27 July 2010
I'm really grateful to Yair (we ran into each other in a discussion elsewhere) for coming over here and giving the Mathematical Universe Hypothesis a bit of a blasting. Instead of just adding to the comments, I thought I would be lazy, and cheat a bit by replying to Yair's comments in a brand new post. Scurrilous, I know, but I had to do a bit of thinking and typing, and that tipped me over the character limit. For the context, please see the comments on the original thread. So here goes... enjoy!
Excellent points; I don't know if I have sufficient philosophical or mathematical background to tackle them, but I'll give it a go... I'm cutting and pasting a bit here; hope this makes sense.
I suspect Godel puts an end to a true TOE.
I have a sneaky suspicion that may be correct - for beings *within* the system. If the MUH (for want of a better term) is correct, then THIS universe has its own rules, but is essentially of the same status as a universe based on the rules of Conway's Game of Life. If you can imagine sentient critters within the GoL, I wonder whether it is possible for them, no matter what experimentation they do, to perceive that they are in fact running in an instance of GoL? For sure, they might hit upon the solution, but could they *prove* it? I don't know. Yet in that case, we can say that a ToE *exists*, and in principle they could know those rules - just (perhaps) not whether they apply to *their* universe.
The MUH says that the right description of reality is not some particular description/TOE, but specifically the most general description we can think of.
Not quite - there is a TOE for THIS universe, and that is quite specific; however, *every* TOE has its set of universes (e.g. the GoL rules will have a set of universes - to all intents and purposes *infinite*). For each universe, its TOE defines *it*. A TOEE (Theory of EVERY Everything) is, as you say, essentially Mathematics itself. At least that's how I understand it...
Third - How did our mathematical faculties evolve? I believe they evolved as they did due to the Boolean nature of classical physics.
Actually, I don't think that's correct. I think we (somehow) hit an evolutionary trajectory wherein it was advantageous for our brains to evolve the capacity for doing mathematics, but we do still find it rather hard, and I think that suggests that our abilities have evolved to solve some of these general problems because the mathematical nature of the universe makes it behave that way, rather than these being an *arbitrary* feature of the universe.
Hmmm, maybe badly phrased again... However, it is really *very* striking that when we move away from classical physics to the quantum world, mathematics is really rather insanely good at describing what we should expect to see - even when it is hugely counter-intuitive. Again, to me that suggests that our brains have hit on an ability to do maths in the abstract, quite apart from how we perceive reality. I admit this is a tricky one, and I do feel (as a medical doc) a little out of my depth.
they capture a feature of physical reality, an order and regularity that can be expressed mathematically - but not the mathematical object itself.
Actually I think you are right here, but I don't think that affects my point. We make models, but the models are at best representations of *subsystems*. The deeper mathematical reality could (I think) be represented mathematically, and if you were able to do this for the universe, you would have a complete description (just like the Game of Life) that would be isomorphic with the "real" thing, so a subsystem within the real universe would have no way of knowing whether its universe was "real" or mathematical, because in either instance it would *feel* real.
Mathematical truths don't hold in the world.
But the *right* ones do, which is the interesting thing.
there is no further requirement that this massive combinatorial construct will match the structure of reality itself.
I'm not sure how this corresponds with Turing's notions of computability; it would seem that with a few very *very* basic axioms (and I admit to assuming these, but they "seem" OK for this purpose), we can pretty much get anywhere we want to, and moreover the universe appears to be the sort of structure that is describable in these terms (although not *easily*, I'll grant you).
I'm working my way (very slowly) through Penrose's "Road to Reality", which is rather good, if hard going for a biologically-minded person. He seems to argue for the "existence" of mathematical truths in quite a hard way.
can you imagine that only a small world exists, where a single intelligent being is busy thinking? He can imagine alternate reality, but by assumption these do not exist in our hypothetical. Why can he imagine worlds that don't exist? Why can't we?
Because I think(!) that when we use mathematics, we gain *access* to alternate realities (read-only). I have already mentioned our worlds that "don't exist", such as GoL instances. Yet if the MUH is correct, they DO "exist", but are only seen as such from the viewpoint of those *inside* them. Like us inside ours.
Thanks again for your input, Yair - much appreciated. As I mentioned, I think I'm running out of philosophical road here; might be good to have Max Tegmark himself (or Roger Penrose) take this on :-)
[Is that you that I found in Google at the HUJ? If you ever find yourself hanging out with any of the evo-devo crowd, I have a very good friend who heads one of the labs there.]
Posted by Shane at 20:48