I am not meant for your religion
In looking over the traffic, the archives, and the responses I have gotten, it's clear that I mistook this list for a place where some of the Healey/Moravec/Egan/Tegmark/etc. ideas of an ensemble view of the universe, with "all topologies models" and "all variants of theorems" sorts of metamathematics ideas might be discussed. I didn't expect my own special interest, which is more along the lines of toposes, lattices, logic, and math, would be the mainstream, of course. But I also didn't expect to find a weird religious cult, with trappings of quantum suicide, roulette ideas takens seriously, and acausal notions about how one should act. Not to mention the grab-bag of weird cosmologies with no support offered, the theories of physics with no backing, the claims that anyone criticizing these models must be part of the corrupt establishment, and so on. Enough. You folks don't like my stuff and I don't like your religion about killing yourself and others in order to "perfect" the Multiverse. I learned more about causal decision theory, but then learned that it's all wrong, that one should do things which are counter to one's own interests so as to perfect the Multiverse. And I heard about the suicide machines, the arguments for and against euthanasia, and the bizarre ideas about making the Multiverse more perfect (for whom?) by taking Moravec's and Tegmark's whimsies seriously. I will say this, you guys will make for some great characters in some future "Star Trek" movie, the one where a cult of Multiversians is setting off special weapons to destroy universes which fail to be perfect in various ways. I'll miss some tidbits of math I discussed with some of you, but I won't miss the rest. Until we meet in another reality, --Tim May
Re: Quantum Decision Theory
n unsuccessful suicide are likely to be even worse is not itself an obvious conclusion. Which is why the usual decision theory process gives about the right answer without appeal to many worlds theories. (More notes: I've also known unsuccessful suicides and have read about others. They are usually not "worse off" in any determinant way for our exercise. They wake up in a hospital room, or the bleeding stops, whatever. Only occasionally do suicides fail in ways that make them worse off. And in those cases, they can always try again, or have a hired killer waiting to finish the job.) The argument wrt superannuation is that standard decision theory should lead one to the same decision as the QTI case (a bit like life insurance, except one is betting that you won't die instead that you will) - however in such a case why would anyone choose lump sum investment options over lifetime pension? Because they think they can grow the sum more than the pension fund managers can. And, since they can always take the grown amount and _buy_ an annuity later if and when they get tired of investing, they lose nothing by trying, if they think they can in fact do better than the managers. (I hope you considered this benefit of the lump sum payout. If you are at all bright, which you clearly are, you can grow the lump sum at least as well as some government/bureaucrat pension fund managers typically do. At some point you can then buy a larger annuity than the one you picked in the conservative option. And if you die before you expect to, you had the use of that lump sum for things of importance to you.) Noone that I could recall came up with a convincing argument against the Euthanasia issue - it would seem that committing euthanasia on someone is actually condemning them to an eternity of even greater misery than if you'd just left things alone - quite the contrary to what one expects in a single universe notion of reality. I don't see this at all, even besides the issue of my own focus on the branch *I* am in. --Tim May
Quantum Eugenics
On Tuesday, January 14, 2003, at 02:45 PM, Hal Finney wrote: Russell Standish refers to his earlier post, http://www.escribe.com/science/theory/m781.html and now writes: Noone that I could recall came up with a convincing argument against the Euthanasia issue - it would seem that committing euthanasia on someone is actually condemning them to an eternity of even greater misery than if you'd just left things alone - quite the contrary to what one expects in a single universe notion of reality. The problem Russell points out is that in the MWI if we try to kill someone, we will succeed in many branches, but fail in some others. And in those in which we fail, we will probably injure the victim. The effect from the point of view of the continuations of the victim is that their quality of life has been worsened. From the point of view of _this_ particular continuation, namely, me, I will choose injury over death. Any attempt to claim that injury means "quality of life has been worsened" as compared to death is...well, I won't say "silly," as that's not a good debating term. Let's turn to simple causal decision theory. A man is approaching me with a large knife. I have two choices: * Choice 1: Attempt to defend myself, probably causing me some amount of injury but possibly saving my life. * Choice 2: Allow myself to be killed. I get the idea from your paragraph above that you think Choice 1 is less desirable, as "The effect from the point of view of the continuations of the victim is that their quality of life has been worsened." Am I misunderstanding your point of view about what I should do? My approach is to optimize for the actual life (reality, branch, world) I am actually experiencing. Arguments that it would be better for "me" to allow myself to be killed, or euthanized, so as to make some average of realities forever disconnected from mine (Everett, De Witt, etc., not contradicted by anyone here, at least not recently) is not convincing to me. I think a counter-argument comes if we look at the larger picture, not just the particular branch in which we are acting. Suppose that the victim has suffered some injury which led to their unfortunate condition, where we are now considering committing euthanasia. In other branches, then, the victim is alive and healthy. If we eliminate them from most of the branches in which they are injured, then in the big picture their average quality of life has been improved. Even though there are a few branches where our attempts to kill them have failed and have made their life worse, those are small in proportion to the set of branches which encompass their entire life. If you believe in this strong form of quantum suicide, then of course the path to "quantum eugenics" is clear: immediately kill children who get less than perfect scores on exams, immediately kill anyone who is outside the societal norms for beauty and charm, and so on. This will mean, so the theory goes, that in some realities there are mostly smart, healthy, beautiful, wise people. Now it may be said that this perspective is invalid, because the injured person here in front of us is the only one we can affect. His consciousness will never re-join the branches where he was not injured. We can only affect him, and therefore we should judge our actions only based on their effects on him and his future continuations. And this is what a person facing attack and death will also choose to do, to optimize his own survival and his own quality of life. Suicide can be a choice in extreme cases, I think many in society would agree. But it has nothing to do with branches of Possibilia forever disconnected from Actuality. --Tim May
Re: Claim: Only one past for a given present
