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REVIEW 4 major objections 4 minor 1 cited by

To Be or Not To Be: Vector ontologies as a truly formal ontological framework

T0 review · 4 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Vector spaces, not categories, make ontology truly formal

desk verdict The paper is a readable philosophical proposal, but the existence predicate is extra structure that breaks the vector-space formalism, so the central formal claim fails on its own terms. read the letter →

arxiv 2505.14940 v1 pith:GVG5AS73 submitted 2025-05-20 cs.AI cs.SC

Kaspar Rothenfusser This is my paper · ORCID
classification cs.AIcs.SC
keywords vectorontologyformalfoundationalspacesapriorifunctionsofexistencemereologyAIinterpretability
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that the ontologies usually called formal—categorical systems built by abstracting from human perception—do not meet the strict criterion of formality set out in the Logical Investigations: validity independent of perception and total absence of content. The author proposes instead that a formal ontology should be a mathematical structure chosen before any ontological content is added, and offers vector spaces as the concrete candidate. A vector ontology defines a domain by a finite set of basis vectors that name quality dimensions; existence is then the membership of a point in the sparse set of real things, and the classic concepts of foundational ontology are recovered as geometric or algebraic features of that space. If the proposal holds, ontology becomes an a priori, content-free framework into which human and machine worldviews can both be poured, making machine representations interpretable and human knowledge machine-readable. The stakes are practical as well as philosophical: many AI systems already compute in high-dimensional vector spaces, so an ontology with vector form could serve as a common interface between human and machine understanding.

What carries the argument

The load-bearing object is the vector ontology $V_{\mathrm{ont}}$, defined as the span of a finite set of basis vectors $\{x_1,\dots,x_n\}$ over a field, where each basis vector names a quality dimension and the field supplies possible values. Existence is encoded by the membership test $v \in V_{\mathrm{reality}}$ for the sparse subset of the space corresponding to actual things, and the main analytical tool is the function of existence $f_e$—a (multi)linear map from the ontology into a binary existence space—which compresses scattered vectors into concepts. Around these sit the derived mechanisms: convex regions model parthood, continuous functions in the time basis model endurance, linear dependence models causation, probabilistic functions model correlation, and reconstruction paths through a small set of interpretable dimensions model similarity and metaphor. The entire argument depends on the choice of basis being interpretable, so that the vector space is navigable rather than a black-box feature space.

What would settle it

Exhibit a single object or concept that cannot be encoded by any finite set of quality dimensions with coordinate values—a normative statement, a universal generalization, or a self-referential thought—and show that no enlargement of the basis repairs the gap; if such a case exists, the finite-basis assumption fails and the expressivity claim collapses.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that the axioms of a vector space constitute a formal ontology in the strict sense, and that nearly all categories of existing foundational ontologies reappear as structural facts about such a space. The basis vectors are universals—quality dimensions—and the field supplies the quale, so an object is a vector; to be is to belong to the sparse subset of the space that records what actually exists, while possible existence is modal membership. Endurants become functions of existence that are continuous in the time dimension, perdurants are the complement; mereological parthood becomes containment of convex regions; causation is linear dependence among vectors; correlation is a learned probabilistic function of existence; similarity is distance; and identity across change is a function of existence that stays continuous or constant in the relevant dimensions. The paper further claims that this structure is not merely convenient: it is already the implicit ontology of neural-network computations, and there is cited empirical evidence that human perception operates in the same geometric manner.

Load-bearing premise

The entire framework rests on the claim that any object of interest can be fully described as a vector in a vector space with a finite number of basis vectors representing quality dimensions; if some objects or concepts—norms, modalities, intentional states—resist finite-dimensional vector representation, the framework cannot express them, and the central claim collapses.

