Contribution
A short position paper, self-described as “largely a synthesis of ideas” (abstract). It proposes a name and a slogan for a research programme: pregeometry is “structureless structure”, meaning the syntactic types of a computational language, and physics is to be built constructively from their compositional rules. It adds no new formal result, model or construction. What it adds is the joining of three threads under one reading: Wheeler’s pregeometry, Leibniz’s monadology (with Pauli–Jung monism), and the formal-language/topos programmes of Isham and of homotopy type theory.
Key insight
The paper’s picture is a chain of three moves: foundational explanation regresses; coherentism stops the regress by making relations basic; formal languages are the mathematics of relations under composition. On that chain, the right “stuff” beneath spacetime is a typed language, with geometry obtained as “sections or projections” of higher-categorical (∞-topos) structure (p. 13). Each link is asserted rather than argued. The thing to remember is that this is a statement of allegiance to a programme (Wolfram-model multiway systems read through HoTT), not an argument for it.
Assumptions
Premises the argument relies on, as the paper states them:
- Regress premise. Inferential justification (“To be justified in believing A on the basis of B one must be (1) justified in believing B, and (2) justified in believing that B makes probable A”, quoted from [51], p. 6) applied recursively to physical explanation yields an infinite regress. This is “particularly relevant if one were to insist on a fundamentally materialist ontology” (p. 6).
- Dichotomy premise. The only responses are foundationalism (self-evident truths halt the regress) or coherentism (mutual justification in a web). A foundationalist substrate would have to be accepted as a “universal self-evident truth”, described as “a rather unsettling predicament” (p. 11).
- Coherentism → relations → formal systems. Coherentism “places precedence on relations rather than objects”, and “the appropriate mathematical framework to formalize a theory of abstract relations … is … a ‘Formal System’” (p. 11). No argument is given for the second conditional.
- Discreteness premise. There is “widespread consensus” that space, time and matter are fundamentally discrete (p. 5). This is qualified on p. 14.
- Definitions (p. 6): a formal language L is a set of well-formed strings over Σ* (the free monoid on an alphabet Σ). A deductive system is an axiom schema and/or inference rules. A computational language is a formal language with a deductive system (p. 7). Computational languages are said to be “inherently constructivist” because their proof systems are intuitionistic (p. 7).
Key results
It argues, and does not prove:
- Four classes of foundational work (§2, pp. 4–5): (i) interpretation of existing concepts; (ii) new mechanisms or phenomenological models; (iii) new physical structures generalizing space, time or matter; (iv) radical reconceptualizations of the “ontological and metaphysical origins of structure and the role of observers”. Wheeler’s pregeometry is the “classic example” of (iv). “Falsifiable theories, at the very least, are marked by a transition from class (iv) or class (iii) to class (ii)” (p. 5).
- Structureless structure (§4, p. 7): “symbolic and relational entities of a formal language … the building blocks of proofs, programs and computations”. Such constructs “necessarily shift away from a materialist ontology of physics”.
- Leibniz as precedent (§5, pp. 9–10): the Monadology is “in all likelihood … the first example of a pre-space theory”. Monads are simples with no structure of their own, so they provide “a kind of structureless structure”. The Wolfram model’s rewriting events are presented as analogous to monads, and its observers as sampling the world “much like a monad” (p. 10).
- Pauli–Jung dual-aspect monism (§5, pp. 10–11): a neutral pre-physical, pre-mental substrate “necessitates pregeometric building blocks”.
- Formal language subsumes approaches (p. 3): the Wolfram model, constructor theory, categorical QM, ZX calculus, intuitionistic mechanics, operator mechanics and assembly theory “can all be seen to be instances of different syntactic formalisms, possibly expressible within a broader language-theoretic framework”. Spin networks, causal sets and the like “can potentially be constructible” within a formal language internal to a higher topos (fn. 1: “soon to be reported”).
- Space from ∞-toposes (§6, p. 13): “the notion of space arises from functorial constructions involving ∞-toposes”. ∞-groupoids model topological spaces via Grothendieck’s homotopy hypothesis ([21]), and cohesive ∞-toposes give synthetic geometry ([79–84]). A computational realization via multiway rewriting systems is cited to [13, 17, 18].
