Axioms and Theorems

(M.C. Escher’s Relativity.)

[This post is intended as a sort of roundabout introduduction. If you find yourself in need of a better one, see the about page here.]

At a base level, humans reason with axioms. That is to say, we all have collections of things that we accept as true without any great need to prove that they actually are. “Being hungry is bad” is an axiom. So is “God exists.” From these axioms, we construct theorems that are capital-t True, at least for us. “Killing is bad” is a theorem, potentially derived from the existence of both the Judeo-Christian God and of the Ten Commandments. In this way, all knowledge is based on what we can’t prove. One way or another, we’ve got to assume some axiom and go from there.

Now, there’s noting particularly wrong with axioms. All formal systems require some of them because needing to have absolute empirical proof for everything is an impossible, maddening task. Think about it: try to prove that two and two is four without presupposing something, even something as simple as the existence of some definable quality of two-ness out there in the universe. Even science, which demands proof for basically everything, presupposes that our observations about the universe represent some immutable truth about it. If you don’t start a chain of reasoning with some assumption, you end up arguing that it’s turtles all the way down.

Where axiomatic thinking really runs into trouble, though, is when you have contradictory axioms. For example, say I’m the President (the horror). A well-known, well-reviled criminal somewhere in the country is about to be executed. I have the power to commute his sentence to life in prison. I also happen to think capital punishment is wrong, based on my belief that all life is precious and should be preserved at all cost. However, I also believe that, because this is a democracy, the will of the people should be the last word on any question. This is inconvenient, because a majority of them definitely think this guy should have been dead yesterday.

So now I’m at an impasse, and it doesn’t look terribly likely to be resolved. I’ve got two axioms in my head, and they contradict. Any answer I come up with is going to violate one of these axioms, making it a logically invalid answer to the question. This is the problem with large sets of axioms. Eventually, they’re going to conflict. At that point, any sort of actual reasoning is going to break down and I’m going to default to dead reckoning. This works okay for small things that don’t matter, but terribly for large things. Witness our previous example.

There’s a pretty simple solution to this problem, though. I can start the construction of my system of  thought with one axiom: “No two axioms can conflict.” That’s it. So, in our example, I have to pick one axiom to keep and one to throw out. I can either think that democracy is absolute or that the death penalty is absolutely wrong, but not both. In cutting my number of axioms, I gained the ability to reason more efficiently and draw better conclusions. Problem solved!


Maybe, because this works just fine for something like science, where you only really need one axiom. It even works fine for personal belief systems, where a relatively small set of internally consistent axioms will get you where you need to go. However, this works terribly for societies. In a society, we’re dealing with constructing a set of basic rules to govern a population of people that are all operating according to their own axioms. Basically, we’ve got to figure out a way that lots of people can live harmoniously while simultaneously differing irreconcilably from each other in millions of ways large and small. The axiom that we used to order our own thinking won’t work for us here, either. Applying it to a society leaves us with a situation where we’d have to choose which axioms are correct and impose those from on high, which is a crappy way to run a railroad. Leaving it out, on the other hand, leads us inexorably towards the worst kind of relativism, where we are forced to conclude that anyone can do anything they damn well please because there’s no true set of axioms to begin moral reasoning from.

So, to recap: no governance is based on a solid axiomatic foundation, no law is legitimate, civilization is crumbling, and we’re all going to die.

Well, probably not. Except for that last thing. We’ve muddled along so far just fine, and I’m pretty sure we can keep doing that. The key issue that this whole axiom mess raises, though, is that there’s a large number of public policy issues that are really hard to hash out effectively. Take marginal tax rates, for example.  A conservative economist might say they’re too high, and a liberal economist might say that they’re too low. Ultimately, they’re both sort of right. Each of their chains of reasoning about marginal tax rates begins with fundamentally different conceptions of the role of the state, and those chains of reasoning don’t get any closer together. Neither of them are wrong, in a purely logical sense, but they can’t both be right. Other controversies in this vein: abortion, capital punishment, etc. We in the US largely solve this problem by having a democracy, where policies are passed through majoritarianism. We don’t need to know who’s right, just who has the most votes.

This is a deceptively simple solution, though. At the end of the day, we still haven’t decided which one of those economists is right. Presumably one of them is, and the world they’re describing is better than the world the other describes. If that’s the case, it would behoove us to figure out who’s right and who is wrong. After all, we want to live in the best world we can. As we’ve established, though, there really isn’t any way to prove that one world is better than the other, and so we remain stuck. They both can’t be right, but we can’t prove either wrong. We’ve defaulted to absolute relativism, and we seem to be stuck there.

This shouldn’t be interpreted as a concession on my part to absolute relativism. On the contrary, in fact. I definitely have a point of view on most issues, and I definitely think I’m right. This is, however, an argument against a certain kind of political discourse. Our takeaway from this should be that governance is actually really complicated. There are no easy answers, and even our own closely-held beliefs rest on potentially shaky ground. All the easy moral and political judgments have already been made. This knowledge shouldn’t paralyze us, but it should humble us. It should make us wrestle with our own beliefs and what they mean in context. It should clarify to us the rules of the game. Good governance may be hard, but pretending it’s easy does no one any good.