Editorial extensions

If this is right

  • If vector ontologies are genuinely formal, then existing categorical systems built by recursive abstraction should be reclassified as foundational rather than formal, and their categories become derived phenomena rather than primitive axioms.
  • An ontology populated inside a vector space is internally consistent by construction, because the axioms themselves cannot generate contradiction; errors surface only as mismatches with observed reality.
  • Because neural networks compute in high-dimensional vector spaces, a vector ontology with interpretable basis vectors would make it possible to extract or align a network's learned ontology, turning the interpretability problem into a design requirement.
  • Interoperability between information artifacts becomes a property of form: any two systems that populate the same vector space can share and navigate each other's content before any domain-specific vocabulary is fixed.
  • Search and analogical reasoning reduce to navigation along a small set of quality dimensions, since reconstruction distance—not raw coordinate distance—is the proposed measure of relatedness.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural next test is to build a bounded vector ontology, train a network on that domain, and check whether the network's learned functions of existence align with the human-selected basis; the author lists this as future work, but it is already well-specified enough to run.
  • The finite-basis assumption suggests an approximation agenda: if domains behave like Fourier spectra, a small dominant set of quality dimensions should account for most of the structure, and identifying that set for real domains becomes an empirical research program.
  • The identification of causation with strict linear dependence predicts that genuinely causal relations are exact algebraic dependencies, which would separate deterministic causation from probabilistic correlation in a sharper way than most current accounts.
  • The formal status of the framework is limited to the bare vector axioms; fixing a domain's basis is a perceptual choice, so the content-free guarantee does not extend to any populated ontology.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 4 minor

Summary. The paper argues that previous "formal ontologies" (Husserl, Guizzardi, BFO, GFO, DOLCE) are not formal in Husserl's sense because they are built by empirical categorization, and proposes instead that a vector space, called a vector ontology (Vont), can serve as a truly formal ontology. It maps central foundational-ontology concepts to vector-space notions: existence as membership in a sparse subset V_reality, endurants as continuous functions of existence, mereology as convex regions, causation as linear dependence, correlation as probabilistic existence, and similarity as Minkowski or reconstruction distance. The paper claims this shows vector ontologies are a priori, independent of perception, and already used by AI systems and humans.

Significance. The proposal is ambitious and potentially valuable: if rigorously established, a vector-space foundation for ontology could offer a principled, interpretable bridge between human conceptualization and machine-learned representations, and would provide a fresh perspective on the long-standing distinction between formal and foundational ontology. The paper deserves credit for engaging with the philosophical literature, for using concrete examples, and for explicitly listing its limitations and future empirical tests in Section 11. However, the central expressivity claim is not demonstrated: the bookkeeping is done by a handful of selected examples, the existence predicate is left formally undefined, and key mathematical descriptions contain errors. The current manuscript is a programmatic essay rather than a formal contribution, and its load-bearing claims are not supported as written.

