Therefore, either the reality on which our space is based must form a discrete manifold or else the reason for the metric relationships must be sought for, externally, in the binding forces acting on it.

It is well known that geometry presupposes not only the concept of space but also the first fundamental notions for constructions in space as given in advance. It only gives nominal definitions for them, while the essential means of determining them appear in the form of axioms. The relationship of these presumptions is left in the dark; one sees neither whether and in how far their connection is necessary, nor a priori whether it is possible.

From Euclid to Legendre, to name the most renowned of modern writers on geometry, this darkness has been lifted neither by the mathematicians nor the philosophers who have laboured upon it.

*On the hypotheses which lie at the foundation of geometry* (1854)

If only I had the theorems! Then I should find the proofs easily enough.

Quoted in I Lakatos, Proofs and refutations (Cambridge, 1976).

JOC/EFR December 2013

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Quotations/Riemann.html