A quick read of the last post, in which I discussed equivalence relations, can’t hurt, if you haven’t already read it, not least because I will be brief if I am more or less repeating material from it. Let’s proceed without further ado (ado can be found in the last two posts).
In one of the examples I discussed previously we supposed that two people are ‘related’ to one another or ‘equivalent’ if they were born in the same month. The main observation of this post is simple and can be summarized thus: This relation, called an equivalence relation, divides the set of all people into twelve different sections, one for each month. I mean this in the obvious way: Each section consists of all the people born in a given month. So me, David Cameron, Julia Roberts and Bill Gates are all in the October section. There are some observations which we ought to note down about these sections. Firstly, within a given section, everyone is ‘equivalent’ to one another. This is a very obvious observation. Secondly, the sections don’t overlap at all. This is obvious because it is impossible to have been born in two different months. Thirdly, everything falls into one of the sections. You have to have been born in some month, so everybody belongs to one of the sections. These sections are the focus of this post and are called the equivalence classes of the relation.
If we suppose that two whole numbers are ‘equivalent’ if they leave the same remainder when divided by seven, then the set of all whole numbers gets divided into seven bits. One bit for all the numbers which are exactly divisible by seven like seven, fourteen, twenty-one etc., one section for all the numbers which leave a remainder of one like eight, fifteen etc… … … and one section for all the numbers which leave a remainder of six like thirteen, twenty etc. So, we say that there are seven equivalence classes.
We’ll now pause for a moment and take some time to think as abstractly as we can about what we know. This will be useful because abstraction allows generalisation. If we understand the idea on an abstract level, we might find it easier to apply the idea to new situations.
So we start off with a collection of objects. Mathematicians tend to refer to basic collections of things as sets and to the things they contain as elements (e.g. a person is an element of the set of all people). So, we start off with a set (e.g. set of all people, all whole numbers). We have a relation on this set, between pairs of elements (e.g. ‘same month of birth’, ‘same remainder on division by seven’). This relation divides or partitions the set into equivalence classes (examples of equivalence classes of the relations I’ve mentioned include ‘people born in October’ and ‘numbers which leave remainder 5 when divided by 7’). The terminology I have just skimmed over can be used to record and express our idea abstractly and succinctly; the way it would be recorded in a maths textbook or in a lecture might go something like: “Let ~ be an equivalence relation on the set X. Then the equivalence classes of ~ partition X”.
But how does all of this relate to numbers? Remember, this all started with me arguing here (the first post of the series) that there was an abstract concept of number which was different to the idea of numbers as quantities of things. We are nearly ready to try to understand my candidate for this abstract concept of number. We’re about to attempt to give a definition of the abstract idea of number. For this we will not be able to use the idea of number, or else our definition will be circular because it will refer to itself. Just to be clear, this isn’t an easy task. We aren’t allowed to use the idea of number in order to define the numbers! So what are numbers?
Let’s start with quantities of things and abstract away from there. Suppose I have some apples:
And some oranges:
How do you know that the quantity of apples is the same as the quantity of oranges? There are two ways of doing this. The first is the way in which you instinctively do it. That is by recognizing that there are three apples and then, separately, recognizing that there are three oranges. Now you know that the quantities are the same because three is equal to three. You could do this if there were ten apples and ten oranges: You could count the set of apples and then count the set of oranges. The second way of recognizing that these two quantities are the same is because you can match up the set of apples with the set of oranges: For each apple there is exactly one orange and for each orange there is exactly one apple. You can simply pair them off. Therefore the quantities must be the same. This second method is of more interest to us because it uses no notion of number. It relates somehow to the identity of the number but without mentioning it explicitly. Perfect. This idea of pairing up the elements of two different sets will be at the heart of our notion of number.
If all we care about is number then the set of oranges in the picture above is equivalent to the set of apples in the picture above, because they both have ‘the same number’ of things in them (whatever that means). So, as far as we’re concerned, the two sets are equivalent. By this I mean that our notion of number should not distinguish between them: The ‘three’ in ‘three apples’ is conceptually the same as the ‘three’ in ‘three oranges’ (I am informed that linguistically speaking, this is not true in every language for every pair of nouns! i.e. some languages use different words depending on what’s being counted…But that’s a different matter!) If you are familiar with the first post in this series you might see where I’m headed with this. Let’s suppose that two sets of things are ‘equivalent’ if their elements can be paired off. I’ll do another example. Here is a set of elephants:
Here is a set of dots:
So, what is going on here? We’d like to think about a big collection of sets. Within this collection, two sets are equivalent if their elements can be paired off in the way I have described. Is this an equivalence relation? – Does it satisfy the criteria outlined at the end of the previous post? If so, what relevance do the equivalence classes have? For now I won’t say any more but these are nice little questions. It is instructive to have a think about them and they will be the subject of my next post.
In the fourth (and hopefully final) post in the series we’ll talk about this relation in more detail, answer the questions and think about whether or not we are any closer to a definition of number!