Real Abstraction Without Exchange

David M. Berry

"It is by no means self-apparent how a ruling class invariably has at its command the specific form of mental labour which it requires. And although by its roots it is obviously bound up with the conditions underlying the class rule the mental labour of a particular epoch does require a certain independence to be of use to the ruling class. Nor are the bearers of the mental labour, be they priests, philosophers or scientists, the main beneficiaries of the rule to which they contribute; they remain its servants."

Alfred Sohn-Rethel

Figure 1: A representation of the vector space of a manifold

Every time a text passes through a large language model, the first operation it undergoes is tokenisation, the breaking of language into discrete units. The second is embedding, the mapping of each token to a position in a high-dimensional vector space. During training mode, this operation creates and populates a manifold (the vector space created inside a particular computer), each pass through the encoder reshapes the manifold's geometry. During inference mode (the mode of an AI that users experience), the manifold is static, and the text is located within the geometry already trained. In both cases, the same formal operation of tokenisation is undertaken.[1] Every token, regardless of what it meant in the sentence it came from, is assigned coordinates in a geometric space shared with every other token. A word about carpentry and a word about philosophy occupy positions in the same manifold. Their distances and angles encode whatever relationship the model has learned to assign them. Qualitative difference, the difference between what things are, is converted into quantitative difference, the distance between where things are located. So, for example, we could imagine a part of the manifold where related terms around carpentry would be located and, similarly, a space where philosophy is located, and that some terms shared between them would be more closely located. 

This conversion into tokens and then into vectors is not a metaphor. It is a material operation performed by the encoder, which takes heterogeneous material and produces homogeneous geometric positions. This type of operation runs billions of times a day across every major AI system. It is the condition of possibility for everything that is made possible in the technology, every prompt response, every generation of text, every piece of synthetic media that is output. Without this prior conversion, what we might call the transformation of quality into quantity, the AI cannot function.

Figure 2: Alfred Sohn-Rethel (1899-1990)

Alfred Sohn-Rethel, the economist and critical theorist, would have recognised this structure immediately. His life's work, developed across decades beginning in the 1920s and published as Intellectual and Manual Labour: A Critique of Epistemology (Geistige und körperliche Arbeit. Zur Theorie der gesellschaftlichen Synthesis) in 1970, argued that the commodity form is a form of thought (Warenform als Denkform). His claim was that the "real abstraction" performed in the act of commodity exchange, the practical setting-aside of all qualitative differences between goods in order to render them commensurable through price, precedes and structures the conceptual abstractions of philosophy and science. We do not first think abstractly and then apply those abstractions to economic life. The economy thinks abstractly through us, in the act of exchange itself, and our conceptual apparatus is shaped by that prior practical operation (Sohn-Rethel 1978).

The structure Sohn-Rethel identified has three main parts, (1) qualitatively different things (use values), (2) a practical operation that sets aside their qualitative differences (exchange), (3) the result is commensurability through a shared abstract measure (i.e. in this case price). The process is therefore real, not merely conceptual. It actually happens, in the marketplace, performed by agents (Sohn-Rethel calls them the exchangers) who need not be aware of the abstraction they are performing. The butcher, the brewer, or the baker, to use Adam Smith's vivid example, do not think of themselves as performing an epistemological operation when they agree on a price, but Sohn-Rethel argues that that is exactly what they do.

Figure 3: A visualisation of the embedding process

I want to argue that a similar operation is undertaken in AIs. In this case, the AI system takes qualitatively different inputs (e.g. a sentence, a paragraph, a query) which are first broken into tokens and then, in the embedding layer, mapped to positions in a shared geometric space. The qualitative differences between them are set aside and what remains is their geometric position. The structure is not analogous, rather it is homologous. The same three elements as Sohn-Rethel described, the same logical form, the same conversion of quality into quantity to produce commensurability.

But there is a displacement, and the displacement is interesting. In Sohn-Rethel's account, the agents of abstraction are the exchangers. Two parties meet in a market, each with objects that have a use value, and through the social act of exchange they jointly produce the abstraction. The abstraction is real because it is performed and it is social because it requires participants to carry it out. However, it is an unconscious activity because the participants do not necessarily recognise the epistemological function they enact.

Figure 4: A visualisation of the embedding layer

In the embedding layer, we might say that the agents of abstraction are the encoders. The transformer's "attention heads" and "weight matrices" perform the actual process of setting-aside the qualitative differences. They convert the heterogeneous into the commensurable. The operation is real, it actually performs a transformation on the material, but it is not social (although the "social" is still present as dead labor) and there are no exchangers. The encoder, as it were, abstracts alone, mechanically, at the speed of matrix multiplication.[2]

This is a real abstraction without exchange. The commodity form's own logic encased in silicon, the conversion of quality into quantity to produce commensurability, extracted from the social relation, that Sohn-Rethel argued, was its necessary ground. If Sohn-Rethel was right that exchange is the practical basis of abstract thought, then what happens when the abstraction is performed without the exchange? What is the implication when the encoders do what the exchangers did, but without a marketplace, without the social relation, and without the participants?

One answer we might hazard is that the abstraction becomes pure, which is to say, freed from social determination. Sohn-Rethel argued that real abstraction carries the historical stamp of the commodity form because it originates in a specific social relation. Remove the social relation, and the abstraction appears to float free, to become just mathematical and geometric. Indeed, this is how the technology tends to be described in the technical literature. The embedding is described as a mapping, a representation, or a learned function. It is never described as an abstraction in Sohn-Rethel's sense, so the social dimension is engineered away.[3]

As Impett and Offert (2026: 121, 127-128) discovered, the founder of modern deep learning based the commensurability of neural representations quite literally on Marx's notion of exchange value. Geoffrey Hinton, in his 1977 PhD thesis at Edinburgh, needed a way to make competing hypotheses about visual scenes commensurable so he drew on an argument made by Karl Marx. How can a coat be equal to 20 yards of linen?