On Tuesday, January 14, 2003, at 12:35 PM, Hal Finney wrote: Tim May writes: This arises with quantum measurements of course. Once a measurement is made--path of a photon, for example--all honest observers will report exactly the same thing. There simply is no basis for disputing the past, for Alice to say "I saw the photon travel through the left slit" but for Bob to say "I saw it travel through the right slit." That's an interesting example, because usually the point of two-slit experiments is that there is no "fact of the matter" about which slit the particle went through. No, _if_ (iff) a measurement of which slit the photon goes through is made, then no interference pattern is observed. This is standard QM at this point. My point was that Alice and Bob will also agree with what the outcome of the measurement was. That's why you get interference from the double-slit. The interference pattern is only seen when no measurement of which slit the photon passed through was made. What would you say about the past in that case? Are there two pasts, one where the particle went through each slit, which have now recombined to form the present? Or just one past, where the particle managed to go through both slits at once? I would say "nothing" about the past prior to a measurement being made. --Tim May
Re: Claim: Only one past for a given present
On Tuesday, January 14, 2003, at 11:43 AM, Tim May wrote: Rereading my paragraphs, maybe they are unclear. It takes entire chapters of books (I like David Albert's book, or Smolin's "Life of the Cosmos" (from whence the cat and dog example was taken), Bub, Hughes, and Barrett) to talk about these things, so my paragraphs are doing the ideas justice. I meant to write "...are NOT doing the ideas justice." (Regrettably, I often hear the emphasis in my head but neglect to type the "not." This is my most annoying typo.) --Tim
Re: Claim: Only one past for a given present
On Monday, January 13, 2003, at 02:40 PM, Jesse Mazer wrote: Tim May wrote: On your point about "many pasts are fundamentally caused by quantum uncertainty in memory devices," I strongly disagree. There is only one past for one present, whether RAMs dropped bits in recording them or historians forgot something, etc. (This is captured by the formalism of observations, as well. Even with Uncertainty, all honest observers will report the same outcome of an experiment. We have not seen a violation of this, nor is one expected. There are various ways to look at this, including the topos-theoretic view of subobject classifiers. But the point is that in our history either an event happened or it did not. This is independent of whether the event was observed, recorded, written about, remembered, etc.) But this is a topic of great fascination for me, and I hope we can continue to discuss it. I am quite strongly persuaded that "many pasts for a particular present" is not a reality. Understand that I am not including "current interpretations," as in "Some historians think the Roman Empire fell because of lead in their plumbing" sorts of theories of the past. I am referring to space-time events. As noted, I also view time and events as a lattice. But lattices have certain properties of importance here. More on this later. --Tim May What do you mean by this, exactly? In a deterministic universe with time-symmetric laws, there'd be only one possible past history for a given present state, and only one future history as well, while in a universe with stochastic laws, or deterministic laws in which paths in phase space could converge, there might be multiple past histories that would lead to exactly the same present state. I think the number of multiple past histories (other than the _actual_ one) which "lead to exactly the same present state" is as close to zero as one would like to calculate. As to why there is only a single past but multiple futures, this is implicit in the measurement process. (I doubt you will find this convincing unless I expand on this.) Consider the "There will be a sea battle next year" proposition, the favorite of the Stoics and Aristotle. Unknown at this time, and few prospects for computation. Determinism is not very supportable, especially at full detail. And yet the proposition "There was a sea battle at Jutland during World War II" is answerable. And all will agree on that answer. The future is not knowable, the past is not disputable. This arises with quantum measurements of course. Once a measurement is made--path of a photon, for example--all honest observers will report exactly the same thing. There simply is no basis for disputing the past, for Alice to say "I saw the photon travel through the left slit" but for Bob to say "I saw it travel through the right slit." (If I am wrong on this, please correct me ASAP!) Honest observers will report the same outcomes of measurements, whether those measurements are of photons in slits or sea battles. This shows up in the formalism of lattices, especially the orthomodular lattices of quantum mechanics. Of course, in quantum mechanics it's not even clear that we can talk about the "present state of the universe" as if it's a well-defined entity, in which case it may not make sense to ask whether "the" present has multiple pasts or multiple futures. A MWI advocate would say we could talk about the present state of the universal wavefunction, but that's different from the present state of an individual "world"--I believe there's a fair amount of controversy about what people even mean by "worlds" in the MWI. With a hidden variables interpretation of QM you can talk about the universe's present state, but the exact details of the present state would always be unknowable. From an instrumentalist point of view, the state of an experiment (or of the universe) is recorded by the set of measurements made. I'm not saying things are not weird...QM by any interpretation gives results not in accord with our "realist" intuition. (E.g., the quantum box with an animal which may have one door opened to reveal whether it's a dog or a cat or the other door opened to reveal whether its a male or a female, but never both at the same time and never even sequentially so that one first opens one door, sees a dog, opens the second door, sees a male. In the ordinary world, both doors are openable, either simultaneously or sequentially, and things are as we would expect them to be. Not so with quantum things, modulo entanglement decoherence, etc.) Even at the "present" the universe will not have a "defined past," as delayed choice experiments show. But I would argue that those delayed ch
Quantum Decision Theory
Answering the last question first, "Do you find this perspective useful?"... I'm not yet convinced of any of the utility of the MWI for any bet or action, but I certainly think you are pursuing something that _might_ be interesting or even useful, with a kind of "quantum decision theory" view. But I've yet to see anything convincing. More comments: On Monday, January 13, 2003, at 02:33 PM, Wei Dai wrote: On Fri, Jan 10, 2003 at 08:54:38PM -0800, Tim May wrote: But in this, the only universe I will ever, ever have contact with, I optimize as best I can. And I assume all the myriad mes are doing the same in their universes, forever disconnected from mine. You're taking the question too personally. The issue here is whether rationality only involves local optimization within the branch that one is in without regard to other branches, or whether one can also take into account what one believes to be happening in other branches. You yourself may be a local optimizer, but the larger question is whether rationality allows global optimization or not. Notice that the latter is more general than the former, because all local optimizers can be modeled as global optimizers with a special form of utility function. I would like to see some better examples of what these "take into account what one believes to be happening in other branches" decisions or optimizations might be. If in fact the branches are unreachable to us, then causally there can be no effect of one branch on another. From the causal decision theory I believe you support (Joyce's book), this is just about a perfect example of where causal decision theory says "no causal link." Now, as I discussed in reply to Hal, there's much evidence that what people _believe_ affects their actions in this world, this branch. But this is true without recourse to many worlds theories. A person's belief in an afterlife usually affects his actions in this life. Examples abound, and were we