensive things other than consciousness). He notes how if we listen to a melody, each new note or moment changes the character of the melody as a whole, just as each new moment of consciousness more generally changes the whole of consciousness on the whole. It seems Bergson gives one way in this melody example how we might divide our consciousness. He says that we can dwell on one note, while the melody continues developing in the moving present, and when we do so, we notice a difference between how the melody sounded back when that note was playing with how it sounds now, after more development has taken place. It seems then we can “divide” consciousness temporally by considering different moments making up present consciousness. What we notice is that each is different qualitatively. (One moment of the melody is not more or less than another. They instead have different characters). I am not sure Bergson would consider consciousness an intensity, but perhaps it is meant to illustrate more the idea of a type of multiplicity, of which Deleuze includes intensities, which change in nature when divided. For, SH’s next point is that the two types of multiplicity (ones that when divided produce parts that are the same in nature and ones that when divided produce parts that are different in nature) “can be mapped onto the extensive and intensive in a relatively straightforward manner” (179)]
Finally, as a third characteristic, ‘intensity is an implicated, enveloped, “embryonised” quantity’ (DR 237/297). This third characteristic is derived from the previous two. We have seen that cardinal numbers are divisible. Deleuze makes the claim in a lecture on Bergson that this is because, since they are a collection of equal units, dividing them is simply an intellectual operation [the following up to citation is Deleuze quotation]:
The divisibility of the unit; for a number is a unity only by virtue of the cardinal colligation, that is to say the simple act of the intelligence that considers the collection as a whole; but not only does the colligation bear on a plurality of units, each of these units is one only by virtue of the simple act that grasps it, and on the contrary is multiple in itself by virtue of its subdivisions upon which the colligation bears. It’s in this sense that every number is a distinct multiplicity. And two essential consequences arise from this: at once that the one and the multiple belong to numerical multiplicities, and also the discontinuous and the continuous. The one or discontinuous qualifies the indivisible act by which one conceives one number, then another, the multiple or continuous qualifying on the contrary the (infinitely divisible) matter colligated by this act. (L 00/00/70)
Qualities, on the other hand, are not divisible. It makes no sense to talk of dividing rationality, or animality, for instance. Now, intensity is not like quality, in that it can be divided. It is not composed of equal elements, however, but is rather a sequence of asymmetrical relations, such as we find with the ordinal numbers (it is asymmetrical in that second is defined by being ‘in between’ first and third, but first and third are not ‘in between’ second). Thus, ‘a temperature is not composed of other temperatures, or a speed of other speeds’ (DR 237/297). If an intensive multiplicity is not simply constituted from pre-existing elements, then division is true division, leading to a change in the nature of what is divided. Here we can turn to Bergson’s alternative form of multiplicity. For Bergson (at least at this stage of his philosophical development), this alternative form of organisation is that which we find in our conscious | states, although for Deleuze this mode of organisation is not simply a feature of our perception of the world, but rather of the world itself. As we can see, Bergson’s account of the perception of a melody presents clearly the way in which dividing non-extensive multiplicities leads to a change in their nature [the following up to citation is Bergson quotation]:
Pure duration is the form which the succession of our conscious states assumes when our ego lets itself live, when it refrains from separating its present state from its former states. For this purpose it need not be entirely absorbed in the passing sensation or idea; for then, on the contrary, it would no longer endure. Nor need it forget its former states: it is enough that, in recalling these states, it does not set them alongside its actual state as one point alongside another, but forms both the past and the present states into an organic whole, as happens when we recall the notes of a tune, melting, so to speak, into one another. Might it not be said that, even if these notes succeed one another, yet we perceive them in one another, and that their totality may be compared to a living being whose parts, although distinct, permeate one another just because they are so closely connected? The proof is that, if we interrupt the rhythm by dwelling longer than is right on one note of the tune, it is not its exaggerated length, as length, which will warn us of our mistake, but the qualitative change thereby caused in the whole of the musical phrase. We can thus conceive of succession without distinction, and think of it as a mutual penetration, an interconnexion and organization of elements, each one of which represents the whole, and cannot be distinguished or isolated from it except by abstract thought. (Bergson 1910: 100–1)
The two different notions of multiplicity can be mapped onto the extensive and intensive in a relatively straightforward manner [the following up to citation is Deleuze quotation:]
Therefore there are two types of multiplicity: one is called multiplicity of juxtaposition, numerical multiplicity, distinct multiplicity, actual multiplicity, material multiplicity, and for predicates it has, we will see, the following: the one and the multiple at once. The other: multiplicity of penetration, qualitative multiplicity, confused multiplicity, virtual multiplicity, organized multiplicity, and it rejects the predicate of the one as well as that of the same. (L 00/00/70)
These two forms of multiplicity can be related to the extensum (extensity in general), and the spatium (the field of intensive difference as a whole).
(178-179)
[Recall the three spatial syntheses from section 5.3. Quoting from our brief summary: “1) Intensive differences are localized by being distributed into various spatial locations, and they move from place to place according to how they interrelate and interact (in thermodynamics, for example, heat moves from its location to where cold is located, normally). 2) The extensive space into which these intensities are distributed and the qualities belonging to those things in extensive space come about somehow by means of intensive depth. 3) These distributions of intensities and their explications into extensive properties and other qualities continues to remain fresh and in a perpetual state of renewal.”] “The three spatial syntheses show how these two multiplicities are related” (SH 179). [It seems the idea is that they are related by the process of explication. I think I grasp what the results of explication are (namely, explicated intensities in the extensive realm, perhaps in the form of numericalized intensive properties, extensive properties, and qualities) but as a process or activity, I still do not quite understand how explication works. I have questions like, where are the deeper intensive differences that become explicated? Are they in the same location as the explicated ones? How does the transition from one realm to the other transpire? Or is it that there are extensive givens, but acting on them somehow are forces of variability, and explication is the effects of the variation?] SH concludes this section by turning us to the next one: “The final question to be addressed is how the intensive multiplicity is | related to the Idea. In answering this question, we will also have to deal with the problem of individuation, or the emergence of the subject from an a-subjective field of intensity” (SH 179-180).
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