Can Exchange Cause Change? Part I. Mathematical Derivation

Can exchange cause change? If Deleuze’s thesis on difference and repetition applies to economic transactions, we have reason to believe it can. But how? Is not to ask whether exchange can cause change another way of asking whether the repetition of an object into its image of value as money can produce difference, true difference, meaningful difference, a repetition that causes a new change in kind? At first glance perhaps it’s not so evident that exchange can cause change. But let us consider this.

i. One of the most, if not the most ontologically basic questions asked in mathematics is whether two given mathematical objects are the same? In the geometer’s hands, the matter of difference in repetition in relation to the question of change is transmuted, indeed elevated, to the status of the crucial question of all mathematics.

Badiou says mathematics is ontology, but this is wrong. Ontology is ontology. Ontology is the study of Being. Being is univocal. Being’s univocity articulates itself in a diffuse set of registers. And mathematics and finance are two such registers –which renders them neither metaphorical nor analogical nor pure; rather, each register is a derivative subset isomorphic to the other; but neither register resembles each other in actuality.

When the geometer maps the transformation of an object into its image, such as

congruent motions, i.e. classical exchange
Figure 1.0. congruent motions, i.e. classical exchange

…we see that a general repetition of the object has occurred. The question is: given that O is now O’, has change occurred?

On the one hand, let us not overthink this question: Is O the same as O’, or are O and O’ different? –and if so, in what respect?

On the other hand, let us not underthink the questions contained inside this question: What do we mean by “the same”? What do we mean by “different”? What do we mean by “change”?

When the mathematician says “the same” she often means “equal”. For example:

Richeson observes:[1]

5 · 4 + 6 – 23 = 18

To say that these two different expressions are “equal”, this means that the expression

5 · 4 + 6 – 23

…and the number “18” –that these two things are the “same”.

Is it possible that it is more than ironic that this equivalence relation persists in the economist’s conception of classical exchange?

If 1lb. of coffee is exchangeable for the price of $5, when that object, the 1lb. of coffee, is transformed into its image of value as money, to say that these two expressions are “equal”, this means that the expression 1lb. of coffee and the amount of $5 –they are “the same”.

Or to cite another example from mathematics, we know and say that the polynomials:

x2 + 3x + 2     and    (x +2) (x +1)

…are “equal”. Even though they are housed in and as two different incarnations, they are said to be the same. And the reason why is that one can move, or more correctly put, ‘transform’ one expression into the other, i.e. one can treat and repeat one expression as the other, while yet no change has occurred.

So too if 1lb. of coffee is worth $5, and 2lbs. of tea is worth $5, even though the two objects are different incarnations, they are regarded as the same. One can transform one expression of $5 into the other, and one can repeat the one expression “coffee” for the other expression “tea”, while yet once again we see no change has occurred.

In classical exchange, then, which involves the immediate settlement of physical objects for their images of value as money, we see that exchange does not cause change.

Figure 1.1. Classical Exchange
Figure 1.1. Classical Exchange
Figure 1.2. Generic Finance
Figure 1.2. Generic Finance

We have said before that classical exchange is to generic finance in exchange what congruence is to similarity in geometric transformations, and with a little reflection on this one can easily see why (also please see the Appendix to Of Synthetic Finance for more on this). Neither congruence nor similarity allow for much change to occur. Different geometric objects may move to and fro, and back and forth, as if they were strung up and running along parallel lines. If they remind you of Cartesian coordinates, this will come as no surprise, for they are numerical multiplicities par excellence. Yes, classical objects and generic financial objects move –of course they move –in space and time. But their motions in this ambient Euclidean space always entail an invariant change, an empty form of difference, a variation that produces no true variation at all. Any and all difference therefore comprises an indifference to difference, a non-differentiation, a generality, a repetition of the same.

Conversely, in Euclidean space, if one object does not repeat itself congruently, or similarly –e.g. if its angles, its shape, and so on, are not the same – when it is transformed into its image, the Euclidean geometer will say that there are two objects that are different, i.e. a change has occurred, and that these two objects do not belong to the same equivalence class of exchange. There is between them no symmetry. There is a difference.

ii. This, of course, is not the end of the matter, for with the Euclidean geometer we cannot agree. Topology has a much more progressive set of criteria for what constitutes difference; for this reason it has a much more relaxed set of invariance requirements on the object transformed into its image. Topology is often much less impressed –dispositionally– by claims made by Euclidean geometric transformations, these self-congratulatory assertions by the Euclidean geometer that an object which does not repeat into its congruent or similar image realizes difference. Topology illustrates that where there is often an alleged, actualized, and apparent difference, there is, in truth, only generality and equivalence.

