If you’ve been reading through my blog, you might have noticed I’ve just done something strange. I’ve argued that platonism is true, and then I’ve argued that platonism is false. So which is it? Well, I’d like to argue here that it can be both. I’d like to argue for another mathematical philosophy, called Dialetheism.
Dialetheism is the belief in Real Contradictions. It’s the belief that there are things about the world that are both True, and False simultaneously. I am going to argue for it a little backwards. First, I will argue that it’s true, then I will argue that it’s possible. It seems mathematically impossible, so it’s important to explain why it seems that way, and why it’s not. But I think you will be more motivated to care whether it’s possible if you believe that it’s true, that an impossible thing is true. You will find yourself standing at the brink of the abyss and feel your epistemological world start to crumble beneath your epistemological feet. Then you will grab hold of the epistomological rope I toss you. Mmm… epistemology.
Arguing that dialetheism is true turns out to be the easy part. My favourite illustration of it is probably in your pocket right now. In your phone is the unholy spawn of contradiction, a diabolically dialetheist device, your GPS. When you open your navigation app and you see that little dot showing where you are, you are tacitly signing on to the dialetheist agenda. Let me explain.
You see, science, and physics in particular, has a problem. There is not yet any unified theory of physics. Most physicists think we will find one eventually, but in the meantime we have general relativity, which is Einstein’s theory of gravity, and quantum mechanics. These two theories are incompatible. In certain situations they give different answers to the same questions.
In a sense, then, scientists are already dialetheists. When they apply quantum mechanics in one context, and general relativity in another, the overarching system they use contains contradictions. As usual in physics, The way they go about applying these theories ensures there are no problems, that the two theories never get close enough to each other to touch. As usual in physics, this isn’t formalized and is simply built into the way things are done.
So let’s return to GPSs. There are electrical circuits in your phone, which are made of components so small that they encounter, and have to be designed around, the quantum mechanical properties of the world. These quantum mechanical circuits are calculating your location based on the positions of satellites orbiting the earth, and those calculations requires such precision that these quantum mechanical circuits must be using general relativity to model the satelites’ trajectories. GPSs beautifully synthesize these two contradictory theories.
The common response among scientists has been to cover our eyes and pretend there’s nothing contradictory going on here. Put more generously, they assert that all of these calculations are just approximations. Eventually we will find a unified theory and then it will all make sense. But in the meantime, we are using a contradictory theory, and it works! That should be shocking enough that we would want to understand it, not sweep it under the rug.
Let me reiterate this for clarity. The overarching theory that physicists use today contains contradiction. The assumption is that at least one of these theories must be wrong, and we will find a theory that matches or exceeds both in the predictions they make, and gives us a clear answer for the cases that the two theories contradict on. We might find such a theory, but what I’m interested in here is our current understanding. That is the Truth by which we live, even as we seek to improve it.
There’s something implicit in my argument so far that I want to make explicit: what is Truth? I developed this more in my last two posts, but in short I would claim that Truth consists of theories, which accurately explain and predict experiences. This empiricism seems to me the foundation of the scientific revolution, and it’s worked pretty well for us as a species. In any case, it’s what I’m relying on here.
I’ve argued that dialetheism is a part of our best theory yet, our current Theory of Everything. Well, there’s still one massive hurdle to clear before we can walk away from this discussion as dialetheists. And that’s that this is not how math works. Math is logical; it doesn’t allow for contradictions. Are we going to throw out math, and just rely on our feelings to tell us what’s true? Is everything “relative” now? Not physics! Not my beautiful physics! This, I would claim, is why scientists cover their eyes and pretend this isn’t happening. Because here lies madness. Unfortunately for you, I’ve uncovered your eyes, and now the abyss is staring back.
Fortunately, the blog post doesn’t end here. I’m going to argue that we can have our math, and eat it too. Before we get there, though, let’s talk about why precisely the laws of mathematics pose such a problem to dialetheism. There may be other reasons that it’s uncomfortable or unintuitive, but Mathematically, the problem starts, and ends, with the principle of explosion. This is an axiom of logic which states that if there is any inconsistency in a mathematical system, then the system explodes, and everything is true. There is only one inconsistent system: the trivial system, in which everything is true. In Latin this principle is called, ex contradictione qudolibet (from contradiction, everything follows), and in mathematical notation, \(P \wedge \neg P \vdash Q\); from a contradiction (\(P\) and not \(P\)) we can prove \(Q\) (literally anything else!).
If we want to claim that dialetheism is a good theory, that it give us a model that is good at predicting outcomes in the real world, then we’re in trouble. Of course a system in which everything is true is going to give us all the right answers, but it’s also going to give us all the wrong answers! That’s not a very good theory at all.
Enter paraconsistent logic, stage right and left. In Why I’m Not A Platonist, I introduced the idea that there are multiple logics one can use to build up mathematics. Classical Logic is the one you’re likely familiar with. Last post I talked about Constructivism, and in this post I’m going to talk about Paraconsistent Logic. It turns out, we can take that bothersome principle of explosion, and just like any axiom, we can throw it out.
Of course, just because we can throw out an axiom, doesn’t mean we should. There’s no guarantee that the new system will be worth looking at. As it turns out, it isn’t; not at first glance. Because of asymmetries in logic, Paraconsistency turns out to be much trickier than Constructivism. We need to introduce new things to our logic to have enough structure to be interesting. But it can be done, and the systems you end up with seem to be rich and deep and viable foundations for mathematics. Because it’s harder, and less intuitive, and because they haven’t seen a need to, mathematicians have largely neglected paraconsistent logic, and less progress has been made towards using it as a foundation for mathematics.
So I can’t claim a full victory. We’re not all the way to a paraconsistent understanding of reality. But neither do we have a consistent understanding of reality. When it comes down to it, science is already operating in a dialetheist way. We can choose to acknowledge that, and work to understand it, or bury our epistemological heads in the sand.
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Daniel Goldman has made a similar argument, which also points to some philosophical dead-ends in science that bayesian statistics runs into, and suggests that paraconsistency could get us past those.