For no particular reason i started thinking about math philosophically recently, and I came upon a thought that … well, I’ll just be honest … intrigues me.
So, being interested in discussion, I guess I’ll just put it out there and see if this resonates with anyone.
I was pondering the question: What is a number? Not in the sense of, what’s the definition of a number? But instead, getting to the philosophical question? In the same way that Plato would ask the big “What is … ” questions, like “What is justice?” “What is being?” “What is life?” So in this sense, in the Platonic sense filled with wonder … I started to explore the question: What is the nature of numbers?
Limiting myself, at first, to thinking about the counting or “natural” numbers, I sensed that this set of numbers is, largely, a tool that humankind has developed to help us get a measure of control over our world. As an illustration, instead of looking at a bunch of sheep and just saying, “Geez, I’ve got a lot of sheep,” we could take the further step and say, “In fact, I have 87 sheep.” Using numbers as a tool in this way gives us control because it lets us know exactly how many of a thing we have. And that lets us compare and contrast how many we have to how many other people might have. And that in turn allows us to know if our flock (in this case) is increasing or decreasing. Almost needless to say, getting this quantitative knowledge over the things that matter to us allows us to have more control over those things. So in this respect numbers — as a whole — are a tool that give us a measure of control over our environment.
And even if we extend the set of numbers to fractions, we could still say that those fractional numbers give us a measure of control over things we care about. It is far more precise to say “I still have three-fourths of a bag of rice” than to say “I have the better part of a bag of rice left.” And so on and so forth. Fractions also give us a precise sense of how much we have, and that knowledge gives us exact information about things that matter to us.
But then — and this is where it gets interesting for me — if we think instead about irrational numbers, it seems that something profound changes. Irrational numbers have the remarkable twin properties that: 1) their decimal expansion both never ends and also has no discernible pattern, and 2) irrational numbers cannot be expressed as a ratio of integers. So when working with irrational numbers, we can never say exactly how much of a thing we have. Yes, we can approximate, but we cannot state the quantity exactly.
So if this is correct, that means that when we are faced with an irrational number, there’s a limit to how precisely we can know the quantity being pointed at. For example, if we are looking at a square whose side length is 1 inches, and if we point at the diagonal of that square, we cannot get a completely controlling handle on how long that diagonal is. That is because the length of the diagonal is the square root of 2 inches, and the square root of 2 is an irrational number. So how long is it? Well, it is more than 1.4 inches, but less than 1.5 inches. And yes, we can be still more precise and say that the value of the square root of two is greater than 1.414 but less than 1.415. Given computer algorithms for carrying out the decimal expansion to n decimal digits, we can carry out this analysis to as many place values as we would like. But at the end of the day, even though we can achieve whatever degree of precision we might desire, we can never fully capture the complete value of this number. We can never know it completely, the we that we can know a rational number like, for example, 3/8.
So if we can never know the value of an irrational like the square root of two, doesn’t that suggest that this number, in some sense, exists on its own, independent of our number system? If you agree with this assessment, then the moment that we discover this irrational quantity, we hit upon some kind of bedrock, some kind of thing, something that has, in a sense, an existence that is independent of our mind. Now instead of us controlling and using the numbers, we are in the subservient position of trying to figure out the exact nature of this quantity — and failing for thousands of years now to completely nail it down. And to that extent, we need to give these irrationals our respect. No, we did not invent the irrationals. And no, the irrationals are not the tool of humankind. Instead, humans have merely encountered the irrationals.
This reminds me a bit of the book/movie Jurassic Park. In the same way that the scientists and park operators believed that they could control dinosaurs they would re-animate, but found out that they could not control them (at all!), we the humans believed for a long time that we were fully in charge of numbers. Yet as soon as we hit upon the irrationals, we found out that we are not in charge of these numbers. They have an existence all their own.
One fairly modern side note to these ramblings comes from the work of George Cantor. This great explorer of infinity, derided in his day but celebrated later, discovered that there are, in fact, two levels of infinity when it comes to number. There is the infinity of the rational numbers, which is a countable kind of infinity, an infinity that bears a one-to-one correspondence with the natural numbers. But then beyond that infinity there is a separate and vastly greater infinity, the infinity of the irrational numbers. Cantor actually discovered that the number of irrational numbers is so much greater than the number of the rationals that its magnitude cannot even be expressed; it can only be hinted at.
And going back in history for one another bookend, another reference to the power of the irrationals … we come to the Ancient Greek mathematician Hippasus. As the legend has it, Hippasus, after entering the Pythagorean brotherhood, made a discovery that greatly affected the Pythagorean lovers of number. Hippasus discovered an irrational number. He was playing around with the diagonal of a square and discovered through a reductio ad absurdum proof, that the diagonal of a square cannot be simplified to a rational number. Up until Hippasus, no one had ever discovered an irrational number. How was Hippasus rewarded by the Pythagoreans for his discovery? According to legend, they took him out to sea for a lovely picnic on a boat. And when the picnic was over, they threw him overboard and sailed away. So much for poor Hippasus. Perhaps, as he was drowning in the sapphire waters of the Aegean Sea, he realized that he was paying a price for discovering that humankind does not have as much power over numbers as we had thought.
But even though the Pythagoreans put Hippasus to death, they could not put his ideas to death. The existence of the irrationals came to light hundreds of years later. It was discovered again, not invented. It has the power to be found because it exists.
So what are numbers? Some numbers are things that give us power. But others, the majority of numbers, have been there all along, and we are just the explorers who came along, noticed, and pointed.