- Outlook (§7, pp. 13–14): primitives of pregeometry are “not merely discretizations of classical structures”. Pregeometry is “about nature constructing itself from abstract computation”. Higher structures are a “recurring theme” that will “likely bridge … constructivist and background-independent approaches”.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | A meta-theory of structures is “incompatible with a metaphysics based on material realism”, because that leads to an infinite regress (p. 3, §3) | weak | informal argument from the inferential-justification principle; why a materialist ontology in particular regresses and a relational one does not is not shown |
| C2 | Coherentism entails that formal systems are the right framework for pregeometry (p. 11) | weak | assertion (“It turns out that the appropriate mathematical framework … is … a ‘Formal System’“) |
| C3 | There is widespread consensus that space, time and matter are fundamentally discrete (p. 5) | weak | two 1995 citations ([52], [45]); qualified by the paper itself on p. 14 |
| C4 | Leibniz’s Monadology is likely the first pre-space theory, and monads are a structureless structure (pp. 9–10) | moderate | historical interpretation leaning on Risi [75] |
| C5 | Many listed formalisms (Wolfram model, constructor theory, ZX, assembly theory, …) are instances of syntactic formalisms within one language-theoretic framework (p. 3) | weak | assertion; “can all be seen to be”, “possibly expressible” |
| C6 | Algebraic quantum-gravity structures (spin networks, causal sets, tensor networks, matrix models, …) can be constructed in the internal language of a higher topos (p. 3, fn. 1) | weak | promised: “soon to be reported in a forthcoming publication” |
| C7 | Spatial structure and geometry can be realized from pregeometric rewriting systems via higher homotopies (p. 13) | moderate for the cited homotopy constructions; weak for “geometry” | cited to the authors’ own [13, 17, 18]; not reproduced or assessed here |
| C8 | A language-theoretic constructivist framing “serves as an important conceptual advance” for computation-and-physics approaches such as the Wolfram model (p. 14) | weak | assertion |
Method
Conceptual synthesis and historical reading. There is no formal method: the paper defines “formal language”, “deductive system”, “computational language” and “structureless structure” (pp. 6–7), then relates them by narrative to Wheeler, Leibniz, Pauli–Jung, Isham–Döring topos theory and HoTT.
Concepts
- Pregeometry — Wheeler’s term for whatever underlies spacetime, meant to “transcend any structural explanation of space, time, matter and even physical law” (p. 7). Here it is narrowed to “symbolic/linguistic structures which do not come endowed with any pre-assigned geometric attributes” (p. 3).
- Structureless structure — the “symbolic and relational entities of a formal language … the building blocks of proofs, programs and computations” (p. 7). The paper does not resolve the tension in the phrase: types have syntactic structure.
- Computational language — a formal language equipped with a deductive system, taken to be intuitionistic and hence “constructivist” (p. 7).
- Constructivist coherentism — the paper’s name for its stance: relations over objects, mutual justification over foundations, and existence as constructibility.
- Class (iv) investigation — foundational work that seeks new “ontological origins” from which existing physical structure is emergent or derivable (p. 4).
Connections
It builds on Wheeler’s pregeometry and “It from Bit” as presented by Misner–Thorne–Wheeler and Patton & Wheeler, with Blum & Furlan for history. It uses the Isham–Döring topos programme ([36–40, 52, 53]) as its closest formal predecessor. It relies on HoTT/univalent foundations ([68, 83, 84]) and Schreiber’s cohesive ∞-topos programme ([79–81]). For the computational realization it relies on the authors’ own Wolfram-model papers ([13, 17, 18]) and Wolfram ([91–95]).
It is by overlapping authors with LIT-133 (Elshatlawy, Rickles & Arsiwalla on observers), which was read in full and Rejected on 2026-09-26. The two share an idiom: a named notion is introduced (there, “minimal observer”; here, “structureless structure”), and its substantive content is deferred to Wolfram-model constructions elsewhere. Its coherentism is epistemological (a response to the regress of justification), but it slides into a metaphysical claim about relations. That is the move LIT-173 (Morganti, from OSR to metaphysical coherentism) makes explicitly, and a reader wanting that move argued should go there. The dossier’s pairing with Carroll’s Hilbert-space fundamentalism (LIT-123) is apt only loosely. Both deny that space is fundamental, but Carroll derives space from a Hamiltonian’s structure, while this paper derives it from syntax.
