Table of Contents
Purpose of Seminar ....................................................................................................................................... 2
Outline of Seminar .................................................................................................................................... 2
Greek Mathematics....................................................................................................................................... 2
Mathematical Proof vs. Scientific Proof .................................................................................................... 3
Euclid ............................................................................................................................................................. 4
Elements ....................................................................................................................................................... 4
Examples from the Elements ........................................................................................................................ 5
Proposition 1 ............................................................................................................................................. 5
Proposition 5 ............................................................................................................................................. 6
Proposition 6 ............................................................................................................................................. 6
Proposition 47 ........................................................................................................................................... 7
Epicurean objection .................................................................................................................................. 7
Application throughout the curriculum ........................................................................................................ 8
Connection to logic ................................................................................................................................... 8
Connection to apologetics ........................................................................................................................ 8
Connection to English class ....................................................................................................................... 9
Connection to all disciplines ..................................................................................................................... 9
Conclusion ................................................................................................................................................... 10


There are different aspects to the purpose this seminar.

I have a specific aim in that I think our Classical Christian education can be greatly enhanced if when teaching geometry we return to teaching from the classic textbook; Euclid’s Elements

I have a general aim in that I think that we can learn a lot about how we ought to teach mathematics
at all levels (not just Geometry) from Greek mathematics and specifically Euclid’s Elements

If we teach our students effectively using Euclid’s Elements and if we can engage our kids at all levels with the reasoning and thinking advocated by the Greek mathematicians we will help our students become better thinkers by; a) Helping them to understand how to give justification for truth b) Helping them see the presuppositional aspect of mathematics c) Helping them to understand how to organize and present mathematical truth
Outline of Seminar

Greek mathematics

Euclid

Elements

Examples from the Elements

Application throughout curriculum

Conclusion
“Greek geometry stands among the great intellectual/artistic movements of Western civilization and in this sense has much in common with Elizabethan drama or French impressionism. Like the impressionists, the Greek geometers shared a general philosophy and style, and though there were as many variants among the Greeks as among the French artists, the deeper, unifying characteristics of an impressionist painting or a Greek theorem are instantly recognizable.” – William Dunham, The
Mathematical Universe

Obviously mathematics had been around since the creation of our world but the Greeks took it to an entirely new level

Before the Greeks, mathematics was simply a means to an end, a tool to solve practical problems
The Egyptians used basic rules of Geometry to reconstruct field boundaries after the Nile flooded the surrounding plains and other such daily problems


The Greeks saw mathematics as an end and sought mathematical truth without interest in its application.

The Greeks thought learning math was valuable in it of itself, not because it helped us practically
A lot of our modern curriculum has fallen into the same thinking as the student of Euclid
Our Geometry text says….
“Why we should learn about trapezoids and kites…. To solve real-life problems, such as planning the layers of a layer cake. Why we should learn to find the bisectors of a triangle… To solve real-life problems such as finding the center of a mushroom ring.”
We will not be fulfilled if we emphasize the importance of learning math because it will help them practically, in their jobs… “but I’m going to be an English teacher… but I’m going to play in the NFL”

The philosophy and style of Greek mathematics exhibited three main characteristics
1.
The impressive use of logical rigor for proving theorems.
2.
The purely Geometric – as opposed to numerical – nature of their mathematics
3.
Their skillful organization in presenting and developing mathematical propositions.

Another critical thing to note is that the Greeks were committed to finding the truths embedded in the mathematical design of nature.
Many ultimately turned, what we as Christians understand to be the means by which God creates, and they turned it into an idol
We as Christians have come to find that God has created a world of order that can be explained with the language of mathematics yet the Greeks saw this language as the thing to be worshiped

Some of the most important mathematicians in Greek mathematics include
Thales, Pythagoras, Euclid, Archimedes, Zeno, Diophantus and one of the first women mathematician recorded in history, Hypatia
Mathematical Proof vs. Scientific Proof

It is absolutely critical for students to understand the difference between scientific proof and mathematical proof
This difference is well explained in the problem of the mutilated chess board
The problem is this, if one were to take 31 dominoes where each domino covers exactly two squares, is it possible to arrange 31 dominos so they cover all the 62 squares on the chessboard
 Scientific approach – attempt to do it in different ways, the scientific method, you experiment over and over and over again, after 5 years of trying, you would probably make the reasonable conclusion that it can not be done (you’re right)
 Mathematical approach (the Greek approach) – the mathematician uses logic and makes logical statements that force a conclusion
 The corners that were removed were both white squares which leaves us with 32 black squares and 30 white squares

