Euclid's Elements

The Elements begins with definitions of geometric terms such as "point", "straight line", and "circle" (more definitions are introduced later on, when needed). Next there are five "Postulates" : statements about geometric terms that are assumed to be true without being proved. Most of these assert, or can be viewed as asserting, that some geometric construction is possible, in any case they are all assertions involving one or more of the geometric terms that have been defined. For example the first postulate states that a straight line can be drawn between any two points. The fifth postulate is the most interesting: and is known as the "parallel postulate". Euclid's wording of this is rather complicated, and an equivalent postulate, known as Playfair's Axiom is substituted. This says "Through a given point there is only one line parallel to a given line". After the five postulates come five "common notions", also to be accepted without proof. These are not phrased in geometric terms, but seem to be intended as generally valid principles of reasoning; for example the first common notion asserts "Things which are equal to the same thing are equal to each other". From this point onwards everything in the Elements is proved in a series of Propositions, 13 books of them, beginning with a proof that an equilateral triangle can be constructed using any given line as one side, and ending with a construction of the five platonic solids: the tetrahedron, cube, octahedron, icosahedron and dodecahedron. Each step in the proof of any proposition is justified by an appeal either to one of the postulates or common notions, or to a postulate that has been proved earlier on.

Euclid is not perfect: there are places where unconscious assumptions have been made, which cannot be justified on the basis of anything that has gone before, so that either more postulates are needed†It is possible, even likely, that some of the common notions were not present originally, and were added by later editors of Euclid's work who had noticed such an assumption. , or the wording of a postulate needs to be clarified. For example the first postulate should probably specify that there is exactly one straight line between two points, rather than saying just that there is a straight line, which could be taken as meaning that there is at least one, leaving open the possibility that there might be two or more. Yet the deficiencies are really only minor quibbles. Euclid presents a highly accurate and extremely useful theory of space. Any deviations of real space from Euclid's theory, at least in our region of the universe, are almost unmeasurably small.

Informal Axiomatic Mathematics

Today most branches of mathematics are usually presented in a similar way. A theory will begin by introducing some primitive concepts. At least some of these concepts must be left undefined, though an attempt may well be made to suggest some intended model or interpretation of the terms. Next some axioms are introduced. In everyday language axiom usually means "self-evident truth", and in Euclid's Elements the postulates were probably intended as such, although many later geometers felt that the fifth postulate was not self-evident, and should be proved. In mathematics today an axiom is somewhat different: it is a statement that is assumed to be true for the purposes of the theory under consideration. I shall say more about this in a moment. Once the axioms have been been stated, the theory will proceed by examining the consequences of the axioms. It might not be the case that the axiom is "true" in the "real world". Mathematicians can, and frequently do, construct theories, which seem to be about similar concepts, but which have contradictory axioms. What I mean by this, is not that the axioms of any one of them give rise to contradictions (such theories are usually considered useless), but that what is an axiom in one theory might be provably false in another theory.

Euclid again

Euclid's fifth postulate has been subjected to just this kind of treatment. As I already mentioned, many later geometers felt it should be possible to prove this postulate. Thomas Heath's edition of the Elements runs through many of these attempts, before coming to the first really significant work on the subject: almost two thousand years after Euclid, in 1733 (the year of the author's death) Gerolamo Saccheri's snappily entitled work "Euclides ab omni naevo vindicatus; sive conatus geometricus quo stabiluntur prima ipsa universae Geometriae Principia" attempted to put Euclid's elements on a firm foundation, proving the fifth postulate using "reductio ad absurdam" arguments. Saccheri formulated alternative axioms (or hypotheses) that contradict the parallel postulate, and he attempts to show that a theory based on these would lead to contradictions. His hypotheses concerned two angles of a certain quadrilateral, now known as a Saccheri quadrilateral in his honour, and his work is discussed in considerable detail in Roberto Bonola's history of non-Euclidean geometry. In brief, under Euclid's postulate these angles should be right angles. Assuming the angles were obtuse Saccheri obtained the required contradiction. On the assumption that the angles were acute Saccheri obtained a number of strange theorems, but he could not actually find a contradiction. Eventually he concluded that the theorems "are repugnant to the nature of a straight line".

Bonola says of Saccheri's work, that after this most determined attempt on behalf of the Fifth Postulate, the fact that no actual contradiction had been found in the alternative hypotheses of the acute angle could not help suggesting that it might in fact be possible to construct a consistent geometry using it. Over the next fifty years several mathematicians, including Gauss, concluded that it would be impossible to prove the parallel posutulate. The first two mathematicians, to actually publish accounts of the alternative geometry were John Bolyai (or Bolyai Janůs to give him his Hungarian name), and Nicholas Lobachevski. It should be noted that they did not really have any more right to assume this geometry was consistent than Saccheri did that it was not. What is now known, is that this geometry is as as consistent as Euclidean geometry - if it is contradictory then so is Euclidean geometry, and if Euclidean geometry is contradictory then so is this alternative geometry.

†This was first proved by Eugenio Beltrami.