Intro to classical Arithmetic, Lesson 07. Of Proofs & Axioms
by William C. Michael
To complete the objectives of this lesson, complete the following tasks:
- Study your Lesson.
- Complete your Memory Work.
- Complete your Lesson Exercises.
- Complete the Lesson Examination.
Thus far in Arithmetic, we have learned the different branches of Mathematics: Arithmetic, Geometry, Music and Astronomy. We have learn about Unity and units. We have learned about Comparison: equalities, inequalities, majority and minority. If you are uncomfortable with these subjects, you should go back and review. In this lesson, we will move on to understand why these subjects are important. We will learn about Proofs and will memorize our first set of mathematical Axioms.
Wise and Foolish Builders
In the world we live in, people are happy to say things like, “What’s true for you is not true for me.”, or “I have my own opinions on this topic.” These statements sound cute, but they really make no sense. When we hear statements like these, we must ask, “What would cause a person to say such things?”. After all, if at that moment someone dumped a can of paint over his head, or kicked his cat, I bet that you would find them suddenly in agreement some things are right and wrong for everyone.
Most people we meet up with have never been trained in the art of reasoning, known as Logic. They do not understand that ideas can be proven to be true or false by clear rules of truth and falsehood. They think that because they can’t prove an idea false that no one can. When they hear that wise men disagree with them, they suggest that the wise men may have been wrong, rather than fearing that they might be right. It is because of pride that they often make errors that cause troubles in every area of their lives and may put their souls in danger of even greater troubles. Their life is built like a house upon sand, for their decisions in life are based on what seems true to them day by day, rather than what is really true everyday. The book of Proverbs warns us: “The fool takes no delight in understanding, but rather in displaying what he thinks.”
On the other hand, if we would be wise, we will never trust in ourselves, but in the guides that God has given us to save us from errors. First of all, we will study and trust the Scriptures, in which God has taught us many things we could never have known on our own. Second, we will study and trust the teachings of the Catholic Church, which is protected by the Holy Spirit from error in teaching us what we should believe and how we should behave. Third, we will study the classical liberal arts to strengthen our ability to reason carefully and truly on every subject. With these three guides, we will be able to build a life that is true, safe and happy. We will be wise builders.
This is what we are seeking to do in every subject, including Arithmetic.
Reason and Proof
One of the greatest gifts God has given us is the gift of Reason. This gift allows us to solve problems and to test whether solutions are true or false.
Every human being has reason, but reason can be ignored by sinful desires and can be weak without training. If we wish to use our reason rightly, we must understand how it works and exercise ourselves in its use. In Arithmetic, reason allows us to prove whether statements made are true or false. When we wish to demonstrate or show that a statement us true or false, we use Proofs.
A proof is a connection of arguments used to prove that a statement is true or false. We must show that what is said follows from truths that have already been proven to us or that need no proof. Statements that need no proof are called Axioms. Over the next few lessons, we will learn the first axioms of Arithmetic. Once these are mastered, we will begin to work with proofs.
What is an Axiom?
The most basic rule of learning is that we use what we know to understand what we don’t know. For example, if we don’t know how long a piece of wood is, we use the units on a measuring tape (whose length we do know) to measure it. We learned that this is the basic idea of measurement: counting how many times a known quantity is contained in an unknown quantity.
In life, we will face many complicated problems that need to be figured out. Here’s an example:
Robert wishes to rearrange his garden area by dividing it between roses and fruit trees. He knows that his garden area is a square with each side measuring 70 ft. in length. He wants to arrange the garden according to a drawing he made, where each black line represents a line of wood fencing. How many feet of wood fencing will Robert need to buy?
Now, you may be able to guess at an answer for this, but I don’t think Robert wants to drive all the way to the hardware store, pay for the fencing and drive it all home to find out your guess was wrong! If you guess too little, he will have to all the way back to the store, wasting time and money on travel. If you guess too much, he will have wasted money and bought more fencing than he needed. Therefore, Robert wants you to prove that your answer is true before he makes the trip. At that point, you would probably have to admit you can’t help him — but not for long.
What you would need to solve a problem like this are a set of axioms, rules and definitions that could be used to prove your computation to be true and not a sloppy guess. Of these, axioms are the most important because they are self-evident. What that means is that axioms are statements that need no proof — because everyone can see clearly that they are true.