talking or arguing in the room I described in my first post today, we might consider a bunch of them. R.I.G. Hughes, in "The Structure and Interpretation of Quantum Mechanics," 1989, discusses this issue of betting in a MWI environment: (late in the book, after much discussion of operators, lattices propositions, measurements, interpretations, probability measures, etc.) "But what, on Everett's account, has become of the world which is actual in Lewis's? If there is no such privileged world, then something odd happens to our conception of probability. For if _all_ (relevant) events with nonzero probability are realized in some world or other, then are not all those events certain of occurrence? (This was pointed out by Healey, 1984, p 593.) And if I wager on what the outcome of a measurement will be, will it not pay "me" to place my bet on whatever outcome is quoted at the highest odds, without regard to the probabilities involved? .. (Before an epidemic of long-odds betting is upon us, however, I should add that even the National Security Council would be hard put to divert funds from my Swiss bank account in one world to its counterpart in another.) "These levities aside, we may ask what new understanding of the measurement process MWI gives us. After a measurement each observer will inhabit a world (for her the actual world) in which a particular result of the measurement has occurred. And the "total lack of effect of one branch on another also implies that no observer will ever be aware of any 'splitting' process" (Everett, 1947, p. 147n). What is this observer to say about the physical process which has just occurred? From where she stands, the wave packet has collapsed no less mysteriously , albeit no more so, than before." "We are still left with the dualism that the interpretation sought to eradicate. As de Witt (1970, pp. 164-165) himself remarks, the many-worlds interpretation of quantum mechanics "leads to experimental predictions identical with the (dualist) Copenhagen view."" (p. 293) (Tim again.) Now we all know this, but it makes the point that probabilities are calculated identically in both interpretations. If Wei can find a way, no pun intended, to show that "quantum decision theory" produces different results OTHER THAN THOSE BASED ONLY ON THE BELIEF ITSELF, this would seem to contradict the "identical experimental predictions" and would be an important contribution. (Excluding results based only on the "BELIEF ITSELF" means that it is not kosher or persuasive to argue that because someone's belief changes their actions it must mean that the belief is correct. If this were allowed, Allah's suicide bombers would be proof that Islam is right, and so on, for every religious and other beli
Ways of Arguing Physics
This is the first of probably (the future is not yet known!) articles I'll do today for this list, responding to the comments of several of you. Yesterday I was reading from a new book, "Faster Than the Speed of Light," Joao Magueijo, about his theory that the speed of light may vary from place to place and, especially, over time. One thing he reminded me of is the culture of sitting in crowded offices doing physics by arguing, drawing pictures, arguing, yelling, laughing, drawing more pictures, shaking heads in despair, and arguing some more. Of the arguments go on until one side gives up, or falls asleep, or leaves in disgust, unconvinced. (Things are no so orderly as in the "decision duels" described by Marc Stiegler in "David's Sling.") He also mentions the late night drinking, the arguments where each side is mumbling and sleepy. Lee Smolin has called this the "Russian way to do math and physics." Well, we don't have this kind of bandwidth, or time. We write our little articles, sans drawings and equations, and we just don't have the time or energy to spend hours debating and arguing and resolving intermediate issues. So it is not surprising that we see even _less_ convergence of views than the office arguers above probably see. So, onward to those replies I need to write. --Tim May
Claim: Only one past for a given present
On Monday, January 13, 2003, at 12:38 PM, George Levy wrote: Tim May wrote If you mean that "many presents" have "many pasts," yes. But the current present only has a limited number of pasts, possibly just one. (The origin of this asymmetry in the lattice of events is related to our being in one present.) I mean one (many?) present has many pasts as well as many futures. Many pasts are fundamentally caused by quantum uncertainty in memory devices; many presents are caused by uncertainty in observation devices; many futures are caused by uncertainty in the controlling devices. The past cannot be ascertained precisely just as the future cannot be predicted precisely. Our consciousness is a fuzzy point in the many world. It has an infinite number of pasts and an infinite number of futures, an everbranching tree toward the past and an everbranching tree toward the future. Taking many observer moments together, I view the many world more as a lattice then as a tree. Thus navigation in the many-world makes sense. On your point about "many pasts are fundamentally caused by quantum uncertainty in memory devices," I strongly disagree. There is only one past for one present, whether RAMs dropped bits in recording them or historians forgot something, etc. (This is captured by the formalism of observations, as well. Even with Uncertainty, all honest observers will report the same outcome of an experiment. We have not seen a violation of this, nor is one expected. There are various ways to look at this, including the topos-theoretic view of subobject classifiers. But the point is that in our history either an event happened or it did not. This is independent of whether the event was observed, recorded, written about, remembered, etc.) But this is a topic of great fascination for me, and I hope we can continue to discuss it. I am quite strongly persuaded that "many pasts for a particular present" is not a reality. Understand that I am not including "current interpretations," as in "Some historians think the Roman Empire fell because of lead in their plumbing" sorts of theories of the past. I am referring to space-time events. As noted, I also view time and events as a lattice. But lattices have certain properties of importance here. More on this later. --Tim May
Re: Many Fermis Revisited
On Monday, January 13, 2003, at 10:47 AM, George Levy wrote:
Tim, Hal, Russell
Since we have several futures ( and several pasts), time travel is
just a particular case of many-world travel.
I somewhat agree...and we are not the first to make this point.
However, we need to be careful about saying we have "several pasts" (I
assume by "several" you mean "many").
The usual modal operators are needed. We have many possible futures,
but our possible pasts are limited by the events which are "necessary"
to produce the world we are actually in. The square operator for "that
which necessarily may be" and the diamond operator for "that which may
be."
If you mean that "many presents" have "many pasts," yes. But the
current present only has a limited number of pasts, possibly just one.
(The origin of this asymmetry in the lattice of events is related to
our being in one present.)
...interesting theory elided...
If there is only a single sequence of events ("a past") which produces
the actual world we are in today, then your time machine will not work,
as one cannot go back to a world where the past was different from what
"actually happened."
(And if one did, then of course one would be an actor in a past that
never happened, a la the usual grandfather paradox in all of its usual
variants. So "returning to the present" would be to a different
present.)
If this idea has any merit this is why space travelers are not
observable either. It provides a form of cosmic censorship. By
reducing their measure through QS and the likes, advanced aliens just
evolve out of existence in our world!
You ought to read "Finity," by John Barnes. He explores a very similar
idea.
