God has no power over the past except to cover it with oblivion. – Pliny the Elder
Predicting the future is hard to do. Just ask any economist, stock trader, or weatherman, and you’ll hear of the horrors of predictions gone wrong. But what about on a more fundamental level? What can we say about prediction? In the past, I (and many others before me!) have argued that entities in universes with finite information per volume and local laws are fundamentally limited in their ability to predict the future (for a fable, see The Library of Laplace, for more clear thoughts see this post.) But what about the opposite direction? What if we ask, instead of whether we can predict the future, whether we can predict the past?
The basic argument I give about why “predicting the future” is hard is as follows. First I should define what I mean by predicting the future! What I mean by this is that you are some physical entity (a person, a computer, etc.) which wants to know what is going to happen in the future. That is you’ve got information about the world around you and you want to use it to make a prediction about what is going to happen in the future. Now suppose that the universe is such that a finite amount of information can only fit into a finite volume of space and the laws of physics are local. The quintessential version of this sort of universe is a cellular automata (quantum, probabilistic, deterministic, I don’t care which) but there are other ways to arrive at such a universe. For instance the holographic principle leads to the amount of information in a volume is proportional to the area of its boundary, thus leading to a tighter bound than that of a cellular automata like universe, but still there is a fundamental “size” associated to a basic unit of information, a “bit.” But let’s take the cellular automata universe because it’s easier to talk about.
So suppose you are beastie in a deterministic cellular automata universe. “You” are a collection of bits describing your state spread across the lattice of the cellular automata. As a being in a universe with local update laws, you also have your own past light-cone: the things that could have influenced your current states. But outside of this lightcone you don’t have access to any information. Now ask the question: can you predict what is going to happen your future? Well there is a problem: in order to predict your future you really need to know initial conditions which are outside of your lightcone: i.e. suppose the universe started a long long time ago with initial conditions