In mathematics we learn a number of axioms, and these are the most important lessons in all of mathematics. What is most important is understanding that mathematics are not based on mathematics, but basic truths that apply to all subjects. When you understand the axioms, you can use them to solve almost any problem. Whenever a problem arises, no matter how complicated it may seem at first, you will simply start to take it apart, piece-by-piece, with the axioms that you know are true. This way, you always know that you are following a sure guide to solving problems, because you know for sure that the statements you are using are true.
Let’s get started with our first axioms!
AXIOM I: ANYTHING MAY BE ASSUMED AS UNITY.
In lesson 02, we learned that “Unity is a known quantity we refer to as one.”. The only requirement that exists for a quantity to be assumed as Unity is that it be known. We said in that lesson that any quantity could be assumed as Unity and then used to number or measure another unknown quantity. This first axiom states that this is true of any thing. Anything may be assumed as Unity, and that should be self-evident.
Let’s take an example to illustrate this axiom. John and his daughter Mary were stopped along a highway and Mary was looking at a map. John asked, “Mary, how far is it from where we are to New York?” Mary, pulled a quarter from her pocket and measured, “It’s about seven quarters from here to New York.”. John complained and said, “That’s not what I mean! I want to know how how far it is in miles on the road, not quarters on the map.” Mary, responded, “Well, then, you should have said so. Don’t you know that anything can be assumed as unity?”
AXIOM II: EVERY QUANTITY IS EQUAL TO ITSELF.
Today, we may look at this and wonder why in the world anyone would need to make this statement. However, in history, there were some troublemakers who tried to deny that it was true! For example, one ancient Greek philosopher used a river as an example to prove this false. He said that it is impossible to say that a certain river is the same as itself because the water in it is constantly changing. Therefore, the river cannot be tomorrow what it is today. Since he believed that all things in the world were like rivers, with their parts constantly changing, he denied that this could be an axiom. Fortunately, mathematics allowed men to see plainly that this is in fact an axiom that needs no proof. While a river’s changes may appear confusing to us, a number like one or five is always equal to itself and allows all to see that every quantity is equal to itself.
Now, remember: the axioms are necessary for proving that an answer in Arithmetic is true and this is why statements like this are important. Once we prove that a statement flows from an axiom, anyone who refuses to accept it will look absurd. This, Aristotle said, is the firmest of all truths — that a thing is the same as itself. This is where we must begin in mathematics. We will see an example using this axiom below.
AXIOM III: QUANTITIES WHICH ARE BOTH EQUAL TO ONE AND THE SAME THIRD ARE EQUAL TO ONE ANOTHER.
In the axiom above we learned that a quantity is equal to itself, which is the same as saying, “If two quantities are equal to one another, they are the same quantity.” In this axiom we take another step forward and say that if there is a quantity and two other quantities are equal to it, then those two quantities are equal to one another.
For example, Jonathan is carrying a bag of peaches and he places it on a balance. When he does so, he finds that it is equal in weight to a certain bag of apples. David comes to the balance with a basket of cherries and when he puts it on the balance, it is equal in weight to the same bag of apples. Jonathan looks at David and says, “Our bags weigh the same!” David asks, “How do you know they weigh the same, when we didn’t put them on the balance at the same time yet?” Jonathan explains, “That’s easy. Quantities which are both equal to one and the same third are equal to one another. Both of our bags were equal in weight to the bag of apples, so they must be equal to one another.”
Note that when we combine this axiom to the one before it, we can say even more. Since the weight of Jonathan’s peaches is equal to the weight of David’s cherries, we can say that the weight of their bags are the same, because “Every quantity is equal to itself.”
AXIOM IV: THE WHOLE IS MORE OR GREATER THAN ITS PART.
This axiom states a basic relationship between a whole and its part, for there can be no whole if there are no parts and each part must be less than the whole from which it is taken. There are many uses of this axiom in Arithmetic if we think about it.
Jonathan and David were arguing one day about the numbers four and seven. David was persuaded that two was more than five. Jonathan laughed every time he heard David say this, but he couldn’t think of how to prove David wrong. Finally, Jonathan thought of what to do. He put two stones on the ground and asked David to count them. David did. Then, Jonathan added three more and told David to count them all. David did. Jonathan then asked, “What’s greater, David, a whole or a part — a whole apple or a part of an apple?” David answered, “That’s easy — a whole is greater than a part.” Jonathan asked, “Are not 2 and 3 parts of 5?” “Yes, so what?”, David replied. “Then 2 must be smaller than 5, since it is a part of 5! A whole is greater than its parts, you know.”
