I greatly enjoyed the feedback I got on my last post. So here’s a little more digital philosophy for you.
Last time, we proposed that all logically-derived sequences less complex than the universe could be said to be real. However, given that we still don’t know how complex our own universe actually is, the total set of things that are ‘real’ is still ambiguous.
You can go one of three ways here:
2: You can take the line that Jurgen Schmidhuber does, and say that only computable mathematics is real. In other words, only those theories of nature that can be represented as computer programs need be considered. (Tegmark appears to have also considered this option.)
3: You can be even more restrictive, and say that we don’t even have proof that all computable systems exist. In this case, we’re essentially saying that the universe can be represented as a really big finite state machine.
Despite the really excellent reasoning done by the chaps promoting options 1 and 2, I’m going to propose that we go with option 3.
To show why, let’s first put all the rules for making logical sequences in order, starting with the simplest, using a definition like this one. Computer scientists in the audience may complain at this point, because ordering this way requires that we pick an arbitrary machine definition as our measuring stick to define ‘simple’. However, we’re not going to be fussy. We’re just going to pick one. Then, we’re going to list out the total set of rules that our language can make and consider the output they produce.
Regardless of what language for making rules we use, we’re going to see certain things happen. For starters, there are going to be output patterns that show up more than once because some rules are going to be equivalent to each other. And some patterns will show up more often than others, because some of the rules that make them are going to contain redundant steps. Thus, as we build out our series of machines, certain kinds of output are going to dominate. And as the series gets larger, they’re going to dominate a lot.
In fact, the simpler the output is, the more frequently you’ll end up with a rule that produces it. Indeed, if you explore the set of simple algorithms from the ground up in the way that Stephen Wolfram and his buddies have done, this is exactly what you see.
Now consider that our universe appears somewhere in this list. The most likely rule for making output that looks like the universe is going to be the simplest one, by far, because its output will be duplicated so many times that it overwhelms all the other possibilities. Furthermore, given that we don’t have proof that things more complex than the universe even exist, the only rule for the universe that we need to bother considering is the simplest one that will do the job properly.
This result is helpful, because it tells us that the scientific habit of preferring minimally complex solutions over more ornate ones exists for a good reason. Of all the physical models that fit the output to a given experiment, the one that’s simplest really is more likely to be true, so long as it doesn’t conflict with other results. (And if it does conflict with other results, it’s wrong!)
This is all nice and useful, though so far we haven’t said anything about continuous models versus discrete ones. But consider this:
Any variable that varies smoothly cannot be represented in a countable number of discrete bits, whereas a single continuous number can code for any number of discrete values at the same time.
In other words, continuous systems aren’t just more complex than discrete ones. By every measure that includes both systems that people have been able to come up with, continuous systems are infinitely more complex. (If anyone has reason to believe to the contrary, I’d love to hear about it!)
This is not to say that you can’t produce continuous functions that produce simple output. However, these functions are isolated instances in a sea of more complex possibilities. And furthermore, all the continuous models that produce this kind of result turn out to equate in complexity to systems that aren’t continuous. In other words, you can always rewrite the functions they generate in terms of machines that don’t need smooth numbers. Thus, for any reasonable definition of simple that covers both cases, discrete models will always precede the continuous ones.
I would go so far as to propose that there is no reasonable way to say that continuous systems are simpler than discrete ones. And what this means is that if there’s even the slightest reason to believe that we can represent physical reality without smooth numbers, we should pursue it.
The burden of proof, therefore, properly lies with the continuum enthusiasts. In order for their models to be plausible, they need to prove that a discrete approximation cannot suffice. However, science doesn’t work that way. Unless people have decent analytical tools that don’t rely on calculus, discrete models simply aren’t going to be explored. This means that anyone hoping to actually get the Theory Of Everything right has about three hundred years of mathematical catching up to do in order to provide a framework that can compete with current expectations. Some actual experimental evidence for quantized spacetime would help too, though maybe we won’t have to wait too long.
Now, so far I haven’t said anything about the difference between Option 2 (computable reality), and Option 3 (even stupider reality), but I think that will have to wait for another post. In the mean time, happy reasoning, and may the force of Ockham’s Razor be with you.