Mathematical Proof and the Principles of Mathematics/History/Euclid

About 300 BCE Euclid wrote The Elements, the most successful textbook of all time. Not only was it still being used 2000 years after it was written, but it had a huge influence, often in ways that Euclid could not have guessed, on the future development of mathematics.

The Elements
The majority of The Elements concerns geometry, though it also contains material on what we would call number theory. Geometry had been studied for thousands of years before Euclid, and while it may have contained some original material (we just don't know), it was certainly meant to be a summary of the more elementary results known at the time for students new to the subject.

What made The Elements different was its standard of rigor. Geometry in previous times was a collection of random facts, rules of thumb, and approximations. Euclid's approach was to start with a small number of "first principles". These consist of axioms, postulates and definitions which were to be accepted as true without question. There are differences between the three terms as Euclid used them, and we don't need to delve into the details, but definitions were intended to clarify the meaning of terms being used, while axioms and postulates where regarded as self-evident, either because they were a product of common sense or because they were obvious from a basic understanding of the universe. Starting with a firm foundation of first principles, further results were added using deductive logic, like stones cemented in place layer by layer to create a magnificent tower.

In hindsight we can see the flaws in Euclid's handiwork. But the ideal of using this method to create a body of knowledge of such scope, and yet with such certainty in every detail, remains. This axiomatic method is a characteristic of mathematics in particular, and the so-called formal sciences in general. In contrast, the scientific method, developing later and over a longer period of time, characterizes the so-called natural sciences.

Euclid's influences
Since very little is actually known about Euclid, it's difficult to say how The Elements came to be structured in the way it is. But there are two philosophers whose influence can be detected. The first is Plato with his theory of ideals. This states that every object in the real world is only an imperfect copy of a perfect version in the world of ideals. The points, lines and planes of Euclid's geometry were idealized versions of what exists in the real world. In the real world a field may be approximately flat and level, but it also has rocks, furrows, and possibly a rabbit hole or old stump in the corner. But a plane in Euclid's geometry is a perfect, flat and featureless version of this.

The structure of The Elements was almost certainly inspired by Aristotle's Organon. Aristotle saw the need for first principles to avoid circular reasoning. In addition, Euclid's use of the terms axiom, postulate and definition are almost exactly in accordance with Aristotle's. The logic Euclid used was much more flexible than that described by Aristotle, but, to be fair, Aristotle's logic wasn't up to the tasks that Euclid demanded.

The use of logical methods to establish geometrical results dates to Thales. As far as we know however, these were isolated theorems, not parts of a unified whole of the type created by Euclid. But his methods almost certainly had an influence on Plato and Aristotle and therefore were the seeds from which grew the Axiomatic method some centuries later.

Similarly, Hippocrates of Chios (not to be confused with Hippocrates the physician) is often credited with being the first to arrange the exposition of geometry into a logical order. Unfortunately, only fragments of this work still exist so it's impossible to tell how logically it was arranged or to what degree Euclid was influenced by it.

Another possible influence wasn't philosophical but cultural. Ptolemy Soter, the ruler of Alexandria at the time Euclid taught there, was very active in encouraging the spread and advancement of knowledge, and established the famed Library of Alexandria. (Historically, this attitude seems to be rather rare, and in general monarchs tend to regard the pursuit of knowledge for its own sake indifferently, though a few regarded it with open hostility.) Some have claimed that the existence of a leisure class and a general distaste at the time for practical matters also played a role.