I’m delighted to kick off the Autumn/Fall with Felicia Jing’s essay here in the Math(s), Philosophy, History reading group.

Felicia Jing just completed a PhD in Political Theory at Johns Hopkins University, is a visiting fellow at NYU’s Remarque Institute this Fall, and will be joining Brown University in the Spring 2027 for a postdoc in the Program of Science, Technology, and Society. The essay we read today is a condensed version of her dissertation’s first chapter that is forthcoming in Political Concepts: A Critical Lexicon.

This essay offers us a new and more materialist angle on the history and politics of computer science. The materiality of the algorithm is, following Marx, not only the economy of the hardware on which it runs, but also the socio-political logic that structures developments in hardware, software, and programming. The substitution of software for hardware when it comes to questions of its economy—a substition spirited in no small part in media studies by Friedrich Kittler’s infamous claim that “there is no software”—displaces rather than addressses the question of the politics and historicity of computing.

Jing shows that there is something about the logic of algorithms—and therefore of software, given code is treated as the materialization of algorithmic logic—that walks hand-in-hand with capital. Indeed, Jing’s gesture to this intimate contact between capital and computation builds on Alfred Sohn-Rethel’s chief insight, namely that one of the most conseqential results of capital’s perceptual physics [Best] on the division of labour is to conceptually separate the head from the hand, and therefore to valorize the labor that issues from the ‘intellect’ in a way that undermines the conscious and egalitarian contract of capital and the wage, that is, that all labour is exchangeable under the aegis of fair compensation.

Jing narrates the courting of these two concepts first by touching on the algorithm as an Althusserian ‘absent concept’, and then by firming it up as a Robinsonnade with reference to the Marxian critique of prehistory:

In a vanishing act of ‘hiding the machine’, algorithms are sanitized not only from the material development of hardware, its dirty industrial history, its dependency on the labor of earlier ‘human computers’ but also—importantly–from the state machine. That is, the centralized bureaucratic state, class antogonisms, and the legal forms of coercion were, at every moment, a site of struggle over and against the political form of an alorithm. (Jing 2026, 8–9)Jing, Felicia. 2026. “Algorithm.” July

Jing goes on to show how the conceptual constitution of the ‘algorithm’ as prehistorical flattens the history of computer science into a crude, teleological account, geared towards a narrative of ever-advancing progress. This flattened conception squeezes the possibility of criticality, politicality, and thus an emancipatory politics from the entire enterprise of computing, readying it to be cast, like a ketchup wrapper, into the dustbins of history as a bone that could never stick in the throat of capital’s voracious appetite.

What I love about this essay is how Jing takes on Donald Knuth just as Marx did Adam Smith, i.e. as the originator of a mythology with real consequences. I think Knuth is the right figure to take on for a number of reasons, not least because it is also Knuth who establishes the idea of ‘literate programming’ as its idea, which is to say that Knuth argues that a program should be conceptualized as a work of art such a a novel, an artefact by analogy of literature chiefly, and not of industry. Jing’s argument shows how the Knuth’s analogical insistence in literate programming might hide another absent concept in the computer program. In its prehistorical guise as architected by a single and singular genius, the computer program becomes—like the novel—a form of expression distinctly fathered by modernity, and therefore by capital.

To mark the genealogical relation between computing and capital in this way is not the same, critically, as consigning computing and computer science to the collection of practices with which we must not dare to engage should we seek an emancipated future. On the contrary, following Marx’s engagement with the political economy of his day, it is to insist that we must read figures such as Donald Knuth again, symptomatically, to articulate more concretely how capital and computing are entangled as logical structures, what power one does and does not have over the other.

It is Jing’s keen engagement with this materiality of computing, a materiality truer than that which results from holding one’s eye up to the end of a fiber optic cable—as friend-of-the-pod Tung-Hui Hu once reported himself doing—that makes her essay an opportune way out of our group’s summer focus on the 19^th century, our 19^th century summer.

One of our group’s recurring themes is how we should think the materiality of mathematics in general, and the materiality of pure mathematics in particular, the algebraic turn that authors we have read such as Jeremy Gray and Erich Hörl have referred to variably as mathematics’ modernism or modernization. Algorithmic materiality, it seems to me, skips hand-in-hand with the materiality of calculus, at least when the song of capital is playing over the speakers. The conceptual divisions we have tracked in previous sessions, between applied and pure mathematics, between pure mathematics and mathematic physics, and between algebra and geometry: all these divisions should be thought in the light of capital’s surreptitously enforced division between head and hand, between intellectual-qua-immaterial and manual-qua-material labour. This is what Jing’s essay illuminates for us, and I’ll now turn over to her to say a few words about it before we open up for any and all questions and comments as usual.

By Lachlan Kermode.

