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Axiomatic Systems Math 161 - Notes Neil Donaldson Spring 2021 1 Geometry and the Axiomatic Method 1.1 The Early Origins of Geometry: Thales and Pythagoras We begin with a condensed overview of geometric history. The word geometry itself comes from the ancient Greek terms geo (Earth), and metros (measure): indeed the ancient Greeks are the culture most associated with the development of geometry. Ancient times (pre-500 BC) Egypt, Babylonia, China, India: basic rules for measuring lengths, areas and volumes of simple shapes. Applications: surveying, tax collection, construction, religious practice, astronomy, navigation. Worked examples without general formulæ/abstraction. Ancient Greece (from c.600 BC) Philosophers such as Thales and Pythagoras began the process of ab- straction. General statements (theorems) formulated and proofs attempted, mving mathematics beyond the practical. Concurrent development of early scientific reasoning. Euclid of Alexandria (c.300 BC) Collected and developed earlier work, especially of the Pythagoreans in the Elements; one of the most important books in Western history and remaining a standard school textbook until the early 1900’s. Began the axiomatic method on which modern mathe- matics is based. Later Greek Geometry Archimedes’ (c.270–212 BC) work on area and volume included techniques sim- ilar to those of modern calculus. Ptolemy (c. AD 100–170) writes the Almagest, a treatise on astronomy which begins the study of trigonometry. Post-Greek Geometry During the European Dark Ages, geometric understanding was developed and enhanced by Indian and Islamic mathematicians who particularly developed trigonometry and algebra. Analytic Geometry The melding of algebra with geometry was completed by Descartes in mid-1600’s France, with the advent of co-ordinate systems. Modern Development Non-Euclidean geometries (e.g. hyperbolic geometry) help provide the mathe- matical foundation for Einstein’s relativity and the study of curvature. Following Klein (1872), modern geometry is highly dependent on group theory. Thales of Miletus (c.624–546 BC) Thales was an olive trader from Miletus, a city-state on the west coast of modern Turkey. Through his trades and travels he absorbed mathematical ideas from nearby cultures including Egypt and Babylon. Thales holds pride of place in mathematics as one of the first philosophers to state theorems. Here are five theorems, at least partly attributed to Thales; the last indeed is still known as Thales’ Theorem. 1. A circle is bisected by a diameter. 2. The base angles of an isosceles triangle are equal. 3. The pairs of angles formed by two intersecting lines are equal. 4. Two triangles are congruent if they have two angles and the included side equal. 5. An angle inscribed in a semicircle is a right angle. Thales’ proofs were not what we’d understand by the term, with his reasoning being partly example- based. The important developement was that his statements were abstract and general: e.g. any circle will be bisected by any of its diameters. α β α β The pictures show Thales’ Theorems 1, 2 and 5. ‘Arguments’ for Theorems 1 and 2 could be as simple as ‘fold.’ Theorem 5 follows from the observation that the radius of the circle splits the large triangle into two isosceles triangles: Theorem 2 says that these have equal base angles (labelled), now check that a + b is half the angles in a triangle, namely a right-angle. The ancient Greek word q#wr#w (theoreo) has several meanings: ‘to look at,’ ‘speculate,’ or ‘consider.’ These early theorems were essentially general principles which were supposed to be fairly clear by looking at the picture the right way! Pythagoras of Samos (570—495 BC) Pythagoras grew up on Samos, an island not far from Miletus in the Aegean sea. He also travelled widely, eventually settling in Croton, southern Italy, around 530 BC where he founded a philosophical school devoted to the study of number, music and ge- ometry. Amongst other mathematical contributions, it has been claimed that the Pythaogreans first classified the regular (Platonic) solids and developed the nusical relationship between the length of a vibrating string and its pitch. While it is difficult to verify such assertions, their obsession with number and the ‘music of the universe’ certainly inspired later mathematicians and philosophers— in particular Euclid, Plato and Aristotle—who believed they were refining and clarifying this earlier work. Indeed Book I of Euclid’s Elements seems to have been structured precisely to provide a rig- orous proof of what we now know of as Pythagoras’ Theorem, perhaps the most famous result in mathematics. 