In topology, a square can be stretched into a cylinder, or an annulus.[2]

 Inline image 1

Figure 1.3

In topology, a square can be stretched and then bent into a torus.

Inline image 1

Figure 1.4

Such objects can be transformed into one another. For the topologist, when this actual transformation happens, no material change is thought to have occurred. There is yes, some difference to the object’s repetition: but once again, it is of the order of generality, it is an invariant form of variation, an empty form of difference, which is to say that no real change has occurred.

To be clear, in topology, if one object can be continuously transformed into another, these objects are considered to be the same. And the symmetrical transformation of an object into its image need not be congruent, and it need not be similar. Bending, twisting, turning, squashing, or stretching the shape of an object does not comprise change, for it does not change the object’s topology, it does not change its topological invariants, no true difference is caused by this transformation. By contrast, puncturing, cutting, and gluing does (or at least can) change the object in kind.

Let us observe that the Deleuzian registers of reality (actuality, potentiality, virtuality) now begin to supervene on our considerations herein. If, following Richeson, we observe a variety of polyhedra, such as in Figure 1.5

Inline image 1

Figure 1.5

….we see that each individuated polyhedral may phenomenally appear quite different than, for example, a sphere; and in actuality they are; but that the nonnactualized, but very real structure of what is possible for them to become in actuality is in fact the same: the three polyhedra immediately to the right of the sphere are not actually a sphere, but they could easily be continuously deformed into a sphere. Their extensive properties are what they are in actuality; but their intensive properties, while not actual, are every bit as real.

Conversely, we see that other polyhedra will also phenomenally appear to be quite different than a sphere; and once again in actuality they are. However, it is also true that in this second instance, an ontological examination of their topological invariants reveals the incorporeal existence of an even deeper structure than merely their intensive properties –a nonactualized, nonprobabilistic structure– to what is possible for them to become. The topologist devises methods by which to examine the ontology of these objects, which in turn allow her to study these objects’ topological invariants. And if the topologist, in the course of her ontological examination of their topological invariants, finds that the invariants are different, in turn such objects are considered qualitatively different in kind.

Already in this rudimentary inaugural consideration of the first elements of topology, we see that the topologist requires an additional register of reality. It is not enough to merely defer to geometric objects’ (a) ‘actuality’, i.e. that which, with their veritable extrinsic properties, they “are” as actualized objects; and their (b) ‘potentiality’, i.e. that which is possible for the objects to become in actuality, and as such, is subject to a probability distribution. She is in need another concept here, for when considering the topological invariants used to categorize geometric objects, she’s already (like Tartini in the Devil’s Trill) tapping another register of reality –something that adheres not to the object’s actual properties, not to their extensive properties, but which more fundamentally comprises the structure to what is possible for their extensive properties to become. And this is neither actual nor potential, nor strictly speaking “in them”, as such. Rather it is an incorporeal structure, another register altogether, that supervenes on the other two registers. The topologist is making recourse to, without naming, the technical concept invoked Deleuze to elaborate this register: it is the register of the virtual.

iii. If, in the course of its repetition from one to another place in space, a geometric figure can be deformed into another geometric figure in 3-dimensional space –and is then ostensibly different in its repetition– the topologist will say ‘these two figures have the same extrinsic topology’. In using this term ‘extrinsic’, the topologist is delineating for us two distinct ontological categories. The first is isotopy; the second is homeomorphism.

If two objects have the same extrinsic topology, they are called isotopic –topically, extrinsically, apparently,  they are actually different and yet potentially the same. For example, the four polyhedra in Figure 1.5 share the same extrinsic topology: they are isotopic. They need not be actually at that moment the same as each other -but they are for the topologist the same as each other, insofar as a simply bending or twisting or squashing or stretching is all that is required to transform the one into the other with yet any change to its topological invariants having occurred.

However, a deeper and more ontologically-fundamental dimension of a geometric figure’s reality is available to thought. For the topologist, two given objects may not appear to be the same, i.e. are not even extrinsically the same, but nonetheless are regarded as the same if they have the same ‘intrinsic topology’. “Do not trust phenomenal appearances!”, the topologist warns. “Don’t put too much stake in the fleeting, temporal shape of a geometric figure!”, the topologist exclaims, running with her lantern clasped tightly in hand, and out into the marketplace. “These extrinsic objects that you see as fixed, these too are actualized incarnations, contingent states of a deeper materiality!”, she urges. “Don’t mistake the immanent for the transcendent!”, she says.

Will the political economist listen to her? What does the topologist mean?

