What do you mean when you say a horse is a mammal or a square is a rectangle? Should we accept Aristotle’s classification of animals where humans are not animals, or should we accept what biology textbooks teach and say humans are types of mammals? On what basis can we make these types of decisions?
Let us listen to a conversation between Fliba, who thinks squares are rectangles, and Lagard, who thinks they are not.
Lagard: My textbook said that squares are types of rectangles. I find that weird. I have always thought that they are two different things. I think the textbook might be wrong.
Fliba: But, squares are types of rectangles.
Lagard: Why do you say that?
Fliba: My teacher told me.
Lagard: For most things we learn in mathematics, either the teacher gives us reasons or we can find reasons in the textbook. I didn’t see any reasons for this in the textbook and my teacher didn’t tell me any. Did yours?
Fliba: No. She just told me that squares are types of rectangles.
Lagard: Maybe there are no reasons. Maybe somebody just decided this is the case. Your and my parents chose very strange names for us. They just liked the sound of the name.
Fliba: But this is mathematics and not somebody’s name. You would hope that there is some thought which went into deciding things. Maybe we can try figuring it out ourselves.
Most mathematics textbooks tend to give proofs or at least some reason to believe certain claims. This is especially true in geometry when dealing with claims like the Pythagorean Theorem or that the angles of an equilateral triangle are equal. However, there are other types of statements made in textbooks for which no reasons are given. For instance, we study particular objects, like right-angled triangles and not others, like triangles with one angle equal 47.3 degrees. We choose certain definitions of terms and not others. Some proofs are considered better than others.
One of the running themes of this Substack will involve exploring these aspects of mathematics which are largely neglected in formal education. A reason for doing this is that some of the considerations we make in mathematics generalize to other areas. For example, you will see later in these articles that the question about squares and rectangles has some elements in common with a question about humans and animals. Some of the reasons we choose one definition over another also transfer to different disciplines and even conversations with friends.
Let us return to Fliba and Lagard and see how they decide to proceed.
Lagard: How do we do that?
Fliba: Why don’t we just try both possibilities and see what happens.
Lagard: Maybe we should start by writing out what a square is and what a rectangle is.
Fliba: Okay. We can start with what the textbook says. It says a square is a quadrilateral with all sides equal and all angles 90 degrees and a rectangle is a quadrilateral with opposite sides equal and all angles 90 degrees.
(For those who don’t know what a quadrilateral is, it is a closed shape with four straight line sides)
Lagard: What does that tell us?
Fliba: It tells us that a square is a rectangle since if a square has all sides equal, it must also have opposite sides equal.
Lagard: Of course that will be the case if we use what the textbook says squares and rectangles are. The writer clearly thinks that squares are rectangles.
Fliba: So, how do you want to define squares and rectangles?
Lagard: We can use the same definition for squares. But, we can add something to the definition of rectangles. We can say that a rectangle is a quadrilateral with opposite sides equal, all angles 90 degrees, and adjacent sides, those which share a corner, not equal.
Fliba: I agree that if we use your definition, then squares are not types of rectangles.
Lagard: So, you agree I’m right? That squares are not types of rectangles.
Fliba: That is not what I said! I said that if we use your definition, then squares are not types of rectangles. However, I think that the textbook definition is better than yours and we should use that.
Lagard: Why?
Fliba: I’m not sure. Maybe we can ask somebody to help us. Your older sister is studying mathematics, right? Maybe we can ask her. Does she also have an odd name like ours?
Lagard: Yes. Her name is Glagalbagal.
If we accept the textbook definitions, it is clear that squares are rectangles. Let us work through the reasoning to convince ourselves:
Let ABCD be a square.
Then, we know that:
AB = BC = CD = DA
This implies that AB = CD and BC = DA.
Now, ABCD is a quadrilateral with all angles 90 degrees and opposite sides equal.
Hence, ABCD is a rectangle.
However, if we accept Lagard’s definition of a rectangle, then it is no longer the case that squares are rectangles since squares have adjacent sides equal.
So, how do we choose between these definitions of a rectangle? Let us listen in again and see if Glagalbagal can help Fliba and Lagard find an answer.