major comments (4)
  1. [Section 5.2.2 and Section 5.2.1] Existence is the central notion of the paper, but it is not defined by the vector-space axioms. Section 5.2.1 defines VontD = span{x1,...,xn}, and the axioms in Section 5 guarantee closure under addition and scalar multiplication, so every linear combination of basis vectors is already an element of VontD. Section 5.2.2 then defines existence as membership in a sparse subset V_reality (via "v ∈ V_domain reality"), but this subset is not a vector subspace, is not closed under the vector operations, and is given no axioms or rules specifying which vectors belong to it. The paper explicitly notes that the full vector space is "continuous or densely populated" while the ontology cares about a sparse population, yet no formal characterization of this sparse population is provided. Consequently, existence is an external, unexplained predicate, and the claim that vector ontologies are a priori formal frameworks from which existence can be "deducted" is not supported; the formalism is a vector space plus an unformalized additional structure.
  2. [Section 5.2.2, "key assumption"] The paper asserts that "any object of interest can be fully described as a vector in a vector space with a finite number of basis vectors representing quality dimensions." This is the load-bearing assumption of the entire framework, but it is merely stated, not argued for. The paper offers no reason why all ontological concepts—including normative, modal, intentional, or qualitative phenomena—should admit finite-dimensional vector-space representation. The analogy to Fourier transforms in the same section is suggestive, but it is not a derivation. Since this assumption is exactly what must be established for the expressivity claim, its status as an unproven premise is a major gap.
  3. [Abstract and Section 11] The abstract characterizes the showcase as "concrete proof" that a class of formal ontology based on vector-space axioms can express most foundational-ontology concepts, and the conclusion states that the author "clearly demonstrated" this. Yet no formal proofs are given. The mappings in Section 5.2.3 are stipulated: endurants are defined as continuous functions of existence, causation as linear dependence, correlation as probabilistic existence, and parthood as convex regions. These are redefinitions of ontological concepts in vector-space terms, not consequences derived from the axioms. Section 11 itself says "since this Paper is theoretical," and lists future empirical tests. The discrepancy between the abstract's "concrete proof" and the actual content is not merely a matter of wording; it obscures the epistemic status of the central claim, which is at best a conjecture supported by examples.
  4. [Section 5.2.1 and Section 5.2.3] Several mathematical statements are incorrect or inconsistent with the formal framework. In Section 5.2.1, the paper suggests using N (natural numbers) for discrete properties and gives an example vector [4,0,0,255] with integer components, but a vector space is defined only over a field, and N is not a field (it lacks additive inverses). The shelf example uses R, but the colored-shapes example uses integer coordinates without specifying a field. In Section 5.2.3, "Functions of Existence" are introduced as linear maps, yet the worked example fe(t) is a piecewise function that is not linear; the term "microcontinuity" is used without a formal definition; and the "Convex subspaces" definition actually defines a convex subset, not a subspace, since it is not closed under addition and scalar multiplication. These inaccuracies undermine the paper's stated goal of providing a rigorous, formal framework.
minor comments (4)
  1. [Throughout] There are multiple typographical and formatting errors, including "V ector" in the Section 5.1 heading, "Existance" in the mathematical comment, "Preconstrcution" in the reconstruction path formula, and "sucessfully" in Section 8.
  2. [Section 9.3] The text refers to "section 4.1.3" when discussing functions of existence, but the relevant discussion appears in Section 5.2.3.
  3. [Section 5.2.2] The notation "Blue Rectangle exists = ⇒ [4, 0, 0, 255] ∈ Vcolored-shapes" is malformed; the symbol "= ⇒" is not standard, and the distinction among Vcolored-shapes, Vdomain, and V_reality is used inconsistently throughout the section.
  4. [References] The name "Gärdenfors" is consistently misspelled as "Gardenfors"; please correct all occurrences.

Circularity Check

3 steps flagged · score 6.0 of 10

Partial circularity: the a priori status and expressivity claims are built from stipulated definitions (V_reality membership, continuous-FOE endurants, linear-dependence causation) rather than derived from vector-space axioms.

  1. self definitional [Section 5.2.2 (Existence), pp. 7-8, expression 'v ∈ Vdomain reality']
    "whether something is or is not is the only information we can directly extract from a vector ontology. ... the existence of an object in a domain is given as the truthfulness of the following expression: v ∈ Vdomain reality. ... A vector ontology, however, is interested in both the dense population, describing all theoretically possible objects, as well as the sparse population of the vector space representing reality, which is what we mean to describe in the expression v ∈ Vdomain reality."

    Existence is defined as membership in Vdomain reality, but that set is in turn characterized only as 'the sparse population of the vector space representing reality.' No axioms decide which vectors are in Vdomain reality. The vector-space axioms themselves make every linear combination a vector in V, so they cannot distinguish actual from possible existence. The actual/possible distinction is therefore an extra, unaxiomatized input, and the sentence that existence is 'the only information we can directly extract' restates the author's choice of where to put ontological information rather than being a consequence of the vector-space axioms.

  2. self definitional [Section 5.2.3 (Continuity), p. 10]
    "Endurants can be defined as FOEs that are continuous across time; Perdurants, on the other hand, are not continuous FOEs, and in fact, they do not form a coherent subset in a vector ontology at all. ... Interestingly, this shows that endurants do indeed have an a priori special property in our formal ontology, given as the continuity."

    The allegedly 'a priori special property' of endurants is placed directly into their definition: endurants are defined as FOEs that are continuous across time. The conclusion that continuity is an a priori property of endurants is therefore a restatement of the definition, not an independent result derived from vector-space axioms. The step also requires adding time as a basis vector, a contentful modeling choice, so the claimed independence from perception is not established by the axioms alone.

