I was cheerfully living my life a couple weeks ago when suddenly I started wondering: do the real numbers actually exist? (I might not have been fully sober.) I spent some time researching this question and found the results to be remarkable, so I’m sharing what I went through and what I learned in the process.
Disclaimer: I am not a mathematician or philosopher; what follows is my best understanding of the literature, and for brevity I’m going to skip putting “In my understanding…” everywhere.
We all have an intuitive understanding that there are abstract numbers that refer to the measure of something, and that these numbers are continuous. We may have seen a proof that the rational numbers in some sense aren’t “enough”, so we need some sort of other kind of number to fill the “gaps”, which can be defined technically.
But we also know that we can’t simply name a property and then consider the set of things with that property. And even if we could, we can’t assume that the set is non-empty. So while we might understand a lot of the properties of the real numbers, do they actually exist?
I did some searching and was pretty disappointed. What I found are proofs that the rationals “aren’t enough” and thus we think about the real numbers, but I couldn’t find anyone talking about whether these things actually exist.
In particular, it seems intuitive that there are countably many descriptions of numbers (assuming that descriptions are finite-length strings of some finite language). And the real numbers are uncountable, so accepting the existence of the reals means that there are numbers that exist but can never be described. And this feels deeply unsettling to me: if we can never interact with these numbers even conceptually, it would be impossible to distinguish between two universes where in one they exist and in the other they don’t. Which to me makes their existence feel tenuous at best.
I decided to try to dig down through the math to determine what actually implies the existence of the real numbers.
I looked at Dedekind cuts, one of the ways to define real numbers. It seems that the existence of real numbers is equivalent to the existence of Dedekind cuts.
But Dedekind cuts are just sets of rational numbers. And I believe the rational numbers exist (though more on that later). So the existence of the real numbers is equivalent to the existence of particular sets of things that I’m assuming exist.
So now we need to dive into set theory. Modern math is based on the Zermelo-Fraenkel set theory axioms, and when combined with the axiom of choice are called “ZFC”. I had vague knowledge that this is what was at the “bottom” of math, but I didn’t know much about it.
It turns out that ZFC is just nine axioms and are superficially straightforward. For example one of them is “if you have two things, there is a set of those two things”.
So now we can see where the real numbers come from: we use the axiom of power set to generate all subsets of the rationals, and within that power set the Dedekind cuts exist.
This makes a sort of sense as an ending point for this investigation: my original worry about the real numbers (that we can’t describe most of them) applies equivalently to the axiom of power set. If we can’t describe all of the elements of a power set, I have the same unease that those elements exist.
I believe my unease here is similar to the idea behind Skolem’s Paradox, which says that there’s a countable model of ZFC. In essence, unless you add additional intuition to the system (which it seems like people are willing to), you can’t even be guaranteed that the axiom of power set will generate an uncountable set, because the axiom of power set only quantifies over things that actually exist, and there might only be countably many things that exist.
But at least this answered my question: the real numbers exist because most mathematicians believe in the axiom of power set (and don’t worry too much about Skolem’s Paradox), which in turn implies the existence of the real numbers.
It’s a bit unsatisfying though: the real numbers exist because we believe they exist?
Now things start getting a bit weird. I had been a bit loose about my ideas of what things “exist”, but now I’m forced to confront it.
It turns out that there is a vibrant disagreement about whether mathematical constructs exist at all. The topic is too wide for me to summarize, but I found the Wikipedia page on the topic to be fascinating.
In particular, some people say that mathematical objects don’t actually exist, we just have particular rules for manipulating formulas that happen to lead to really nice real-world applications.
Within the other group of people, the ones who say that mathematical objects do exist, there is disagreement about (in my view) what existence means. While I don’t fully understand the nuances of space, I vaguely understand my unease about the reals as putting me in the Constructivist camp.
So while in the end it is a bit unsatisfying — the existence of the real numbers is a philosophical question which in some sense can probably never be answered — I learned that this is well-trodden ground, and how my views fit within the taxonomy of beliefs.