Math 161 - Notes

Neil Donaldson

Spring 2021

1 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 (): 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 () formulated and proofs attempted, mving 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 .

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 (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 . 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’ .

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. ‘’ 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 α + β is half the angles in a triangle, namely a right-angle. The ancient Greek word θεωρεω (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 , 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 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 or .

2. /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 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 G and a ∗. 2. Axioms/Postulates: (A1) Closure: ∀a, b ∈ G, a ∗ b ∈ G. (A2) Associativity: ∀a, b, c ∈ G, a ∗ (b ∗ c) = (a ∗ b) ∗ c. (A3) Identity: ∃e ∈ G such that ∀a ∈ 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 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 .

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 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.

Consistency To demonstrate this, it is enough to provide a concrete model3 of an axiomatic system. An abstract model only demonstrates relative consistency, dependent on that of the underlying system; in most mathematical cases this is all that can be achieved. While an inconsistent system is essentially useless from a mathematical perspective, demonstrating the inconsistency of a system might be very challenging.

Independence An axiom can be seen to be independent by exhibiting two models; one in which all the axioms are true, the other in which only the considered axiom is false.

Completeness This is very unlikely to hold for most useful axiomatic systems in mathematics (see Godel’s¨ theorems below), though examples do exist. To show incompleteness, an undecidable4 statement is required; exhibit two models where the statement is true in one and false in the other. Alternatively, the undecidable can be treated as a new independent axiom of an enlarged axiomatic system.

3The above model of a monoid is strictly an abstract model in that the integers themselves are defined in terms of another axiomatic system. A concrete model is one that can be embedded in reality, where contradictions are assumed impossible(!!). Thus a single dot • on the page is a concrete model of a monoid when equipped with the operation • ∗ • = •. 4A famous example of an undecidable statement from standard is the , which states that there is no with strictly smaller than that of the real numbers. Both the hypothesis and its negation have been proven to be independent of the standard axioms of set theory.

5 Example 1.6. The axiomatic system for a monoid is:

Consistent We have a (concrete) model.

Independent Consider three models:

• (N, +) satisfies axioms A1 and A2 but not A3. • ({e, a, b}, ∗) defined by the following table satisfies A1 and A3 but not A2

∗ e a b e e a b e.g. a ∗ (b ∗ b) = a ∗ a = e 6= a = a ∗ b = (a ∗ b) ∗ b a a e a b b b a

• (Z \{1}, +) satisfies axioms A2 and A3 but not A1.

Incomplete The proposition ‘A monoid contains at least two elements’ is undecidable just from the axioms. For instance, ({0}, +) and (Z, +) are models with one/infinitely many elements.

Alternatively, we could ask if all elements have an inverse. That this is undecidable is the same as saying that the following axiom is independent of A1, A2, A3. (A4) Inverse: ∀g ∈ G, ∃g−1 ∈ G such that g ∗ g−1 = g−1 ∗ g = e.

The new system defined by the four axioms is also consistent and independent: this is the struc- ture of a group. Even this is incomplete however; consider a new axiom of commutativity.. .

G¨odel’s incompleteness theorems In 1931, the German logician Kurt Godel¨ proved two revolu- tionary results about axiomatic systems.

Theorem 1.7. 1. Any consistent system containing the natural numbers is incomplete.

2. The consistency of such a system cannot be proved within the system itself.

Godel’s¨ theorems have caused a great deal of anguish since we can’t do much without the natural numbers! He essentially says that there is no ultimate consistent complete axiomatic system; there will always be undecidables.5 Godel’s¨ second theorem fleshes out the difficulty in proving the con- sistency of an axiomatic system. If a system is sufficiently complex to describe the natural numbers, its consistency can at best be proved relative to some other axiomatic system. We are only scratching the surface of axiomatics here; many subtleties lie beyond the level of this course. Our goal is to cover just enough about axiomatic systems to understand what Euclid did and how he motivated later mathematicians. If you want a more thorough introduction to this topic, with many more examples, try the following site: http://web.mnstate.edu/peil/geometry/C1AxiomSystem/AxSysWorksheet.htm

If you really want to dive down the rabbit hole, consider taking a in formal or .

5Perhaps this is reassuring; mathematics will never be finished! However, the undecidables cooked by Godel¨ are ana- logues of the famous liar paradox: ‘This sentence is false.’ It is a matter of debate whether considering such is truly useful!

6 Exercises. 1.2.1. In the game of Nim, two players are given several piles of coins, each pile having a finite number of coins. On each turn a player picks a pile and removes as many coins as they want from that pile, as long as they remove at least one coin. The player who takes the last coin wins. Suppose that there are two piles where one pile has more coins than the other. Prove that the first player to move can always win the game. 1.2.2. Consider a system where children in a classroom choose different flavors of ice cream. Sup- pose we have the following axioms: (A1) There are exactly five flavors of ice cream: vanilla, chocolate, strawberry, cookie dough, and bubble gum. (A2) Given any two distinct flavors, there is exactly one child who likes these. (A3) Every child likes exactly two flavors of ice cream. (a) How many children are in the classroom? Prove your assertion. (b) Prove that any pair of children likes at most one common flavor. (c) Prove that for each flavor, there are exactly four children who like that flavor. 1.2.3. Consider an axiomatic system that consists of elements in a set S and a set P of pairings of elements (a, b) that satisfy the following axioms: (A1) If (a, b) is in P, then (b, a) is not in P. (A2) If (a, b) is in P and (b, c) is in P, then (a, c) is in P. (a) Let S = {1, 2, 3, 4} and P = {(1, 2), (2, 3), (1, 3)}. Is this a model for the axiomatic system? Why/why not? (b) Let S be the set of real numbers and let P consist of all pairs (x, y) where x < y. Is this a model for the system? Explain. (c) Use the results of (a) and (b) to argue that the axiomatic system is incomplete. I.e., think of another independent axiom that could be added to the axioms A1 and A2 for which S and P in part (a) is a model, but for which S and P from part (b) is not a model. 1.2.4. The undefined terms of an axiomatic system are ‘brewery’ and ‘beer’. Here are some axioms. (A1) Every brewery is a non-empty collection of at least two beers (every brewery brews at least two beers!). (A2) Any two distinct breweries have at most one beer in common. (A3) Every beer belongs to exactly three breweries. (A4) There exist exactly six breweries. (a) Prove the following theorems. i. There are exactly four beers. ii. There are exactly two beers in each brewery. iii. For each brewery, there is exactly one other brewery which has no beers in common. (b) Prove that the axioms are independent. (When negating A1, you should assume that a brewery is still a collection of beers, but that any such could contain none or one beer)

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