Note 86 — Introductory material

Lotze opens with a somewhat startling statement, which, though philosophically worthy to be true, does not appear to be historically borne out. Euclidean Geometry has been chiefly shaken, he says, by the Kantian notion of the exclusive subjectivity of space—if space is only our private form of intuition, to which there exists no analogue in the objective world, then other beings may have other spaces, without supposing any difference in the world which they arrange in these spaces (p. 233). This certainly seems a legitimate deduction from the subjectivity of space, which, so far from establishing the universal validity of Euclid, establishes his validity only after an empirical investigation of the nature of space as intuited by Tom, Dick or Harry. But as a matter of fact, those who have done most to further non-Euclidean Geometry—with the exception of Riemann, who was a disciple of Herbart—have usually inherited from Newton a naïve realism as regards absolute space. I might instance the passage quoted from Bolyai in Chapter I., or Clifford, who seems to have thought that we actually see the images of things on the retina[101], or again Helmholtz's belief in the dependence of Geometry on the behaviour of rigid bodies. This belief led to the view that Geometry, like Physics, is an experimental science, in which objective truth can be attained, it is true, but only by empirical methods. However, Lotze's ground for uncertainty about Euclid is a philosophically tenable ground, and it will be instructive to observe the various possibilities which arise from it.