“All is number” – Pythagorean School
In Part 1 I attempted to present some of the philosophical reasons why we may think infinity is a pretty dodgy concept. I now want to address the second and third questions I posed
What do we mean when we say mathematics exists?
What do we mean when we say the universe exists?
Much has been said about reality and existence, probably way too much. I’m not trying to introduce new ground here, just present a view that I think goes nicely with the idea that everything is finite. That view is known as the Mathematical Universe Hypothesis (MUH) an idea originated by Max Tegmark. It states that our universe is nothing more nor less than a mathematical structure. It implies that any mathematical structure exists in the same sense that our universe exists. I won’t elaborate on this in great detail (I hope to in a later post) but I’d like to give some brief motivation to this idea.
We’ve all seen the idea of being in a simulated reality such as in the movie The Matrix (and yes it has to be in the top ten best sci-fi movies ever). In that story humans exist in the real world but are wired up to a computer simulated world, that they feel is real, by direct connection to the sensory inputs of their central nervous system. In fact, the only way they can tell it’s not real is when the laws of physics are a bit different. They can’t actually tell the difference by how the matrix ‘feels’. Suppose we go a step further and create a computer simulation within which complex organised forms can evolve, and that over time they come to satisfy our definition of life. Suppose they go even further and evolve intelligent behaviour and perhaps even self awareness, like we did. As far as they are concerned, the reality of their world feels as convincing as the reality of ours.
One conclusion that might be drawn from this is that we are in a computer simulation created by intelligent beings of another universe (a ‘real’ universe). Nick Bostrom came up with a logical argument to suggest that this is in fact likely. But even though I like his argument I don’t buy it. In fact I don’t like this conclusion at all for various reasons. There is a much better one. I find the opposite far more compelling: rather than saying our universe is like theirs, we can say theirs is like ours, that the creatures in our simulation really exist and we’ve just happened to stumble on the algorithm that reproduces them in a simulation. We both exist by virtue of the fact that our universes are constructible from a finite algorithm.
There is a key point here so I want to emphasise it more: complex algorithm-derived forms exist whether or not someone simulates then. They are not invented, they are discovered. Consider the decimal expansion of pi, 3.14159…, an apparently complex series of random numbers (more precisely they appear to have the property of being normal). Pi’s complexity exists regardless of whether we develop mathematics.
Consider a more concrete finite discrete universe, such as the one depicted in the picture banner at the top of this blog: the digital universe of the game of chess. Sixty-four discrete locations, thirteen discrete states of each location (empty, black king, white pawn etc) and a highly arbitrary set of rules that define all possible changes (Queen to g4) and a strategy (take the king). From this very simple universe emerge complex higher properties (control the middle section, advance pieces early, develop an attack down one side). It was only by historical accident that we developed the particular rules that define chess. But these emergent properties would still have always existed in the chess universe.
This argument may convince you for a while but when you shake your head and come back to reality you may feel it’s incomplete. You may feel that something more convincing is needed to doubt the realer reality of the universe. I’d say it’s the other way around. Show me the evidence for the higher reality of our universe. Magic? Religion? It’s obvious? That our sensory input is more real because we can pick up a rock and feel it? None of this is scientifically meaningful. I think the proposition that our universe is more real than mathematical structures is the one that’s from left field and needs justifying. Previously we held a geocentric view of the universe, that the earth was at its centre and everything revolved around it. Currently most still think that our universe is unique and is the only universe that exists. I think it’s the same geocentric fallacy and will eventually be discarded.
I can now give you my personal answer to the questions I posed:
Mathematics and the universe exist in exactly the same sense. They are real by virtue of their constructability and their reality is enjoyed only by creatures that exist within it.
In my final post I will explain how the presumptions I’ve discussed lead naturally to a universe that begins with a Big Bang. And how they lead to a quantum mechanical universe. And to a preposterously large universe.