On Friday, November 29, 2002, at 02:44 AM, Marchal Bruno wrote:
Bruno is right that I would emphasize the mathematics over the "COMP" aspects. Computations are kinds of mathematics: mappings, iterations, theorem provings, even topological operations of various kinds. Not all mathematics is easily implemented on computers, but the principle is clear.Stephen Paul King <[EMAIL PROTECTED]> wrote:I agree completely with that aspect of Bruno's thesis. ;-) It is theBut then you need to explain what "implemention" are. Computer scientist
assumption that the 0's and 1's can exist without some substrate that
bothers me. If we insist on making such an assuption, how can we even have a
notion of distinguishability between a 0 and a 1?.
To me, its analogous to claiming that Mody Dick "exists" but there does
not exists any copies of it. If we are going to claim that "all possible
computations" exists, then why is it problematic to imagine that "all
possible implementations of computations" exists as well.
have no problem with this. There are nice mathematical formulation of it.
Tim would say that an implementation is basically a functor between categories.
You seen to want a material preeminent level, but this is more a source
of difficulty than an explanation. What is that level?
I suppose I am partly a Platonist, in that I believe there's more to mathematics than merely symbol manipulation (the Formalist) school. Computers are exciting because they give us another way to make real (or reify) the abstractions of mathematics. I believe, for example, that categories (e.g., HILB or VECT) are in some sense "real," that we can send our minds and our computers as robot explorers into these "scapes" of Platonia, into the ideosphere, into noespace, or whatever. (Sorry for waxing poetic...)
There is a sense in which the Platonist point of view is consistent with the Chaitin/Wolfram notion that mathematics will become largely explorational. Arguably, this has been what mathematics has _always_ been, that the process of discovering truths is not about proving theorems from postulates, at least not exclusively. Even geometry got its start not from considering abstractions out of a pure ideosphere, but from issues of measuring the earth (geo-metry), of building pyramids, of dividing farmlands, of measuring grain storage, and so on. Later mathematics was also guided at least partly by the practical, whether the study of differential equations or elliptic functions. Of all of the possibly-provable truths, laid out like stepping stones in a vast marsh of as-yet-unproved and possibly-unprovable truths, which stepping stones are followed, and are laid in the marsh as new proofs are obtained, is often shaped by engineering and physics considerations. Even in the purest areas of mathematics, such as number theory. The Chaitin argument that computers will be used increasingly to explore this landscape is, I think, certainly correct. (Personally, I am tremendously excited to think about what future versions of Mathematica, for example, will look like when running on computers 100 or 1000 times faster than my current Mac and running with immersive VR graphics systems. At Moore's Law rates of progress, I'll have this here in my home within the next 10-15 years or so. This is my main interest, more so than speculating on whether the universe runs on a computer or not. But Everything issues touch on this...)
OK, so which is it, really, Platonism or Formalism? Paul Taylor makes a good case in "Practical Foundations of Mathematics" that category theory in general and topos theory in particular provide the unification of these two points of view. Mathematical objects live in a universe of categories, with certain rules for moving between categories, and that various universes exist as toposes. We as humans can manipulate these rules, learn how these objects behave, and thus explore these spaces.
Now whether it makes sense (or "is really the case") to say that Reality is some kind of computer program is not all clear to me. Like many others, I have problems with the notion that reality is a program running on some kind of metacomputer. Perhaps computation is woven into the fabric of spacetime at a deep enough level, and perhaps there are alternative "state machine" rules which could be imagined in other universes (or even in different parts of our universe, e.g., changes in rules at very high energies, or near singularities, etc.
I'm not--at this time--much engaged by the "universe as a computer program" idea. A useful hypothesis to have--the Zuse/Fredkin/Lloyd/Schmidhuber/Wolfram/etc. thesis, in its various forms--but a long, long way from being established as the most believable hypothesis. To me, at least.
(I think Egan gives us a fairly plausible, fictional timeline for figuring this stuff out: a workable TOE by the middle of this century, i.e., within our lifetimes. That is, a theory which unifies relativity and QM, and which is presumably also brings in QED, QCD, etc. Perhaps involving a mixture of string/brane theory, spin foams and loop gravity, etc. Lee Smolin has some plausible speculations about how these areas may come together over the next several decades. This TOE is of course not expected to be truly a theory of everything, as we all know: the phrase TOE is mostly about the unification of the two major classes of theories noted above.
Then perhaps several centuries of very little progress, as the energies to get to the Planck energy are enormous (e.g., compressing a mass about equal to a cell to a size 20 orders of magnitude smaller than a proton). Egan plausibly describes an accelerator the length of a chunk of the solar system, using the most advanced "PASER" (the solid-state lasing accelerators proposed recently), to accelerate particles to the energies where discrepancies in models (computer programs??) might show up. In one of his novels ("Diaspora") he has this happening a few thousand years from now. This sounds about "right" to me. (I'll be happy to give some of my reasons for "pessimism" on this timetable if there's any real interest.)
Of course, breakthroughs in mathematics may provide major new clues, which is where I put my efforts.)
I take the "Everything" ideas in the broader sense, a la Egan's "all topologies model," a la the "universes as toposes" (topoi) area of study, etc. My focus is more on logic and the connections between topology, algebra, and logic. It may be that we learn that at the Planck scale (approx. 10^-35 m) the causal sets are best modeled as computer-like iterations of the spin graphs. But this is a long way from saying consciousness arises from the COMP hypothesis, so on this topic I am silent. As Wittgenstein said, "Whereof one cannot speak, one must remain silent." Bluntly, don't talk if you have nothing to say.
Which is why I have little to say about the COMP hypothesis. I'll be excited if evidence mounts that there's something to it. If the COMP hypothesis has engineering implications, e.g., affects the design of AI systems, this will be cool.
This seems to be assuming the conclusion. Gleason's Theorem and Kochen-Specker are about the properties of Hilbert spaces. But the reason we use the Hilbert space formulation for quantum mechanics, as opposed to just using classical state spaces, is because the Hilbert space formulation (largely of Von Neumann) gave us the "correct" noncommutation, uncertainty principle, Pauli exclusion principle, etc., things which were consistent with the observed properties of simple atoms, slit experiments, etc. In other words, the Planck/Einstein/Heisenberg/Schrodinger/Bohr/etc. results and successful models (e.g., of the atom) gave us the Hilbert space formulation, which Gleason, Bell, Kochen, Specker, etc. then proved theorems about.Could we not recover 1-uncertainty from the Kochen-Specker theorem of QM itself?Probably so.
I don't think it would be kosher to assume reality has aspects of the category HILB and then use theorems about Hilbert spaces to then prove the Uncertainty Principle.
(My apologies if this was not what was intended by "recover 1-uncertainty.")
This is a good example, by the way, of how the physics applications of Hilbert spaces incentivized mathematicians to study Hilbert spaces in ways they probably would not have had Hilbert spaces just been another of many abstract spaces. Gleason had many interests in pure math, so he probably would have proved his theorem regardless, but Bell, Kochen, and Specker probably would not have had QM issues not been of such interest.