Weyl and the Problem of Space

This book provides a good, and varied, set of introductions to the subject of how Hermann Weyl analysed the problem of the nature of space, and to the considerable literature that already exists on the subject. It will not be possible to do the fifteen essays equal justice, so I merely note that there are four essays on the broad intellectual and philosophical context in which Weyl worked, and two on Weyl’s ideas about intuitionism and infinitesimals (these connect to his work in mathematical physics through his ideas about the continuum and epistemology). These leaves nine essays on Weyl’s deep involvement with the general theory of relativity and the issues arising from it. I shall try to indicate the content of these essays by tying them to their place in the development of Weyl’s thought.

The space of Newtonian mechanics is a bare stage in which the characters (particles and solid bodies) interact with each other, by impact or, more mysteriously, by forces acting at a distance; space itself is inert and exerts no effect on the particles. We in the audience impose Cartesian rectangular coordinates at will, and convert observations from one coordinate system to another by global, linear transformations that are isometries with respect to the familiar Euclidean metric. In many ways, the space of special relativity (Minkowski’s space-time) is little different. There is now an indefinite metric, but coordinate transformations between observers are linear and act globally; space-time is still inert.

The universe of general relativity is very different, even if empty space is taken to be a four-dimensional manifold isomorphic to \(\mathbb R^4\) as a vector space. Coordinate transformations (diffeomeorphisms) act locally, only the tangent spaces at each point have the structure of Minkowski’s space-time; space acts on the objects in it — that is the core of the new theory of gravity. What is it to do geometry in such a space? And, if you think you have sorted that out, what is it to do physics? These are deep problems, and might perhaps have come up in pure mathematics first, before physics seemed to need them, but they did not, and the result was a great jumble of ideas, in which few if any of the best mathematicians and physicists of the time made no serious mistakes. Prominent among them was Hermann Weyl.

In 1916, at the age of 31, Weyl resumed his Professorship at the ETH in Zürich after performing his military service. Among other important tasks, he decided to rewrite Einstein’s theory in the most up-to-date form of differential geometry that he could manage, thus tackling the first of the deep problems just mentioned. This was already a major achievement, as the revisions of the five editions of his Raum Zeit Materie from 1918 to 1923 demonstrate. There was, for example, no book on four-dimensional differential geometry at the time, and much of the mathematical work that had been done on Riemannian geometry was a heavily algebraic study of tensors (Einstein’s word) and their transformations. But the problem of doing physics in the new setting remained, and Weyl tackled that too.

The problems that chiefly occupied Weyl are not to do with rewriting the physical laws of flat space-time in the new setting and solving the equations that arise. Plain rewriting was not difficult. Einstein had, of course, already begun to do so, and Weyl could start from there. Rather, Weyl took up the problems of interpreting physics conceptually using the new mathematical language. So, components of the metric tensor are now interpreted as “physical phase quantities, to which there corresponds something real, namely, the ‘metrical field’.” Physical laws should be expressed in tensorial terms, thus ensuring their invariance under coordinate transformations. This makes the role of linear and affine connections (introduced by Levi-Civita in 1917 and Weyl, independently) hard to understand, if the connection coefficients are also to have physical meaning.

In his Raum Zeit Materie, Weyl first discussed the problem of how matter determines the gravitational field, and therefore of solving simple cases of Einstein’s field equations. Then he turned to cosmological considerations, and the finiteness of the universe. Then, as is well-known, he sought to unify gravitational and electro-magnetic forces (at the time, the only known forces of nature) in a single theory. This work is also the origin of the gauge-theoretic approach to physics. He was led to make a significant criticism of Riemann’s theory of geometry, because it seemed to him that there was no a priori reason to assume that the length of a vector was invariant under parallel translations (any more than direction is). Einstein objected that such a theory would contradict the evidence of spectral lines from distant stars, and Weyl retreated, but this hints at the question of what one means by such a basic quantity as length in the new theory.

