Contribution
A short paper in formal ontology. It names a two-way typology of spatial boundaries and argues that the classical puzzles of contact and division (which half of a cut sphere owns the cut, which state owns the border) come from treating every boundary as one kind. Bona fide boundaries get ordinary closure topology, mereologised; fiat boundaries get a Brentanian theory of coinciding boundaries. The formal part is a sketch: “Our aim will be illustrative”. Its one surprising formal result is that the bona fide theory needs the fiat one, because self-connectedness has to be defined by fiat connection.
Key insight
A boundary is bona fide when there is something in the world for it to follow: a gap, or a change of material. Otherwise it is fiat, and exists because something demarcates it: an act of perception, a word, a treaty, a definition. The test is whether the discontinuity is present. Who or what made the discontinuity is not part of the test.
Assumptions
- Spatial, synchronic, macroscopic. Variables range over “all spatial entities, both extended and nonextended”, and the axioms are read synchronically (fn. 15). There is no time, no process and no dynamics in the formal theory. Non-spatial boundaries are not treated.
- Common-sense level of description. The authors grant that physics shows “macroscopic physical objects are not continuous and that they do not have boundaries of the sort countenanced by common sense at all”. The theory is a theory of the common-sense world, defended as needed for “the fiat entities carved out by ordinary discourse”.
- No vague reality. Vagueness “is a conceptual and semantic matter that pertains ontologically only to the fiat world”; every candidate aggregate of molecules is “precisely determinate”.
- Extensional mereology with unrestricted fusion (A≤4, A≤5), chosen for simplicity, not argued for.
- Material objects do not coincide; fiat boundaries can, because they have no bulk.
Key results
- The typology (§1).
- Inner boundaries: “Inner boundaries involving spatial discontinuity (holes, fissures, slits) or qualitative heterogeneity (of material constitution, texture, electric charge) we shall call bona fide boundaries”. The rest are fiat (the hemispheres of a homogeneous sphere).
- Outer boundaries too: planet surfaces are bona fide, national borders fiat “at least in those cases where they lie skew to any qualitative differentiations or spatial discontinuities”.
- Kinds of fiat: social (generically dependent on beliefs and customs, “relatively isolated from causal change”), individual (“individually dependent (on these acts, taking place now)”), perceptual, linguistic (mass and count nouns: “cow” and “cows” differ in “the different sorts of boundary that are imposed upon this material”), and mathematical (the equator).
- An object with any fiat portion of its boundary is a fiat object: the North Sea, and also “A mountain, a bay, the branches of a tree, or the stem of a champagne glass are all fiat objects in this sense.”
- Fiat objects are not arbitrary: “real properties of the underlying factual material … both enable the drawing of fiat borders and constrain the types and location of fiat borders which can be drawn.”
- “while we can reasonably assume that all bona fide objects are connected, fiat objects may be scattered” (Polynesia, the constellation Orion).
- Contact between bodies is impossible (§1.2, §2). “As far as bona fide outer boundaries are concerned, however, no genuine contact or coincidence of boundaries between John and Mary is possible at all.” A kiss is a real basis for a fiat demarcation, not contact.
- Cutting (§2.1). Cutting reveals no inner surface; the outer surface deforms until it splits, like an oil drop. A boundary made by cutting is new, not a fiat boundary turned bona fide. Likewise for fences built along a border: “this is not a process of transformation. The categorial distinction between fiat and bona fide boundaries is absolute.”
- Coincidence (§2.2). A fiat border is owned by both sides: “each hemisphere has its own equator, and the two equators coincide”. Asymmetric boundaries (only one of the pair present) are allowed: the old inner-German border.
- Bona fide theory (§3.3). Primitive B(x, y), “x is a bona fide boundary for y”. Closure c(x) := x + b(x), with the Kuratowski axioms mereologised (AB1–AB3). Bona fide boundaries are transitive, dissective and symmetric: B(x, y) → B(x, −y) (TB3). Contact is external connection, and TB4: EC(x, y) → (Cl(x) → ¬Cl(y)). So John’s head is not in bona fide contact with his body.
