A point has no physical significance

A quotation I want to keep hold of:

I maintain that the mathematical concept of a point in a continuum has no direct physical significance. It has no meaning to say the value of a coordinate x... has a value x = sqrt(2) in. or x = π cm. [...]

Modern physics has achieved its greatest successes by applying the methodological principle that concepts which refer to distinctions beyond possible experience have no physical meaning and ought to be eliminated … The most glaringly successful cases are Einstein's foundation of relativity based on the rejection of the concept of aether ... and Heisenberg's foundation of quantum mechanics .. I think that this principle should be applied also to the idea of physical continuity

Max Born, quoted in Brillouin (1962, p303). Though I'm wondering whether this is any more true of irrational than rational numbers. A coordinate of 2 cm is just as much a point in a continuum as π cm, is it not?

Brillouin, L.

Science and Information Theory

Academic Press, 2nd ed. 1962