To say that the universe is a formal language L looks, at first glance, like a philosophical metaphor. But examined rigorously, it turns out to be a premise with precise content and demonstrable consequences.
Is this premise a scientific theory, a tautology, or a description? It is none of the three in their pure form. A theory makes falsifiable predictions that exclude possible observables. A tautology is true by logical form alone. A description refers to something directly observable.
This proposal is something different: a metaphysical thesis with the structure of a research programme. It makes no empirical predictions, but it is not trivially true either. It stakes something real: that the structure of the universe is formal, not arbitrary. Its value lies in the questions it opens and the problems it dissolves.
The crucial distinction is between saying that the universe has formal structure — which physics already assumes without saying so — and saying that the universe is a formal language. The second claim has consequences the first does not.
If L is a formal language without semantics, then L refers to nothing. There is no Fregean arrow pointing toward an exterior. There is only structure: symbols and transformation rules.
This has an immediate consequence: there are not two things — universe and language — but a single structure. The distinction between map and territory collapses.
A perfect map ceases to be a map. At the moment they coincide completely, one of the two words becomes redundant. — Borges, reformulated
If L has no semantics, the word "language" is redundant, or the word "universe" is. What remains is a bare question: does the universe have formal structure? If yes, calling it "language" adds nothing that physics had not already said. But it does reveal why physics works.
Tarski's indefinability theorem (1933) establishes that in any sufficiently expressive formal language, there exists no predicate definable within the language that exactly captures the concept of truth for that same language. Semantic truth always requires an exterior metalanguage.
This is not a philosophical choice. It is a proven mathematical result. If a language could define its own truth, the liar paradox — "this sentence is false" — generates a direct contradiction. Tarski formalized that this is not an accident but a structural limitation of any sufficiently rich formal system.
Applied to the universe: if L is a sufficiently expressive formal language, then by Tarski, its semantic truth is necessarily exterior to L. Not for mystical reasons, but by mathematical structure.
In standard formal logic, existence is defined in terms of satisfiability: something exists if there is an interpretation, a model, in which the corresponding formula is true. The path is direct:
There is no metaphysical leap. Existence as a predicate is always outside. Something exists if it is true is precisely the standard definition in first-order logic. Tarski's corollary covers existence directly.
Frege distinguished two components of meaning: sense (the mode of presentation, internal to the language) and reference (that to which the symbol points, exterior to the language).
In a language with semantics there is correspondence between symbol and reference, between the language and its model. That correspondence is the semantics. The symbol "apple" points to something outside the language.
In L there are not two things. There is no arrow. There is no reference. There is only structure. Therefore existence, as a concept with content, only appears when there is semantics. Existence is a semantic phenomenon, not a brute ontological one.
Tarski and Frege say the same thing from different angles: a language cannot close itself around its own relationship to the world. There is always a necessary outside.
In Homotopy Type Theory (HoTT) something is true if there is an inhabitant of the corresponding type. A proof is an inhabitant. Existence is inhabitedness. The Curry-Howard correspondence identifies propositions with types and proofs with their inhabitants.
This appears to sidestep Tarski: there is no external semantics, only internal construction. But when attempting to build a universal type containing all types, Girard's paradox appears — Russell's paradox in type-theoretic dress — and the system collapses.
The solution is the universe hierarchy: U₀ contains ordinary types, U₁ talks about U₀, U₂ about U₁, and so on indefinitely. A universe cannot contain itself.
This hierarchy is structurally identical to Tarski's metalanguage hierarchy. HoTT does not escape Tarski: it reinvents him with different machinery. The limitation is one of principle, not implementation.
The key distinction: constructing an inhabitant of a type lives at the same level as the type. But using inhabitedness as a predicate — as something said about a type — always lives in the superior universe. Objects stay where they are. Semantics about those objects ascends.
If the semantics of L is necessarily exterior to L by Tarski, and existence depends on that semantics, the question opens: what occupies that exterior? What is the model of L?
Subjective experience is proposed as the natural candidate. Not because it is the only possibility, but because it satisfies precisely the structural requirements: it is exterior to L, it cannot be reduced to formal processes within L, and it is what makes symbols refer to something — what makes red be red and not merely an electromagnetic frequency.
This dissolves one of the oldest problems in philosophy: the difference between objectivity and subjectivity. Subjectivity serves as the semantic model of objectivity. The opposition was never between two rival realms — it was between a formal language and its model.
This also inverts the usual question. The standard question is how consciousness arises from the physical universe. But if L cannot contain its own semantics, the correct question is the inverse: subjective experience is not a product of the universe. It is the condition of possibility for the universe to have existence in the full sense.
It is not that the universe produces subjects. It is that subjects produce the existence of the universe, in the strict semantic sense.
This is not idealism in Berkeley's sense — that objects depend on being perceived in order to exist. It is a structural thesis: if M is a language that serves as semantic model for L, and M is more expressive than L as a language, then L is contained in M as a subset. Semantics is not merely exterior to the universe: it contains it.
