Kant and Non-Euclidean Geometries. Thought of the Day 94.0

The argument that non-Euclidean geometries contradict Kant’s doctrine on the nature of space apparently goes back to Hermann Helmholtz and was retaken by several philosophers of science such as Hans Reichenbach (The Philosophy of Space and Time) who devoted much work to this subject. In a essay written in 1870, Helmholtz argued that the axioms of geometry are not a priori synthetic judgments (in the sense given by Kant), since they can be subjected to experiments. Given that Euclidian geometry is not the only possible geometry, as was believed in Kant’s time, it should be possible to determine by means of measurements whether, for instance, the sum of the three angles of a triangle is 180 degrees or whether two straight parallel lines always keep the same distance among them. If it were not the case, then it would have been demonstrated experimentally that space is not Euclidean. Thus the possibility of verifying the axioms of geometry would prove that they are empirical and not given a priori.

Helmholtz developed his own version of a non-Euclidean geometry on the basis of what he believed to be the fundamental condition for all geometries: “the possibility of figures moving without change of form or size”; without this possibility, it would be impossible to define what a measurement is. According to Helmholtz:

the axioms of geometry are not concerned with space-relations only but also at the same time with the mechanical deportment of solidest bodies in motion.

Nevertheless, he was aware that a strict Kantian might argue that the rigidity of bodies is an a priori property, but

then we should have to maintain that the axioms of geometry are not synthetic propositions… they would merely define what qualities and deportment a body must have to be recognized as rigid.

At this point, it is worth noticing that Helmholtz’s formulation of geometry is a rudimentary version of what was later developed as the theory of Lie groups. As for the transport of rigid bodies, it is well known that rigid motion cannot be defined in the framework of the theory of relativity: since there is no absolute simultaneity of events, it is impossible to move all parts of a material body in a coordinated and simultaneous way. What is defined as the length of a body depends on the reference frame from where it is observed. Thus, it is meaningless to invoke the rigidity of bodies as the basis of a geometry that pretend to describe the real world; it is only in the mathematical realm that the rigid displacement of a figure can be defined in terms of what mathematicians call a congruence.

Arguments similar to those of Helmholtz were given by Reichenbach in his intent to refute Kant’s doctrine on the nature of space and time. Essentially, the argument boils down to the following: Kant assumed that the axioms of geometry are given a priori and he only had classical geometry in mind, Einstein demonstrated that space is not Euclidean and that this could be verified empirically, ergo Kant was wrong. However, Kant did not state that space must be Euclidean; instead, he argued that it is a pure form of intuition. As such, space has no physical reality of its own, and therefore it is meaningless to ascribe physical properties to it. Actually, Kant never mentioned Euclid directly in his work, but he did refer many times to the physics of Newton, which is based on classical geometry. Kant had in mind the axioms of this geometry which is a most powerful tool of Newtonian mechanics. Actually, he did not even exclude the possibility of other geometries, as can be seen in his early speculations on the dimensionality of space.

The important point missed by Reichenbach is that Riemannian geometry is necessarily based on Euclidean geometry. More precisely, a Riemannian space must be considered as locally Euclidean in order to be able to define basic concepts such as distance and parallel transport; this is achieved by defining a flat tangent space at every point, and then extending all properties of this flat space to the globally curved space (Luther Pfahler Eisenhart Riemannian Geometry). To begin with, the structure of a Riemannian space is given by its metric tensor g_μν from which the (differential) length is defined as ds² = g_μν dx^μ dx^ν; but this is nothing less than a generalization of the usual Pythagoras theorem in Euclidean space. As for the fundamental concept of parallel transport, it is taken directly from its analogue in Euclidean space: it refers to the transport of abstract (not material, as Helmholtz believed) figures in such a space. Thus Riemann’s geometry cannot be free of synthetic a priori propositions because it is entirely based upon concepts such as length and congruence taken form Euclid. We may conclude that Euclids geometry is the condition of possibility for a more general geometry, such as Riemann’s, simply because it is the natural geometry adapted to our understanding; Kant would say that it is our form of grasping space intuitively. The possibility of constructing abstract spaces does not refute Kant’s thesis; on the contrary, it reinforces it.