--Tim May
"They played all kinds of games, kept the House in session all night,
and it was a very complicated bill. Maybe a handful of staffers
actually read it, but the bill definitely was not available to members
before the vote." --Rep. Ron Paul, TX, on how few Congresscritters saw
the USA-PATRIOT Bill before voting overwhelmingly to impose a police
state
Re: Many Fermis Revisited
On Sunday, January 12, 2003, at 06:54 PM, Russell Standish wrote: (I'll limit myself to only commenting on the last, and most interesting, point.) This is where I lose your argument. I can't see why an MWI communication capable civilisation should be able to spread throughout our universe any faster than a non-MWI communication capable one. And even if its true, all it does is place tighter bounds on how difficult it is to create such civilisations. I agree that I didn't spend as much time as I could have on this point. Consider what would happen if MWI communication/travel happened in our timeline. First, let's distinguish between what I'll "weak MWI communications" and "strong MWI communications" (or travel, which is essentially isomorphic to communication): * Weak MWI communication. Strange, cryptic, ghostly sorts of communications, somewhat like the "I Ching" pentagrams and fleeting glimpses of "close" worlds in Dick's "The Man in the High Castle," Echoed in the James Hogan novel from 1997, "Paths to Otherwhere, and in a time travel version in Greg Benford's "Timescape," where a future/branch a century in the future attempts to communicate via particle physics with the "present" to stop/alter an outcome. * Strong MWI communication. Full communication with other branches, including substantive exchanges of information. Heinlein's "Glory Road" is a somewhat fantasy-oriented take on this, but the notion is clear: "Queen of the Nine Billion Universes," etc. Imagine what will happen if strong MWI communication happens in our universe, our branch: -- presumably access to all of the manifold knowledge from every universe which has done science, engineering, etc. -- vast amounts of technology (as some universes are "ahead" because the Newtonian revolution happened in 535 A.D., etc.) -- like a quantum computer, every calculation run a bunch of times, answers already known A summary of the Hogan book captures a bit of the impact: "The well-worn sf notion of parallel universes receives a computer-driven update in Hogan's latest novel. Berkeley research scientists Hugh Brenner and Theo Jantowitz are just beginning to make startling progress in siphoning information from other universes by means of sophisticated computer technology when their funding disappears. Fortunately but not fortuitously, they are recruited by a secret Defense Department research arm to continue their work under the umbrella of Project Octagon. Joined by a motley team of brilliant minds, including a Buddhist philosopher, the two quickly develop the means to shift their awarenesses to other versions of themselves in the "multiverse" and to preview thereby future outcomes for their home universe ..." This is why is it seems reasonable to me that MWI communication would dramatically a civilization's technology. (Not saying this knowledge would cause them to colonize the universe...maybe they'd give up in despair, or contemplate their navels, whatever. But contact with 10^10^N other worlds sure could be a kick in the pants, a kick that puts a civilization drastically ahead of any civilization which is still evolving "unitarily.") >From Hal's reference to Mike Price's document (which I read several years ago, so I'd forgotten or had not read his bit about MWI and Fermi), it looks like Price reached the same conclusion. --Tim May
Re: Many Fermis Revisited
On Sunday, January 12, 2003, at 05:38 PM, Russell Standish wrote: The key assumption here is whether advanced technological civilisation (such as ourselves) is easy or difficult on the timescale of the age of the universe (10^10 years). Assuming that this is difficult (contra to your comments below), solves the standard Fermi paradox (namely other advanced civilisation are too far away to have reached us yet, and probably too far away to ever reach us, unless the universe starts contracting again. So your Throgians are as good as mythical. I cannot understand why you would say "contra to your comments below" when in several places I discussed this issue of how common life is: "... estimates of likelihood of advanced civilizations elsewhere. If we are the only form of life in our timelike region of the universe, i.e., within a few billion light-years, then of course this makes the odds of another receiver-builder nil." "My hunch is that alien civilizations may well exist, but are not abundant " I made no assumptions of nondifficulty (to use your phrasing). This is in fact why I picked the Thogians a few hundred million light-years from us. Now perhaps you think advanced civilizations are even rarer than in this example, there have not yet been any civilizations reaching our level, except ourselves, anywhere within a billion light-years or so of us. Arguing this one way or another was not my point. Rather, it is that the effects of MWI communication (or time travel) would likely be enormous and that such a civilization would be expected to expand and show themselves in the (likely) billion or more years they would have had to expand, build Dyson spheres and other cosmic artifacts, send signals, etc. This also implies that such technological civilisations are also rather diffuse within the Multiverse, _excepting_ of course those which share part of their history with ours (eg the Nazis which won WWII). We have some predictive power as to what those people would be like, since they will be similar to us. I'm not following this at all. Why do you think that communication with (or actual travel to) worlds is dependent on our ability to _predict_ things about them? I can see an argument to be made that only close worlds can be communicated with, and some folks have argued this, but this argument was not made by you here. So, I for one, would not discount Hal Finney's point. What I said was that the point that we have not yet built a receiver or portal says nothing about what others have done. And if there are other civilizations out there and building such receivers or portals is possible, one would expect a fair number of them to have done so. Since the implications of building such portals are, I think, enormous, I would expect a civilization which has built such things to have expanded even more rapidly through their part of the universe than without such things. --Tim May
Many Fermis Revisited
rmany won the Second World War. For if such other branches _are_ accessible, then if we are not in some special place in the universe these other branches were _also_ accessible to the Throgians in that galaxy in Coma Berenices, enough time ago for the radical implications of MWI travel/communication to produce a massive infusion of knowledge and engineering capabilities. Hal's argument that we have not seen either time travelers or MWI traveler's because we have not yet built receivers yet. My argument is "But someone would have by now." And unless we are in a narrow band of outcomes where some civilizations have done this and are even now expanding toward us but have not yet reached us or produced engineering feats we can observe, they should have shown themselves by now. My hunch is that alien civilizations may well exist, but are not abundant (else we'd see the Galactic Federation already here, etc.), and that neither time travel nor MWI travel/communication is possible. --Tim May
Re: Science
