Math 161 - Notes
Neil Donaldson
Spring 2024
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 comes from the an-
cient Greek geo (Earth), and metros (measure). Measurement (of distance, area, height, angle) had
obvious practical benefits with regard to construction, taxation, commerce and navigation. Astron-
omy provided a related cultural driver of ancient geometry.
Ancient times (pre-500 BC) Egypt, Mesopotamia, China, India: basic rules for measuring lengths,
areas and volumes of simple shapes. Applications: surveying, tax collection, construction,
religious practice, astronomy, navigation. Typically worked examples without general for-
mulæ/abstraction.
Ancient Greece (from c. 600 BC) Philosophers such as Thales and Pythagoras began the process of
abstraction. General statements (theorems) formulated and proofs attempted. Concurrent de-
velopment of early scientific reasoning.
Euclid of Alexandria (c. 300 BC) Collected and expanded earlier work, especially that of the Pytha-
goreans. His compendium the Elements is one of the most important books in Western history
and remained a standard school textbook in to the 1900’s. The Elements is an early exemplar of
the axiomatic method at the heart of modern mathematics.
Later Greek Geometry Archimedes’ (c. 270–212 BC) work on area and volume included techniques
similar to those of modern calculus. Ptolemy (c. AD 100–170) writes the Almagest, a treatise on
astronomy which covers the foundations 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 In mid 1600’s France, Descartes and Fermat melded algebra with geometry with
the advent of co-ordinate systems (axes).
Modern Development Non-Euclidean geometries help provide the mathematical foundation for
Einstein’s relativity and the study of curvature. Following Klein (1872), modern geometry is
highly dependent on group theory.
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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 trading and travelling, he absorbed mathematical ideas from
nearby cultures including Egypt and Mesopotamia. Here are five results partly attributable to 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.
The last is still known as Thales’ Theorem. Thales’ arguments were not rigorous by modern standards.
His real innovation was to state abstract, general principles: any circle is bisected by any of its diameters.
The Greek word θεωρεω (theoreo), from which we get theorem, has several meanings: ‘to look at,’
‘speculate,’ or ‘consider.’ Thales’ results were supposed to be clear just by looking at a picture.
β
α
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 α + β is half the angles in a triangle, namely a right-angle.
Pythagoras of Samos (570—495 BC) Pythagoras grew up on Samos, an island in the Aegean Sea
not far from Miletus. 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. It has been claimed that the Pythagoreans first classified the regular (Platonic) solids and
developed the musical relationship between the length of a vibrating string and its pitch. While it
is difficult to verify such assertions, the Pythagorean obsession with number and the ‘music of the
universe’ certainly inspired later mathematicians and philosophers—particularly Euclid, Plato and
Aristotle—who believed they were refining and clarifying this earlier work.
Of course, Pythagoras is best known for the result that bears his name.
Theorem 1.1 (Pythagoras). The square on the hypotenuse of a right triangle equals the sum of the
squares on the remaining sides.
Two important clarifications are needed for modern readers.
1. By square, the Greeks meant an honest square! There is no algebra, no numerical lengths, and
the equation a
2
+ b
2
= c
2
won’t be seen for another 2000 years.
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2. The word equals means equal area, though without a numerical concept of such. The Greeks
meant that the large square can be subdivided into pieces which may be rearranged to produce
the two squares on the remaining sides.
The result suddenly seems less easy! The pictures below provide a simple visualization.
A simple proof of Pythagoras’ Theorem
The problem with this ‘proof’ is that it relies on subtracting the areas of the four congruent triangles
from a very large square, whereas the Greeks idea of area was essentially additive. Book I of Euclid’s
Elements seems to have been structured precisely to correct this and provide a rigorous constructive
proof. Indeed it is possible (though very ugly!) to apply the 47 results leading up to and including his
proof of Pythagoras, in such as way as to explicitly subdivide the hypotenuse square and rearrange
it into the two smaller squares as required.
Much has been written about Pythagoras’ Theorem, including many, many proofs.
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It is often
claimed that Pythagoras himself first proved the result, but this is generally considered incorrect—
the ‘proof’ most often attributed to the Pythagoreans is based on contradictory ideas about numbers
which were debunked by the time of Aristotle. Moreover, other cultures, particularly ancient China,
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are known to have used the result, at least in example form, several hundred years before Pythago-
ras. Regardless, an argument over attribution is fruitless without first agreeing on what constitutes a
proof. This means that we need to spend some time considering Axiomatic Systems. . .
Exercises 1.1. 1. Let the side-lengths of the above triangles be a, b, c. Can you rephrase the proof
algebraically?
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’. . . )
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Including a proof by former US President James Garfield. Would that current presidents were so learned. . .
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In China, Pythagoras’ Theorem is known as the gou gu, which refers to the two non-hypotenuse sides of the triangle.
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1.2 Axiomatic Systems
Arguably the most revolutionary aspect of the Euclid’s Elements was its axiomatic presentation.
Definition 1.2. An axiomatic system comprises four types of object.
1. Undefined terms: Concepts accepted without definition/explanation.
2. Axioms: Logical statements regarding the undefined terms which are accepted without proof.
3. Defined terms: Concepts defined in terms of 1 & 2.
4. Theorems: Logical statements deduced from 1–3.
Examples 1.3. Here are two systems viewed informally in this framework. In each case we provide
only examples of each type of object, not a full description of an axiomatic system.