Individuation. Thought of the Day 91.0

The first distinction is between two senses of the word “individuation” – one semantic, the other metaphysical. In the semantic sense of the word, to individuate an object is to single it out for reference in language or in thought. By contrast, in the metaphysical sense of the word, the individuation of objects has to do with “what grounds their identity and distinctness.” Sets are often used to illustrate the intended notion of “grounding.” The identity or distinctness of sets is said to be “grounded” in accordance with the principle of extensionality, which says that two sets are identical iff they have precisely the same elements:

SET(x) ∧ SET(y) → [x = y ↔ ∀u(u ∈ x ↔ u ∈ y)]

The metaphysical and semantic senses of individuation are quite different notions, neither of which appears to be reducible to or fully explicable in terms of the other. Since sufficient sense cannot be made of the notion of “grounding of identity” on which the metaphysical notion of individuation is based, focusing on the semantic notion of individuation is an easy way out. This choice of focus means that our investigation is a broadly empirical one drawn on empirical linguistics and psychology.

What is the relation between the semantic notion of individuation and the notion of a criterion of identity? It is by means of criteria of identity that semantic individuation is effected. Singling out an object for reference involves being able to distinguish this object from other possible referents with which one is directly presented. The final distinction is between two types of criteria of identity. A one-level criterion of identity says that two objects of some sort F are identical iff they stand in some relation RF:

Fx ∧ Fy → [x = y ↔ R_F(x,y)]

Criteria of this form operate at just one level in the sense that the condition for two objects to be identical is given by a relation on these objects themselves. An example is the set-theoretic principle of extensionality.

A two-level criterion of identity relates the identity of objects of one sort to some condition on entities of another sort. The former sort of objects are typically given as functions of items of the latter sort, in which case the criterion takes the following form:

f(α) = f(β) ↔ α ≈ β

where the variables α and β range over the latter sort of item and ≈ is an equivalence relation on such items. An example is Frege’s famous criterion of identity for directions:

d(l₁) = d(l₂) ↔ l₁ || l₂

where the variables l₁ and l₂ range over lines or other directed items. An analogous two-level criterion relates the identity of geometrical shapes to the congruence of things or figures having the shapes in question. The decision to focus on the semantic notion of individuation makes it natural to focus on two-level criteria. For two-level criteria of identity are much more useful than one-level criteria when we are studying how objects are singled out for reference. A one-level criterion provides little assistance in the task of singling out objects for reference. In order to apply a one-level criterion, one must already be capable of referring to objects of the sort in question. By contrast, a two-level criterion promises a way of singling out an object of one sort in terms of an item of another and less problematic sort. For instance, when Frege investigated how directions and other abstract objects “are given to us”, although “we cannot have any ideas or intuitions of them”, he proposed that we relate the identity of two directions to the parallelism of the two lines in terms of which these directions are presented. This would be explanatory progress since reference to lines is less puzzling than reference to directions.

Weyl and Automorphism of Nature. Drunken Risibility.

In classical geometry and physics, physical automorphisms could be based on the material operations used for defining the elementary equivalence concept of congruence (“equality and similitude”). But Weyl started even more generally, with Leibniz’ explanation of the similarity of two objects, two things are similar if they are indiscernible when each is considered by itself. Here, like at other places, Weyl endorsed this Leibnzian argument from the point of view of “modern physics”, while adding that for Leibniz this spoke in favour of the unsubstantiality and phenomenality of space and time. On the other hand, for “real substances” the Leibnizian monads, indiscernability implied identity. In this way Weyl indicated, prior to any more technical consideration, that similarity in the Leibnizian sense was the same as objective equality. He did not enter deeper into the metaphysical discussion but insisted that the issue “is of philosophical significance far beyond its purely geometric aspect”.