- Fiat theory (§3.4). Primitive B*(x, y). AB1: B(x, y) → x ≤ y. “the main distinguishing feature of fiat boundaries is that they are necessarily parts of the entities they bound”. So there is no closure operator and no open/closed distinction; fiat boundaries are not symmetric (“oriented boundaries”). Connection by fiat boundary: C*(x, y) := O(x, y) ∨ ∃z∃w(B*(z, x) ∧ B*(w, y) ∧ z ≈ w), with ≈ spatial coincidence, an equivalence relation.
- Dependence (§3.5). All boundaries are dependent particulars, needing some host “of the appropriate kind” (generic, not specific, dependence). The naive thesis (AB4, AB4) is trivially satisfied by a scattered host, so a self-connected host is required, and self-connection is defined by fiat connection: “A self-connected entity is one all of whose parts are separated at most by fiat”, Cn(x) := ∀y∀z(x = y + z → C*(y, z)). Hence “the theory of bona fide boundaries presupposes the theory of fiat boundaries”.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Spatial boundaries divide into bona fide (discontinuity or heterogeneity present) and fiat (demarcated without one), and the division is absolute | moderate | examples and the cutting argument; a stipulated typology, not derived |
| C2 | Bona fide boundaries obey ordinary closure topology; fiat boundaries do not support the open/closed distinction | moderate as formal proposal | AB1–AB3, TB1–TB4; AB*1 and its consequences; no models or consistency proof given |
| C3 | The puzzles of contact and division dissolve once the two kinds are separated | moderate | §2, argued case by case (sphere, disk, equator, Mason–Dixon line) |
| C4 | Distinct bodies are never in genuine contact; apparent contact of bodies is a fiat matter | moderate | appeal to physics and to TB4; informal |
| C5 | Vagueness belongs only to fiat objects; bona fide reality is precise | weak | asserted, with Quine 1985; the supervaluation-style argument is a paragraph |
| C6 | The bona fide theory presupposes the fiat theory, through self-connectedness | moderate | AB5, AB5, DB5; “the formal implications of which have still to be examined” |
Concepts
- Bona fide boundary. One that follows a spatial discontinuity or a qualitative heterogeneity in the world. The authors also gloss it as “physical”.
- Fiat boundary. One that follows no such discontinuity and owes its existence to demarcation; it is part of what it bounds (AB*1).
- Fiat object. An object any part of whose outer boundary is fiat. It may be scattered.
- Individual and social fiat. Dependent on particular acts now, against generically dependent on a practice that can outlast any one act.
- Coincidence (≈). Sameness of spatial location, possible for boundaries, not for material bodies.
- Generic dependence. A boundary needs some host of the right kind, not any particular one; the host depends specifically on its boundary.
Connections
- Builds on Brentano’s theory of the continuum (through Chisholm and Smith 1997) for the fiat side, on Bolzano and Kuratowski for the bona fide side, and on Smith 1994 (“Fiat Objects”), where the terms were introduced.
- Varzi’s SEP entry on mereology (LIT-124) is the neighbour already held; this paper is the mereotopological extension of that framework.
- Casati & Varzi on holes and location (cited in fn. 8, 19) supply the location/occupation distinction coincidence relies on.
Bearing on the record
The LIT asked for four checks.
1. Does fiat/bona fide map onto CLAIM-194’s produced/imposed? No. The two cut across each other. Bona fide is a test of whether a discontinuity is there; produced is a test of who makes and keeps the boundary. A cell membrane is bona fide and produced. The beam joining two pendulums is bona fide and imposed: it is a body, and the paper counts human-made things such as border fences as bona fide boundaries once they exist, while denying that anything was transformed. The equator is fiat and imposed. Nothing in the paper is fiat and produced, and the paper has no place for a boundary the bounded thing draws for itself: fiat is drawn by cognition or convention from outside, bona fide is found in the world. So the distinction sorts the membrane and the beam together, exactly as CLAIM-194 says the gap criterion does. That is indirect support for the counter’s point that a criterion indifferent to who made a boundary cannot separate the two cases. It is not a way of answering it.