usiasm. His novels "Distress," "Quarantine," "Diaspora," "Permutation City," "Schild's Ladder," etc. are wildly imaginative and thought-provoking. Best of all, he gets to develop his ideas at length and with fictional characters to explicate the details, WITHOUT people like us to point out obvious flaws or lack of observational details. (I'll add that I think he's the most realistic author I've seen in a while on how some of the physics may unfold. For example, in some of his novels (SL, Diaspora) he has "new physics" only being discovered a century or so from now, which I think is a plausible timeline. And, even with a new TOE, it takes another thousand years of AI-enhanced thinking before new energy regimes are adequately probed (via an accelerator that is roughly the size of the solar system, to probe Planck scales).) The point is this: anyone proposing a "wild theory" here or any other realtime list is going to need to expect folks taking potshots and pointing out inconsistencies and flaws. For most of science, this works very well. (The case of Wolfram's "new kind of science" is an excellent example to discuss in this connection. Maybe in another post.) We are like the Caltech students that Niven described in the early 70s: they demolished the physics of "Ringworld" and pointed out ways that it could and could not work. How could we be otherwise? I suppose your line: I cannot understand your point here. But if the "several" who were once here are no longer posting, I am not stopping them.< refers to my phrase "'well composed' edifice of the scientific doctrines..." (discounting the personal defensive) - maybe if you care to glance at my 'older' essay (http://pages.prodigy.net/jamikes/SciRelMay00.html) that would release me from lengthy explanations - subject maybe to my newer miscraftings here - OK, I just checked out your URL and scanned your essay. It looks to be about religion and memes. I'm not sure what it has to do with analysis of "Everything" theories a la Tegmark, Egan, Schmidhuber, Fredkin, Zuse, etc. Again, I have no idea what you are talking about here<) Idea I have, wording is hard. I may mention some key-phrases without contextual explanations (and without asking Wei Dai to reformulate the list in favor of these ) as stirring around lately in select speculations: -- "complexity-thinking", -- extending the limits of reductionism: induction-buildup, to deduction-analysis, -- extending the limited models of reductionist science, -- natural systems as networks of networks, -- total interconnectedness -- etc., but I am afraid that whatever I mention opens another Pandora's box of worms. We (working in these lines) have still arguments how to understand (then formulate) concepts like impredicative, endogenous, emergent, etc., beside the re-identification of 'older' terms galore. I certainly encourage you to more fully explicate your ideas. But understand that I (and others, I think) will "compare and contrast" theories with what has been observed, what appears to be solidly known, etc. This is what we would do if Vinge were to post ideas here, like I said. I'm quite skeptical that much of the "complexity-thinking" is as important as some think it is. (I know about Chaitin, and introduced him to the "Extropians" list in 1993, as Hal can confirm. I've also corresponded with him, and I went to the first Artificial Life (A-LIFE) conference in Los Alamos in 1987 largely because I'd read that John Holland, Greg Chaitin, and several others that I wanted to meet would be there...as it turned out, Chaitin cancelled. I've also read the usual complexity theory stuff. Close links with computation and cryptology, of course. But drawing overbroad conclusions, as I think Prigogine does, is why I am skeptical.) (To add another comment. At this first A-LIFE I also had a lot of time to talk to Stuart Hameroff about his "nanotubules" and "cytoskellular consciousness" theories. Strange stuff. A perfect example of my novelization point: were Hameroff to develop his ideas in a novel, a well-written and engaging novel, we might be able to say "Weird, but interesting!." But when I see Hameroff's ideas in essays on the Web, or Penrose's vague claims that gravity may have something to do with quantum weirdness, I remain intensely skeptical.) I think there's more than plenty of fascinating new physics being vigorously discussed in the modern physics community. The arXive site is fun to browse. When a theory is so weird that it is not even discussable by workers in the field, then our skepticism meters must reflect this. Your comments did not "aggravate" me...they simply prodded me to set down some of the many ideas percolating in my head. This may make me seem like a "conservative" here, but it's my nature to analyze and critique, to compare and contrast. Extraordinary claims require extraordinary proof. --Tim May
Re: Science
On Saturday, January 11, 2003, at 03:11 PM, John M wrote: This list - several years ago - took a free approach, alas lately more and more conventional opinions slip in, regrettable for me, because I hold that the conventional "science establishment" holds feverishly to old addages, acquired in times when the epistemic cognitive inventory was much less than available today (which is much less than that of tomorrow). Even the "topics of the future" build on ancient observations and their explanations (formalism), in order to conform with the scientists' earlier books, teachings, pupils, discussions. Given that there is no moderation, no censorship, it is clear that talk about "this list...took" is missing the point. "This list" is really "the comments of those subscribed and contributing." As always, if you believe people are talking about the wrong things, your best approach to is to persuasively make your own points which you believe fit your conception of what subscribers to the list "should" be talking about. I have no understanding of what you mean by saying "alas lately more and more conventional opinions slip in." If you think my views are too conventional, for example, or that I should not be posting to this list, I suppose you can ask Wei Dai to remove me. I believe nearly all of my posts are in the spirit of the list's charter, discussing as I do MWI, Tegmark/Egan, possible worlds, modal logic, etc. (I seldom if ever discuss the Schmidhuber thesis, and the "COMP" thesis, as these are not currently interesting to me. I notice plenty of other people discussing them, and I read their comments with _some_ interest, anticipating the eventual day when the COMP stuff is more germane to me.) In MOST cases the methodology works in practical ways, builds technology, up to the point when "understanding" comes in. This is a many negated term, many so called scientists satisfy themselves with practical results (for tenure, awards, etc.) Few researchers take the stance to "free" their mind from learned prejudice and check the 'well composed' edifice of the scientific doctrines for sustainability under the newly evolved vistas. There were several on this list. I cannot understand your point here. But if the "several" who were once here are no longer posting, I am not stopping them. The new ideas were quickly absorbed into the existing formalistic mill - calculative obsolescence and semantic impropriety, which confused many. New science is like Tao: who says "I developed a theory within it" does not know what he talks about. Science is on the crossroad: (I wold not say bifurcation, because I have negative arguments against this concept) and we know only that something 'new' is in the dreams, we need more thinking before we can identify "what". Again, I have no idea what you are talking about here. Speaking of "science" usually means "old science". This list started out to serve the "new science". It woulod be a shame to slip back into the conventionalities. Talk to Wei Dai. I write what I think is true and important. --Tim May
Science
had some inkling that they could incorporate "fudge factors" into their theories, parameters left blank until they could be filled in, but NO EXPERIMENTS and NO OBSERVATIONS needed such fudge factors. Within the accuracy of the experiments, there were no epicycles or orbital corrections needed (not for a long time, not until slight deviations from theory in the orbit of Mercury showed up, as one example). And this correspondence is not accidental. This is not like saying the theory of epicycles is a limiting case of classical mechanics and gravitation, which it most certainly is not. Let Planck's constant go toward zero and classical mechanics happens. Let v << c and classical mechanics happens. or you can say, no we're about as good at it as always, maybe a little more refined in method but not much, and we'll continue to get fundamental scientific revolutions even in areas we see as sacrosanct theory today. It took better tools to see into the regimes where the older theories failed. It will probably take substantial increases in accelerator energies to see into the regimes where quantum gravity is evident...some say that short of probing at Planck scales (of length, energy, time), we will have no way to distinguish amongst competing theories. More optimistic folks think we may see some kind of evidence within our lifetimes. Certainly we are not in a solipsistic situation where new theories are easily formulated and knock off older theories without experimental evidence or unexplained phenomena. I don't understand your "secret cause of asymmetry in the universe" point. We understand some things about symmetry breaking in particle physics theories, via gauge theories and the like. If you want more than this, you'll have to expand on what you mean here. It is a Koan (kind of). A self-referential, absurd example of a notion that an imbalance in a formal symbol system (the words I'm using, and the quotes) could possibly be the cause of asymmetry in the physical universe. Probably not, but I won't get into a debate with you on this. --Tim May