Thus, we see that this axiom applies to any discussion of a whole and its parts, whether we mean a whole thing and its parts or any quantity and the units it contains.
AXIOM V: THE WHOLE IS EQUAL TO ALL OF ITS PARTS TAKEN TOGETHER.
This axiom provides us with the answer to many questions. As we said in the last lesson, anything that can be said of a whole can also be said of the collection of all of its parts. Let’s take another example.
Jonathan and David were standing on the deck of their swimming pool one day, when David asked, “I wonder how much all the water in this pool weighs. Maybe a million, billion tons?” “No, not that much.”, Jonathan replied, “Do you want to measure it?”. “Measure the swimming pool?! How in the world could we get this whole swimming pool onto a scale and weight it? It must be one of those mysteries we learned about in Catechism.”, said David. Jonathan laughed and said, “It’s not a mystery, David. We have to use our reason here. I know that a whole is equal to all of its parts taken together. I also know that a gallon of water weighs about seven pounds, so if we emptied the pool one gallon at a time and counted all of the gallons of water, Dad could tell us how many pounds the water in the pool weighed when full.” “You’re right Jonathan,” Mom said, “but don’t you think about emptying this pool.”
This works in the opposite way also. Anything that is true of the collection of parts is also be true of the whole those parts belong to.
As you develop a collection of axioms, definitions and rules in Arithmetic, you will need to be able to start writing proofs. The process is very simple.
We begin by stating what it is that needs to be demonstrated (demonstrandum). This will be a statement known as a theorem. From the theorem, you will simply work backwards to show how the theorem flows logically from the axioms, rules and definitions you have already learned. Once we have completed our proof, we end with the phrase, “which was to be demonstrated”.
Example 1. Theorem: Demonstrate that 4 cups is equal to 2 pints.
Proof: We know that 4 cups is equal to 1 quart and that 2 pints is also equal to 1 quart. Axiom III states that quantities which are both equal to one and the same third are equal to one another. Since 4 cups and 2 pints are both equal to 1 quart, then 4 cups must be equal to 2 pints, which was to be demonstrated.
Example 2. Theorem: Demonstrate that one foot is greater than one inch.
Proof: We know that one foot contains twelve inches, which means twelve inches collected together form a whole of one foot. Thus, one inch is a part of one foot. Axiom IV states that a whole is more or greater than its parts. Since one inch is a part of one foot, one foot must be greater than one inch, which was to be demonstrated.
This is a proof and, as we will learn, you can write them in a number of different ways. For now, we will simply work to understand how to use our axioms to reason out our proofs.
In this lesson, we begin to understand why all of the definitions and rules in past lessons are important. We will begin to use these and the axioms to demonstrate the truth or falsehood of any statements made in Arithmetic. We learned that demonstration is made by proofs which begin with a theorem to be demonstrated and move step-by-step through the reasoning that joins the theorem (which is unknown) to the most sure axioms (which are known). In your exercises below, you will get your first attempt to write your own proofs and in your exam prove your mastery of the lesson. From now on, get used to the challenge: Prove it!
- What is a proof, or demonstration?A proof, or demonstration, is a connection of arguments used to demonstrate the truth or falsehood of a statement.
- What is a theorem?A theorem is a statement that needs to be demonstrated and is called in Latin demonstrandum.
- What is an axiom?An axiom is a self-evident statement, that is, one that does not need to be demonstrated.
- State Axiom I.AXIOM I. Everything may be assumed as Unity.
- State Axiom II.AXIOM II. Every quantity is equal to itself.
- State Axiom III.AXIOM III. Quantities which are both equal to one and the same third are equal to one another.
- State Axiom IV.AXIOM IV. The whole is more or greater than its part.
- State Axiom V. AXIOM V. The whole is equal to all of its parts taken together.
Supply proofs for the following statements using your axioms, definitions and rules learned in this course. Use the examples in your lesson above to guide you. If you need help, please contact us.
- Theorem: Show that 1,760 yards equals 5,280 feet.
- Theorem: Show that 8 gallons equals 4 pecks.
- Theorem: Show that 1 mile is greater than 100 feet.
- Theorem: Show that 1 ton is greater than 100 pounds.