Defining Freedom

by pku 2 min read20th Dec 20187 comments


I’ve always found the concept of Freedom confusing.

There’s a level on which it makes sense. When William Wallace is talking about being free from the yoke of Edward Longshanks, it’s obvious what he means – the English king and his goons regularly come to town, order people around, take their stuff and beat them up. Being free of English rule means they don’t do that anymore, and William Wallace and his friends can just go around farming and living their peaceful lives without having to worry about anything worse than the soul-crushing depression of living in Scotland.

On the other hand, what the heck does the word free mean in the context of “the land of the free and the home of the brave”? Is there even a reasonable definition? The original song is about being free from English rule, but it’s been two and a half  centuries since Edward Longshanks and his goons were ruling America, and not-being-ruled-by-the-King isn’t really a core property of most Americans’ identities these days.

I settle the first one by thinking of freedom as defined relative to a constraint. You can be free of something if you don’t have to worry about it when making decisions. This matches the common use of freedom – you’re free of having to worry about parking if you don’t have a car, free of a tyrant if you aren’t constrained by him telling you what to do, etc. This also explains why the second use seems so weird – it’s trying to use a fundamentally relative term without using it in relation to anything. So my response to the second use used to be to roll my eyes at people throw around empty deep-sounding terms.

But now I think we can resolve this. We start with the mathematical definition of degrees of freedom – your total freedom level at a given time is the number of options you have available to you at that time (if you want to sound all mathy, you can call this the local dimension of your options space or something)[1].

But there’s a fundamental problem here: At the end of the day you’ll only pick one of the options you had, because you can only pick one – once you eliminate the big constraints, you’ll just be subject to smaller and smaller constraints until you’re down to one option. Even if there’s no law forcing you to wear black socks, you’ll end up wearing black socks by the constraint that they were a dollar cheaper on Amazon and you were too lazy to scroll down. If you actually incorporate every single constraint you have, you end up having one action. Can we solve this using more math?

Yes. Let f(x) be the utility function on the space of possible positions you can be in. In this definition, the choice you make in a given position is simply to take a step in whichever direction increases f(x)the most. We generally think of constraints as cliffs in the utility function – if you disobey the tyrant he’ll probably kill you, so the “disobey tyrant” direction of decision space has a massive drop in f(x). Smaller constraints, like “white socks are a dollar more and involve scrolling down on Amazon”  are only minor dips in f(x).

In this case, we can think of absolute freedom as a measure of flatness of f – Some measure of how many directions you can go without falling down too big a cliff. There are a lot of ways to formally measure this, but the idea is that this should be a value that goes down more by wider or steeper cliffs (although steepness has diminishing returns – the difference between a specific option getting you badly injured or killing you is fairly minor if you can easily just not take that option). In a sense, this is just a measure of resilience – how many non-terrible routes do you have? Because your position and options are constantly changing, there’s a lot of value in having multiple non-terrible paths.

Finally, note that under this definition freedom definitely isn’t all we want – we also like having opportunities to massively increase our utility (instead of just saving ourselves from decreasing it too much). We can think of Welfare vs. Freedom as increasing our EV vs. reducing our variance.

[1] There’s an issue here with how to count/measure options – e.g. “go to the ball/don’t go to the ball” is clearly a freer choice than “go to the ball wearing white socks/go to the ball wearing black socks” – but like most measure issues you can mostly just ignore it and use your intuition.