Re: Possible Worlds, Logic, and MWI
On Saturday, January 11, 2003, at 01:39 AM, Eric Hawthorne wrote: This strict "anonymous symbols" interpretation is how one must treat formal logic and propositions expressed in formal logic too. Every time I read someone bemoaning how logic has difficulty with expressing "what is going to happen in future", I think, why would you expect a formal system of symbols to have anything to do with future time in reality? There are excellent reasons to expect a formal system of symbols to correctly predict future time in reality: the operation of machines, chips, programs. Of enormous complexity, iterating for trillions of steps in time, the outcomes are consistent and predictable. As for someone "bemoaning how logic...future," temporal logic is an active research area. Arthur Prior has written much about the logic of time. Modal logic is essentially about this kind of reasoning. Pace the point below about comets hitting planets, a formal symbol system is not going to predict something dependent on events we cannot see (yet) or model (yet). It would be unreasonable to expect a logic of time to somehow predict events from outside our "knowledge cone" (like a light cone, but for knowledge). As far as I know, there is no good formulation of a formal connection between a formal system and """"""reality""""" <-unbalanced quotes, the secret cause of asymmetry in the universe. How's that for a "quining" paragraph? We analyze Reality in bits and pieces, in facets. We analyze planetary motions, and now we have logical symbol models which are enormously accurate and far-reaching in time. Granted, models of future planetary positions cannot predict events outside the model, such as collisions with comets not yet charted, and so on. But this is not a plausible goal of any model. I don't understand your "secret cause of asymmetry in the universe" point. We understand some things about symmetry breaking in particle physics theories, via gauge theories and the like. If you want more than this, you'll have to expand on what you mean here. Is there? For example, "truth" is defined in formal logic with respect to, again, formal models with an infinite number of formal symbols in them. It is not defined with respect to some vague "correspondence" with external reality. Actually, science is just about such correspondences with external reality. I haven't argued that logic alone is a substitute for science, measurement, experimentation, refutation, correction, adjustment, model-building. Someone was writing about "correspondence theory" with this goal in mind many years back, and that sounded interesting. I haven't read Tegemark et al. What do they say about the formalities of how mathematics extends to correspond to, or to be? external reality? To me, there is still a huge disconnect there. Again, I don't understand what you mean by "there is still a huge disconnect there." If you are refuting Tegmark, you should read his articles first. If you are saying that much still needs to be done, this is of course true, fortunately. --Tim May
Re: Possible Worlds, Logic, and MWI
On Friday, January 10, 2003, at 08:54 PM, Tim May wrote: Wei suggested that in the context of a many-worlds universe (not just the quantum MWI but even for a broader set of possibilities), you might not make this same decision. You know that when the coin flips, the universe is going to effectively branch and both possibilities are going to be actualized. Let us suppose that in addition to slightly preferring apples to oranges, you have a strong value preference for diversity. You like variety and you dislike having everything the same everywhere. In that case, you might rationally choose to receive an apple on heads but an orange on tails. While this slightly reduces your average pleasure level in terms of tasting the fruit, this could be more than compensated by your increased pleasure at knowing that you are enjoying diverse experiences in the two worlds. To add something to my last comment, there is a huge difference between these two situations: 1. Alice believes in the MWI, whether it is true or not. 2. The MWI is true, whether Alice believes in it or not. I fully accept that Situation 1, where Alice believes in the MWI, can and likely will alter her choices. She may alter her risk assessment model, she may change what she believes about religion, and so on. I think, Hal, that in your language above you are confusing the issue of Alice's faith in MWI with the actual reality or nonreality of MWI. Your comments about "You know that when the coin flips" and "knowing that you are enjoying diverse experiences in the two worlds" are not statements about what is actually happening but, importantly, about what Alice _believes_ will happen. This is paralleled in religion: "Alice knows that when she prays to Baal, he listens and smites her enemies. She knows that Baal has prepared a place for her in his party room in the afterlife. She knows that dying young for Baal will only take her to Baal that much sooner. She awakens every day with hope and expectation." No doubt that belief alters behavior. But it doesn't make either the existence of MWI or Baal any more real. (Understand of course that I am not putting belief in MWI on the same level as belief in YHWH or Allah or Baal or Yog-Sotteth. But we must be careful in using language like "You know that your choice does such and such.") --Tim May
Re: Possible Worlds, Logic, and MWI
in each world. We're not just acting to maximize the expected outcome in each world averaged across all of them, we're acting to maximize the utility of the "big picture", the entire set of worlds affected by our acts, considered as a whole. Why would there be any reason to try to maximize the utility of this "big picture"? For those of us who don't even strive for "the greatest good for the greatest number" in a single-branch universe, why would striving for more good (whatever "good" is) in 10^300+ branches be interesting or important? In any case, if MWI is correct, then there is every type of universe imaginable, consistent only with the laws of physics and math, and every decision for the good or the bad or whatever has been made countless times in countless ways. By any calculus of the multiverse, the sheaf of universes in which Tim May or Hal Finney even exist is of measure approaching zero. Meanwhile, I'm _here_. --Tim May "Dogs can't conceive of a group of cats without an alpha cat." --David Honig, on the Cypherpunks list, 2001-11
Possible Worlds, Logic, and MWI