Glagalbagal: That is an interesting question. I have never thought of asking that before. But, maybe I can help investigate this by giving you a few tools. First, let us look at some ways in which we can represent these ideas. The first thing I am going to introduce you to are tree diagrams. See these:
I apologize for my handwriting. If we look at these diagrams above, when you see an arrow, that means that the object at the pointy end of the arrow is a type of the object at the tail of the arrow. So, the first diagram is saying that squares and rectangles are quadrilaterals, but is not saying that squares are type of rectangles. The second is saying that squares are types of rectangles. The third diagram is saying that humans are not types of animals while the fourth is saying that humans are types of mammals which are types of animals. So, humans are types of animals.
Lagard: What about the arrows which have nothing at the pointy end?
Glagalgabal: They just mean that there are other things which we don’t care about for what we are doing. For example, there are quadrilaterals which are not squares or rectangles. Those are clubbed into the ‘other’ category.
Lagard: So, the first diagram is the one I want and the second is the one Fliba wants.
Glagalgabal: Yes.
Fliba: So, how do we use these?
Glagalbagal: That is for you to figure out. Before that, let me introduce you to another representation for the same thing - Venn diagrams.
Fliba: I’ve seen those. Lots of circles.
Glagalbagal: Yeah. Let me show you how to use them to represent the same thing as above.
In the first two, the outer circle represents quadrilaterals while the outer circle represents living things in the other two. When one region is inside another, it is a type of the other. For example, the third diagram says that mammals are types of animals but humans are not types of animals (or mammals).
At some point, I will do an article on these representation systems and will hopefully remember to link it here. That will go into it in more detail. However, what Glagalbagal introduced should be sufficient for this article. So let’s get back to the conversation.
Fliba: I still don’t know how to figure this out. Any other suggestions?
Glagalbagal: Yes. So far, you have only thought about squares and rectangles. Let us bring in other quadrilaterals - parallelograms, rhombuses and trapeziums.
Fliba: I remember learning about those but I forgot. Can you tell me what those are?
Glagalbagal: There are a few ways to define a parallelogram. You can define it as a quadrilateral with opposite sides parallel or as one with opposite sides equal. I suggest you use the equal definition since you are dealing with equality of sides. The following is a parallelogram:
A rhombus is a quadrilateral with all sides equal. Here is an example of a rhombus:
Fliba: So, a square is a type of rhombus and a rhombus is a type of parallelogram?
Glagalbagal: That is what I’m asking you to figure out just as you are trying to do with squares and rectangles. Just as with squares and rectangles, you will see that with some definitions, there is this relationship while not with other definitions. Why should we choose one type of relationship over another?
Fliba: Okay. You still have to tell us what a trapezium is.
Glagalbagal: It is a quadrilateral with one pair of opposite sides parallel. Here is an example of a trapezium:
Now figure this out for yourselves and let me know. I also want an answer but I don’t want to do the work!
Try to define the different types of quadrilaterals in different ways so that there are different relationships between them. The two types of representations Glagalbagal introduced will help with that. Also, think about the same thing in biology. I will pick up these threads in Part 2. However, if you want to learn, it is essential you spend some time thinking about this on your own.
I love this article! It does a nice little dive into questioning how math is categorized and defined. It gives a great jumping off point into the decisions that might be made while presenting math and in any other categorical activity all while being accessible! The structure of conversation and pictures really helps with understanding what is going on. I look forward to what you post in the future!
I noticed a couple of typos if you want to fix them:
"You will see in later these articles that the question about squares and rectangles some some elements in common with a question about humans and animals." The word 'some' is written twice and there needs to be a 'has' after rectangles.
"Glagalbagal: That is for you to figure out. Before that, let we introduce you to..." The word 'we' doesn't make sense here.
"Fliba: I remember learning about those but I forgot. Can you tell me that those are?" The word 'that' should be 'what' instead.
I love this article! It does a nice little dive into questioning how math is categorized and defined. It gives a great jumping off point into the decisions that might be made while presenting math and in any other categorical activity all while being accessible! The structure of conversation and pictures really helps with understanding what is going on. I look forward to what you post in the future!
I noticed a couple of typos if you want to fix them:
"You will see in later these articles that the question about squares and rectangles some some elements in common with a question about humans and animals." The word 'some' is written twice and there needs to be a 'has' after rectangles.
"Glagalbagal: That is for you to figure out. Before that, let we introduce you to..." The word 'we' doesn't make sense here.
"Fliba: I remember learning about those but I forgot. Can you tell me that those are?" The word 'that' should be 'what' instead.