 Each domino covers two neighboring squares and neighboring squares are always different colors
 Thus the first 30 dominos placed will cover exactly 30 white squares and 30 black squares
 This leaves you with one domino and two black squares
 But we know from the second logical statement that each domino can only cover two squares of different colors so no matter how we try, the board can not be covered.
A mathematician, a physicist, and an engineer are riding a train through Scotland.
The engineer looks out the window, sees a black sheep, and exclaims, "Hey! They've got black sheep in Scotland!"
The physicist looks out the window and corrects the engineer, "Strictly speaking, all we know is that there's at least one black sheep in Scotland."
The mathematician looks out the window and corrects the physicist, "Strictly speaking, all we know is that at least one side of one sheep is black in Scotland."
“If Euclid failed to kindle your youthful enthusiasm, then you were not born to be a scientific thinker.” -
Einstein

Little known of Euclid because we have no firsthand accounts of his life or work

However he is universally considered the Father of Geometry

Euclid born in 330 BC, first head of the Mathematics department at the Library of Alexandria

Like I said above regarding the Greeks and their desire for mathematical truth without regard for its application. There is a story told of Euclid that illustrates this well and one I like to use with my students often.
It is said that one time a student of Euclid’s asked the purpose of the mathematics he was learning… Euclid turned to his slave and said “give the boy a penny since he desires to profit from all that he learns.” The student was dutifully expelled.
 If I followed that example with all the students that asked me the same question, I wouldn’t have any left
 Right now I am a “roaming” teacher without a room but when I do get my own room, that quote will definitely be prominently displayed on my wall
 My emphasis on this way of understanding and appreciating mathematics really has limited the “Why do I have to learn this?” mentality

What made Euclid’s name synonymous with the subject of geometry was his authoring of the most famous math textbook of all time, the Elements
“At the age of eleven, I began Euclid, with my brother as my tutor. This was one of the great events of my life, as dazzling as first love. I had not imagined that there was anything so delicious in the world.” -
Bertrand Russell

Euclid’s book the Elements is arguably the most successful textbook in history



Until this century it was also the second best selling book in the world after the Bible
In addition, it is estimated to be second to the Bible in the number of editions published
Considered the most successful and influential textbook ever written
The Geometry text book for more than 2,000 years
Its logical progression is unparalleled by any other math text
The Elements is made up of 13 books where by the use of constructions and logic, Euclid proves 465 propositions.
The text is beautiful in its layout
What makes it so significant is its ability to progress from such basic principles (parallel lines never intersect, one can construct an equilateral triangle) to the sophisticated consequences that result
The first book starts with 23 definitions, 5 common notions, and 5 postulates
The best way to think of the Elements is like an empty box
Here Euclid is trying to prove mathematical truths with as little as possible
With these definitions, notions and postulates
However, to build this structure, Euclid needed some tools (like a hammer and saw, etc.)
In Euclid’s tool box were only two weapons.
He takes a compass and straight edge and then builds an unbelievable edifice of mathematical truth o The compass had to be collapsible o The straight edge could not be marked (thus a ruler was out)
Construct and prove, construct and prove, construct and prove and soon you have a remarkable building that set the stage for the great mathematics that would come over the next 2,500 years
I use the Elements in my geometry class and we construct and prove and go through the first book… I have achieved great results from use of this text and it is my recommendation that we return to its use in our geometry classes
Proposition 1

Purpose of showing this proof – basic introduction to the thinking and reasoning Euclid uses to prove propositions.
It is this text and type of reasoning that would establish what would be mathematical proof

Before starting this proof, remember, Euclid wants to assume as little as possible so all we have at our disposal, the only weapons we can use are 23 definitions, 5 postulates, 5 common notions, a collapsible compass and a straight edge

Think if the daunting dilemma before Euclid, where does he start?

As we know, a journey of a thousand miles begins with a step
Euclid’s first step was the proposition that it is possible to construct an equilateral triangle.