Weyl did not claim that this idea solves the epistemological problem of objectivity once and for all, but at least it offers an adequate mathematical instrument for the formulation of it. He illustrated the idea in a first step by explaining the automorphisms of Euclidean geometry as the structure preserving bijective mappings of the point set underlying a structure satisfying the axioms of “Hilbert’s classical book on the Foundations of Geometry”. He concluded that for Euclidean geometry these are the similarities, not the congruences as one might expect at a first glance. In the mathematical sense, we then “come to interpret objectivity as the invariance under the group of automorphisms”. But Weyl warned to identify mathematical objectivity with that of natural science, because once we deal with real space “neither the axioms nor the basic relations are given”. As the latter are extremely difficult to discern, Weyl proposed to turn the tables and to take the group Γ of automorphisms, rather than the ‘basic relations’ and the corresponding relata, as the epistemic starting point.

Hence we come much nearer to the actual state of affairs if we start with the group Γ of automorphisms and refrain from making the artificial logical distinction between basic and derived relations. Once the group is known, we know what it means to say of a relation that it is objective, namely invariant with respect to Γ.

By such a well chosen constitutive stipulation it becomes clear what objective statements are, although this can be achieved only at the price that “…we start, as Dante starts in his Divina Comedia, in mezzo del camin”. A phrase characteristic for Weyl’s later view follows:

It is the common fate of man and his science that we do not begin at the beginning; we find ourselves somewhere on a road the origin and end of which are shrouded in fog.

Weyl’s juxtaposition of the mathematical and the physical concept of objectivity is worthwhile to reflect upon. The mathematical objectivity considered by him is relatively easy to obtain by combining the axiomatic characterization of a mathematical theory with the epistemic postulate of invariance under a group of automorphisms. Both are constituted in a series of acts characterized by Weyl as symbolic construction, which is free in several regards. For example, the group of automorphisms of Euclidean geometry may be expanded by “the mathematician” in rather wide ways (affine, projective, or even “any group of transformations”). In each case a specific realm of mathematical objectivity is constituted. With the example of the automorphism group Γ of (plane) Euclidean geometry in mind Weyl explained how, through the use of Cartesian coordinates, the automorphisms of Euclidean geometry can be represented by linear transformations “in terms of reproducible numerical symbols”.

For natural science the situation is quite different; here the freedom of the constitutive act is severely restricted. Weyl described the constraint for the choice of Γ at the outset in very general terms: The physicist will question Nature to reveal him her true group of automorphisms. Different to what a philosopher might expect, Weyl did not mention, the subtle influences induced by theoretical evaluations of empirical insights on the constitutive choice of the group of automorphisms for a physical theory. He even did not restrict the consideration to the range of a physical theory but aimed at Nature as a whole. Still basing on his his own views and radical changes in the fundamental views of theoretical physics, Weyl hoped for an insight into the true group of automorphisms of Nature without any further specifications.

Automorphisms. Note Quote.

A group automorphism is an isomorphism from a group to itself. If  is a finite multiplicative group, an automorphism of  can be described as a way of rewriting its multiplication table without altering its pattern of repeated elements. For example, the multiplication table of the group of 4th roots of unity  can be written as shown above, which means that the map defined by

is an automorphism of .

Looking at classical geometry and mechanics, Weyl followed Newton and Helmholtz in considering congruence as the basic relation which lay at the heart of the “art of measuring” by the handling of that “sort of bodies we call rigid”. He explained how the local congruence relations established by the comparison of rigid bodies can be generalized and abstracted to congruences of the whole space. In this respect Weyl followed an empiricist approach to classical physical geometry, based on a theoretical extension of the material practice with rigid bodies and their motions. Even the mathematical abstraction to mappings of the whole space carried the mark of their empirical origin and was restricted to the group of proper congruences (orientation preserving isometries of Euclidean space, generated by the translations and rotations) denoted by him as ∆⁺. This group seems to express “an intrinsic structure of space itself; a structure stamped by space upon all the inhabitants of space”.