The metaphysical thesis is anti-materialist constructivism: what exists at bottom are rules of construction (types and their compositional rules), and space, time, matter and law are built from them (“Nature constructing itself”, p. 10). The paper presupposes that an ontology can be read off from the foundations of mathematics, “the metaphysics of spacetime geometry directly draws from formal advances in metamathematics” (p. 14). The metaphysics tag is justified.
Bearing on the record
It supports no THEORY document and should produce none. At most it is a pointer to a programme (Wolfram-model pregeometry via HoTT) whose content lives in other papers. The dossier’s open question, whether any concrete construction of spatial structure from type theory is given or only promised, is answered: only promised or cited. It carries nothing for ML practice. The words “computational language” and “compositionality” do not make it a claim about learning systems, and nothing here belongs in the Anthology of the SOTA.
Limitations
- No construction, model, theorem or prediction. The paper says it is “a precursor” and “an initial metaphysical exploration” (abstract; p. 13). The abstract’s “evidence suggesting that … homotopy type theory provides the conceptual building blocks” is never presented as evidence in the body.
- The argument from regress to formal languages has unargued steps (C1, C2). The foundationalism/coherentism dichotomy is imported from epistemology and applied to ontology without comment.
- “Structureless structure” is not given criteria that would distinguish it from ordinary mathematical structure.
- The discreteness “consensus” (C3) is overstated, and the paper half-retracts it on p. 14.
- The Wolfram-model framing is not assessed independently. The paper’s affiliations include Wolfram Research, and it thanks Stephen Wolfram.
- Falsifiability is invoked (p. 5) for class (iv) work, but the paper gives no route by which its own proposal would reach class (ii).
Open questions
- Does the promised construction of spin networks, causal sets or tensor networks inside a higher topos’s internal language (fn. 1) exist? The Routledge chapter version (2025) or a later paper would settle whether C6 was delivered.
- What distinguishes “structureless structure” from structure? A criterion would be needed before the notion could do metaphysical work.
- Is there any observable consequence of reading pregeometry as a typed language rather than as, say, a causal set? Without one, the proposal stays in class (iv) by its own classification.
Corrections to the seeded skim
- The dossier says §§6–7 “link the proposal to HoTT/univalent foundations and to computational models of physics”, which suggests a link the paper develops. It develops none. §6 (pp. 11–13) summarizes Isham’s topos programme, the internal-language/topos correspondence ([54]) and the cohesive ∞-topos programme ([79–84]). It then cites the authors’ own [13, 17, 18] for a type-theoretic construction of ∞-groupoids from Wolfram-model multiway systems. No construction is carried out in this paper. The claim that algebraic quantum-gravity structures can be built “internal to an appropriate (higher) topos” is deferred to “a forthcoming publication” (fn. 1, p. 3).
- The dossier’s §5 bullet leaves out that §5 (pp. 9–11) also presents Pauli–Jung dual-aspect monism as a second precedent. The section closes that such monism “necessitates pregeometric building blocks for our perceived reality” (pp. 10–11), which is asserted.
- The dossier leaves out the paper’s most exposed empirical-sounding claim: “there is widespread consensus on the view that space, time and matter are fundamentally discrete” (p. 5, citing only Isham 1995 [52] and a 1995 bibliography [45]). The paper itself qualifies this on p. 14, where operator algebras give geometry that is “neither continuous nor discrete”.
- The dossier summary says geometry is built “from relations rather than a material substrate”. That is right about the aim, but the paper places “the ontological weight on the very rules of construction themselves” (p. 10), not on relations as such. The slogan is “Nature constructing itself” from “abstract computation” (pp. 10, 14).
- Funding (FQXi / Fetzer-Franklin, John Templeton Foundation grant 62106, ARC DP210100919) and thanks to Stephen Wolfram are on pp. 14–15, alongside the Wolfram Research affiliation the dossier already flags.
- metaphysics tag: justified. The paper is explicitly “an initial metaphysical exploration” (p. 13) whose thesis is about what is fundamental: relations, or rules of construction, rather than material substrate.
- primary topic: metaphysics (keep). The thesis is ontological. philosophy-of-science is a fair second tag, and mathematics a justifiable third for the HoTT/topos material.