What did Kant really mean in viewing Euclidean geometry as the correct geometrical structure of the world? It is widely known that one of the main goals that Kant pursued in the First Critique was that of unearthing the a priori foundations of Newtonian physics, which describes the structure of the world in terms of Euclidean geometry. How did he achieve that? Kant maintained that our understanding of the physical world had its foundations not merely in experience, but in both experience and a priori concepts. He argues that the possibility of sensory experience depends on certain necessary conditions which he calls a priori forms and that these conditions structure and hold true of the world of experience. As he maintains in the “Transcendental Aesthetic”, Space and Time are not derived from experience but rather are its preconditions. Experience provides those things which we sense. It is our mind, though, that processes this information about the world and gives it order, allowing us to experience it. Our mind supplies the conditions of space and time to experience objects. Thus “space” for Kant is not something existing – as it was for Newton. Space is an a priori form that structures our perception of objects in conformity to the principles of the Euclidean geometry. In this sense, then, the latter is the correct geometrical structure of the world. It is necessarily correct because it is part of the a priori principles of organization of our experience. This claim is exactly what Poincaré criticized about Kant’s view of geometry. Poincaré did not agree with Kant’s view of space as precondition of experience. He thought that our knowledge of the physical space is the result of inferences made out of our direct perceptions.
This knowledge is a theoretical construct, i.e, we infer the existence and nature of the physical space as an explanatory hypothesis which provides us with an account for the regularity we experience in our direct perceptions. But this hypothesis does not possess the necessity of an a priori principle that structures what we directly perceive. Although Poincaré does not endorse an empiricist account, he seems to think, though, that an empiricist view of geometry is more adequate than Kantian conception. In fact, the idea that only a large number of observations inquiring the geometry of physical world can establish which geometrical structure is the correct one, is considered by him as more plausible. But, this empiricist approach is not going to work as well. In fact Poincaré does not endorse an empiricist view of geometry. The outcome of his considerations about a comparison between the empiricist and Kantian accounts of geometry is well described by Sklar:
Nevertheless the empiricist account is wrong. For, given any collections of empirical observations a multitude of geometries, all incompatible with one another, will be equally compatible with the experimental results.
This is the problem of under-determination of hypotheses about the geometrical structure of physical space by experimental evidence. The under-determination is not due to our ability to collect experimental facts. No matter how rich and sophisticated are our experimental procedures for accumulating empirical results, these results will be never enough compelling to support just one of the hypotheses about the geometry of physical space – ruling out the competitors once for all. Actually, it is even worse than that: empirical results seem not to give us any reason at all to think one of the other hypothesis correct. Poincaré thought that this problem was grist to the mill of the conventionalist approach to geometry. The adoption of a geometry for physical space is a matter of making a conventional choice. A brief description of Poincaré disk model might unravel a bit more the issue that is coming up here. The short story about this imaginary world shows that an empiricist account of geometry fails to be adequate. In fact, Poincaré describes a scenario in which Euclidean and hyperbolic geometrical descriptions of that physical space end up being equally consistent with the same collection of empirical data. However, what this story tells us can be generalized to any other scenario, including ours, in which a scientific inquiry concerning the intrinsic geometry of the world is performed.
The imaginary world described in Poincaré’s example is an Euclidean two dimensional disk heated to a constant temperature at the center, whereas, along the radius R, it is heated in a way that produces a temperature’s variation described by R2 − r2. Therefore, the edge of the disk is uniformly cooled to 00.
A group of scientists living on the disk are interested in knowing what the intrinsic geometry of their world is. As Sklar says, the equipment available to them consists in rods uniformly dilating with increasing temperatures, i.e. at each point of the space they all change their lengths in a way which is directly proportional to temperature’s value at that point. However, the scientists are not aware of this peculiar temperature distortion of their rods. So, without anybody knowing, every time a measurement is performed, rods shrank or dilated, depending if they are close to the edge or to the center. After repeated measurements all over the disk, they have a list of empirical data that seems to support strongly the idea that their world is a Lobachevskian plane. So, this view becomes the official one. However, a different data’s interpretation is presented by a member of the community who, striking a discordant note, claims that those empirical data can be taken to indicate that the world is in fact an Euclidean disk, but equipped with fields shrinking or dilating lengths.
Although the two geometrical theories about the structure of the physical space are competitors, the empirical results collected by the scientists support both of them. According to our external three-dimensional Euclidean perspective we know their bi-dimensional world is Euclidean and so we know that only the innovator’s interpretation is the correct one. Using our standpoint the problem of under-determination would seem indeed a problem of epistemic access due to the particular experimental repertoire of the inhabitants. After all expanding this repertoire and increasing the amount of empirical data can overcome the problem. But, according to Poincaré that would completely miss the point. Moving from our “superior” perspective to their one would collocate us exactly in the same situation as they are, i.e.in the impossibility to decide which geometry is the correct one. But more importantly, Poincaré seems to say that any arbitrarily large amount of empirical data cannot refute a geometric hypothesis. In fact, a scientific theory about space is divided in two branches, a geometric one and a physical one. These two parts are deeply related. It would be possible to save from experimental refutation any geometric hypothesis about space, suitably changing some features of the physical branch of the theory. According to Sklar, this fact forces Poincaré to the conclusion that the choice of one hypothesis among several competitors is purely conventional.
The problem of under-determination comes up in the analysis of dual string theories with two string theories postulating two geometrically inequivalent backgrounds, if dual, can produce the same experimental results: same expectation values, same scattering amplitude, and so on. Therefore, similarly to Poincaré’s short story, empirical data relative to physical properties and physical dynamics of strings are not sufficient to determine which one between the two different geometries postulated for the background is the right one, or if there is any more fundamental geometry at all influencing physical dynamics.