Note 77 — Introductory material

As regards the first point, it is to be observed that people certainly had some conception of space before Riemann invented the notion of a manifold, and that this conception was certainly something other than the common qualities of all the points, lines or figures in space. In the second place, Erdmann's view would make it impossible to conceive God, unless one were a polytheist, or the universe—unless, like Leibnitz, one imagined a series of possible worlds, set over against God, and none of them, therefore, a true Universe—or, to take an instance more likely to appeal to an empiricist, the necessarily unique centre of mass of the material universe. Any universal, in short, which is a bond or unity between things, and not merely a common property among independent objects, becomes impossible on Erdmann's view. We cannot, therefore, unless we adopt Mill's philosophy intact, regard the conception of space as demanding a series of instances from which to abstract. But even if we did so regard it, Riemann's manifolds would leave us without resources. For Euclidean space still appears as unique, at the end of his series of determinations. We have instances of manifolds, but not instances of Euclidean space. Thus if Erdmann's theory of conceptions were correct, he would still be left searching in vain for the conception of Euclidean space.