1 more flagged steps
  1. renaming known result [Section 5.2.3 (Linear dependence and Correlation), pp. 12-14]
    "I propose to define causation as a strict linear dependence between vectors representing concepts. Similarly, I propose to understand Correlation as a probability function of existence. This also makes it evident that causation has a drastically different meaning than correlation and can not be investigated with the same methods."

    Causation and correlation are not derived from independent ontological data or from the vector-space axioms; they are stipulated as names for linear dependence and probabilistic membership. The subsequent claim that causation and correlation 'have a drastically different meaning' follows by construction from those stipulations. This is a relabeling of known linear-algebra concepts as ontological categories rather than a demonstration that the vector framework predicts or explains the distinction.

full rationale

The paper contains no fitted parameters, no external benchmark, and no load-bearing self-citation chain, so most standard circularity patterns are absent. The circularity is definitional and located in the central expressivity claim. Existence is defined as membership in Vdomain reality, which is itself introduced only as 'the sparse population of the vector space representing reality'; no axioms govern that subset, and the vector space's closure makes every linear combination a member of V. Thus the 'to be or not to be' content is read into the framework rather than deduced from it. Similarly, endurants are defined as continuous functions of existence and then presented as having an a priori special property, with the special property being exactly the defining one. Causation and correlation are renamed as linear dependence and probabilistic existence, so the asserted difference between them follows from the proposed definitions. The author is transparent that the mappings are proposals and even concedes that basis choice introduces bias, which mitigates any suggestion of deception but not the logical point: on inspection, the ontological content being 'captured' is largely the chosen definitions themselves. The paper is coherent as a proposed vocabulary, but the stronger claim that vector ontologies yield foundational-ontology content as a priori occurrences independent of perception is not supported by the derivation; it is supported only by stipulation. This warrants a score of 6: partial circularity, where several claimed representations reduce by construction to the definitions adopted.

Assumptions & free parameters 5 free parameters · 5 assumptions · 4 invented entities

The framework rests on standard vector-space axioms plus several domain assumptions about objecthood, sparse reality, and perception as multilinear maps. The major free choices are the per-domain basis vectors and fields, which the paper calls contextual but which introduce the empirical content that the 'formal' claim is meant to exclude. No numbers are fitted to data, but the basis choices are hand-selected for each example.

free parameters (5)
  • Domain basis vectors = e.g., B_shelves = {height, width}
    Chosen by the modeler per domain (Section 5.2.1); the choice of quality dimensions is empirical and determines the ontology's content.
  • Field F of the vector space = e.g., R for continuous qualities, N for discrete
    Chosen per quality dimension (Section 5.2.1); the field determines which values a quality can take and is a modeling choice.
  • Minkowski metric exponent r = not specified (often r=2)
    Similarity distance in Section 5.2.3 is parameterized by r; no principled value is derived, taken from Gardenfors.
  • Reference vector v_origin = not specified
    Reconstruction paths are defined from a chosen reference vector (Section 5.2.3); the choice changes all relation distances.
  • Number of basis vectors per domain = tens to hundreds (suggested)
    Section 8 assumes a finite and manageable number of dimensions with asymptotic resolution; this is a claimed empirical fact, not derived.
assumptions (5)
  • ad hoc to paper Any object of interest can be fully described as a vector in a finite-dimensional vector space of quality dimensions
    Explicitly stated as 'the key assumption of vector ontology' in Section 5.2.2; if false, the framework collapses.
  • standard math Axioms of a vector space (commutativity, associativity, identity, inverses, distributivity)
    Standard mathematical axioms used as the formal structure (Section 5).
  • ad hoc to paper Reality corresponds to a sparse subset V_reality of the full vector space
    Introduced in Section 5.2.2; contradicts the vector-space axioms since a sparse subset is not closed under the vector-space operations.
  • domain assumption Basis vectors can be chosen before and independent of perceptual content
    Section 9.1 claims the structure exists before the basis; this premise underpins the 'a priori' claim and is not argued for.
  • domain assumption Perception of object boundaries and concepts corresponds to learning multilinear maps (functions of existence)
    Asserted in Section 5.2.3 as evidence that ANNs and humans use vector ontologies; no empirical support is provided in this paper.
invented entities (4)
  • Vector ontology Vont
    purpose: A formal ontology defined as a vector space whose basis vectors are quality dimensions
    A re-description of a vector space with an ontological interpretation; no new mathematical content and no falsifiable predictions beyond the mapping itself.
  • Functions of existence (FOE)
    purpose: Linear maps that unify patterns of existence into concepts
    Introduced in Section 5.2.3; equivalent to ordinary linear maps and not empirically distinguished from other function classes.
  • Reconstruction path / reconstruction distance
    purpose: A path in the vector space representing relations between objects
    Defined in Section 5.2.3 as P_reconstruction = v_origin + sum of a*x_i; no evidence that cognition uses this specific representation rather than any other metric.
  • Probability function of existence fpe
    purpose: A probabilistic function modeling correlation
    Proposed in Section 5.2.3; equivalent to standard probabilistic models, no new handle.