2 A simple proof of Pythagoras’ Theorem1 Much has been written about Pythagoras’ Theorem, including many, many proofs.2 It is often claimed that Pythagoras himself first proved the result, but this is generally accepted to be absurd given that the ‘proof’ most often attributed to the Pythagoreans is based on contradictory ideas about numbers. Other cultures’ frequent use of the result suggests that the statement was known several hundred years before Pythagoras; the earliest known reference comes from China where the state- ment is known as the gou gu. Any argument over attribution is somewhat moot, given that we first need to agree on what ‘proof’ means. As far as modern mathematics is concerned, this means that we need to spend some time considering Axiomatic Systems. Exercises. 1.1.1. Label the second picture above so that each triangle has side lengths a, b, c: now use algebra to give a simple proof of Pythagoras’ Theorem. 1.1.2. A theorem of Euclid states: The square on the parts equals the sum of the squares on each part plus twice the rectangle on the parts By referencing the above picture, state Euclid’s result using modern algebra. (Hint: let a and b be the ‘parts’. ) 1The second picture will animate if you click on it while open in a full-feature pdf reader like acrobat. 2Including a proof by former US President James Garfield: would that current presidents were so learned. 3 1.2 The Axiomatic Method It is believed that a primary motivation for Euclid’s writing of the first book of the Elements was to rigorously prove Pythagoras’ Theorem. In so doing, he essentially invented the axiomatic method. Before considering Euclid’s approach, we describe the revolutionary structure Euclid followed, that of a deductive or axiomatic system. Definition 1.1. An axiomatic system comprises four pieces: 1. Undefined terms: concepts accepted without definition/explanation. In the setting of basic ge- ometry these might include line or point. 2. Axioms/Postulates: logical statements regarding the undefined terms which are accepted with- out proof. For example: There exists a line joining any two given points.a 3. Defined terms: concepts defined in terms of 1 and 2. For instance a triangle could be defined in terms of three non-collinear points. 4. Theorems: logical statements deduced from 1–3. aIn Euclid, an axiom is considered to be somewhat more self-evident than a postulate, but the distinction is not impor- tant to us. For instance, one of Euclid’s axioms is, ‘Things which are equal to the same thing are also equal to one another,’ essentially the transitivity of equality. This feels more general than the postulate regarding a line joining two points. In the strictest sense of the word, a proof is an argument demonstrating the truth of a theorem within an axiomatic system. Given that the Pythagoreans did not follow such a system, they cannot really be said to have proved anything! Examples 1.2. We consider two simple examples of a deductive system: the first is a game—many such can be considered deductive systems—while the second is more traditionally mathematical. Chess 1. Undefined terms: Pieces (as black/white objects) and the board. 2. Axioms/Postulates: Rules for how each piece moves. 3. Defined terms: Concepts such as check, stalemate or en-passant. 4. Theorems: For example, Given a particular position, Black can win in 5 moves. Monoid 1. Undefined terms: A set G and a binary operation ∗. 2. Axioms/Postulates: (A1) Closure: 8a, b 2 G, a ∗ b 2 G. (A2) Associativity: 8a, b, c 2 G, a ∗ (b ∗ c) = (a ∗ b) ∗ c. (A3) Identity: 9e 2 G such that 8a 2 G we have a ∗ e = e ∗ a = a. 3. Defined terms: Concepts such as square a2 = a ∗ a, or commutativity a ∗ b = b ∗ a. 4. Theorems: For example, The identity is unique. Given that the concept of a set and that much notation is needed, one could argue that many more axioms are really required! 4 Definition 1.3. A model of an axiomatic system is an interpretation of the undefined terms such that all the axioms/postulates are true. Example 1.4. (G, ∗) = (Z, +) is a model of a monoid, where e = 0. The big idea of models and axiomatic systems is this: Any theorem proved within an axiomatic system is true in any model of that system. Mathematical discoveries often hinge on the realization that two seemingly separate discussions can be described in terms of models of a common axiomatic system. Properties of Axiomatic Systems Definition 1.5. Certain properties are desirable in an axiomatic system: Consistency The system is free of contradictions. Independence An axiom is independent if it is not a theorem of the others. An axiomatic system is independent if all its axioms are. Completeness Every proposition within the theory is decidable; can be proved or disproved. In practice, it is very unlikely that one has a system which satisfies all three desirable properties. We unpack slightly what is required to demonstrate or contradict each. By necessity, our descriptions are vague, since many notions need to be clarified (such as what is meant by a valid proposition within a theory) before these notions can be made rigorous.
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