Imagine an arrow in flight. At any given instant in time, the arrow is not moving, because an instant is a snapshot. If it cannot move in an instant then it cannot move in any instant and motion is therefore impossible.
This is one of Zeno’s Paradoxes! Here is another:
Suppose that Homer is running for a bus. Before he gets to the bus he must get halfway there. Before he gets halfway there he must get a quarter of the way there and before that an eighth of the way there and so on. Completing the journey requires completion of an infinite number of tasks. Is this possible? Zeno maintained that it was not.
Mathematical thought has something to say about these paradoxes. We will by no means lay to rest every possible concern of the philosopher but in this post and the next we will investigate the mathematical ideas which are brought to light by these things and then move on to discussing the continuum hypothesis.
In elementary school, we learnt a lot about how whole numbers were written. We learnt that, for example. the number “three hundred and twenty-seven” is written with a ‘3’ in the hundreds’ column”, a ‘2’ in the tens’ column” and a ‘7’ in the units’ column. This knowledge formed the basis of how to add and subtract: You’d start on the right with the units column and you ‘carry’ if you need to, i.e. if the units column adds up to more than nine. Let’s think about another number, say 2735. Using this knowledge of how to write numbers down we can observe that
2735 = 2 x 1000 + 7 x 100 + 3 x 10 + 5 x 1,
and writing it in this way really highlights the decimal nature of our number system. When we’re a little more experienced we extend this idea to decimal fractions. We say that the number 24.56 has a ‘5’ in the tenths’ column and a ‘6’ in the hundredths’ column. Again, using our knowledge of decimal notation, we observe that
24.56 = 2 x 10 + 4 x 1 + 5 x 1/10 + 6 x 1/100.
Now, we are nearing our big question. It is well known (or you can take it from me now!) that the decimal representation of the fraction 1/3 is “nought point three recurring”, which means that it cannot be written down in full because the threes in it’s decimal representation (0.333333333333…) would go on forever! A recurring decimal is something which one would typically meet pretty early on in school, but I want to draw your attention to the fact that there is something left unexplained. After all, using the same idea as above we observe that
1/3 = 3 x 1/10 + 3 x 1/100 + 3 x 1/1000 + 3 x 1/10000 + . . . . .
What we have here is an infinite sum. This is called a series.
How does one make sense of a series? I needn’t worry about the concept of “nought point three recurring” per se, because that is just shorthand for “one third” and obviously I can cope with one third. However, if we are unable to make mathematical sense of this infinite sum, then the whole system of decimal notation is a bit rubbish because it can’t actually write down one third properly. As I highlighted in my earlier examples, surely the whole point of the tenths’ column, hundredths’ column etc. is that equations like the one above should be true?
Now, clearly some sort of sense has been made out of this particular series above, because we are under the impression that it should be equal to one third, but how? By what process or through what rigorous mechanism is meaning attributed to an infinite sum? Even if we can attribute the correct meaning to this series and explain convincingly why it is equal to one third, will we then be able to use our method to deal with other series? How about 1-1+1-1+1-… or 1 + 1/4 + 1/9 + 1/16 + 1/25 + … (here I am going through the square numbers). This is a great example of some ‘proper’ mathematics because it is actually about sums, a very basic idea that everyone is familiar with, but in this context our focus is not on the answer (e.g. one third – we know the answer!) but on the argument. Why is it equal to one third? You cannot actually compute the sum because it goes on forever so for your school teachers to claim that one can make sense of it mathematically, for one to claim that this sum actually adds up to one third and doesn’t just ‘represent’ it somehow, one needs a convincing argument.
There is a convincing argument which I will mention only briefly (partly because it is not the main topic of the post and partly because it becomes increasingly convincing the more you learn about its consequences and the more you use it, so I couldn’t possibly convince a sceptic outright in one post). I should say that each number that makes up the infinite sum is called a term of the series. The argument is based on the idea that if one were to keep adding on successive terms of the series, eventually the cumulative total will be very close to one third. The crucial point is that if you add up enough the series (like “the first 1000 terms”), you can get the total as close as you like to one third. By ‘as close as you like’ I mean that if you gave me a very very small number, I could add up loads and loads of terms of the series: enough so that the difference between my running total and the unobtainable goal of one third would be smaller than your very very small number!
It is indeed true that this process doesn’t work for every series, but that does not diminish its usefulness. For example, take 1-1+1-1+1 -… The running totals for this series somehow don’t ‘settle down’ enough. It just goes between one and zero forever.
So, when I describe a decimal representation of a number which goes on forever (like 0.23232323…) we now know what that actually means: It means that the series which the decimal expansion defines (2×1/10 + 3×1/100 +…) actually sums up to the number being represented (in the way described in the previous section).
A number represented by a decimal which eventually repeats itself, like a whole number or 3.56 (where it is implicit that the rest of the entries after the ‘6’ are zeroes) or 900934.33333…. is called a rational number. A number represented by a decimal which does not exhibit this behaviour is called an irrational number.
The rational numbers and the irrational numbers together form the real numbers: The number line consists of precisely these numbers and nothing more. Do remember the excuse I am using for bringing up the real numbers in this post: It is that they are uncountable. So, there are vastly more real numbers than there are natural numbers. It is perhaps of interest to note that there are only countably many rational numbers, so the irrationals make up ‘most’ of the real numbers.