Proposition 5

Proposition 5 is what we know today as the Base Angles Theorem stating that if two sides of a
 triangle are equal, then the angles opposite these sides are equal
Tradition says that it became know as the Pons Asinorum or the bridge (pons) of asses. o This is because dullards, or asses, were able to accomplish the first 4 propositions but this was the first real test in the Elements o If you were unable to cross this bridge, the pons, then you were probably not intellectually capable of progressing beyond proposition 5 and further into the geometric promised land of the Elements
Proposition 6

Proposition 6 is significant because it is Euclid’s first use of what is called a Reductio Ad Absurdum or what we know as a proof by contradiction

This is truly a clever and ingenious method of proving mathematical proof. Hardy says;
“Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or even a piece, but a mathematician offers the game.” - G.H. Hardy, A Mathematician’s Apology

The idea of a proof by contradiction is as follows o In order to prove something true, you first assume the opposite of what you desire to prove true o Therefore you are assuming something not true to be true o You then progress logically through a proof trying to prove this proposition o At some point you will find a logical absurdity, something that can not be true as it breaks a previously understood truth o When you find this logical absurdity, you progress backwards through the proof to find out where you went wrong o You will inevitably find that your logic was all correct except for your original assumption which means it must be wrong and you have then proved the converse of your original assumption

To see exactly how a reduction ad absurdum works, we will go through proposition 6 in Euclid’s first book

Another proof by contradiction and one of Euclid’s most famous was his demonstration of the existence of irrational numbers.

How does a reduction ad absurdum relate to our Christian faith? It has a very important relationship. o In the same way that we reason above, we can follow the line of reasoning that results from unbelief o If we assume the opposite of what we believe to be true, mainly that there is no God and the atheistic evolutionary worldview is true, what does this assumption require?

o If there is no God, then the Bible is not true. o If the Bible is not true we are not made in the image of God. o If we are not made in the image of God, then man is an animal. o If man is an animal, then moral absolutes make as much sense as they do in the animal kingdom. o If moral absolutes are meaningless, then there is no good and evil or right and wrong. o Few unbelievers would be willing to accept such a conclusion, much less agree that’s where the logic flows, but where is the logic wrong? (the original assumption)

We also see the apostle Paul use this brilliant tool, on his visit to Greece (what better place!?) in 1
Corinthians 15:13-19
“But if there is no resurrection of the dead, not even Christ has been raised; and if Christ has not been raised, then our preaching is vain, your faith also is vain. Moreover we are even found to be false witnesses of God, because we testified against God that He raised Christ, whom He did not raise, if in fact the dead are not raised. For if the dead are not raised, not even Christ has been raised; and if Christ has not been raised, your faith is worthless; you are still in your sins.
Then those also who have fallen asleep in Christa have perished. If we have hoped in Christ in this life only, we are of all men most to be pitied.”
Is this not a brilliant illustration of the usefulness of a proof by contradiction, Paul understood it’s effective power (especially with the Greeks!) and we need to teach its effective power to our students that they might reason like Paul
Proposition 47

We will now take a look at proposition 47 in Euclid’s first book.

This comes at the end, the second to last proposition of book I and is the most popular theorem in math history…. the Pythagorean theorem.

Sadly, most students know this theorem by its algebraic expression of a2 + b2 = c2.

The Greeks would not have recognized that representation of the Pythagorean Theorem.

They know it by its geometric representation.

The Pythagorean Theorem is truly a remarkable truth.

An early 20th century professor Elisha Scott Loomis collected and published 367 known proofs of the Pythagorean Theorem.

It is strongly suggested that, as time permits in your class room, the teacher do several of these with their students. It is a fun and instructive exercise.

In 1876, an unlikely participant in the field discovered an additional proof, James Garfield who would later become President of the United States joined the illustrious group of mathematicians known for providing a unique proof to the Pythagorean Theorem.

How beautifully elegant the logic is in proving the Pythagorean Theorem
Epicurean objection

The Epicureans of the day ridiculed the rigorous logic required of Euclid and other Greeks claiming that no such proof was needed for some of the propositions that Euclid proved.


An example of one proposition that the Epicureans felt unnecessary to prove was proposition 20 of book 1

Proposition 20 states that any two sides in a triangle are greater than the third. In essence, the proposition states that the quickest way from point A to point B is a straight line. o The Epicureans argued that even a donkey knows this to be true when it seeks straw, it will go straight to the straw and not along the longer two sides of the triangle o What idiot doesn’t know this? Why do we need such a proof?