To better illustrate this –and especially to illustrate the ontological isomorphism between topology and finance in relation to the virtual, as well as the consequences of the univocity of Being for the speculative materialist political economist– let us take this a step further.

Let us consider an instance of cutting and gluing –or as the speculative materialist would rather put it when examining the ontology of financial objects: let us consider an instance of differentiating and dedifferentiating, or dividing and reassembling. (Note: We will later discuss that in securitization this process goes by the terms ‘pooling’ (dedifferentiation) and ‘tranching’ (redifferentiation): it is a division (cutting) that occurs, but we wish to consider whether it can and does produce a new change in kind?)

Inline image 1

Figure 1.6

It is the case that cutting and gluing very well can and may effect change to a geometric figure. It is possible that cutting and gluing may change the topology of an object. But it is also true that this is not always necessarily the case.

For example, Richeson points out that if we cut a shape, but then glue ‘the severed pieces so that the cuts line up exactly as they did before’, in this case no change has occurred, for the geometric figure’s topology has not changed.

This is the case with the knotted torus in Figure 1.6. Here if we cut the tube of a torus in half, we have created a cylinder. If we stretch out the cylinder, but then reglue the cylinder in the manner illustrated in Figure 1.6, i.e. as a knot, the resulting shape is still topologically the same as that of the original torus. It is true that the two objects –O and O’– are not isotopic, i.e. they are not extrinsically the same; for the knotted torus cannot actually be obtained, or derived, from the original torus through its deformation in 3-dimensional space. The intrinsic topology of the two tori are the same, their extrinsic topology is not. Therefore, again for the topologist no change has occurred. There is no difference that has been made.

How then can we know whether in a given transformation a true change has occurred? How do we know difference when and where it occurs? In short, what is different and what is the same?

For the topologist, two objects are the same, i.e. they have the same intrinsic topology, if there is a one-to-one correspondence between the topological properties of the two objects that preserve their closeness –or as Richeson puts it, ‘if nearby points in one shape correspond to nearby points in the other.’

Mobius, who was one of the founders of topology, named such a relation a homeomorphism.

iv. So herein lies our first speculative materialist concern with topology: we wish to know more about the homeomorphisms that supervene on extrinsically different financial assets? For this reason we must turn more directly to finance to address this question.

For example, when Modigliani and Miller observe the occurrence of a nondifferentiation between debt and equity for the capital structure of a firm, is this observation predicated on the assumption of a fundamental homeomorphism between debt and equity -that, yes, extrinsically and in actuality debt and equity are different, but intrinsically they are the same?

When Black-Scholes-Merton provide a formula (which we know is ‘wrong’, but for this reason is inverted and still functionally-used) that allows for continuous zero-beta hedging with a combination of partial objects (options) and whole objects (treasuries), i.e. which are extrinsically different, is not their silent premise that the economic properties of a variety of financial objects can be continuously arranged so as to achieve absolute nondifferentiation at every tick, i.e. that ostensibly different objects’ economic properties can be cut and then glued, divided and reattached in a variety of manners so as to always achieve a homeomorphism between assets and liabilities –which is, of course, when incessantly plugged into BSM, is supposed to perpetually realize the promise of a zero-beta portfolio?

And when today, after implied volatility (which is a cold slap in the face to performativity), Ayache gives Deleuzian PED’s to the BSM athlete, who then dynamically replicates at every tick, to what great topological feat do we bear witness? -The answer is, namely, that when differentiation gives you a glimpse of itself, right away the market maker must move to dedifferentiate. If partial objects can be used to continuously recalibrate, it is because of the increasing plasticity of certain financial assets; it is because certain financial assets have now acquired an augmented pliability, a hyperfungibility, an increased symmetry, i.e. their properties are isolable, they can be stretched and bent and squashed and torqued. Some financial assets are, in a word, isotopic, and some are homeomorphic.

Now, then, we wish to know (a) which financial objects are extrinsically the same, which are intrinsically the same and (b) what, if anything, this increasing regressive differentiation of financial assets means for the historical-materialist trajectory of finance as a dynamical system, as  a set of classes of exchange, as a series of markets, as well as the progressive differentiation of the objects populating them?

Indeed, it does seem that the whole edifice of modern finance is greatly illuminated when refracted through the ontology of topology. It for this reason that in Part II we will examine financial exchange under its light.

Outline of IV Parts

Part I. Mathematical Derivation

Part II. Financial Derivation

Part III. Dynamical Systems Derivation

Part IV. Financial Derivation

[1] David Richeson, Euler’s Gem (pg. 173). All mathematical examples are drawn from Richeson.

[2] Euler’s Gem. All subsequent figures are drawn from Richeson.


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