Platonic Mathematics: Are Numbers Real?
I used to be convinced that mathematical objects are in some sense real. That numbers have an existence independent of humans knowledge of them. Recently, in a Wittgensteinian turn, I’ve come to the realization that the question “are numbers real” is a bad one.
The heart of the problem is that the question is ambiguous. Math is a set of models that attempt to describe phenomena we observe in reality. Those models are obviously invented. We didn’t discover them carved into a rock somewhere. The phenomena in the world they are describing are obviously discovered. So the answer to the (badly phrased) question is both. “Numbers” are a feature of a model humans invented to describe a physical phenomena. The physical phenomena that numbers model is something humans discovered in the world. The only way we have access to those things is through the intermediary that is our models. But this introduces a new problem.
Back at the dawn of mathematics, we discovered new mathematical truths by studying the world. If you play with pebbles, you will quickly notice that for some sets of pebbles, you can divide them into two piles of equal size, and for some other sets one group will always have one more pebble than the other. These, of course, are even and odd numbers. If you play with them a little more, you may find that for some sets of stones, you can divide them into multiple piles of the same size, and for other sets, you cannot break them into multiple piles of equal sizes, other than one pile with all of them, or a number of piles with a single stone in each. Congrats, you’ve just discovered prime numbers.
So then you set out to design a formal system that captures these facts abouts collections of stones. You come up with a set of axioms describing some facts about them, some rules about how you are allowed to manipulate those facts to generate new facts, and an interpretation of these symbols and rules. If you are cleaver enough, you can come up with a system that exactly describes everything you know about collections of pebbles. This is what we call a model.
But now something interesting happens. Instead of playing with pebbles, we can play with our model to discover new truths. For a long time, it was assumed that if your model was good enough, then anything that was true of pebbles could be figured out by playing with your model. Well, turns out that’s not true. This is one of the things Godel taught us. So there are some things that are true of pebbles, but you can’t discover form your system. At least we still know if we discover it in our model, its true of the pebbles… right?
Unfortunately, it seems the answer is no.
You see, there is this funny thing called Skolem’s Paradox. Its a result of a theorem called the Lowenheim - Skolem theorem.
To explain the thrust of the paradox, I have to elaborate slightly on what a “model” in the technical sense is.
A model is simply an interpretation of a formal system, in which every theorem of the formal system comes out true. A formal system is simply a collection of uninterpreted symbols along with rules for manipulating those symbols. For example, to study the manipulations of pebbles, we came up with a formal system of rules and symbols, and then a model that interprets those symbols in such a way that all pebble manipulations work correctly in the formal system. The domain of a model is the set of objects we can interpret the symbols of the model as referring to. In the pebble example, the domain would be a potentially infinite pile of pebbles, and the ways we can combine, subdivide and otherwise manipulate groups of pebbles.
What Skolem’s Paradox says is that for a single formal system, we can come up with two different models that disagree about the size of the domain. The paradox in particular refers to the real numbers. Cantor proved that there are uncountably many real numbers (don’t worry two much about what that means). So any model we make of the real numbers should tells us that there are uncountably many real numbers. But Skolem’s Paradox assures us that there is at least one interpretation that says there are countably many real numbers.
This is a little disconcerting. One of the theorems in the formal system we use to study the real numbers states that there are uncountably many real numbers. But there is another interpretation of the system where that theorem is still true, but there are only countably many real numbers. Obviously, in this other interpretation the English translation of the formal string would not be “there are uncountably many reals.” Exactly what it would mean is unclear. The take home point, though, is that we want to think that the real numbers have an existence independent of us, and hence that they have properties that aren’t the result of our conceptions of them. Surely something as basic as how many there are should be one of those properties. But it turns out that we cannot develop a formal system that captures that property in an unambiguous way.
Now, what makes a formal system formal is that there are no unambiguous meanings assigned to the symbols. The symbols shouldn’t have any meaning at all. After all, the “formal” part of the name doesn’t refer to tuxedos, but to form, as in shape. So it seems from this point of view that Skolem’s Paradox doesn’t tell us anything we didn’t know already. What interests me, though, is that it says that there are two different models that are not isomorphic to each other.
Isomorphic just means that there is a one to one mapping between two different objects, systems, etc. For example, the text of this post is isomorphic to the sound of it read allowed, where each word on the page is mapped to a certain chunk of sound. If two different models are isomorphic, they are essentially equivalent, and can be turned into each other in an algorithmic way. If two models are NOT isomorphic, then it must be the case that they differ in some tangible way. Maybe one has objects that do not map to another, or there are rules in one that have no counterpart in the other.
So here is where the cookie crumbles. There are things out there called the real numbers. We think they exist independent of us. If they do, they have certain properties. To study them, we develop formal systems and models of them that capture their properties. But we can develop different models of the formal system that supposedly captures their properties which disagree with each other as to what those properties are!
The paradox seems to force us to make a choice between two unpalatable options. Either that the theory of real numbers we have developed is inconsistent, and therefore has no model, or that there is some property of the real numbers that cannot be captured in any (first order*) formal system. Option 1 makes the theory of first order numbers worthless for figuring out what properties the real numbers have. Option two tells us there is some property the real numbers have that we will never discover. So either way, the real numbers have properties we cannot know about.
Now, when we are making models of trivial things like adding and subtracting, we could always go back to the pebbles to see which model “really” captures the thing in the world. But mathematics has gotten so rarefied, so far removed from raw sense data, that it is simply no longer possible to physically verify the results of our model manipulations. So if the real numbers do exist, but the only way we have access to them is through an ambiguous model, how can we be sure they have the properties we think they do? More importantly, if our models imply the existence of some object, how can we be sure the object really exists, if it is possible there is a different model that does not imply the existence of said object?
The obvious place to go from here is to physics, where this question can become extremely pointed, but I’ll save that for a later date.
* Note that Skolem’s Paradox only applies to first order systems. Second order systems have other issues that make them unpalatable to most logicians, even if they are free of this particular problem.