2.7 Levels of reality
In the previous Step 2.6 we saw that you can implement a UTM in the Game of Life, and that a UTM can emulate any computer. Therefore it must be possible to program a UTM running in the Game of Life to run another Game of Life! That means that the Game of Life containing UTM-2 in space -2 can be a program that is in fact being run by a UTM called “UTM-1”, contained in space -1 run by another Game of Life. So, you can think of space -1 as a “higher” space than space -2. In this scenario, not only does UTM-1 run the Game of Life containing UTM-2, it can also be self-aware in the same sense as is UTM-2, with probes coming from UTM-1, just like those coming from UTM-2.
Our argument that UTM-2 is self-aware and that it perceives its environment as real is not affected by the fact that the Game of Life in which it appears is run by UTM-1. However, if we could see UTM-1 itself running step-by-step, say on a desktop computer monitor, there would be no recognizable trace of UTM-2 inside UTM-1, nor would we see UTM-2’s probes or anything else in the environment of space -2, because all of that is encoded in the data and program running in UTM-1. Nevertheless, the essence – the mathematical structure – of UTM-2 in space -2 is still contained within UTM-1’s program and data, and we can consider the mathematical structure of space -2, including its UTM-2, as being a mathematical substructure of space -1.
When the Game of Life was first developed in 1970, it wasn’t run by computers at all: it had to be played either with pencil and paper or, more practically, with counters on a large grid like a Chinese Go board. The configurations of the live cells at each time-step would be the positions of the counters on the board, analogous to the screen on a modern computer monitor. So, the “computers” in the 1970s were computers in the original sense of the word – people! But, with a sufficiently large pattern, such as the UTM, we’d need to dedicate a whole army of people to the task of moving the counters to their proper places, if we wanted to make any measurable progress with the Game of Life. But, in such a case, where many different people are involved in executing the program, where, exactly, is the UTM’s continuing self-awareness taking place? It’s not on the board itself, just as the computer monitor isn’t the cradle of UTM’s self-awareness. But it can’t be in the minds of the people moving the counters, because they could each be allocated different sections of the board, so that nobody would see the overall picture. Our conclusion is that the self-awareness doesn’t arise within the computers running the Game of Life program. All that computers do is to simulate the Game of Life. The reality is within the mathematical structure itself.
That’s all very well (I hear you say), but suppose I run the Game of Life on my desktop computer and suppose that particular Game of Life features UTM-1 being self-aware and running the Game of Life with UTM-2 in it. In that case (you say), I can actually see the Game of Life with UTM-1 running on my desktop’s monitor screen, so, unlike space -2, space -1, which is displayed on my monitor screen, is real.
Well, yes, the display on your monitor screen is real, but think about it – you can’t see the actual probes coming from UTM-1 – they’re encoded in the program language used by the desktop: C++, for example. All you can see is the simulation of the mathematical structure of space -1 displayed on your monitor. (Incidentally, in this scenario, you and I, and your desktop and the monitor are all in space 0. Everything in space 0 is real – at least, to us, who are part of space 0.)
This shows the Game of Life running on a desktop computer within x- and y- parameters that we call space -1. On the desktop computer monitor, we can see that a UTM, called UTM-1, is running within the Game of Life. What we can’t see directly, though, is that the program that UTM-1 is running is in fact another Game of Life, which generates space -2, including a UTM, labelled UTM-2.
So, it seems that, if a computer is programmed to simulate a sufficiently sophisticated mathematical model of a universe, the reality within that universe can nevertheless be perceived only within the mathematical structure itself: what the programmers see is merely a simulation of that reality. Moreover, the computer running the simulation has no effect on the mathematical structure, which is completely independent of the computer simulating it.
Well, in that case (you continue), suppose, in our “real”(space 0) world, I build a robot with cameras and microphones, laser radar and other sensors to enable its central computer to build a model of its environment and enable it to move around, and look in the mirror and be aware that it is looking at itself. Then, from what we have said about self-awareness, that robot would be crudely self-aware. So, surely this contradicts the assertion that the central computer is merely simulating self-awareness and that self-awareness is not generated inside the computer?
OK (I say), let’s examine this. Let us accept that your robot is indeed self-aware, at least on its own terms, because, with appropriate substitutions of eyes for cameras, ears for microphones, a brain for the central computer modelling its environment and so on, we can make the same case about ourselves – we know that we are self-aware, and so, similarly, we can say the same thing about your robot. And yet I talked earlier about computers merely simulating self-awareness.
The difference is that your robot is not running a program to simulate a whole space containing a self-aware UTM in the way that UTM-1 is running a program in space -1 to simulate UTM-2 in space -2. The correct analogy is for your robot to be sending out probes in its own space (our space, i.e., space 0) in the same way that UTM-1 sends out probes in space -1. (The probes your robot sends out include the light pulses from the laser radar.)
So, if we understand that your robot, we ourselves, and everything in our environment are all part of the same mathematical structure, namely space 0, the dilemma dissolves. This is consistent with what we learned from the thought experiment to test the block universe: our universe is a fixed, four-dimensional structure. For emphasis, we could add “…in which the quantum outcomes of all events are embedded.” We may say that reality within a mathematical structure is the complete definition of that structure.
Notice that the levels of reality are asymmetric: while we in space 0 can access space -1 (after all, we can simulate the Game of Life on our monitor screen), there is no way for UTM -1 in space -1 to access our own space 0, or for UTM-2 in space -2 to access any part of space -1. This is a general rule.
In this section, I have tried to make it more plausible that a pattern, or part of a pattern – i.e., a mathematical structure or mathematical substructure – can, on its own terms, be “aware” of its environment, even when that environment includes the mathematical substructure itself. We have seen that such a pattern, call it space -2, can in fact be part of a larger structure (call it space -1) within a nested hierarchy.
Of course, we saw this earlier when we investigated the block universe, and found that events in your future are already in someone else’s past if they are moving towards you. Since the past can’t be changed for anybody, then neither can your future. To regard our universe as a block universe or to think of it as purely a mathematical structure are two sides of the same coin.
Note that the mathematical structure of our universe doesn’t gradually evolve with time such that, at any moment, while the past and present structure is fixed, the future structure is not yet decided. No, you mustn’t imagine the block universe as a kind of brick slowly crystallizing from the bottom up, as it were, with the upper part not yet solidified. That would be the wrong way round. It is wrong because the four dimensions of space and time emerge from the structure itself. This means that the structure can’t use time in which to evolve, because time is one of the dimensions that emerges from the structure. It would be like levitating by pulling upwards on your shoelaces. Instead, we must visualize the whole structure as one (enormous) fixed pattern.
This, of course, presents us with a dilemma. We saw in Step 1 that the universe is absolutely indeterministic – future events are undecided – and yet, in this Step 2, we have found that, if we accept the evidence that the universe is purely a mathematical structure, then the universe is completely determined through its structure. In the following Step 3, we shall see how these opposed positions are resolved.
