>From Osher Doctorow [EMAIL PROTECTED] Sunday Dec. 1, 2002 0958 I agree again with Tim May.
I also think that category theory and topos theory at least in its definition as a branch of category theory are too restrictive, largely because they are more abstract than concrete-oriented in their underlying formulations. In fact, perhaps this is a key problem with computers. Most human beings whom I know have enormous difficulty in finding a Golden Mean between abstraction and concreteness insofar as the concrete reality and abstract reality are concerned if you get my meanings. The problem is only slightly less prevalent in academia. Computers seem to be nowhere near solving this problem - in fact, the more similar to human beings they get, the more difficult it may be for them to solve the problem. I am not even sure that most human beings in or out of academia think that there should be a Golden Mean between abstraction and concreteness [exclamation mark - several of my keys are out including that one]. In fact, I would conjecture a PRINCIPLE OF RECURSIVE SUBSTITUTION. This says that a computer or person will replace some simple operation [addition, subtraction, multiplication, division, limits, composition, non-commutative multiplication, etc.] by another simple operation if the result of using one operation predominantly and/or other operations secondarily is too slow by some standard that might be varied. Actually, don't replace them permanently - keep older methods indefinitely because they might eventually pull ahead for example, but at least move the new operations into a prominent position. My opinion is that important subtraction or subtraction-addition results are far more rapid in pure and applied applications across different fields and disciplines than composition and functors and objects in categories for example. Osher Doctorow ----- Original Message ----- From: "Tim May" <[EMAIL PROTECTED]> To: <[EMAIL PROTECTED]> Sent: Saturday, November 30, 2002 11:45 AM Subject: Re: The universe consists of patterns of arrangement of 0's and 1's? > > On Friday, November 29, 2002, at 02:44 AM, Marchal Bruno wrote: > > > Stephen Paul King <[EMAIL PROTECTED]> wrote: > > > >> I agree completely with that aspect of Bruno's thesis. ;-) It is the > >> 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. > > > > But then you need to explain what "implemention" are. Computer > > scientist > > 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? > > 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. > > 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. > > > > > >> Could we not recover 1-uncertainty from the Kochen-Specker > >> theorem of QM itself? > > > > Probably so. > > > > 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. > > 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. > > > > > --Tim May >