Equivalence and Abstraction
The forms of value presented so far contain an asymmetrical or polar relation, whereby the property of symmetry and consequently the equivalence relation, which always includes back-reference, can already be established for the simple form of value by means of the operation of reversing the respective “equation”. The axiom of symmetry thus proves to be absolutely necessary for the representation of the equivalence relation of classical commodities. Hans-Dieter Bahr writes: “The logical form of an equation initially consists of an abstract mapping or projection that expresses the word. But without a further rule of operation, this mapping is indistinguishable from a metaphor: Canvas is like skirt, both like gold. It is only through the axiom of symmetry that this so – like gains the structure of an equivalence relation, an exactly so – like. 20 cubits of canvas are exactly equal to one skirt, if one skirt is equal to 20 cubits of canvas.” (Bahr 1983: 379) The value expression “commodity x A is worth commodity y B” (sentence: commodity A expresses its value in commodity B) is thus to be understood as equivalent to the value expression “commodity y B is worth commodity x A” (sentence: commodity B expresses its value in commodity A). If equivalence applies, then the two propositions can certainly differ with regard to the perspective of observation – which is always also an observation of difference inherent in all first-order observations – but their indifference and thus their symmetry remains guaranteed by the fact that the unity of difference can be established, insofar as the unity of difference here denotes a binary observation difference without one of the two sides of the observation difference being marked. The second-order observer, for example, who observes that a first-order observer describes a value expression, can describe the first-order observation as an observation that uses the unity of difference of value/object of use. From his point of view, this applies to both first-order observers, who can of course also take on the role of the second-order observer. The mode of equality of observation (the first-order observer), which is established by the second-order observer, reflects on the one hand the symmetry of the equivalence relation, which on the other hand is established by an assignment rule/axiom.
For Marx, the equivalence relation is supposed to indicate, among other things, that different commodities have a single common property (value), namely as a dimension for determining the elements belonging to a quantity, with which commodities can be compared as quantities in the first place. This property is quantity per se, which Sohn Rethel called “quantity in abstracto”, whereby for him the relational determination of this quantity always implies a practical realization. (Cf. Sohn Rethel 1970: 55f.) In this context, Sohn Rethel rightly said that the equivalence of the so-called exchange equation does not mean equality, but rather equivalence, although this statement should by no means be aimed at any kind of mediation between exchange and value expression (value form), as Sohn-Rethel himself believes. (Ibid: 57) Rather, this is an equivalence that is purely immanent to the value form, so that it must not be interpreted as a (capitalist) exchange relationship, because money is ultimately required for this. In the capitalist commodity economy, the relationship between relation and relata is a very special one, meaning that we are not dealing with a relation that remains external to the relata, but rather, as will be shown, value is usually understood by Marxist economists as one and the same common property of the relata in the relation of equality, more precisely the equivalence (internal relation) of classical commodities, i.e. of heterogeneous objects. i.e. of heterogeneous objects which, despite their qualitative difference in terms of value, are considered equal, with their property of symmetry representing quasi images of value in the value form by representing value in a homogeneous economic space, which is itself characterized by a rigid isometric movement of objects, i.e. the absolute non-variability of the properties of objects. And what tends to disappear with commodification/capitalization is what Deleuze calls the anti-sterile or clothed repetition, which only ever repeats the new or the variation and thus remains bound to a single case, to a singularity that always returns differently. (Deleuze 1992a: 43f.) This singularity is not only confirmed, but affirmed by difference thinking, in that it knows that with repetition one does not add a second and third time to the first, but raises the first time to the nth power. Deleuze writes in the introduction to Difference and Repetition: “As a mode of behavior and as a point of view, repetition concerns an irreplaceable, irreplaceable singularity.” (Deleuze 1992a: 15) It is precisely this non-identical kind of non-replaceability that gradually disappears with the contemporaneous internal history of capital (with whose comprehensive grasping (Althusser) the enormous tenacity, solidity and stability of capital is ensured to the point of fluid solidification) (although the problem of non-identical repetition emerges in a new form in synthetic financial instruments), whereby even the very last thought of the non-identical and the open is supposed to vanish into thin air, as if capitalism actually possessed the evolutionary tenacity and life expectancy of a crocodile. (Pohrt 2012: 69)
