Flaws in Euclid

The Five Axioms

Given two distinct points P and Q, there is a line through P and Q.Any line segment can be extended indefinitely.Given two distinct points P and Q, a circle centered at P with radius PQ can be drawn.All right angles are congruent.Given any line l and a point P not on l, there exists a line through P that is parallel to l.

Using these axioms, try to prove that points and lines exist.

Given two distinct points P and Q, there is a line through P and Q.Any line segment can be extended indefinitely.Given two distinct points P and Q, a circle centered at P with radius PQ can be drawn.All right angles are congruent.Given any line l and a point P not on l, there exists a line through P that is parallel to l.

Using these axioms, try to prove that given a line l, there exists a point lying on l.

Given two distinct points P and Q, there is a line through P and Q.Any line segment can be extended indefinitely.Given two distinct points P and Q, a circle centered at P with radius PQ can be drawn.All right angles are congruent.Given any line l and a point P not on l, there exists a line through P that is parallel to l.

Using these axioms, try to prove that given a line l, there exists a point NOT lying on l.

Given two distinct points P and Q, there is a line through P and Q.Any line segment can be extended indefinitely.Given two distinct points P and Q, a circle centered at P with radius PQ can be drawn.All right angles are congruent.Given any line l and a point P not on l, there exists a line through P that is parallel to l.

Strictly, under Euclidean Geometry, we have no guarantee

that points and lines exist.

It is also just based on our (unfounded) assumptions that

lines contain points.

A simple abstract axiomatic system

Undefined terms: Fe's Fo's, and the relation belongs toAxiom 1: There exist exactly three distinct Fe's in this system.Axiom 2: Two distinct Fe's belong to exactly one Fo.Axiom 3: Not all Fe's belong to the same Fo.Axiom 4: Any two distinct Fo's contain at least one Fe that belongs to both.

Prove the following Theorems.Fe-Fo Theorem 1. Two distinct Fo's contain exactly one Fe in common.Fe-Fo Theorem 2. There are exactly three Fo's.Fe-Fo Theorem 3. Each Fo has exactly two Fe's that belong to it.

In proving the results in the previous slide, you may have

imagined dots or dashes on a sheet of paper. We will take the point of view that these dots and dashes

are a model for the given geometry.

More generally, given an axiom system, we can interpret the undefined terms in some way (does not have to be dots or

dashes). If the axioms are satisfied based on our interpretation, then

we can call our interpretation a model.

Sample Model?

Designate the Fe's as people and the Fo's as committees.Axiom 1: There exist exactly three distinct people.Axiom 2: Two distinct people belong to exactly one committee.Axiom 3: Not all people belong to the same committee.Axiom 4: Any two distinct committees contain at least one person who belongs to both.

Sample Model?

Designate the Fe's as books and the Fo's as bags.Axiom 1: There exist exactly three distinct books.Axiom 2: Two distinct books belong to exactly one bag.Axiom 3:Not all books belong to the same bag.Axiom 4: Any two distinct bags contain at least one book that is on both.

The use of modelsAll theorems in the system are also correct statements in the model.Fe-Fo Theorem 1. Two distinct Fo's contain exactly one Fe in common.Fe-Fo Theorem 2. There are exactly three Fo's.Fe-Fo Theorem 3. Each Fo has exactly two Fe's that belong to it.

The use of modelsAll theorems in the system are also correct statements in the model.Fe-Fo Theorem 1. Two distinct committees contain exactly one person in common.Fe-Fo Theorem 2. There are exactly three committees.Fe-Fo Theorem 3. Each committee has exactly two persons that belong to it.

The use of models

Models are laboratories for experimenting with the axiomatic system.Mathematicians often discover that one model has applications to completely different models.

Back to Euclids Elements

Construct an equilateral triangle [Proposition 1] based on the following [Euclidean] axioms.1.Given two distinct points P and Q, there is a line through P and Q.2.Given two distinct points P and Q, a circle centered at P with radius PQ can be drawn.

The first proposition

1.To construct an equilateral triangle on a given finite straight line.

There is a flaw in the logic because of the assumption that the two circles will have a point of intersection.

Quite a few of Euclids proofs, such as the construction of an equilateral triangle or the SAS theorem, are based on reasoning from diagrams. However, as weve shown, diagrams can be deceptive.A larger system of explicit axioms is needed.One of the first set of axioms introduced to remedy the defects of Euclids work was given by Moritz Pasch in 1882.Note: This was around 2000 years after Euclids Elements.

David Hilbert (1862-1943)

Presented not the first, but perhaps the most intuitive set of axioms, closest in spirit to Euclids.

One must be able to say at all timesinstead of points, lines and planestables, chairs and beer mugs.

- David Hilbert

Euclids axioms in this sense

1.Given two distinct tables P and Q, there is a chair through P and Q.2.Given two distinct tables P and Q, a circle centered at P with radius PQ can be drawn.Can you know use these to construct an equilateral triangle?

Euclid still relied on visual reasoning.

Point that which has no partLine breadthless lengthSurface that which has length and breadth only

In an axiomatic method

Points, lines and planes are undefined terms.We use models to represent these terms.In this system, points are not necessarily represented as dots, and lines are not necessarily represented as long straight strokes.

The Five sets of Hilberts axioms

Axioms of IncidenceAxioms of BetweennessAxioms of ParallelsAxioms of CongruenceAxioms of Continuity

An exercise

Consider the following proof (found in some high school geometry texts) that the base angles of an isosceles triangle are congruent.Let the bisector of C meet AB at D.ACD is congruent to BCD by SAS.A is congruent to B. Find the flaw in the proof.

How do we know that D lies between A and B?

We need Betweenness axioms to prove that the bisector of C does meet AB in a point between A and B.

Betweenness Axioms (Hilbert)Notation A * B * C to mean point B is between points A and C

1. If A * B * C, then A, B, and C are three distinct points lying on the same line, and C * B * A.

[fills a gap in one flaw above]

Betweenness Axioms (Hilbert)2. Given any two distinct points B and D, there exists points A, C, and E lying on BD such that A * B * D, B * C * D, and B * D * E.

[ensures that there are points between B and D, and that the line BD does not end at B or D]3. If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other two.

[ensures that a line is not circular]

Exercise

Given two points A and B, consider the two rays AB and BA. a.Draw diagrams to show that AB BA = AB, and AB BA = AB. b.Prove these formulas. Use the definitions of segment and ray, as given on the next slide.

Given two points A and B. The segment AB is the set whose members are A, B, and all the points that lie on line AB that are between A and B.The ray AB is the following set of points on AB: all points on segment AB and all points C on AB such that A * B * C.

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