In my last post, I discussed the existence of an abstract concept of number, separate from concrete realisations of numbers as quantities of things.
In this post I would like to talk more about this abstract concept. What is it? I hope to explain a very important idea in mathematics, the application of which (to the issue of what numbers are) will not appear until the next post. For now, just read and enjoy.
The reader will be familiar with the idea of equality in arithmetic. For example, conceptual nonsense aside, everybody is aware that four plus one is equal to five and that three multiplied by two is equal to six. There are three facts about this idea of equality which I would like to bring to your attention. They are all very simple and may well seem trivial to the reader, but do bear with me.
The first is that any thing is equal to itself. It is perfectly legitimate to say that one is equal to one, because it is. Or that two times three is equal to two times three, because it is.
The second fact may be illustrated thus: “Two plus one is equal to three. Therefore three is equal to two plus one”. This statement is also almost too obvious to have any content. It is just the fact that when we have an equality between two arithmetical expressions we can flip it over (left/right, across the middle) and it is still true.
The third fact is something we often use when doing arithmetic in our heads. Suppose I ask you to multiply eight by six and then add thirteen. Naturally, you would do this in two stages: First you would muliply eight by six, leaving you with the sum “fourty-eight plus thirteen”. Now you would compute this sum to get sixty-one. I want you to think about what happens now, after you’ve done the hard work but just before you announce the answer. Having done these two, separate calculations, you then concatenate them into a single equality. This is the important fact about equality which I want to draw your attention to. Remember, the calculations which you directly verified were (deep breath) “eight times six, plus thirteen is equal to fourty-eight plus thirteen”… and (deep breath), “fourty-eight plus thirteen is equal to sixty-one”. From these two equalities you deduce one equality. To summarize using letters instead of numbers: If a is equal to b and b is equal to c, then a is equal to c.
These three simple facts about equality are part of a bigger idea which is very important throughout mathematics. In order to facilitate the coming abstraction, let’s think about what we just did in more general terms. What we had was a collection of objects (arithmatical expressions like “five plus one” etc.) and a possible relation between pairs of these objects (equality). It was the case that for any pair of objects, either they are related to one another (i.e. “five plus one” is equal to “two times three”) or they are not related to one another.
Levels of Discernment
Let’s look at a completely different example in order to illustrate the width of the applicability of the principle.
I share my birthday with Bill Gates.
I’m now going to make an identical series of trivial observations.
Firstly, any person is born in the month in which they were born in. For example, it is logically legitimate to say that Bill Gates was born in the month that Bill Gates was born in.
Secondly, if Bill Gates was born in the same month as Julia Roberts…
…then it follows that Julia Roberts was born in the same month as Bill Gates. I’ve just switched round the names.
Thirdly, if Bill Gates was born in the same month as Julia Roberts, and Julia Roberts was born in the same month as David Cameron…
What we have here is a collection of objects (people) and a possible relation between pairs of these objects (sharing month of birth). It was the case that for any pair of objects, either they are related to one another (me sharing the month of my birth with David Cameron) or they are not related to one another.
The relation of ‘sharing month of birth’ is not the same as the equality relation of the previous section: I am not equal to David Cameron (nor is my birthday equal to his birthday) and yet, as far as this relation is concerned, we are the same, because we were born in the same month. I am somehow being a lot less stringent when deciding when two things are equivalent. I’m saying “well…it doesn’t matter if you’re two different people, I only care about the month of birth being the same”.
Another example is furnished by considering the collection of whole numbers and asserting that two numbers are equivalent if they leave the same remainder when divided by, say, 24. Now twenty-five is equivalent to one. Forty is equivalent to eighty-eight. Again, the same three ‘obvious’ facts are true and again it is a case of somehow being far less discerning about when two things are ‘the same’. After all, there is a sense in which we think of 1pm today as being the same as 1pm tomorrow, even if they are clearly different times because there are twenty-four hours between them.
So, I think we might be getting the point by now. Let’s finally make all of this abstract so that we can see the big picture.
I am describing something which mathematicians call an equivalence relation. Here comes the maths. Suppose I have a collection of objects. There need not be finitely many of them (it might be the collection of all numbers, for example). An equivalence relation is a special type of relation between pairs of objects. Firstly, two objects are either related or they are not related to one another. Secondly, a relation is only an equivalence relation when three additional conditions are met. At the risk of bombarding the reader with silly terminology, the three points are:
- [Reflexivity] Any object is always related to itself.
- [Symmetry] If a is related to b, then b is related to a.
- [Transitivity] If a is related to b and b is related to c, then a is related to c.
As alluded to in the first two examples.
Because non-examples are also sometimes helpful: The relation of ‘being siblings’ satisfies 2. and 3. but not 1., because you are not your own sibling.
To get an idea of where these ideas fit in: It’s the kind of thing one learns during the first week of a maths degree. (I was asked about transitivity at my interview, though I was not expected to have seen it before).
The idea is that an equivalence relation is a relation which is like equality but usually ‘weaker’. At times it is very useful to say that two things, which aren’t actually equal to one another, are the same as far as you’re concerned. For example if you were challenged to find somebody who was born in October, then as far as you’re concerned (i.e. for the purposes of this (bizarre) challenge), I am equivalent to Julia Roberts. More relevantly, one often comes across situations in which two mathematical objects ought to be considered equivalent, even though they are not actually the same. One such situation is for answering a problem like “What time will it be 400 hours from now?”. The point here is that for the purposes of this (less bizzare) challenge, 400 is equivalent to sixteen (i.e. 400 leaves a remainder of sixteen on division by 24).
In order to make precise this notion of two things being ‘the same as far as you’re concerned’ we have equivalence relations.
In the next post we’ll finally get on to talking about what numbers are! We will use equivalence relations.