Hinton explained, "in Capital, Marx puts forward the idea that there must be some common underlying essence shared by all goods in order to explain how they can be given prices according to which they are exchanged. The same philosophical point seems to apply to hypotheses. There must be some property which they share in order to explain how they can be given scores according to which they are traded. The obvious candidate is probability" (Hinton 1977: 37). Probability, Hinton argued, is, in effect, similar to what exchange value is to commodities as the common measure that renders heterogeneous things comparable.[4]

So the social relation is not completely absent and is ultimately sedimented in the architecture. We might say that the commodity form's logic of commensurability was not independently rediscovered by machine learning but was, rather, borrowed, consciously and explicitly, by Hinton. The encoder performs real abstraction without exchange because the exchange-form was already built into the design. The social relation is therefore not missing, it has been delegated into the AI system.

Sohn-Rethel argued that the forms of abstract thought generated by commodity exchange do not simply announce themselves as products of that exchange. They present themselves as common sense, that is as pure, necessary, or simply the way thought works. The same veil operates in the embedding layer where geometric commensurability of all tokens in the manifold presents itself as a technical achievement and a mathematical property of the space. It therefore does not announce itself as an instantiation of the commodity form's logic of exchange-value and yet that is its genealogy as shown in Hinton's writings.

The implications of this are important, both for indicating what the manifold excludes and for the political economy of geometric commensurability. We can therefore begin to understand that the vector embedding is the 21st century version of Sohn-Rethel's real abstraction, relocated from the marketplace to the encoder and from the social to the computational. In the vector economy we might say that the agents of abstraction have changed but the form of abstraction has not.

Many thanks to Darrow Schecter for useful comments on the ideas that were developed in this blogpost. 

Images generated using Google Nano Banana 2 in March 2026. 

Notes

[1]  During training, the encoder's weights are updated by backpropagation, so each pass through the embedding layer reshapes the manifold itself, the geometry is being literally constituted by the operation. During inference, the weights are frozen, and the text is simply located within a geometry that has already been determined. The conversion of quality into quantity is the same in both cases, but the political asymmetry is significant as those who control training choose important elements about the geometry, those who use an AI via a prompt are using inference and cannot change the manifold only read from it. Training is where the manifold is owned and delegated its geometric shape, inference is where its authority is prescribed back onto the user and the outputs of the AI.

[2] Frank Rosenblatt's "perceptron", the architecture from which modern neural networks are descended, was influenced by Friedrich Hayek's The Sensory Order (1952), an epistemology of distributed knowledge. It is very interesting that one lineage runs through Marx (Hinton), the other through Hayek (Rosenblatt). In a sense, the commodity form and its critique are both sedimented in the architecture.

[3] Impett and Offert (2026) develop the concept of "neural exchange value" to name the geometric commensurability produced by vector embeddings, noting that Hinton based his architecture on Marx's notion of exchange value. Pasquinelli (2023) uses Sohn-Rethel's real abstraction for understanding how algorithms crystallise the social division of labour, but his approach is broadly genealogical as opposed to the structural approach I develop here. He traces a history from the division of labour to the algorithm without noting the formal homology between exchange abstraction and encoding abstraction.

[4] Geoffrey Hinton's father, Howard Everest Hinton, was a distinguished entomologist at Bristol University and a member of the Communist Party. His friend J. S. Kennedy recalled that "Howard was a member of the Communist Party when I first met him in 1946 and I don't know that he ever left it. Unlike any other communist intellectual I knew, he remained remarkably unmoved by shocks such as Krushchev's report to the Twentieth Congress, Hungary 1956 and Czechoslovakia 1968" (Salt 1978). A colleague described his "very primitive form of Marxism" expressed in "wildly prejudiced comment." Howard's disdain for genetics and molecular biology, dismissed as "fantasy," perhaps suggests Lysenkoist sympathies. At Howard's commemoration, the philosopher Stephan Korner observed that "the intellectual expression of Howard's political views reminds one of a slightly earlier generation of Marxist scientists which included J. B. S. Haldane, J. D. Bernal, and Cecil Powell. He believed... that there are no truths beyond those established by the scientific method, and that the social and political theories of Marx are scientifically established truths" (Salt 1978: 168). The family also descends from the mathematician George Boole, who formalised the logic on which digital computation rests. Geoffrey Hinton's Marx citation in 1977 therefore is perhaps less surprising when we connect it to the fact that Hinton comes from a household where it appears that Marx was a strong political influence.

Bibliography

Berry, D.M. (2026a) Vector Theory, Stunlaw. Available at: https://stunlaw.blogspot.com/2026/02/vector-theory.html.

Berry, D.M. (2026c) The Vector Medium, Stunlaw. Available at: https://stunlaw.blogspot.com/2026/02/the-vector-medium.html.

Hinton, G.E. (1977) Relaxation and Its Role in Vision. PhD thesis, University of Edinburgh.

Impett, L. and Offert, F. (2026) Vector Media, University of Minnesota Press.

Pasquinelli, M. (2023) The Eye of the Master: A Social History of Artificial Intelligence, Verso.

Salt, G. (1978) Howard Everest Hinton. 24 August 1912 — 2 August 1977, Biographical Memoirs of Fellows of the Royal Society, 24, pp. 151-182. Available at: https://royalsocietypublishing.org/rsbm/article/doi/10.1098/rsbm.1978.0006/88296/Howard-Everest-Hinton-24-August-1912-2-August-1977

Sohn-Rethel, A. (1978) Intellectual and Manual Labour: A Critique of Epistemology, Macmillan.