I’m delighted today to introduce Matt Handelman, who is an Associate Professor of German, a member of the Core Faculty in Digital Humanities, and Affiliated Faculty in Jewish Studies at Michigan State University. Today we’re discussing a chapter from his book The Mathematical Imagination: On the Origins and Promise of Critical Theory, published by Fordham University Press in 2019. Matt argues that critical theory caricatures mathematics as “a mere instrument of reason akin to a form of political quietism” (Handelman 2019, 40)Handelman, Matthew. 2019. “The Trouble with Logical Positivism: Max Horkheimer, Theodor W. Adorno, And the Origins of Critical Theory.” In The Mathematical Imagination: On the Origins and Promise of Critical Theory, The Mathematical Imagination: On the Origins and Promise of Critical Theory. Fordham Univ Press through a reductionist equation of mathematics with a total rationalization of the world, words put in its mouth by logical positivism. Critical theory—read principally through a close and critical engagement with Horkheimer and Adorno’s magnum opus Dialectic of Enlightenment—treats logical positivism, in turn, as a school of thought that is “a synecdoche for an exclusionary type of meaning” (Handelman 2019, 39)Handelman, Matthew. 2019. “The Trouble with Logical Positivism: Max Horkheimer, Theodor W. Adorno, And the Origins of Critical Theory.” In The Mathematical Imagination: On the Origins and Promise of Critical Theory, The Mathematical Imagination: On the Origins and Promise of Critical Theory. Fordham Univ Press and guilty of a “stubborn adherence to and repetition of the given” (Handelman 2019, 47)Handelman, Matthew. 2019. “The Trouble with Logical Positivism: Max Horkheimer, Theodor W. Adorno, And the Origins of Critical Theory.” In The Mathematical Imagination: On the Origins and Promise of Critical Theory, The Mathematical Imagination: On the Origins and Promise of Critical Theory. Fordham Univ Press. Critical theory in the wake of Horkheimer and Adorno has failed to separate mathematics as a discipline, a culture, and a field of thought from logical positivism’s strong-arming of it into empiricism’s true form of appearance, the ideal vocabulary for objectively accounting for the world, and so has turned a blind eye to mathematics’ political and critical potential. Though Matt argues that the critical account of logical positivism is merited, he recognizes how it has given mathematics as a whole a bad name, mischaracterizing it as a bad infinity with which critical philosophy should not engage.

The resolution of this squabble, Matt proposes, is a project that he calls negative mathematics in the shadow of Adorno’s notion of a negative dialectics. Negative mathematics overcomes the Frankfurt School objection by seeing mathematics not as an ideology that flattens the world out into a set of protocol sentences, but as a discourse which already has style, space for interpretation, and contingency. What Matt calls negative mathematics is what some of the mathematicians we have read in this group, such as Giuseppe Longo, would probably just call ‘mathematics’. The mistreatment of mathematics by the critical and theoretical humanities is a theme with which we in this group have engaged before, and before handing over to Matt to say something if he would like to, I want to draw out just two ways in which the necessity of negative mathematics seems to be appearing in affairs proximate to us today.

Recounting Benjamin’s investment in them, Matt paraphrases critical theory’s advocacy for language and philosophy with the statement:

Language and philosophy both depend on their content as much as on their form: the power of language lies in the dialectical interaction between what I say and how I do it. (Handelman 2019, 39)Handelman, Matthew. 2019. “The Trouble with Logical Positivism: Max Horkheimer, Theodor W. Adorno, And the Origins of Critical Theory.” In The Mathematical Imagination: On the Origins and Promise of Critical Theory, The Mathematical Imagination: On the Origins and Promise of Critical Theory. Fordham Univ Press

If we replace ‘language and philosophy’ here with ‘mathematics’ and drop the word ‘dialectical’, we have what I see as quite a similar stance to the mainstream reaction to OpenAI’s production of a Lean formalization resolving the Navier–Stokes existence and smoothness problem with over 11,000 AI agents in approximately 88 hours. As two prominent mathematicians put it in a co-authored blog post in response to the looming question of whether OpenAI’s result would sunset mathematics:

Mathematics is clearly a much broader enterprise than just problem solving. Mathematicians strive to develop new concepts and theories, to ask and answer new questions, to unify disparate areas, to educate and sustain scholarly communities, and to produce work that is valued for its beauty and depth (De Toffoli and Duede 2026)Toffoli, Silvia De, and Duede. 2026. “After Math.” September.

This might be evidence that what we call mathematics in the academic-institutional sense today is quite different from what Horkheimer and Adorno called mathematics. It is also true, I think, that we have more extensive vocabularies in the twenty-first century to distinguish different kinds of mathematical enterprise, in no small part due to the proliferation of the computing sciences. There are aspects of Adorno and Horkheimer’s critique that still ring piercingly true when we substitute ‘computing’ for ‘mathematics’, such as the idea that it “underpin[s] the categories of economic utility, scientific advancement, rationalization, and, ultimately, the domination of nature as a prime example of instrumental reason” (Handelman 2019, 29)Handelman, Matthew. 2019. “The Trouble with Logical Positivism: Max Horkheimer, Theodor W. Adorno, And the Origins of Critical Theory.” In The Mathematical Imagination: On the Origins and Promise of Critical Theory, The Mathematical Imagination: On the Origins and Promise of Critical Theory. Fordham Univ Press. Anything that the artificial intelligentsia cannot reduce to numbers they dismiss as irrelevant to the essential function of language, of image and video, and of life (Handelman 2019, 37)Handelman, Matthew. 2019. “The Trouble with Logical Positivism: Max Horkheimer, Theodor W. Adorno, And the Origins of Critical Theory.” In The Mathematical Imagination: On the Origins and Promise of Critical Theory, The Mathematical Imagination: On the Origins and Promise of Critical Theory. Fordham Univ Press.

Just as an empiricism rooted in protocol sentences ultimately flowers as the crudest kind of mathematical idealism when we look at it in the rearview mirror of critical theory, the data empiricism that we see blossoming today in the atmosphere of rhetorical grandiosity, in artificial general intelligence and world models, seems the most immature kind of scientific realism. Critical theory’s suspicion of mathematical logic also seems a weed we are yet to properly root out: mathematics and the computing sciences hardly mix with the humanities, except at the mixers where the mathematical sciences and the humanities are lined up on either side of the conference venue and asked to move arm-in-arm, courting each other.