A word to watch: the paper says boundaries are “imposed” on material by cognition (“the systematic imposition of boundaries of many different sorts”). That imposition is an observer’s demarcation, the thing THEORY-071’s individuality condition rules out. CLAIM-194’s “imposed on it” is something else: a physical constraint maintained by something other than the system. The record should not read one as the other.
2. Is the beam pair’s boundary fiat? The LIT asked whether the boundary around the coupled pair, drawn by the analyst’s target, is fiat “only if a target counts as demarcation”. On the paper’s own terms the answer is yes, and for a stronger reason than the target. Distinct bodies are never in genuine contact (C4): pendulums and beam are separate bodies, so the assembly is not one bona fide object, and anything bounding it as one is at least partly fiat. The same holds for every combined form of distinct bodies the record has considered: the dancing couple (CASE-046) is a scattered fiat object like Polynesia. A target-relative block (CLAIM-157, relative as CLAIM-152 requires) fits the paper’s individual fiat: dependent on a choice made now, enabled and constrained by real properties of the material, which here are coupling strengths. That last clause is the paper’s nearest thing to the gap criterion.
3. What this means for QUESTION-035. Since every combined form of separate bodies comes out fiat, the distinction cannot discriminate among the candidates the question cares about: the couple dancing and the couple after the song are both fiat. It classifies the question’s answers rather than supplying one. It also fixes a vocabulary: whatever criterion answers the question (gap, entrainment CLAIM-172, compensation), it draws a fiat boundary grounded in real differences, and the paper’s clause that real properties enable and constrain fiat borders is the right description of how such a boundary can be non-arbitrary without being bona fide. The record may want that sentence in CLAIM-157’s defence; the use is the record’s, not the paper’s.
4. Where the resemblance fails. The paper is spatial and synchronic. Its heterogeneity is of “material constitution, texture, electric charge”, not of interaction strength, and it has no dynamics. Reading a near-decomposability gap as a “qualitative heterogeneity” that makes a block boundary bona fide would extend the paper, not apply it. Its dependence relation is ontological (a boundary needs some host), not the causal maintenance that THEORY-072’s production and Weber & Varela’s self-production (LIT-1018) are about. Nothing in it bears on CLAIM-174’s agency conditions or on THEORY-073’s Markov blankets beyond the point in check 1.
No machine-learning instruction is carried; no anthology topic holds it.
Limitations
- The formal theory is admittedly illustrative: no models, no consistency or completeness result, and the authors say the consequences of AB5 and AB*5 “have still to be examined”.
- The typology is stipulated by examples. Borderline cases between the two kinds, such as a gradient of material with no sharp change, are handled only by the claim that vagueness is fiat.
- Time is absent. Boundaries that come and go, or are maintained by a process, are outside the theory; cutting and fencing are discussed only informally.
- The physics is gestured at, not used: the paper concedes that at the physical level bodies have no common-sense boundaries, which leaves the status of “bona fide” itself level-relative in a way the paper does not discuss.
Open questions
- Is “bona fide” level-relative? If at the physical level there are no common-sense boundaries, then a membrane is bona fide only at a level of description, and the absoluteness of the distinction needs restating.
- Is there a third kind: a boundary neither found in the world nor drawn by an observer, but drawn and maintained by the bounded system? The autopoietic literature says yes. A formal theory of it would have to add time and a maintenance relation the paper does not have.
- Can a dynamical (non-spatial) gap be treated as heterogeneity, giving a bona fide boundary in a state space? A mereotopology of variables rather than regions would be needed to say.