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I recently saw Christopher Nolan‘s latest blockbuster Inception. I thought it was exceptional and it may well have jumped straight into my favourite films’ shortlist. But, imagine the compounded joy when I began to realise it had some links with one of my favourite books of all time, namely Douglas Hofstadter‘s incredible Pulitzer Prize-winning Gödel, Escher, Bach: An Eternal Golden Braid, hereafter GEB . Hofstadter’s book weaves together the works and ideas of these three men in a beautiful way, touching on many many themes throughout, the overall complexity of which is some sort of metaphor for or representation of “how cognition and thinking emerge from well-hidden neurological mechanisms”.
The analogies in this post are tenuous at best and the braid I weave is thin and frayed in comparison but it’s all fun, so let’s weave away.
“This sentence is false.”
Attempting to decide whether or not the previous sentence is true or false yields a paradox. This paradox has been known for centuries and is called the Liar paradox. It is a very simple manifestation of a far-reaching idea, which will, ultimately, ruin our attempt to define numbers via the route we have been going down.
Enjoyment of this post really does require the reader to have understood the previous post.
Remember, at this stage we are trying to define an abstract notion of number, and in order to do so we cannot use numbers!
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. Read the rest of this entry »
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.
Read the rest of this entry »
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.
This is the second of two posts about the number line and I will assume that you’ve at least skimmed through the first. In this post I will be getting to the root of the awesomeness of the number line. If you follow it until the end you will genuinely have understood more than just the gist of the idea.
Collections of Numbers
Picture the number line. Let’s think about numbers which are on the left hand side of the number ‘two’. The number ‘one’ is on the left-hand side of two. As is ‘three-quarters’. As is ‘negative fourteen point three’ etc. Numbers that are to the left of ‘two’ are said to be ‘less’ than two. So, we are currently thinking about numbers which are less than two. Just so we’re all agreed and we know what we’re dealing with, imagine colouring a number blue if it is less than two and black if it isn’t. A portion of your number line should now looks like this:
What is there to say about this collection of numbers, the blue portion of our number line? Read the rest of this entry »
GCSEs are a type of public exam taken by school leavers in the UK when they are around 16 years of age. Included in the standard syllabus is some familiarity with a thing called the number line. It is an object which harbours some ideas which are very important to mathematics at a higher level, but these ideas would not generally have not been taught to GCSE candidates.
I’m going to try to explain what the number line is and why I think it is so great. In this first post (hopefully one of two), we will construct the number line and talk about it a bit. In the second we’ll get to the root of its awesomeness. We’ll start very slowly, try to keep things simple and we’ll try to keep things abstract!
As it is almost certain that this blog will include a lot of mathematics and a lot of talk about mathematics, I decided to set myself the somewhat ridiculous task of deciding what mathematics is…
[I had most of these thoughts in the last few days and so they might be a bit rough, but I felt that I did make some sort of progress. I have also (in a radical coup) deliberately avoided the use of examples, to see if I can explain what maths is without actually showing it to you directly, because, although this post will hardly serve as a terribly practical alternative, I am often irritated by having to explain what it is I do by saying “well, imagine a 3-dimensional sphere…”]
After some (light) research, I found out that in philosophy spiel I am much more of a quasi-empiricist, a fallibilist, humanist and relativist than an absolutist or a foundationalist when it comes to my views on mathematics (both the epistemology thereof and its place in society).
What on earth do these silly terms all mean? I am no expert and am throwing them around willy-nilly to poke fun at them myself. I will explain in a rough manner what I am getting at, and with a lot of bias toward my own point.
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