Zurich University of the Arts · Immersive Arts Space
What Escapes the Model? The Limits of Formalisation in AI
October 29, 2026 – Zürich, Switzerland
Formalization Is the Limit
For a Critical Formalism after Machine Learning
Unbelievable progress
2012/2014
2024/2026
What is critique?
Anti-dogmaticity
Knowledge does not bear on substances,
but on phenomena
Conditionality
Knowledge is not absolute,
but relative to very precise conditions
Limitedness
The legitimacy of knowledge is not unbounded,
but circumscribed within fairly precise limits
The birth of contemporary critique
“In some remote corner of the universe, flickering in the light of the countless solar systems into which it had been poured, there was once a planet on which clever animals invented cognition. It was the most arrogant and most mendacious minute in the ‘history of the world’…”
On Truth and Lying in a Non-Moral Sense (Nietzsche, 1873)
The formalization of language theory
Logical Syntax of Language
(1934)
Logic
↓
Semantics
Information Theory
(1948)
Mathematics
↓
Pragmatics
Generative Linguistics
(1957)
Computer Science
↓
Syntax
Nietzsche against forms
“their eyes merely glide across the surface of things and see ‘forms’; nowhere does their perception lead into truth; instead it is content to receive stimuli and, as it were, to play with its fingers on the back of things.”
“…the great edifice of concepts exhibits the rigid regularity of a Roman columbarium, while logic breathes out that air of severity and coolness which is peculiar to mathematics.”
“…the only things we really know about them [the laws of nature] are things which we bring to bear on them: time and space, in other words, relations of succession and number.”
LLMs as programs computing functions
Neural LM
❓
🧠❓
🤖❓
🦜❓
💻❓
🗺️❓
\(f\)❗
Function
\(f\colon \mathbb{R}^n \overset{f_1}{\longrightarrow} \mathbb{R}^{n_1} \overset{f_2}{\longrightarrow}\cdots \overset{f_K}{\longrightarrow} \mathbb{R}^{n_K} \overset{g}{\longrightarrow}\mathbb{R}^m\)
Computable
Function
\(f\colon \mathbb{R}^n \overset{f_1}{\longrightarrow} \mathbb{R}^{n_1} \overset{f_2}{\longrightarrow}\cdots \overset{f_K}{\longrightarrow} \mathbb{R}^{n_K} \overset{g}{\longrightarrow}\mathbb{R}^m\)
Turing
Machine
(Turing, 1937)
Recursive
Function
(Gödel, 1934/1986)
Lambda
Term
(Church, 1936)
\(\Longleftrightarrow\)
\(\Longleftrightarrow\)
credit: Nynexman4464
\(\lambda m . \lambda n . \lambda f . \lambda x . m f ( n f x)\)
Example: characters in Wikipedia
\(\color{red}{W} = \{\texttt{-},\texttt{/},\texttt{0},\texttt{1},\texttt{2},\texttt{3},\texttt{4},\texttt{5},\texttt{6},\texttt{7},\texttt{8},\texttt{9},\texttt{=},\texttt{a},\texttt{b},\texttt{c},\ldots,\texttt{w},\texttt{x},\texttt{y},\texttt{z},\texttt{é}\}\)
\(\color{blue}{C} = W \times W = \{\texttt{(-,-)},\texttt{(-,/)},\texttt{(-,0)},\ldots,\texttt{(é,z)},\texttt{(é,é)}\}\)
\(\begin{aligned} M_{\color{red}{w}\color{blue}{c}} &= \mathrm{pmi}(\color{red}{w},\color{blue}{c})\\ &= \log \frac{p(\color{red}{w},\color{blue}{c})}{p(\color{red}{w})\,p(\color{blue}{c})} \end{aligned}\)
SVD of a PMI matrix
\(U\)
\(\Sigma\)
\(V^T\)
Truncate
\(U \times \Sigma\)
\(\hat{U} \times \hat{\Sigma}\)
Embed
\(\hat{U} \times \hat{\Sigma}\)
But why?
Embeddings as functions over sets
\(\color{red}{W} = \{\texttt{-},\texttt{/},\texttt{0},\texttt{1},\texttt{2},\texttt{3},\texttt{4},\texttt{5},\texttt{6},\texttt{7},\texttt{8},\texttt{9},\texttt{=},\texttt{a},\texttt{b},\texttt{c},\ldots,\texttt{w},\texttt{x},\texttt{y},\texttt{z},\texttt{é}\}\)
\(\color{blue}{C} = W \times W = \{\texttt{(-,-)},\texttt{(-,/)},\texttt{(-,0)},\ldots,\texttt{(é,z)},\texttt{(é,é)}\}\)
\[\begin{aligned} M \colon {\color{red}{W}} \times {\color{blue}{C}} &\to \mathbb{R}\\ ({\color{red}{w}},{\color{blue}{c}}) &\mapsto \mathrm{pmi}({\color{red}{w}},{\color{blue}{c}}) \end{aligned}\]
\[\begin{aligned} M_w \colon {\color{red}{W}} &\to \mathbb{R}^{\color{blue}{C}}\\ {\color{red}{w}} &\mapsto {\color{blue}{M(w,-)}} \end{aligned}\]
\[\begin{aligned} M_c \colon {\color{blue}{C}} &\to \mathbb{R}^{\color{red}{W}}\\ {\color{blue}{c}} &\mapsto {\color{red}{M(-,c)}} \end{aligned}\]
\(M^* \colon \mathbb{R}^{\color{red}{W}} \to \mathbb{R}^{\color{blue}{C}}\)
\(M_* \colon \mathbb{R}^{\color{blue}{C}} \to \mathbb{R}^{\color{red}{W}}\)
Embeddings as functions over sets
\({\color{red}{M_* M^*}} \colon \mathbb{R}^{\color{red}{W}} \to \mathbb{R}^{\color{red}{W}}\)
\({\color{blue}{M^* M_*}} \colon \mathbb{R}^{\color{blue}{C}} \to \mathbb{R}^{\color{blue}{C}}\)
\[\begin{aligned} \{{\color{red}{u_1}}, \ldots, {\color{red}{u_m}}\} &\subset \mathbb{R}^{\color{red}{W}}\\ \{{\color{blue}{v_1}}, \ldots, {\color{blue}{v_n}}\} &\subset \mathbb{R}^{\color{blue}{C}}\\ \{\lambda_1, \ldots, \lambda_{\min(m,n)}&, 0, \ldots, 0\} \end{aligned}\]
\({\color{red}{M_* M^* u_i}} = \lambda_i {\color{red}{u_i}}\)
\({\color{blue}{M^* M_* v_i}} = \lambda_i {\color{blue}{v_i}}\)
The \({\color{red}{u_i}}\) and \({\color{blue}{v_i}}\) are (linear)
fixed points!