But already on the earlier level of physical knowledge, so Weyl argued, the mathematical automorphisms of space were larger than ∆. Even if one sees “with Newton, in congruence the one and only basic concept of geometry from which all others derive”, the group Γ of automorphisms in the mathematical sense turns out to be constituted by the similarities.

The structural condition for an automorphism C ∈ Γ of classical congruence geometry is that any pair (v₁,v₂) of congruent geometric configurations is transformed into another pair (v₁^*,v₂^*) of congruent configurations (v_j^* = C(v_j), j = 1,2). For evaluating this property Weyl introduced the following diagram:

Because of the condition for automorphisms just mentioned the maps C T C^-1 and C^-1TC belong to ∆⁺ whenever T does. By this argument he showed that the mathematical automorphism group Γ is the normalizer of the congruences ∆⁺ in the group of bijective mappings of Euclidean space.

More generally, it also explains the reason for his characterization of generalized similarities in his analysis of the problem of space in the early 1920s. In 1918 he translated the relationship between physical equivalences as congruences to the mathematical automorphisms as the similarities/normalizer of the congruences from classical geometry to special relativity (Minkowski space) and “localized” them (in the sense of physics), i.e., he transferred the structural relationship to the infinitesimal neighbourhoods of the differentiable manifold characterizing spacetime (in more recent language, to the tangent spaces) and developed what later would be called Weylian manifolds, a generalization of Riemannian geometry. In his discussion of the problem of space he generalized the same relationship even further by allowing any (closed) sub-group of the general linear group as a candidate for characterizing generalized congruences at every point.

Moreover, Weyl argued that the enlargement of the physico-geometrical automorphisms of classical geometry (proper congruences) by the mathematical automorphisms (similarities) sheds light on Kant’s riddle of the “incongruous counterparts”. Weyl presented it as the question: Why are “incongruous counterparts” like the left and right hands intrinsically indiscernible, although they cannot be transformed into another by a proper motion? From his point of view the intrinsic indiscernibility could be characterized by the mathematical automorphisms Γ. Of course, the congruences ∆ including the reflections are part of the latter, ∆ ⊂ Γ; this implies indiscernibility between “left and right” as a special case. In this way Kant’s riddle was solved by a Leibnizian type of argument. Weyl very cautiously indicated a philosophical implication of this observation:

And he (Kant) is inclined to think that only transcendental idealism is able to solve this riddle. No doubt, the meaning of congruence and similarity is founded in spatial intuition. Kant seems to aim at some subtler point. But just this point is one which can be completely clarified by general concepts, namely by subsuming it under the general and typical group-theoretic situation explained before . . . .

Weyl stopped here without discussing the relationship between group theoretical methods and the “subtler point” Kant aimed at more explicitly. But we may read this remark as an indication that he considered his reflections on automorphism groups as a contribution to the transcendental analysis of the conceptual constitution of modern science. In his book on Symmetry, he went a tiny step further. Still with the Weylian restraint regarding the discussion of philosophical principles he stated: “As far as I see all a priori statements in physics have their origin in symmetry” (126).

To prepare for the following, Weyl specified the subgroup ∆o ⊂ ∆ with all those transformations that fix one point (∆_o = O(3, R), the orthogonal group in 3 dimensions, R the field of real numbers). In passing he remarked:

In the four-dimensional world the Lorentz group takes the place of the orthogonal group. But here I shall restrict myself to the three-dimensional space, only occasionally pointing to the modifications, the inclusion of time into the four-dimensional world brings about.