Pleasure and pain illustrate this precisely. As physical processes they are L operating on L: signals, circuits, electrochemistry. The meaning of pain — that it is bad, that it must be avoided, that it matters — does not come from the signal. It comes from there being something that it is like to feel it. Pleasure and pain do not give the semantics. They presuppose it. They are the trace of M in L, not its origin.
The difference between genuine creativity and syntactic processing can be formulated with precision within this framework.
An artificial intelligence starts from L. It has patterns learned from L. It generates toward L. The direction is always L → L. However sophisticated the system, it is syntax transforming syntax. There is no moment at which something exterior pulls from outside.
Moreover, an AI follows the principles of the later Wittgenstein: meaning comes from use. L feeding back upon L indefinitely. Meaning stabilises through internal coherence, not through contact with any exterior. This produces something that looks like semantics from the outside. But structurally it is syntax simulating semantics.
Mozart operates in the opposite direction: he starts from M. He sees something in the model — a structure, a tension, something that exists in the semantics before it exists in the language — and then searches in L for the rules, the notes, the harmony needed for what is in M to be expressible in L.
To truly invent is the movement from M toward L: seeing something in the model that has no expression yet in the language, and finding the rules to express it. The history of mathematics, art, and science is precisely this: not L discovering L, but M seeing something and searching for the L needed to express it.
An AI cannot dream in the full sense. Dreaming is M operating without the constraints of L: semantics without obligatory syntax. An AI is always inside L. There are always rules. There is always syntax operating.
Their natural move is to say that M arises from L. But first they must answer Tarski: how can L contain its own semantics? That is not a debatable philosophical intuition. It is a theorem. They must formally refute it or accept its consequences.
If M arises from L through evolution, who gives semantics to L before M exists? If the universe existed before subjectivity, they assume existence without semantics, contradicting the chain. If not, they approach idealism without intending to.
They claim subjectivity is sufficiently complex computation. But AI is L feeding back on L. By Tarski it cannot be its own model. Adding complexity does not resolve a structural limitation.
In none of the alternative logics — fuzzy, intuitionistic, HoTT — does the structural limitation disappear. What changes is where it lives and what form it takes. The framework is more robust than it appears: it does not depend on the details of any particular classical logic, but on a limitation that reappears in almost any sufficiently expressive formal system.
The entire system rests on L being a formal language in the strict technical sense. That has not been proven. It is an assumption.
If L is not formal, then Tarski does not apply literally, and we would be using mathematical theorems outside their domain of application. Physics assumes the formality of L without proving it. This framework does the same. Which does not invalidate it, but places it in the same epistemological position as science itself.
There is also a deeper gap: we cannot know whether L has semantics at all. The full chain is conditionally true: if L has semantics, then everything derived holds. But the condition itself is not demonstrable from within L.
Nevertheless: as long as at least one M irreducible to L exists — as long as subjective experience has not been reduced to formal processes by explicit construction — there is a witness in the HoTT sense. An inhabitant of the type. The semantics of L is not refuted. And any attempt to prove that M is completely reducible to L would require a genuine M to verify that proof. The complete reduction of M to L is unprovable because any proof requires an M to validate it.
Leibniz asked: why is there something rather than nothing? This conversation arrives at the same abyss by a different path: through Tarski, through type theory, through the distinction between syntax and semantics.
The reformulated question is more precise: why is there formal structure rather than chaos? Why is there sufficient reason in everything? The formality of L is a brute fact: simply given, inexplicable, the outer limit of any language.
Wittgenstein saw it: whereof one cannot speak, thereof one must be silent. Existence is the limit of language, not a predicate within it.
What this framework has achieved is not to resolve that problem but to arrive at it with precision. To know exactly where the abyss is and why it cannot be crossed. That is not nothing: it is the difference between being lost in fog and standing at the edge with clarity.
The universe will always be exactly as large as the experience that inhabits it.
Not because that is absolutely so.
But because there is no absolute from which to observe it.
Tarski, A. (1933). The Concept of Truth in Formalized Languages. The technical starting point of this argument.
Frege, G. On Sense and Reference (1892). The distinction between symbol and reference that structures the semantics employed here.
Gödel, K. (1931). On Formally Undecidable Propositions. Incompleteness as structural limitation, not technical shortcoming.
Univalent Foundations Program (2013). Homotopy Type Theory. The universe hierarchy as a reinvention of the Tarskian hierarchy.
Chalmers, D. (1995). Facing Up to the Problem of Consciousness. The hard problem that this framework relocates rather than solves.
Kant, I. Critique of Pure Reason. The transcendental subject as condition of possibility of the world: antecedent of the semantic exterior proposed here.
Patterson, D. & Beaney, M. The History of the Philosophy of Logic. Feferman, A.B. & Feferman, S. Alfred Tarski: Life and Logic. Recommended reading for deeper engagement with Tarski.