My introduction to Euclid's Elements: Book I
This is my first experience with the formal study of mathematics in an attempt to fundamentally understand ideas that feel so dim to me. As I understand now, in Book I, Euclid introduces three sets of foundational statements that serve as the basis for the geometric principles and proofs that follow.
Definitions are the first set of foundational statements that clarify the meanings of fundamental geometric terms. This helped me understand the rules behind what seemed previously like obvious conclusions. It struck me that a figure is essentially a description of a set of rules. For example, of the trilateral figures, an equilateral triangle has three equal sides; an isosceles triangle has two equal sides; and a scalene triangle has three unequal sides. However you adjust their lengths, they will always be descriptions of these rules.
Postulates are assumed truths within the framework of geometry, and ensure logical consistency in the system. Something to keep in mind, though, is that these postulates are true in the context of geometry, but are not universal truths. The most interesting of all five is the fifth, which states: If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which the angles are less than two right angles. This sucker drove mathematicians mad for centuries because they thought it was too complex to be an axiom, but possibly a theorem that could be derived from the other four. This led to the discovery of non-Euclidean geometry and other systems.
Common notions were introduced last, and are assumed truths within the framework of mathematics. These rules, together with the postulates, are used to develop and prove geometric theorems. It crossed my mind that these notions were similar to the ‘laws of thought’, although it later became clear that they differ in scope and application.
- Definitions clarify the meanings of fundamental geometric terms.
- Postulates are assumed truths within the framework of geometry.
- Common notions are assumed truths within the framework of mathematics.