Space and Mathematics

Abstract

In the question of the concept of space, even more evidently than in the question of the real numbers, the problem of the relation between mathematics and the so-called real world is posed. Newton formulated his position as follows: “Geometry is founded in practical mechanics, and indeed is no more than that part of mechanics as a whole, which originates in and is confirmed by the art of measurement.” Or Gonseth: “Geometry is the physics of arbitrary space.” The sense of geometry may therefore lie in finding a solid basis for the art of measurement (one that imposes a duty): mathematical consequences of axioms about space ought to be verifiable in actual surveying (hence comes the name of this science). Like physicists always we are then disposed to buy the ideal situation and to treat discrepancies as “incidental” and not “systematic” mistakes of measurement.