Let us now turn to the concept of abstraction, which Peter Ruben wants to be understood in a genuinely Marxian sense as a logical operation that is directly linked to the establishment of the equivalence relation. In a completely different context, Dirk Baecker has referred to Korzybski’s “structural differential”, which comes close to Ruben’s argument and contains the following differentiations: “[…] between events, E, whose properties are infinitely many, objects, O, which select a few of these properties, and names, N, which consider even fewer properties to be essential […] Structures, S, are abstractions that can be written as follows in terms of set theory: S = N ⊂ O ⊂ E. ” (Baecker 2012) And similarly, the value form analysis involves the investigation of a structural differential relationship: If one wants to prove the equivalence relation R(x,y) with the structural properties R(xx) (reflexivity), R(x,y)→ R(y,x) (symmetry) and (R(x,y) & R(y,z)→R(x,z) (transitivity) for a basic set M, then one has to pick out an element a from the basic set M with which it is possible to form a class of equivalent objects with respect to a property. Ruben writes: “If it is possible to form a class of equivalent objects with respect to a characteristic for an element from a basic set, then an equivalence relation exists in the basic set with respect to this characteristic; if an equivalence relation exists, i.e. can be assumed in the basic set, then equivalence classes (abstraction classes) can be formed with respect to the characteristic in question (which is always expressed by a two-digit predictor). In this sense, abstracting means that equivalence classes are formed.” (Ruben 2007: 12) Ruben understands the logical operation of abstraction as the construction of the equivalence relation and the formation of equivalence classes and thus of course immediately denies that the concept of abstraction could be equated with the concept of generalization, for example, which is based on or appeals to a given insight, or that abstraction is even to be understood analogously to the peeling of an onion in order to penetrate step by step to the core.
With the equivalence relation, equivalence classes are defined at the same time, the elements of which are certain commodity quanta, whereby with Michael Heinrich it is also possible to speak of a quotient set with regard to the overarching or common features of these classes, in which in fact all commodities are integrated. With regard to commodities and their relation to a logical third party, no recourse is made at all to the concept of value substance, but rather only to the determination of commodities with regard to their belonging to a quotient set, whereby a product assumes commodity status in the first place. For Harald Strauß, in the case of commodities we must assume “actualizations of the virtuality of value in prices”, which is not a given, because not all products are sold on the markets and thus transformed into commodities; in other words, no matter how much labour is objectified in the products, only through their sale do they acquire commodity status in real terms and thus realize profit. (Strauß 2013: 232)
Let us summarize: If for any given commodity there is a value relation to two other commodities (which can be quantitatively specified as exchange values), then the axiom of transitivity is guaranteed for commodities A,B,C: W(A,B), W(B,C) and W(A,C) apply. The fact that Marx does not explicitly demonstrate this deduction in Capital, although it is already implicit in the unfolded value form, can probably be attributed to the fact that the axiom of transitivity is already included due to the justified assumption of the quantity of commodities as an intensional quantity and the logically flawless formation of equivalence classes that is thus possible, Otherwise, in capitalism we would never be dealing with any form of equivalent commodity-money transactions, but always with some singular form of profiteering, usury or overreaching, through the execution of which profits may be made. profits may be made. We will return to this problem later with the question of equivalent exchange and/or mercantilism. If the conditions of symmetry/equivalence, which according to Marx must be fulfilled for it to be even approximately possible to speak of an economic structural connection in capitalism, are now fulfilled within the framework of the concatenation of commodity-money transactions, then in principle any commodity can occupy the initial or final position in a series, whereby there are always several shifting series that ensure the syntax of commodity concatenations in relation to money. Accordingly, if a conjunctive value relation is written as follows: Wn (A,B,C, …)= W (A,B) and W(B,C) and W (C, …) and so on, the following relation can also be written due to the transitivity of W: Wn (A,B,C, …) = W(A,B) and W(A,C) and W (A, …) and so on; so here already the specific functionalization of the conjunction “and” makes it clear that exactly this syntax is equivalent to a system of relations that is open in genetic terms. Insofar as all commodities, if the axioms of reflexivity and symmetry apply, are in a relationship of equivalence to one another, something like a purely logical – systemic connection is constituted, but ultimately only insofar as objects are placed in relation to one another, which are realized as commodities in their reference to money. (Cf. Kirchhoff/Reutlinger 2006: 210)