How was Riemann's essay on the foundations of geometry misjudged by Russell?

Question

Jost's recent book on Riemann's famous 1854 essay claims in the introduction that Bertrand Russell misjudged Riemann's work. However, Jost does not elaborate. The index at the end of the book does not even contain an entry for Russell. What is known about the reception of Riemann by Russell?

Answer 1

Russell wrote a historico-philosophical Essay on the Foundations of Geometry (1897), which Jost cites on p.127 and which is freely available on Gutenberg. Russell's reception of Riemann is already clear from the section titles concerning him:

60. Riemann regarded space as a particular kind of manifold, i.e. wholly quantitatively
61. He therefore unduly neglected the qualitative adjectives of space
62. His philosophy rests on a vicious disjunction
63. His definition of a manifold is obscure
64. And his definition of measurement applies only to space
65. Though mathematically invaluable, his view of space as a manifold is philosophically misleading

Aside from young Russell being a little too hard on Riemann, what this makes clear is that his qualms were more philosophical than mathematical. It helps to remember that Riemann was a Kantian, he explicitly names Kant's successor at Königsberg, Herbart, as an influence in his lecture, the only other named influence is Gauss, see more in Which school of philosophy motivated thinking about spaces of higher dimension? And, as we know from Russell, "it was towards the end of 1898 that Moore and I rebelled against both Kant and Hegel", to found analytic philosophy.

The Riemann's "vicious disjunction" is that "either the axioms must be consequences of general conceptions of magnitude, he thinks, or else they can only be proved by experience". In Russell's opinion, we must ask instead "What axioms, i.e. what adjectives of space, must be presupposed, in order that quantitative comparison of the parts of space may be possible at all?" And not asking that, to Russell, was Riemann's cardinal sin:

His philosophy is chiefly vitiated, to my mind, by this fallacy, and by the uncritical assumption that a metrical coordinate systern can be set up independently of any axioms as to space-measurement. Riemann has failed to observe, what I have endeavoured to prove in the next chapter, that, unless space had a strictly constant measure of curvature, Geometry would become impossible; also that the absence of constant measure of curvature involves absolute position, which is an absurdity.

What is most ironic about this is that the soon to be anti-Kantian Russell is criticizing a Kantian Riemann for... not being Kantian enough. Namely, overlooking the constitutive principles that have to be adopted a priori for empirical measurements to become possible. The reason Russell puts such a premium on constant curvature is that only with that is transitive metric-preserving action possible, and hence measurement in his 1897 opinion. And that of course is a replication of Kant's similar thesis about the Euclidean geometry. Einstein will dispel such narrow construal of physical geometry soon enough, and credit Riemann's study of the foundations of geometry "in an even more profound way" for providing a language of general covariance in 1912, see How was Einstein led to make a contact with Differential Geometry for his theory of General Relativity?

Jost recommneded commentary on Russell on Riemann is Toretti's Philosophy of Geometry from Riemann to Poincaré, see also this year's Space, Number, and Geometry from Helmholtz to Cassirer by Biagioli.