What exactly are numbers? How can one or should one think about numbers or picture them? How many of them are there?

This is the first in a series of posts on numbers.

I’d like to avoid becoming bogged down in philosophy, but in my recent posts about the number line, I failed to actually explain what I meant by ‘numbers’. If you have read the number line posts this fact may have passed you by completely, but, in order not to get sidetracked, an assumption was made about the readers’ notion of number, namely that it was the same as my own. This could have led to some confusion, though of course I hope that it did not. Even if you have not read the posts, the idea of explaining what numbers are may still seem silly and not worth worrying about. By the end of this series of posts I hope to have convinced you otherwise. In this first post we won’t really be doing any maths, we’ll just be discussing the concept of number. In the next post we’ll start to talk about numbers on a more mathematical level.

I should stress that I am not attempting to define number. I am taking for granted the fact that we have some notion of number already and will then challenge the reader to think hard about what that notion is. Some interesting points about the definition of number are outlined on this wikipedia page.

The Concept of Number

I’d like to introduce the idea of an abstract concept of number. I will argue that we use an abstract idea of number much more than we do a ‘concrete’ one, even for thinking about fairly small numbers. [I will be making some generalisations here based on my own experiences of how people think about numbers but if you have a sufficiently different approach to numbers then I’d be interested to hear about it! Leave a comment!]

I saw two elephants on my road yesterday. By reporting this to you, I am giving you the ability to understand precisely the quantity of elephants which I saw on my road yesterday. I could be lying to you, and it wouldn’t even matter, because you weren’t there to see the elephants anyway! You could still understand the quantity I was talking about but either way you are only imagining these elephants. Two of them.

Suppose another elephant arrives. All in your head, I would like you to actually visualize another elephant joining them. There are two elephants to begin with. Imagine another one walking over and joining them. In your mental picture, count how many there are now.

To see where I’m going with this, read on. Last week I saw thirteen elephants on my road. I would like you to imagine this group of thirteen elephants, standing on my road. Try hard to see them. Now suppose that three more elephants arrive. Again, purely in your mind, I would like you to actually visualize three more elephants joining them. How many are there now?

My guess is that you were able to answer easily in both cases. The answers were ‘three’ and ‘sixteen’. The point of this discourse on elephants is, however, that if you followed the exercise as intended, you were able to genuinely picture one elephant joining a group of two, but struggled to picture three joining a group of thirteen. In the first instance you would have been able to visualize things accurately and actually count the number of elephants directly from your mental picture, as if I had shown you a photograph of three elephants and asked you to count them. If you didn’t do this then give it a go now if you like. In the second instance, you will almost certainly have used some sort of shortcut or trick to avoid having to do this, because it is difficult. It is remarkably difficult to actually think about thirteen things. Were you honestly picturing thirteen elephants or just ‘many’ elephants? Can you ‘see’ all sixteen elephants at once and actually count them or did you give up and work out the answer some other way? What was that other way? [If you had no trouble with thirteen then try slightly larger numbers. Most people run into trouble fairly quickly. If you do not then you may have incredible savant-esque skills.]

In a similar vein, how many dots are there here?

How many are here?

And here?

Not many people would actually need to count how many there are in the first two pictures. They can simply recognise the quantity immediately. Most people would need to do some counting, in one way or another, in order to work out how many dots there are in the third picture.

Hopefully I am illustrating to you that we often deal with numbers as small as thirteen differently from the way in which we deal with really low numbers such as ‘two’,  ‘three’ and ‘five’. Indeed there are thirteen dots in the last picture. If you needed to stop and count to work this out, then you weren’t able to recognise the quantity. You don’t know what thirteen things looks like. Perhaps not ‘knowing’ what thirteen things look like prevented you from picturing thirteen elephants accurately just moments ago.

I am trying to convince you that there is an abstract concept of number. It is a concept which you have a decent understanding of, unlike concrete ‘versions’  or ‘realisations’ of numbers as quantities, such as ‘thirteen elephants’ or ‘thirteen dots’, because although one may find it hard to picture thirteen specific things, one is still at ease with the idea because one is able to fall back on and rely on one’s understanding of the concept on an abstract level. For example, you probably made use of this when you were counting the total of thirteen elephants and three elephants; you were able to avoid counting directly by performing the mathematical ‘sum’: 13 + 3.

Less profound discussion in this vein can be found at xkcd.

In the next post I will begin to give some insight into how a mathematician might try to think about what this abstract concept of number actually is