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Cite this review

Pith. "Pith review of To Be or Not To Be: Vector ontologies as a truly formal ontological framework." pith.science (2026). https://pith.science/paper/GVG5AS73

@misc{pith2026250514940,
  author       = {Pith},
  title        = {Pith review of: To Be or Not To Be: Vector ontologies as a truly formal ontological framework},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GVG5AS73}},
  note         = {Machine review of arXiv:2505.14940}
}
read the original abstract

Since Edmund Husserl coined the term "Formal Ontologies" in the early 20th century, a field that identifies itself with this particular branch of sciences has gained increasing attention. Many authors, and even Husserl himself have developed what they claim to be formal ontologies. I argue that under close inspection, none of these so claimed formal ontologies are truly formal in the Husserlian sense. More concretely, I demonstrate that they violate the two most important notions of formal ontology as developed in Husserl's Logical Investigations, namely a priori validity independent of perception and formalism as the total absence of content. I hence propose repositioning the work previously understood as formal ontology as the foundational ontology it really is. This is to recognize the potential of a truly formal ontology in the Husserlian sense. Specifically, I argue that formal ontology following his conditions, allows us to formulate ontological structures, which could capture what is more objectively without presupposing a particular framework arising from perception. I further argue that the ability to design the formal structure deliberately allows us to create highly scalable and interoperable information artifacts. As concrete evidence, I showcase that a class of formal ontology, which uses the axioms of vector spaces, is able to express most of the conceptualizations found in foundational ontologies. Most importantly, I argue that many information systems, specifically artificial intelligence, are likely already using some type of vector ontologies to represent reality in their internal worldviews and elaborate on the evidence that humans do as well. I hence propose a thorough investigation of the ability of vector ontologies to act as a human-machine interoperable ontological framework that allows us to understand highly sophisticated machines and machines to understand us.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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    GPT-4o-mini places music genres consistently and roughly accurately in an 8D Spotify-feature-based vector ontology, but validation is limited by a model-derived ground truth and weak baselines.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · cited by 1 Pith paper

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    Arp, R., Smith, B., & Spear, A. D. (2015). Building ontologies with basic formal ontology . Mit Press. Borgo, S., Ferrario, R., Gangemi, A., Guarino, N., Masolo, C., Porello, D., Sanfilippo, E. M., & Vieu, L. (2022). Dolce: A descriptive ontology for linguistic and cognitive engineering. Applied ontology, 17 (1), 45–69. Gardenfors, P. (2004). Conceptual s...

    work page 2015

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    Husserl, E. (2001a). Logical investigations volume 1 (D. Moran, Ed.). Routledge. Husserl, E. (2001b). Logical investigations volume 2 . Routledge. Lopes, J. (2023). Can deep cnns avoid infinite regress/circularity in content constitution? Minds and Machines , 33 (3), 507–524. Wilkinson, M. D., Dumontier, M., Aalbersberg, I. J., Appleton, G., Axton, M., Ba...

    work page 2001

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Reviewed August 7, 2026 · model on record in the stance chip above.