I generally think of time as discrete, since as a practical matter, any clock we use to measure time will have some click rate, which will serve as the ultimate limit on our ability to subdivide any internal of time.

This means that from any given moment in time, we can count, and enumerate, all possible future states of the Universe that are, e.g., 10 clicks out from the present. Of course, as a practical matter, we can’t actually do this, but we can do something close to it at a macroscopic level.

For example, for a game like chess, we can certainly enumerate all possible future states of the game that are 10 moves out from a given state. Obviously, reality is much more complex than a game of chess, but if you quantize time, then you can at least conceive of a model of time where all possible states exist along a graph that commences from a node representing the present, and extends some fixed number of clicks into the future, where each node in the graph represents some future state of the Universe. One node would be connected to another node in this graph if the second node might follow the first node immediately in time, in some path through time. So e.g., when you wake up on a given day, it might or might not rain at 1:30 PM, and those two distinct nodes (i.e., “rain”, or “no rain”) of the Universe would be connected to the first node of “waking up”.

So in short, you wake up, and there’s either rain, or there isn’t.

Because a board game like chess is literally part of the Universe, objectively distinct arrangements of the board represent objectively distinct arrangements of the Universe. This might seem trivial, and obvious, but it has a shocking and bizarre consequence, which motivated me to take the time to actually write this note:

In particular, the game of chess is always computable, so long as the game is finite. This means that the path through time from the present state of the game, to the end state of the game, is computable, since we can write a program that represents the positions of the pieces on the board, and proceeds in one-to-one correspondence with the paths of those pieces. Put in less technical terms, we can write a program that models any finite game of chess.

But now imagine a sequence that is not computable, and consider, for example, the answer to a question that is known to be non-computable. If an oracle shows up, and writes down the question, and then writes down the answer to that question, then the path through time that represents that sequence of events is non-computable, and therefore, there is no program that can run in one-to-one correspondence with that sequence of events. Stated differently, you cannot ask a UTM the answer to the question, and get the answer to the question, since it is by definition non-computable.

A corollary of this is, if we imagine the state of the world when the question is posed as a node in time, then without the oracle, the future state where the question is answered is not accessible from the present node. In practical terms, no one can answer the question, since it’s non-computable. In theoretical terms, there is no path from the present node to the future node where the answer to the question is stated.

Bizarrely, this means that if there are people that can solve non-computable problems, they will literally alter the set of future possibilities of the entire Universe, simply by answering these questions correctly. This must be the case, since without people that can answer non-computable questions, those future states of the Universe where those questions are answered are simply inaccessible. In simpler terms, we can’t get to the point where we know the answer to a non-computable question, without a person capable of answering the question. This means, that in a world where there are no such people, those futures are simply impossible to reach from the present.

So, in the most real, physical sense, people that can solve non-computable problems, are a bridge to otherwise impossible futures.