Conventionalism about Geometry

The empirical turn in the philosophy of geometry — geometry as a contingent fact discovered by measurement — runs into a stubborn obstacle: we never measure geometry directly. Every geometrical measurement uses physical objects and processes — rigid rods, light rays, freely falling bodies, clocks — each governed by physical laws. So what any measurement tests is not geometry alone but the conjunction of a geometry and a physics, and a surprising result can always be accommodated by adjusting either. This underdetermination is the basis of conventionalism: the thesis, urged by Henri Poincaré and developed by Hans Reichenbach and Adolf Grünbaum, that the geometry we ascribe to physical space is in part a matter of convention — a stipulation adopted for convenience — rather than a fact forced on us by the world.

This page states the conventionalist argument in its three main versions and the realist replies; it presupposes the epistemology of geometry and connects to the conventionality of simultaneity, its temporal analogue.

Poincaré: geometry as convention

Poincaré argued with a famous thought experiment. Imagine a disc-world whose inhabitants' measuring rods expand and contract with temperature, which varies across the disc in a specific way; as they carry their rods toward the edge, the rods shrink (and the inhabitants shrink with them), so that the edge, though finite to us, is infinitely far to them. Do these beings live in an infinite hyperbolic (non-Euclidean) space, or a finite Euclidean disc with a distorting temperature field? Poincaré's answer: there is no fact of the matter forced by experience. The two descriptions — (a) non-Euclidean geometry with rods that report true distances, and (b) Euclidean geometry with a universal "force" that distorts the rods — save all the same observations. One chooses between them, Poincaré held, on grounds of simplicity and convenience, not truth. Geometry is like a choice of coordinates or units: a convention.

His slogan: no experiment can pronounce on geometry in isolation, only on the combination "geometry + physics." Confronted with anomalous measurements, we may always keep Euclidean geometry and modify the physics (postulate forces), or adopt a curved geometry and simplify the physics. Poincaré expected physicists would always keep Euclid for its simplicity — a prediction general relativity falsified in practice, but which does not touch his in-principle point.

Reichenbach: coordinative definitions and universal forces

Reichenbach sharpened conventionalism into a precise doctrine. Before geometry can be tested, one must lay down coordinative definitions: stipulations linking geometrical terms to physical procedures — e.g. "a rigid rod, suitably corrected, measures equal lengths when transported," or "light travels in straight lines." Only relative to such stipulations does "space is Euclidean" acquire a determinate truth value; the stipulations themselves are neither true nor false but chosen.

His key device is the universal force. Reichenbach distinguished differential forces (which affect different materials differently — e.g. thermal expansion, detectable by comparing rods of different substances) from a universal force (which affects all materials identically, and so is undetectable by any comparison, and cannot be screened off). He argued that one can always retain a preferred geometry $G$ by postulating a suitable universal force $F$ that distorts all rods in concert; the empirical content is exhausted by the combination $G+F$, and the split into "geometry" and "force" is conventional. The natural stipulation, he proposed, is to set universal forces to zero — and then the geometry is fixed by measurement. But that "then" hangs on a convention. So the geometry of space is factual only modulo the coordinative definition "there are no universal forces."

$$\text{empirical content}=\underset{\text{geometry}}{\underbrace{G}}+\underset{\text{universal force}}{\underbrace{F}},F\equiv 0\text{by convention}\Rightarrow G\text{determinate}.$$

Grünbaum: the metrical amorphousness of the continuum

Grünbaum grounded conventionalism in the nature of the continuum itself, connecting it to Zeno's problem of composition. Space, he argued, is intrinsically metrically amorphous: a continuum of points, being dense (uncountably many points, none adjacent to a next), contains no intrinsic metric — nothing built into the bare point-set fixes the distance between two points. Unlike a discrete set, where one could count the points between two others and read off a distance, a continuum offers no intrinsic standard of "equal intervals." The metric must therefore be imposed from outside, by a congruence standard (a stipulation about which spatially separated segments count as equal — typically "what a transported rigid rod marks off"). Since the standard is extrinsic and could have been chosen otherwise, an ineliminable element of convention enters the metric of any continuous space. For Grünbaum this is not an epistemic limitation but a fact about what a continuum is.

The realist replies

Conventionalism has been vigorously contested; the main lines of reply:

• Underdetermination is cheap; simplicity is not. Grant that one can always rescue a preferred geometry with universal forces. But the resulting theory is grotesquely more complex: it posits undetectable, unmotivated forces that conspire, for every material alike, exactly to mimic curvature. The realist argues that when one theory attributes to geometry what another attributes to a mysterious universal force affecting everything identically, ordinary standards of theory choice (simplicity, unification, absence of undetectable posits) decisively favour the geometry — and that these are the same standards that adjudicate any scientific inference, not a mere taste. The "force" story is not an equal alternative but a degenerate one, in the way a geocentric astronomy with enough epicycles is not really on a par with heliocentrism.
• Universal forces are ad hoc and idle. A force that acts identically on everything, is unmeasurable in principle, and exists solely to preserve a chosen geometry violates the same methodological scruples that exclude other undetectable posits. Once such devices are barred (as they are everywhere else in science), general relativity's curved geometry is the geometry, not a convention.
• The Quine–Putnam point: convention collapses into ordinary underdetermination. The conventionalist's "geometry + physics is what gets tested" is just the general Duhem–Quine thesis that theories confront experience holistically. But that thesis applies to all science; it does not make geometry specially conventional. If everything is "conventional" in this sense, the word loses its bite, and geometry is as factual as anything else — which is to say, factual.
• Against Grünbaum's amorphousness. Critics (Putnam, Friedman) deny that the density of the continuum entails the absence of intrinsic metric relations. A Riemannian manifold comes equipped with a metric tensor as part of its structure; there is no reason the physical world's spatiotemporal structure could not include intrinsic metric facts, dense point-set notwithstanding. "No adjacent points to count" does not entail "no fact about distance."

The mainstream verdict, associated especially with Michael Friedman's Foundations of Space-Time Theories, is a qualified realism: there is a genuine, if subtle, conventional element in setting up the framework of a spacetime theory (echoing Reichenbach's relativised a priori), but once the framework and ordinary standards of theory choice are in place, the geometry of spacetime is a substantive empirical fact, not a free stipulation. General relativity does not merely describe curvature conveniently; it discovers it.

Where this sits

Conventionalism is the sceptical foil to the empirical epistemology of geometry, and the debate over it is a case study in the general problem of theory and underdetermination that runs through the whole philosophy of physics. It rests on the structure of the continuum (Grünbaum) and it has an exact temporal analogue in the conventionality of simultaneity, where the same question — factual or stipulated? — is asked of distant simultaneity in special relativity. Its resolution bears on the substantivalism debate, since a geometry that is a genuine empirical fact is more plausibly the geometry of a real dynamical field. The next page turns to a further structural feature of space that resists conventional dissolution — its dimensionality.