Friday, May 21, 2021

Euclid Book 1 Props I-VIII - a foundation for geometry | Sociology and Pure Maths

Modern pure mathematics is based largely on the historically vital example of Euclid, in particular the first Books of his classic work The Elements. Even non-mathematical people can gain an understanding of the logical orientation of the subject by looking carefully at some of the early Propositions in Book I, which is what we do here. We look at Props I --- V, and explain what Euclid is trying to do and how he sets out doing this.

Constructions play an important role, using both the straight-edge (a ruler without markings) and a compass. However there are also purely theoretical results, for example the first Theorem of Prop IV, which gives the famous side-angle-side congruence condition, but stated in a rather laborious way.

In order to understand this, we will have to discuss the role of angle in Euclidean geometry, which notably did not involve the Babylonian angle measurement system (or any other angle measurement system).

And then we are in a position to ask some serious questions about the logical validity of Euclid, especially when regarded as a foundation for modern geometry. Could it be that Euclid was really more of an applied mathematician??


We look at Propositions VI to VIII of Book 1 of Euclid's Elements, perhaps the first place where proofs by contradiction arise in mathematics. The proofs are not entirely transparent however, and a reasonable question arises as to the suitability of Euclid as a foundation for modern geometry. Is this being discussed? If a major branch of mathematics has historical foundations that are no longer suitable for modern times, are we able to speak up and say so? What are the alternatives?

No comments:

Post a Comment