Wednesday, December 21, 2005

Math, Matter & Mind

I found a neat paper on the arxiv recently:

On Math, Matter and Mind

by Piet Hut, Mark Alford, and Max Tegmark. They discuss the metaphysical foundations of reality, using an apparent paradox by Roger Penrose as a focal point (amusingly enough I also found the paradox while skimming through his new book). Here's the paradox: physics makes a compelling case that Matter is a particular type of Math, it is also abundantly clear that the Mind is equivalent to a particular arrangement of matter, and finally it is reasonable to conclude that Math is a one type of idea that the Mind can create. Thus we have that Mind ⊂ Matter, Matter ⊂ Math, and Math ⊂ Mind: each phenomenon in the triangle is a proper subset of the previous one!

Here's generally how the three authors resolve the circular paradox. Tegmark takes the 'Fundamentalist' view point, arguing that it is valid to assert both that matter ⊂ math and mind ⊂ matter but not that math is an invention of the human mind - arguing rather that humans are merely discovering mathematical structures that already exist. Alford takes the 'Secularist' view - to him it is clear that both matter and minds are real phenomenon, although he doesn't care to hypothesize which is more fundamental, and he thinks that math is an abstract creation of the mind. Hut takes the 'Mystic' viewpoint: he doesn't claim that any of the three Ms is a subset of another, but instead conjectures that all three are equally important facets of some deeper and yet undiscovered structure.

My views lie somewhere between Tegmark's and Hut's. Tegmark makes the same two assumptions about reality that I made in the 3 questions posts: first, that the universe is a mathematical structure, and second that all mathematical structures exist (name this vast collection the ensemble). Working in this framework it is thus clear that our material universe is just one type of mathematical structure among many, and our minds are just one type structure in our universe - thus matter ⊂ math and mind &sub matter. This explains, as Wigner put it, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences" (I found it amusing that they use this great phrase a lot in their paper, since I had also included it in a new paper of mine that I'm writing).

My thoughts start to diverge with Max after that however. He thinks it is plausible that we could find a final theory of everything - a final, exact mathematical structure in the ensemble that is our universe. I come to the opposite conclusion with regards to the ensemble - I suspect that we will be forever finding that our current laws are in fact only approximations to yet more complex structures, since the more complex structures vastly outnumber the simpler ones. Thus, for statistical reasons, there will never be a final theory. In Max's view, our's is a rather simple universe, and in order to help explain this fact he proposes that only Gödel-complete (fully decidable) mathematical structures have physical existence, a restriction that dramatically shrinks the ensemble. This seems like madness to me (although it deserves more carefull thought), and furthermore it seems against the spirit of the ensemble, at least as I understand it: there can be no distinction between mathematical structures and physically realized structures. I strongly suspect that there is no upper bound to the complexity of structures in the ensemble - in fact, going even further, I suspect that there isn't even an upper bound to the number of axioms that will eventually be used to build interesting structure - and this incredible structure will come into play as we dig ever deeper into the substrate of reality. Let me quote Hrbacek and Jech from their Introduction to Set Theory:

"As Gödel assures us in his Incompleteness Theorem, no axiomatic theory can decide all statements of arithmetic or set theory. We can thus feel confident that the enterprise of getting closer and closer to the ultimate truth about the mathematical universe will continue indefinitely."

I also think that observers play an absolutely key role in the mathematical ensemble. The Observer Class Hypothesis proposes that observers form by far the largest class of information in the ensemble, precisely because observers can study any other type of structure. This thus explains why we are observers. Since we do happen to be observers (or minds that extract information from exterior sources) we tend to take being observers for granted, but it must be explained. The Observer Class Hypothesis is testable - it requires that there is no upper bound to the complexity of information that we will eventually be able to process, which in turn requires that scientific and technological progress never halt, so that we can construct arbitrarily complex computers (aka minds). Thus in the end it would be valid to say that math ⊂ mind, and indeed this is why we are minds!

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