Geometry and Space

Abstract

Euclid’s Elements, written c. 300 bc, is a rich and complicated work, which can be read in various ways. Its surprisingly logical structure, with its opening assumptions, basic definitions and elaborate network of deductions of many theorems, invites one to treat it as a purely mathematical or logical work. The assumptions and definitions themselves mostly codify what one tends to believe about simple, geometrical figures in physical space. Taken together, these two aspects ensure that the work presents many theorems in a way which suggests they are true statements about the world we inhabit. Indeed, it can seem that the veracity of these statements is a matter of logic, and not at all the result of an early exercise in mathematical modelling. Nonetheless, it was clear to at least some Greek mathematicians and philosophers that their study of geometry was based on some explicit hypotheses about space. For example, Aristotle presciently remarked in his Physica (II 9 200a 16–19) that: If the line is what we recognise it to be from our visual intuition, then the angle sum of a triangle is two right angles.