What actually is a number? Do numbers and mathematical objects actually exist, or have they been invented by mankind with no independent existence in the real world?
There are many different philosophical standpoints to take on this but the one I want to talk about is Platonism. Plato was an ancient Greek philosopher but his ideas still resonate today: many of the greatest mathematicians over the years have been, and still are, ardent Platonists.
In order to get an intuitive handle on Plato’s ideas, we’re first going to look at some mathematical “picture proofs”, that is, mathematical proofs that are not done in the standard way. Normally, in maths we start from what are called “first principles”, which are a set of the most basic rules, such as 1+1=2 and 0+1=1, and then we say that a particular theorem is “proved” if we can arrive at that theorem having started from these first principles and used only logic to arrive at the theorem.
However, we are now going to look at some intuitive and so called “non-standard proofs” of mathematical statements, which will in turn give us an insight into what Platonism is all about.
PICTURE PROOF 1
For the first picture proof we’re going to look at the sum, labelled S, of the odd integers up to some arbitrary odd integer which we call 2n-1. If you’re wondering why we have chosen 2n-1 it’s because an even number must take the form of some integer, n, multiplied by two, so to get an odd number we take 2n and subtract 1.
Before we go through the proof, look at the diagram below and use it to see if you can work out what the sum of the odd integers is equal to.
We can see what is going on by taking the sum one term at a time. To start with let’s look at just the first term, where we have S is equal to 1. To represent this we will put a single dot on the screen, coloured red in the diagram below. Now let’s take the first two terms. S is now equal to 1 plus 3, so we have to add three (blue) dots to our original dot. You will notice our dots now form a 2 by 2 square.
Next we add in the third term, and so we have to add five (green) dots to what we already have, and now we have a 3x3 square. And this process will continue with each subsequent term we add, namely, with each term we add, we are building a bigger and bigger square of dots.
Suffice to say that when we have n terms in the sequence, our dots will form a square of side length n. Or to put it another way the total number of dots, and therefore the sum of the odd integers up to the nth term, is equal to n multiplied by n, or n squared.
So here we have used very little (“traditional”) maths, but instead a picture to arrive at a non-obvious mathematical result.
PICTURE PROOF 2
This next picture proof I will not prove but leave you to think about.
PLATO & THE MATHEMATICAL REALM
So what is Platonism in Maths? What Plato says is that mathematical objects are entirely real and they exist independently of us. That is, maths is not something that has been invented by people; it is not something we have come up with ourselves. Rather, mathematical objects like numbers are perfectly real, and they do have a genuine existence outside of humans and human thought.
To clarify, when we say mathematical objects are real we do not mean that they are physical objects, in the same way that a house or a cat or a tree is physical. We are not going to go walking in the woods and suddenly trip over the number 73. Rather, numbers and objects in maths exist outside of space and outside of time, they exist in what we might call the “mathematical realm”, but nevertheless they do still exist, they are still “real things”.
And because Maths is “real” and not merely a set of rules invented by us, that means that there is such a thing as mathematical truth: a statement is true in maths because it either is or isn’t true, and a statement could still be true even if it has not been proved in a traditional way.
This is the key point: Platonism, more than any other account in maths, is open to the possibility of an endless variety of investigative techniques. We have not “defined” the mathematical realm, we are simply “exploring” it, just as there are numerous ways we might learn something new about the physical world, such as observing something directly, or using thought experiments, or hypothesising and then testing the consequences. In the same way, Platonism can be similarly liberating for mathematical research.
In Plato’s view, although mathematical objects are not physical, we still have access to them. We have a kind of access to the mathematical realm that is something like our perceptual access to the physical realm. Mathematical entities can be perceived and grasped with, for want of a better phrase, the mind's-eye.
This somewhat explains why we can intuit certain mathematical objects, it explains why people feel the compulsion to believe that 2+3=5. It’s like the compulsion to believe that grass is green. In each case we see in some sense the relevant objects under discussion.
The picture proofs shown may have felt more like pretty pictures than thorough mathematical proofs, inferior compared to the “traditional” proofs where we start from first principles and write down a sequence of equations. But within a Platonistic account, there is no reason to doubt the existence and effectiveness of using non-standard means for learning about the mathematical realm.
There are certainly serious misgivings with taking a Platonist view to Mathematics, the main problem being the following question: if mathematical objects exist independently of us, outside of space, and outside of time, then how could they in any way be accessible to us? How could we ever come to know anything about them? There is a lot to say on this, as you can imagine, but what we can say at this stage is that perhaps some pictures are not really pictures at all, but rather windows into Plato’s mathematical realm.
Written by William Brooke, Director of Witherow Brooke