Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.
Axiomatic systems
Based on ancient Greek methods, an axiomatic system is a formal description of a way to establish the mathematical truth that flows from a fixed set of assumptions. Although applicable to any area of mathematics, geometry is the branch of elementary mathematics in which this method has most extensively been successfully applied.
There are several components of an axiomatic system. Primitives (undefined terms) are the most basic ideas. Typically they include objects and relationships. In geometry, the objects are things like points, lines and planes while a fundamental relationship is that of incidence – of one object meeting or joining with another. The terms themselves are undefined. Hilbert once remarked that instead of points, lines and planes one might just as well talk of tables, chairs and beer mugs. His point being that the primitive terms are just empty shells, place holders if you will, and have no intrinsic properties. Axioms (or postulates) are statements about these primitives; for example, any two points are together incident with just one line (i.e. that for any two points, there is just one line which passes through both of them). Axioms are assumed true, and not proven. They are the building blocks of geometric concepts, since they specify the properties that the primitives have. The laws of logic. The theoremsIn this context no distinction is made between different categories of theorems. Propositions, lemmas, corollaries, etc. are all treated the same. are the logical consequences of the axioms, that is, the statements that can be obtained from the axioms by using the laws of deductive logic.
An interpretation of an axiomatic system is some particular way of giving concrete meaning to the primitives of that system. If this association of meanings makes the axioms of the system true statements, then the interpretation is called a model of the system. In a model, all the theorems of the system are automatically true statements.
Properties of axiomatic systems In discussing axiomatic systems several properties are often focused on: The axioms of an axiomatic system are said to be consistent if no logical contradiction can be derived from them. Except in the simplest systems, consistency is a difficult property to establish in an axiomatic system. On the other hand, if a model exists for the axiomatic system, then any contradiction derivable in the system is also derivable in the model, and the axiomatic system is as consistent as any system in which the model belongs. This property (having a model) is referred to as relative consistency or model consistency. An axiom is called independent if it can not be proved or disproved from the other axioms of the axiomatic system. An axiomatic system is said to be independent if each of its axioms is independent. If a true statement is a logical consequence of an axiomatic system, then it will be a true statement in every model of that system. To prove that an axiom is independent of the remaining axioms of the system, it is sufficient to find two models of the remaining axioms, for which the axiom is a true statement in one and a false statement in the other. Independence is not always a desirable property from a pedagogical viewpoint. An axiomatic system is called complete if every statement expressible in the terms of the system is either provable or has a provable negation. Another way to state this is that no independent statement can be added to a complete axiomatic system which is consistent with axioms of that system. An axiomatic system is categorical if any two models of the system are isomorphic (essentially, there is only one model for the system). A categorical system is necessarily complete, but completeness does not imply categoricity. In some situations categoricity is not a desirable property, since categorical axiomatic systems can not be generalized. For instance, the value of the axiomatic system for group theory is that it is not categorical, so proving a result in group theory means that the result is valid in all the different models for group theory and one doesn't have to reprove the result in each of the non-isomorphic models.
Euclidean geometry
Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described (although non-rigorously by modern standards) in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.
For over two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other geometries which are not Euclidean are known, the first ones having been discovered in the early 19th century.
Euclid's Elements
Euclid's Elements is a mathematical and geometric treatise consisting of 13 books written by the ancient Greek mathematician Euclid in Alexandria . It is a collection of definitions, postulates (axioms), propositions (theorems and constructions), and mathematical proofs of the propositions. The thirteen books cover Euclidean geometry and the ancient Greek version of elementary number theory. With the exception of Autolycus' On the Moving Sphere, the Elements is one of the oldest extant Greek mathematical treatises, and it is the oldest extant axiomatic deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science.
Euclid's Elements has been referred to as the most successfulEncyclopedia of Ancient Greece (2006) by Nigel Guy Wilson, page 278. Published by Routledge Taylor and Francis Group. Quote:"Euclid's Elements subsequently became the basis of all mathematical education, not only in the Romand and Byzantine periods, but right down to the mid-20th century, and it could be argued that it is the most successful textbook ever written." and influential textbook ever written. Being first set in type in Venice in 1482, it is one of the very earliest mathematical works to be printed after the invention of the printing press and was estimated by Carl Benjamin Boyer to be second only to the Bible in the number of editions published, with the number reaching well over one thousand.The Historical Roots of Elementary Mathematics by Lucas Nicolaas Hendrik Bunt, Phillip S. Jones, Jack D. Bedient (1988), page 142. Dover publications. Quote:"the Elements became known to Western Europe via the Arabs and the Moors. There the Elements became the foundation of mathematical education. More than 1000 editions of the Elements are known. In all probability it is, next to the Bible, the most widely spread book in the civilization of the Western world." For centuries, when the quadrivium was included in the curriculum of all university students, knowledge of at least part of Euclid's Elements was required of all students. Not until the 20th century, by which time its content was universally taught through other school textbooks, did it cease to be considered something all educated people had read.From the introduction by Amit Hagar to Euclid and His Modern Rivals by Lewis Carroll (2009, Barnes & Noble) pg. xxviii: Geometry emerged as an indispensable part of the standard education of the English gentleman in the eighteenth century; by the Victorian period it was also becoming an important part of the education of artisans, children at Board Schools, colonial subjects and, to a rather lesser degree, women. ... The standard textbook for this purpose was none other than Euclid's The Elements.
The Elements are mainly a systematization of earlier knowledge of geometry. It is assumed that its superiority over earlier treatments was recognized, with the consequence that there was little interest in preserving the earlier ones, and they are now nearly all lost.
Books I–IV and VI discuss plane geometry. Many results about plane figures are proved, e.g., If a triangle has two equal angles, then the sides subtended by the angles are equal. The Pythagorean theorem is proved.Euclid, book I, proposition 47
Books V and VII–X deal with number theory, with numbers treated geometrically via their representation as line segments with various lengths. Notions such as prime numbers and rational and irrational numbers are introduced. The infinitude of prime numbers is proved.
Books XI–XIII concern solid geometry. A typical result is the 1:3 ratio between the volume of a cone and a cylinder with the same height and base.
Near the beginning of the first book of the Elements, Euclid gives five postulates (axioms) for plane geometry, stated in terms of constructions (as translated by Thomas Heath):
Although Euclid's statement of the postulates only explicitly asserts the existence of the constructions, they are also assumed to produce unique objects.
The success of the Elements is due primarily to its logical presentation of most of the mathematical knowledge available to Euclid. Much of the material is not original to him, although many of the proofs are supposedly his. Euclid's systematic development of his subject, from a small set of axioms to deep results, and the consistency of his approach throughout the Elements, encouraged its use as a textbook for about 2,000 years. The Elements still influences modern geometry books. Further, its logical axiomatic approach and rigorous proofs remain the cornerstone of mathematics.
A critique of Euclid The standards of mathematical rigor have changed since Euclid wrote the Elements. Modern attitudes towards, and viewpoints of, an axiomatic system can make it appear that Euclid was in some way sloppy or careless in his approach to the subject, but this is an ahistorical illusion. It is only after the foundations were being carefully examined in response to the introduction of non-Euclidean geometry that what we now consider flaws began to emerge. Mathematician and historian W. W. Rouse Ball put these criticisms in perspective, remarking that "the fact that for two thousand years [the Elements] was the usual text-book on the subject raises a strong presumption that it is not unsuitable for that purpose."
Some of the main issues with Euclid's presentation are: Lack of recognition of the concept of primitive terms, objects and notions that must be left undefined in the development of an axiomatic system. The use of superposition in some proofs without there being an axiomatic justification of this method.