Weyl profoundly hoped that general relativity, properly understood, was not just a good working theory but would be, in a sense to be elucidated, the correct theory, perhaps by being provably the only kind of theory than can deliver the knowledge we have. In the 1760s Immanuel Kant had argued powerfully that the Euclidean geometry was imposed necessarily by the mind on our understanding of raw experience. In his terminology, it was a synthetic a priori form of knowledge — synthetic because it enabled us to know about the world, but a priori because it was necessary if knowledge is to be possible at all (a tricky combination to maintain). Weyl rejected this idea, not least on the grounds that the discovery of non-Euclidean geometry made it no longer convincing, but he wanted good reasons to believe that some form of differential geometry had the chance of standing out so far from all the rest that it could be taken as a new a priori standard. Here, he focussed on the intuitive idea that a metric in geometry is best understood as a quadratic form, and made a profound study of the groups for which a given form is the isometry group.

This formulation of the problem of space is analysed here by Erhard Scholz, who has written extensively on the topic. He shows here that for a time Weyl was satisfied with the result that there is a unique affine connection consistent with a Pythagorean metric. This allowed him to claim that even if the metric field varies from point to point because of the presence of matter, space itself is still homogeneous, i.e. the same at every point. (This question had been considered for space by Helmholtz and Lie.) However, the discovery of the Dirac spinor fields in the 1930s challenged this formulation, and Scholz concludes that ultimately Cartan’s alternative approach via connections with torsion has led to a theory of gravity that may offer a better analysis of the now-unresolved question of the homogeneity of space.

Bernard’s analysis of this question, formulated as the plasticine ball argument (pba), looks at how Weyl showed that too-rigid a definition of how matter determines the metric would show that no motion was possible. The diffeomorphism invariance of space-time might seem to accommodate all change, so Weyl sought a physical interpretation of diffeomorphisms, and the pba asks if, when space (the plasticine) is squashed, has anything moved? Weyl reformulated this problem repeatedly, and Bernard shows that Weyl’s resolution of it ultimately relied (after several earlier formulations) on the recognition that the Einstein field equations only make good sense when given boundary conditions. This means that matter does not uniquely determine the metric, and the pba problem is dissolved, although subtleties in the interpretation of diffeomorphisms remained.

Silvia De Bianchi analyses the homogeneity question with particular emphasis on Weyl’s investigations towards the end of his life on why space-time has four dimensions. She shows that Weyl did not reach a definitive answer to this question, but became optimistic that a further analysis of how topological considerations might determine the kind of physical laws that we have might provide an answer. In her essay, Francesca Biagoli looks at the reception of Weyl’s various ideas, with a focus on his views on the subjective nature of cognition. (This ties to Weyl’s ideas on the continuum discussed elsewhere in this book.)

The nature of space is, of course, still a major question today. Alexander Afriat supplies a historically sensitive but modern reworking of Weyl’s ideas, carefully sorting out the groups involved and their inter-relations, up to the advent of spinors and Yang-Mills, and concentrating how Weyl tried to find a place for electricity in a unified theory of physics. Luciano Boi’s essay covers some of the same ground, but goes further in indicating how they play into modern gauge-theoretic interpretations of physics.

Three final essays look, as some of the opening ones did, at Weyl’s philosophical ideas and especially his involvement with Husserlian phenomenology. The term can have smart, practical people running for cover, but the questions at stake cannot be avoided if you want to be able to say that physics is correct or true, because they are questions about how we know about the world. If you want to get beyond the idea that equations seem to describe and predict, and ask how or why they do, you are asking about what lies beyond the formal descriptions (Weyl’s realm of symbolic constructions) and justifies them, and how we might know about it.

The idea that our being in the world has some connection with how we know about it cannot be easily shaken off, and once accepted requires careful thought. Weyl thought that there was subjective knowledge (each individual’s experience of the world, which is absolute and personal) which was converted into inter-subjective or objective knowledge, often in a mathematised form. The interplay of these two forms of knowledge constituted our intellectual place in the world, as Pierre Kerszberg explains. Jario José de Silva shows how this took Weyl towards Husserlian ideas before he finally retreated somewhat from them, and Benoit Timmermans examines the role of the possible in Weyl’s thought. This includes the domain of mathematical or symbolic constructions and the idea that scientific knowledge is by nature an ongoing process, never to be completed.

It is clear that throughout his life Weyl engaged deeply with questions in mathematics, physics, and the philosophy of mathematics and science. The essays here provide very useful introductions to a great deal of what he wrote about them.

Jeremy Gray is Professor Emeritus, Department of Mathematics and Statistics, The Open University, Milton Keynes, MK7 6AA, U.K.