The first weak point was detected almost immediately. It is even believed that Euclid himself regarded the fifth postulate as different from the other four, in that it was not as self-evident. He was probably forced to add it when he realized that certain propositions towards the end of Book I of the Elements could not be proved without it. Throughout history, most scholars who attempted to remedy this situation followed one of the following two strategies: i) they struggled to “prove” the fifth postulate using only the first four; ii) they attempted to introduce a new postulate that seemed more self-evident than the fifth, and from which, in addition to the other four, they could derive it. Ptolemy (100 A.D.-170 A.D., Alexandria), Proclus (410 A.D.-485 A.D., Athens), Ibn al-Haytham (965 A.D.-1039 A.D., Cairo), Nasir al-Din al-Tusi (1201 A.D.-1274 A.D., Persia), Sadr al-Din (son of Nasir al-Din al-Tusi), Giordano Vitale (1610 A.D.-1711 A.D., Italy), Girolamo Saccheri (1667 A.D.-1733 A.D., Italy), Johann Lambert (1728 A.D.-1777 A.D., Switzerland), are the most eminent followers of the first approach. Omar Khayyám (1050 A.D.-1123 A.D., Persia), John Playfair (1748 A.D.-1819 A.D., Scotland) are among the most famous mathematicians who adopted the second approach.
The “proofs” provided by the ones who followed approach i) were subsequently shown to be wrong, usually because their authors had unconsciously used an “obvious” fact which turned out to be equivalent to the fifth postulate. This makes their arguments ultimately dependant on the fifth postulate itself.
Followers of approach ii) never succeeded in finding a postulate as self-evident as the first four from which they could derive Euclid’s fifth axiom. Many authors did find postulates with this property, but as non self-evident as the fifth.
This state of affairs changed abruptly in the first half of the nineteenth century with the independent realization by Gauss (1817), Lobachevsky (1829), and Bolyai (1831), of the existence of geometries satisfying the first four postulates but not satisfying the fifth. The existence of such geometries constitutes irrefutable proof that the fifth postulate cannot be derived from the first four, in other words, that the fifth postulate is independent from the other axioms. This is considered one of the most important scientific discoveries of all time, having a profound impact in our understanding of how the human mind apprehends reality. In particular, it made evident the distinction between formal discourse (theory) and the objects it intends to describe (models), starting the development of one of the central branches of mathematical logic, known today as Model Theory. The discovery of non-euclidean geometries, together with the work of Gauss on curved surfaces, initiated a process, mainly led by the great german mathematician Bernhard Riemann, that vastly generalized the subject of geometry, by defining Riemannian Manifolds, and regarding them as the central object of study in geometry. This development constituted the mathematical framework for the formulation of Einstein’s Theory of General Relativity where Space-Time is actually conceived as a Pseudo-Riemannian Manifold, a slight variation of Riemann’s original concept.
Let us now talk about the other weak point found in Euclid’s work, namely the occasional departure from modern standards of rigor, and even from his own standards. This criticism started with the revision of the foundations of geometry motivated by the discovery of non-euclidean geometries. The criticism was centered around the following issues:
1. Lack of recognition of the necessity of having primitive terms, i.e., objects and notions that must be left undefined.
2. The use of the “superposition method” without any axioms backing it up.
3. Lack of a concept of continuity needed to prove the existence of some points and lines that Euclid constructs. This happens already when proving Proposition 1 of Book I!
4. Lack of clarity on whether a straight line is infinite or boundaryless in the second postulate.
5. Lack of the concept of betweenness, making some arguments depend on the figure.
Different authors have found different ways to remedy this situation. Like David Hilbert, by rigorously filling in the gaps in Euclid’s work; some others, like George David Birkhoff, by entirely remodelling the theory, formulating axioms around different concepts.
Let us consider Hilbert’s approach. In 1899 Hilbert published his book “Grundlagen der Geometrie” (The Foundations of Geometry). In this book, he proposes an axiomatic system for solid geometry, one from which every theorem can be derived by following a strict sequence of rules of inference, starting from a fixed set of formal assumptions stripped of any intuitive content. For Hilbert, relying on figures, using any intuitions about the nature of geometric objects, or introducing any extra assumptions lying beyond the strict syntactical concepts, is completely ruled out. The book has figures, but they are only used as a heuristic guide, and could be dispensed of without affecting the content of the book. Grundlagen der Geometrie presented geometry for the first time in history, in a purely formal way, i.e., in which the meaning given to the objects in question plays no role whatsoever. The only place where intuition plays a role is in the choice of the axioms themselves. Once the axioms are chosen, the original meaning of the objects can be forgotten without compromising in the least the development of the theory. It can be said that Hilbert presents solid geometry so that it can be understood by lawyers (no offense intended), in that it is not necessary to associate geometrical images to the discourse, because the discourse is authentically independent of any interpretation. Hilbert presented his axiomatic system in groups of axioms, each group concerning an aspect of solid geometry. Although Hilbert’s axioms formalize solid geometry, it is possible to extract from it a subset of axioms for plane geometry. The first group is formed by eight axioms, the so called Axioms of Incidence, which capture the laws governing the incidence relations between points, lines and planes in space. Only three of them are necessary for doing plane geometry. The second group is formed by four axioms called Axioms of Order (or Betweenness). These govern the behaviour of the intuitive notion that a point lies between two other points. The four of them are necessary for doing plane geometry. The third group is formed by six axioms, the Axioms of Congruence, which capture the laws governing the behaviour of congruence of segments and congruence of angles. These six axioms are necessary for developing plane geometry.
For Hilbert’s program, the main goal is not only the formalization of Euclidean geometry, but of mathematics as a whole. For him, the most important problem of all mathematics was the foundation of mathematics itself on a solid basis. This meant to Hilbert to reconstruct his own discipline as a purely formal science. This is known as Hilbert’s Formalization Program. He dreamed of presenting all of Mathematics in the same way he had presented Solid Geometry. Any formal system, as Hilbert envisioned it, must have two fundamental properties: Consistency and Completeness. Consistency means that it is not possible to derive within the theory some statement P and its negation. Completeness, on the other hand, means that for each statement P expressible within the system, either P or its negation ¬P can always be derived. Consequently, a formal system is Consistent and Complete if for each statement P expressible within the system, either P or ¬P can be derived from the axioms, but not both. In his Grundlagen der Geometrie, Hilbert proves both the consistency of this axiomatization, and the nonredundancy of the axioms, by constructing models of his system.
1.2 What you will and will not learn in this book
Although this is a book about plane geometry, it only contains very basic results. The most sophisticated results appear in the last chapter, in which many of the propositions of Book I of Euclid’s Elements are proved. For example, you will not find any mention of the Pythagorean theorem. The emphasis is in the rigorous development of the material, following Hilbert’s axiomatic system. Many results are presented which are “intuitively obvious”, and whose proofs are rather involved, pointing out the price one has to pay for deriving everything from the axioms through pure reasoning. In this book you will also learn about plane non-euclidean geometry from the very beginning. This is made possible by the method adopted of thinking of plane geometries as set theoretical structures in which a certain collections of axioms hold.