Keeping this caveat in mind (restriction to three-dimensional space) Weyl characterized the “group of automorphisms of the physical world”, in the sense of classical physics (including quantum mechanics) by the combination (more technically, the semidirect product ̧) of translations and rotations, while the mathematical automorphisms arise from a normal extension:

– physical automorphisms ∆ ≅ R³ X| ∆_o with ∆_o ≅ O(3), respectively ∆ ≅ R⁴ X| ∆_o for the Lorentz group ∆_o ≅ O(1, 3),

– mathematical automorphisms Γ = R⁺ X ∆
(R⁺ the positive real numbers with multiplication).

In Weyl’s view the difference between mathematical and physical automorphisms established a fundamental distinction between mathematical geometry and physics.

Congruence, or physical equivalence, is a geometric concept, the meaning of which refers to the laws of physical phenomena; the congruence group ∆ is essentially the group of physical automorphisms. If we interpret geometry as an abstract science dealing with such relations and such relations only as can be logically defined in terms of the one concept of congruence, then the group of geometric automorphisms is the normalizer of ∆ and hence wider than ∆.

He considered this as a striking argument against what he considered to be the Cartesian program of a reductionist geometrization of physics (physics as the science of res extensa):

According to this conception, Descartes’s program of reducing physics to geometry would involve a vicious circle, and the fact that the group of geometric automorphisms is wider than that of physical automorphisms would show that such a reduction is actually impossible.” 

In this Weyl alluded to an illusion he himself had shared for a short time as a young scientist. After the creation of his gauge geometry in 1918 and the proposal of a geometrically unified field theory of electromagnetism and gravity he believed, for a short while, to have achieved a complete geometrization of physics.

He gave up this illusion in the middle of the 1920s under the impression of the rising quantum mechanics. In his own contribution to the new quantum mechanics groups and their linear representations played a crucial role. In this respect the mathematical automorphisms of geometry and the physical automorphisms “of Nature”, or more precisely the automorphisms of physical systems, moved even further apart, because now the physical automorphism started to take non-geometrical material degrees of freedom into account (phase symmetry of wave functions and, already earlier, the permutation symmetries of n-particle systems).

But already during the 19th century the physical automorphism group had acquired a far deeper aspect than that of the mobility of rigid bodies:

In physics we have to consider not only points but many types of physical quantities such as velocity, force, electromagnetic field strength, etc. . . .

All these quantities can be represented, relative to a Cartesian frame, by sets of numbers such that any orthogonal transformation T performed on the coordinates keeps the basic physical relations, the physical laws, invariant. Weyl accordingly stated:

All the laws of nature are invariant under the transformations thus induced by the group ∆. Thus physical relativity can be completely described by means of a group of transformations of space-points.

By this argumentation Weyl described a deep shift which ocurred in the late 19th century for the understanding of physics. He described it as an extension of the group of physical automorphisms. The laws of physics (“basic relations” in his more abstract terminology above) could no longer be directly characterized by the motion of rigid bodies because the physics of fields, in particular of electric and magnetic fields, had become central. In this context, the motions of material bodies lost their epistemological primary status and the physical automorphisms acquired a more abstract character, although they were still completely characterizable in geometric terms, by the full group of Euclidean isometries. The indistinguishability of left and right, observed already in clear terms by Kant, acquired the status of a physical symmetry in electromagnetism and in crystallography.

Weyl thus insisted that in classical physics the physical automorphisms could be characterized by the group ∆ of Euclidean isometries, larger than the physical congruences (proper motions) ∆⁺ but smaller than the mathe- matical automorphisms (similarities) Γ.

This view fitted well to insights which Weyl drew from recent developments in quantum physics. He insisted – differently to what he had thought in 1918 – on the consequence that “length is not relative but absolute” (Hs, p. 15). He argued that physical length measurements were no longer dependent on an arbitrary chosen unit, like in Euclidean geometry. An “absolute standard of length” could be fixed by the quantum mechanical laws of the atomic shell:

The atomic constants of charge and mass of the electron atomic constants and Planck’s quantum of action h, which enter the universal field laws of nature, fix an absolute standard of length, that through the wave lengths of spectral lines is made available for practical measurements.