On Friday, January 10, 2003, at 12:34 PM, George Levy wrote: This is a reply to Eric Hawthorne and Tim May. (Tim comment: the quoted text below is partly a mix of my comments and partly George's.) Lastly, like most "many worlds" views, the same calculations apply whether one thinks in terms of "actual" other worlds or just as possible worlds in the standard probability way (having nothing to do with quantum mechanics per se). Good point. Or so I believe. I would be interested in any arguments that the quantum view of possible worlds gives any different measures of probability than non-quantum views give. (If there is no movement between such worlds, the quantum possible worlds are identical to the possible worlds of Aristotle, Leibniz, Borges, C.I. Lewis, David Lewis, Stalnaker, Kripke, and others.) Interesting. I don't know how to proceed in this area. I've been meaning to write something up on this for a long time, but have never gotten around to it. I'll try now. FIRST, let me say I am not denigrating the quantum mechanics issue of Many Worlds. I was first exposed to it maybe 30 years ago, not counting science fiction stories about parallel worlds, and even Larry Niven's seminal "All the Myriad Ways," which was quite clearly based on his MWI readings. Also, I am reading several recent books on QM and MWI, including Barrett's excellent "The Quantum Mechanics of Minds and Worlds," 1999, which surveys the leading theories of many worlds (the bare bones theory, DeWitt-Graham, Albert and Loewer's "many minds," Hartle and Gell-Mann's "consistent histories," and so on. Also, Isham's "Lectures on Quantum Theory," and I've just started in on Nielsen and Chuang's "bible of quantum computers" massive book, "Quantum Computation and Quantum Information," from whence I got the funny Hawking quote about him reaching for his gun when Schrodinger's cat gets mentioned. So I am deeply interested in this, more so for various reasons than I was 30 or 20 or 10 years ago. SECOND, my focus is much more on the tools than on any specific theory. I may be one of the few here who doesn't some wild theory of what the universe is! (I'm only partly kidding...we see a lot of people here starting out with "In my theory...universe is strand of beads...embedded...14-dimensional hypertorus...first person awarenesss...causality an illusion...M-branes are inverted..." sorts of theories. Some have compared our current situation to the various and many theories of the atom in the period prior to Bohr's epiphany. Except of course that various theories of the atom in the 1900-1915 period were testable within a few years, with most failing in one spectacular way or another. Today's theories may not be testable for 1000 years, for energy/length reasons. (One hopes some clever tests may be available sooner...) When I say tools I mean mostly mathematics tools. I'm a lot more interested, for instance, in deeply understanding Gleason's Theorem and the Kochen-Specker Theorem (which I do not yet understand at a deep level!) than I am in idly speculating about the significance of QM for consciousness or whatever. (No insult intended for those who work in this area...I just don't see any meaningful connections as yet.) And the mathematical tools of interest to me right now are these: lattices and order (posets, causal sets), the connections between logic and geometry (sheaves, locales, toposes), various forms of logic (especially modal logic and intuitionistic logic), issues of time (a la Prior, Goldblatt, causal sets again), and the deep and interesting links with quantum mechanics. I'm also reading the book on causal decision theory that Wei Dai recommended, the Joyce book. And some other tangentially related things. A lot of what I am spending time on is the basic topology and algebra I only got smatterings of when I was in school, along with some glimpses of algebraic topology and the like. I'm using category theory and topos theory not as end-alls and be-alls, but as the lens through which I tend to view these other areas. Frankly, I learn faster and more deeply when I have some such lens. If this lens turns out to be not so useful for what I hope to do, I'll find another one. But for now, it gives me joy. I wrote a fair amount here last summer about topos theory, intuitionistic logic, notions of time evolution, and the work of Baez, Smolin, Markopoulou, Crane, Rovelli, and about a half dozen others. This remains a core interest, with some interesting (but not worked out, IMO) connections with QM (cf. the papers of Isham and Butterfield, and I. Raptis, and even some Russians). Bruno is more advanced than I am on the logic, as I have only gotten really interested
Re: Quantum Suicide without suicide
On Thursday, January 9, 2003, at 08:22 PM, George Levy wrote: OK. Let's consider the case of the guy dying of cancer and playing the stock market simultaneously.. In real life, the hard part is to get meaningful probability data. For the sake of the argument let's assume the following scenario: ..scenario elided, not to mislead, but because I will not be using any details of the calculation... As we can see, the rate of return for Alice is 4.8 times that of Bob. Alice will make a profit, but not Bob. Conclusions: All this involves really basic probability theory. The first person perspective probability is identical to the probability conditional to the person staying alive. The probability of the event in question (stock going up) must be tied to the person staying alive ( a cure for cancer). In the case of a "conventional" QS suicide to world conditions matching the requested state: ie. winning one million dollars. In the deathrow case one could imagine a scenario in which the event in question (DNA test discovery) is tied to a reprieve from the governor coming because of a DNA test exhonerating the prisoner. The prisoner could bet on DNA testing as a good investment. The airline case is similar. The hard part is figuring the probability of very unlikely saving events such as a scientific discovery, ET landing on earth or the coming of the messiah :-) How is this different from standard life insurance arguments, where buying a policy is betting one will die and not buying a policy is betting one will live? If one has no heirs to worry about, no concern about the world if and after one dies, then it has been known for a long time that the "smart" thing to do is not to buy life insurance. If one dies, the policy payoff is worthless (to the dead person), but if one lives, the money has been saved. Similar calculations are very simple for going into a dangerous situation: take a bet, at nearly any odds, that one will live. If the odds of survival in going into a combat situation are one in a hundred, and betting odds reflect this, bet everything one can on survival. If one dies, the $10,000 lost is immaterial. If one lives, one has a payout of roughly a million dollars. The scenario with cancer cures and doctors and quackery and all just makes this standard calculation more complicated. And instead of couching this in terms of bets (or stock investments), one can phrase it in standard terms for high risk jobs: "Your chance of succeeding is one in a hundred. But if you succeed, one million dollars awaits you." (I doubt many would take on such a job. But with varying payouts, we all take on similar sorts of jobs. For example, flying on business.) It's a reason some people take on very risky jobs. They figure if they succeed, they'll be rich. If they fail, they'll be dead and won't care. (Certainly not many people think this way, but some do. But "betting on yourself" is not "quantum suicide" in any way I can see. It's just a straightforward calculation of odds and values of things like money (of no value if dead, for example) in the main outcomes. Lastly, like most "many worlds" views, the same calculations apply whether one thinks in terms of "actual" other worlds or just as possible worlds in the standard probability way (having nothing to do with quantum mechanics per se). Or so I believe. I would be interested in any arguments that the quantum view of possible worlds gives any different measures of probability than non-quantum views give. (If there is no movement between such worlds, the quantum possible worlds are identical to the possible worlds of Aristotle, Leibniz, Borges, C.I. Lewis, David Lewis, Stalnaker, Kripke, and others.) --Tim May "How we burned in the prison camps later thinking: What would things have been like if every security operative, when he went out at night to make an arrest, had been uncertain whether he would return alive?" --Alexander Solzhenitzyn, Gulag Archipelago
Re: Quantum suicide without suicide
From: Tim May <[EMAIL PROTECTED]>
Date: Thu Jan 9, 2003 1:22:32 PM US/Pacific
To: [EMAIL PROTECTED]
Subject: Re: Quantum suicide without suicide
On Thursday, January 9, 2003, at 12:32 PM, George Levy wrote:
As you can see, suicide is not necessary. One could be on death row -
in other words have a high probability of dying - and make decisions
based on the probability of remaining alive.
