Platonism is a philosophical position that originated with Plato’s theory of forms. It is the belief that abstract entities such as mathematical objects are “real”, and independent of human thought, and that they “actually exist” outside of time and space. This philosophical position is hotly debated, and that’s largely because those words I put in quotes are themselves hotly debated. What does “real” mean? What does it mean to “exist”?
Well, it seems that following the success and popularity of the scientific method, many people subscribe to an empiricist view of such things. They believe that we only know things about the world by means of sensory experience, and that the only things which “exist” in the world, and are “real” are those things which affect that sensory experience, those which are observable. By this understanding of reality, it would seem that mathematical objects don’t exist. The very idea of an object existing outside of time and space is meaningless, since anything in our experience is going to fall into the realm of time and space. I’m going to argue that there’s something fundamentally lacking with this picture of reality.
Let’s start from a simple place: what does it mean that the chair in front of me exists? How do I know it exists? Empiricists would like to start from the obvious intuition that I can perceive that chair. They reason from there that the chair exists, and that I know that fact (at least with a high degree of certainty). Let’s make the picture a little more complicated: what if I now leave the room? I can still reasonably conclude that the chair exists, even though I am not directly experiencing it. Empiricists will now expand the realm of knowledge and existence, and allow me to reason about what exists based on my past experience. That makes sense. In a similar way we can conclude that the atoms which make up that chair exist, even though we cannot directly experience them, since we know from other experiences that they must.
But let’s return once more to the original question of the chair right in front of me. The whole framework rests on the idea that I perceive that chair. What does that mean? There is a lot of complex processing going on in our brain which results in the experience of a chair. Complex processing that is sometimes wrong; after all, people hallucinate things that aren’t there, and they can mistake one object for another. If we want to base our understanding of reality on the fundamental sensory experience, we have to break it down all the way to the raw stream of information that our brains get, and sensory information does not by itself suffice to give us knowledge or define the world. So we have to ask the question how do we get from that raw stream of information to a chair?
Let’s take a step back. I started all of this by pointing out the effectiveness of the scientific method, because I think it’s the impetus for the rise of empiricism. Let’s revisit that discussion. As Feynman explains, the scientific method consists of theory, prediction, and experiment. We guess a rule of the universe (theory), calculate predictions from that rule, and then test those predictions against experiment. People often focus on the last step of that process, and forget about theory, which I think is exactly what’s happened with empiricism. In particular, it has things backwards. It takes perception as primary, and theory as built on that perception. This is the same as thinking that in the scientific method experiment comes first. Experiments serve to test our theories; we must first have a theory to test. In the scientific method, theory is primary.
So what does it mean that I perceive a chair? It means that I theorize the existence of a chair, and the predictions of that theory are borne out in experiment; my input stream of experience agrees with that theory. I predict that if I move my hand out, I will bump into the chair; and that’s exactly what happens. I predict that I will see light in just such a way, and I do. I predict that other people will say they also see the chair. So let’s make things more complicated again: what does it mean that the chair exists when I have left the room? Once again, the chair is a theory about the world, and whenever I make predictions from that theory they are borne out in my experience. I predict that if I return to the room, I will perceive the chair, that if I ask someone to get me that chair from that room, they will be able to, and these predictions are verified. Atoms are also a theory, borne out in experiment. There is no additional work needed to explain why things that we don’t directly perceive, nonetheless exist.
So now we can return to mathematical realism. I claim that the objects in front of me are mathematical objects, just as much as they are physical objects. Just as I theorize that there is a chair in front of me, and that theory generates certain predictions about how I will experience things, so too I theorize that there is a 1 in front of me. And I predict that if I take another 1 of the same type, and put them together, I will have a 2. If I add another 2, and then take away 1, I will have the same as if I take three 1s and put them together. That’s a prediction I make based on my theory of these mathematical objects. These physical objects are integers in exactly the same sense that they are chairs and tables. Another example is a rubik’s cube, which operates under the mathematical laws of a group. When I interact with that cube, I am interacting with a group. That is why I predict it will be solved when I execute a series of steps. Just as I predict I will see it, and hear it if it hits another object. Experience and theory are both key to our knowledge of, and the reality of, the world around us.
So far I’ve argued that the things which inhabit space and time are also mathematical in nature. I promised something stronger though; platonism claims the real existence of abstract mathematical objects, outside of space and time. These are real in a different sense, but I will argue they are still real. We’ve uncovered a picture where theory and experience together build our knowledge of reality, and reality itself. But what about each of those components independently? We consider our experience of reality to be real, so what about our theories of reality? When we thought of experience as primary and theory as derived, it made sense to consider experience real and theories human constructions. Now, though, we consider theory as basic to reality as experience, so they must both be real.
Interestingly, this view of the world solves another long-standing philosophical problem: the brain in a vat. The question is, how do we know we’re not just a brain in a vat plugged into a simulation that is creating this illusion of a world. The answer is essentially: who cares? Our reality is anyways a model we create that explains the sensory experience we have. That model is as good as correct until there is some experience which disagrees with a prediction we’ve made.