\[M = U\Sigma V^T \qquad \begin{aligned} U &:= [{\color{red}{u_1}}, \ldots, {\color{red}{u_m}}]\\ V &:= [{\color{blue}{v_1}}, \ldots, {\color{blue}{v_n}}]\\ \Sigma &:= \left[\begin{smallmatrix} \sqrt{\lambda_1} & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & \sqrt{\lambda_r} \end{smallmatrix}\right] \end{aligned}\]
\(M_* M^*\) as a covariance matrix
\(M^*\)
\(M_*\)
\({\color{red}{M_* M^*}}\)
Eigenvectors as fixed points
\({\color{red}{M_* M^* u}} = \lambda {\color{red}{u}}\)
\(\times\) 
\(=\) \(\begin{bmatrix} \lambda_1 & \lambda_2 & \lambda_3 \end{bmatrix}\)
Eigenvectors (linear fixed points)
(credits: Jacopo Bertolotti, CC0
via Wikimedia Commons)
Embedding dimensions as fixed points
fixed point!
Chladni figures
“One can conceive of a profoundly deaf human being who has never experienced sound or music; just as such a person will gaze in astonishment at the Chladnian sound-figures in sand, find their cause in the vibration of a string, and swear that he must now know what men call sound — this is precisely what happens to all of us with language.”
Structural features
Eigenvectors of \({\color{red}{M_* M^*}}\):
Eigenvalues of \({\color{red}{M_* M^*}}\) and \({\color{blue}{M^* M_*}}\):
Eigenvectors of \({\color{blue}{M^* M_*}}\):
Words
Lévi-Strauss and the mathematics of man
In the past, the great difficulty has arisen from the qualitative nature of our studies. If they were to be treated quantitatively, it was either necessary to do a certain amount of juggling with them or to simplify to an excessive degree. Today, however, there are many branches of mathematics — set theory, group theory, topology, etc. — which are concerned with establishing exact relationships between classes of individuals distinguished from one another by discontinuous values, and this very discontinuity is one of the essential characteristics of qualitative sets in relation to one another and was the feature in which their alleged ‘incommensurability’, ‘inexpressibility’, etc., consisted.
Lévi-Strauss (1954)
The power of the formal
Derrida (1976)
“Within cultures practicing so-called phonetic writing, mathematics is not just an enclave. […] This enclave is also the place where the practice of scientific language challenges intrinsically and with increasing profundity the ideal of phonetic writing and all its implicit metaphysics (metaphysics itself)”
Deleuze (1994)
“Differential calculus is not the unimaginative calculus of the utilitarian, the crude arithmetic calculus which subordinates thought to other things or to other ends, but the algebra of pure thought, the superior irony of problems themselves — the only calculus ‘beyond good and evil’”
Foucault (2002)
“There is perhaps only one science for which one can neither distinguish these different thresholds, nor describe a similar set of shifts: mathematics […] Hence the fact that their establishment is both so enigmatic […] and so valid […]”
Russian formalism as a generative device
“The purpose of art, then, is to lead us to a knowledge of a thing through the organ of sight instead of recognition. By ‘enstranging’ objects and complicating form, the device of art makes perception long and ‘laborious’. The perceptual process in art has a purpose all its own and ought to be extended to the fullest. Art is a means of experiencing the process of creativity. The artifact itself is quite unimportant.”
Shklovsky (1990)
Bourdieu in embedding space
“If it is true […] that, first, the dominant class constitutes a relatively autonomous space whose structure is defined by the distribution of economic and cultural capital among its members, […] then these structures should be found in the space of life-styles […]. To endeavour to establish this, the whole set of survey data was subjected to correspondence analysis.”
Structuralist mathematics
Architectural genius
“…all peoples have just such a mathematically divided firmament of concepts above them […] Here, one can certainly admire humanity as a mighty architectural genius who succeeds in erecting the infinitely complicated cathedral of concepts on moving foundations, or even, one might say, on flowing water; admittedly, in order to rest on such foundations, it has to be like a thing constructed from cobwebs, so delicate that it can be carried off on the waves and yet so firm as not to be blown apart by the wind.”