Basic Geometry 1. Line and point.
2. There exists a line joining any two given points.
3. A triangle may be defined in using three non-collinear points.
4. Thales’ and Pythagoras’ Theorems.
Chess 1. Pieces (as black/white objects) and the board.
2. Rules for how each piece moves.
3. Concepts such as check, stalemate or en-passant.
4. For example, Given a particular position, Black can win in 5 moves.
A proof is a logical argument demonstrating the truth of a theorem within an axiomatic system. In
practice, this is an ideal to which we aspire, and a proof is simply a convincing logical argument.
Definition 1.4. A model is a choice/definition of the undefined terms such that all axioms are true.
Models are often abstract in that they depend on another axiomatic system. In a concrete model, the
undefined terms are real-world objects (where contradictions are impossible(!)). The big idea 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 seemingly separate discussions can be
described in terms of models of a common axiomatic system.
Example 1.5. Monoid: If you’ve studied group theory, this should seem familiar.
1. A set G and a binary operation ∗.
2. (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, a ∗ e = e ∗ a = a
3. Concepts such as square a
2
= a ∗ a, or commutativity a ∗ b = b ∗ a.
4. For example, The identity is unique.
(G, ∗) = (Z, +) is an abstract model, where e = 0. If you really want a concrete model, consider a
single dot • on the page, equipped with the operation • ∗ • = •!
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Definition 1.6. 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 valid proposition within the theory is decidable: can be proved or disproved.
We unpack these ideas slightly. By necessity, our descriptions are vague. Many notions need to be
clarified (e.g., what is meant by a valid proposition) before these ideas can be made rigorous.
Consistency May be demonstrated by exhibiting a concrete model. An abstract model demonstrates
relative consistency, dependent on the consistency of the underlying system. An inconsistent
system is essentially useless.
Independence To demonstrate the independence of an axiom, exhibit two models: one in which all
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, though
examples do exist. To show incompleteness, an undecidable
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statement is required, which can
be viewed as a new independent axiom of an enlarged system.
Example (1.5, cont). 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
a a e a
b b b a
e.g. a ∗ (b ∗ b) = a ∗ a = e = a = a ∗ b = (a ∗ b) ∗ b
• ( 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.
We could also ask if all elements have an inverse. That this is undecidable is the same as saying
that a new 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
structure of a group. Even this new system is incomplete; for instance, consider a new axiom of
commutativity. . .
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A famous example of an undecidable statement from standard set theory is the Continuum Hypothesis, which states that
there is no uncountable set with cardinality strictly smaller than that of the real numbers.
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Example 1.7 (Bus Routes). Here is a loosely defined axiomatic system. Discuss the questions with
your classmates.
Undefined Terms: Route, Stop
Axioms: (A1) Each route is a list of stops in some order. These are the stops visited by the route.
(A2) Each route visits at least four distinct stops.
(A3) No route visits the same stop twice, except the first stop which is also the last stop.
(A4) There is a stop called downtown that is visited by each route.
(A5) Every stop other than downtown is visited by at most two routes.
1. Construct a model of this system with three routes. What is the fewest number of stops you
can use?
2. Your answer to 1 shows that this system is: complete, consistent, inconsistent, independent?
3. Is the following a model for the Bus Routes system? If not, determine which axioms are satisfied
by the model and which are not?
Stops: Downtown, Walmart, Albertsons, Main St., CVS, Trader Joe’s, Zoo
Route 1: Downtown, Walmart, Main St., CVS, Zoo, Downtown
Route 2: Main St., CVS, Zoo, Albertsons, Downtown, Main St.
Route 3: Walmart, Main St., Downtown, Albertsons, Main St., Walmart
4. Show that A3 is independent of the other axioms.
5. Demonstrate that ‘There are exactly three routes’ is not a theorem in this system by finding a
model in which it is not true.
We are only scratching the surface of axiomatics. If you really want to dive down the rabbit hole,
consider taking a class in formal logic or model theory. As an example of the ideas involved, we
finish with two results proved in 1931 by the German logician Kurt G
¨
odel.
Theorem 1.8 (G¨odel’s incompleteness theorems).
1. Any consistent system containing the natural numbers is incomplete.
2. The consistency of such a system cannot be proved within the system itself.
G
¨
odel’s first theorem tells us that there is no ultimate consistent complete axiomatic system. Perhaps
this is reassuring—there will always be undecidable statements, so mathematics will never be fin-
ished! However, the undecidable statements cooked up by G
¨
odel are analogues of the famous liar
paradox (‘This sentence is false’), so the profundity of this is a matter of debate.
G
¨
odel’s second theorem fleshes out the difficulty in proving the consistency 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. While an inconsistent system might be essentially
useless, good luck showing that what you have really is consistent!
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Exercises 1.2. 1. Between two players are placed several piles of coins. On each turn a player takes
as many coins as they want from one pile, as long as they take at least one coin. The player who
takes the last coin wins.
If there are two piles where one pile has more coins than the other, prove that the first player
can always win the game.
2. Consider a system where children in a classroom choose different flavors of ice cream. Suppose
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.
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.
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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