I think it’s important we bring up the Epicurean argument as this is a common argument brought up in our culture. o I hear such justifications for a variety of different arguments
 One example is our world’s argument for Global warming, people with little or no education on the subject will claim “any idiot knows it’s true and that it is caused by humans and that it is only negative in its effects… I have no time to explain such an evident truth to such idiots….”
 Another example is found in some Christian’s justification for their belief that the end of the world is imminent… “any idiot can take a look around and see that the world is going to pot and will end any day now… if you can’t see this, you are simply blind…”

It would be insulting to think Euclid did not know the obvious nature of proposition 20 but his argument was that it should be the goal of the scientist/mathematician to prove anything and everything they can

If he could begin with no assumptions, he would have done so
Connection to logic

The subject that can most easily be linked to geometry is logic.

In comparing mathematics and logic, Bertrand Russell asserted;
“…it has become wholly impossible to draw a line between the two; in fact, the two are one.”

The reasoning and justification used in the geometry class, the reasoning and justification to go through Euclid’s Elements is identical to that required in the logic class

This connection must be made and collaboration between your logic and geometry classes will greatly benefit your students
Connection to apologetics

The idea of proof, the essential idea to the geometer is another term for justification

You are justifying why something is true

This is the central idea to apologetics, the justification of your belief in the Christian God
“…always being ready to make a defense to everyone who asks you to give an account for the hope that is in you, yet with gentleness and reverence.” - 1 Peter 3:15


Euclid’s Elements is an incredibly effective tool for students to develop the cognitive ability to give a reasonable defense of the faith

Our Christian culture encourages our students to give an emotional defense for their faith… “Why do you believe in Jesus Christ?” “Because He loves me.”

There is nothing wrong with such a response as long as it is accompanied by a reasoned defense similar to what the apostle Paul did on his missionary journeys
“So he was reasoning in the synagogue with the Jews and the God-fearing Gentiles, and in the market place every day with those who happened to be present.” – Acts 17:17 o It’s a curious thing to find the same Epicureans we just talked about opposing the teachings of apostle Paul in the next verse (“And also some of the Epicurean and Stoic philosophers were conversing with him. Some were saying, “what would this idle babbler wish to say?”…”
Connection to English class

I believe the student that excels in understanding how to give justification in Euclid’s Elements will have a better understanding of how to give justification for an argument in an English paper or a history paper

Is not the English paper similar? o You give your theses, your argument and then you go about giving justification for an argument o This is the essence of the Elements o I believe three angles of a triangle sum up to two right angles… now allow me to give you the justification
Connection to all disciplines

I have had many a student argue to the nth degree why math is worthless yet I still have yet to hear a student argue that thinking is a worthless endeavor

Not a single student of mine has ever expressed a sincere desire for ignorance

Mathematics is mental weight lifting preparing them mentally for every area of life

We are teaching them to be Christian problem solvers in the math class

Do not allow your students to think the problem solving in the math class does not apply to any other area of life

The student that creatively finds solutions in the math class will be the same creative problem solver in family disputes, the same person that finds a solution to a business dilemma, that finds a creative method to finding justice in the court room, that brings peace to a church in turmoil… The determination, the will to find a solution that a teacher can cultivate in a classroom can not help but over flow into other areas of the student’s life

I still have a hard time understanding how a student can fail at understanding how to develop a proof in the geometry class yet develop a strong logical argument in an English paper.


Even Lincoln understood the value benefits of mathematical reasoning skills when he said in an autobiographical sketch the following;
“I said, “Lincoln, you can never make a lawyer if you do not understand what demonstrate means”; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what
“demonstrate” means, and went back to my law studies.” – Abraham Lincoln
The first six books of the Elements contain 173 propositions, it’s hard to believe but who would doubt honest Abe

Even though some of you might disagree with Lincoln’s ideas of “demonstrate” he rightly understood the potency of the critical thinking required to think through Euclid’s Elements

Mastery in the math classroom can only help our students in whatever area they will venture in the future.

What to be done
Follow the logical progression of the Elements and demand from our students a reason for every step in geometry.
Why? needs to be the most often asked question. Not what?

Our goal should be to create great thinkers.
“And do not be conformed to this world, but be transformed by the renewing of your mind, so
that you may prove what the will of God is, that which is good an acceptable and perfect.” –
Romans 12:2

Students that can reason like St. Paul, like John Calvin, like Rene Descartes, like Johannes Kepler

I believe that if we are successful in teaching our students how to think through Euclid’s Elements, we will be; a) Helping them to understand how to give justification for truth b) Helping them see the presuppositional aspect of mathematics c) Helping them to understand how to organize and present mathematical truth

Such thinkers will influence our culture for the cause of Christ in every arena of life even if there vocation or focus is not in the practical application of mathematics.