Being on death row, dying of cancer, travelling on an airline, or
sleeping in our bed involve different probability of death... These
situations only differ in degrees. We are all in the same boat so to
speak. We are all likely to die sooner or later. The closer the
probability of death, the more important QS decision becomes.
The guy on death row must include in his QS decision making the factor
that will save his life: probably a successful appeal or a reprieve by
the state governor.
No, this is the "good news" fallacy of evidential decision theory, as
discussed by Joyce in his book on "Causal Decision Theory." The "good
news" fallacy is noncausally hoping for good news, e.g., standing in a
long line to vote when the expected benefit of voting is nearly nil.
("But if everyone thought that way, imagine what would happen!" Indeed.)
The guy on death row should be looking for ways to causally influence
his own survival, not consoling himself with good news fallacy notions
that he will be alive in other realities in which the governor issues a
reprieve. The quantum suicide strategy is without content.
As you see, suicide is not necessary for QS decisions.
No, I don't see this. I don't see _any_ of this. Whether one supports
Savage or Jefferys or Joyce or Pearl, I see no particular importance of
"quantum suicide" to the theory of decision-making.
It would help if you gave some concrete examples of what a belief in
quantum suicide means for several obvious examples:
-- the death row case you cited
-- the airplane example you also cited
-- Newcomb's Paradox (discussed in Pearl, Joyce, Nozick, etc.)
-- stock market investments/speculations
--Tim May
Re: Quantum suicide without suicide
On Wednesday, January 8, 2003, at 10:58 AM, George Levy wrote: In the original verision of Quantum Suicide (QS), as understood in this list, the experimenter sets up a suicide machine that kills him if the world does not conform to his wishes. Hence, in the branching many-worlds, his consciousness is erased in those worlds, and remains intact in the worlds that do satisfy him. Is it possible to perform such a feat without suicide? What is the minimum "attrition" that is required and still get the effect of suicide? Hawking had a good line: "When I hear about Schrodinger's Cat, I reach for my gun." Slightly modify the QS conditions in another direction: instead of dying immediately, one goes onto death row to await execution. Or one is locked in a box with the air running out. And so on. This removes the security blanket of saying "Suicide is painless, and in all the worlds you have not died in, you are rich!" In 99....99% of all worlds, you sit in the box waiting for the air to run out. I don't know if there are other worlds in the DeWitt/Graham sense (there is no reason to believe Everett ever thought in these terms), but if they "exist" they appear to be either unreachable by us, or inaccessible beyond short times and distances (coherence issues). In particular, it seems to me there's a "causal decision theory" argument which says that one should make decisions based on the maximization of the payout. And based on everything we observe in the world around us, which is overwhelmingly classical at the scales we interact in, this means the QS outlook is deprecated. Consider this thought experiment: Alice is facing her quantum mechanics exam at Berkeley. She sees two main approaches to take. First, study hard and try to answer all of the questions as if they mattered. Second, take the lessons of her QS readings and simply _guess_, or write gibberish, killing herself if she fails to get an "A." (Or, as above, facing execution, torture, running out of air, etc., just to repudiate the "suicide is painless" aspect of some people's argument.) From rationality, or causal decision theory, which option should she pick? All indications are that there are virtually no worlds in which random guessers do well. (The odds are readily calcuable, given the type of exam, grading details, etc. We can fairly easily see that a random guesser in the SATs will score around 550-600 combined, but that a random guesser in a non-multiple-choice QM exam will flunk with ovewhelming likelihood.) What should one do? What did all of you actually do? What did Moravec do, what did I do, what did Tegmark do? --Tim May
No infinities needed
On Tuesday, December 31, 2002, at 07:02 AM, Joao Leao wrote: I don't agree with Tim's suggestion that infinite-dimensional Hilbert spaces are somewhat "ancilliary" in QM and that all systems are calculable in finite dimensional modes. In fact infinite sets of spaces are the rule in QM and the finite dimensional subspaces only serve as toy systems. I said it is often done. Many of the details of the infinite case are just not needed. And QM is often taught this way, with no loss of rigor, provided any subtleties are pointed out to the student. For example, here are some fairly typical lecture notes for a course on QM: "2.2. Hilbert Space Hilbert spaces are mentioned in most textbooks on quantum mechanics and functional analysis [3] . Therefore we will only mention some features, which are not found almost everywhere. We will also not have to go into the subtleties of topologies, continuous spectra, or unbounded operators, because throughout this course, we can assume that all Hilbert spaces are finite dimensional. Modifications in the infinite dimensional case will be mentioned in the notes. Our standard notation is <\phi ,\psi > for the scalar product of the vectors \phi ,\psi \in H, ||\phi ||=<\phi ,\phi >1/2 for the norm, and B(H) for the algebra of bounded linear operators on H. Of course, all linear operators on a finite dimensional space are bounded anyway, and the B is used mostly for conformity with the infinite dimensional case. " http://www.imaph.tu-bs.de/qi/lecture/qinf21.html In nearly every area of physics, the issue of "infinity" is phrased in terms of sequences or structures approaching or growing towards the infinite or infinitesimal. For example, a test mass is assumed to be small enough not to perturb the curvature tensor. But actual infinite spaces are not needed, not even in thermodynamics. This dispenses with a lot of the mathematical cruft, alluded to above (continuous spectra, compactness, etc.). That cruft contains a lot of beautiful math, but physics just doesn't need it, at least not very often. I have no axe to grind on this. For those who want to study only the completely general, infinite-dimensional cases, cool. But a good understanding of finite-dimensional vector spaces (e.g., the Halmos book) provides the math one needs for QM, especially at the level we usually discuss it at here. (As many here perhaps already know, Halmos was Von Neumann's assistant, writing up his lectures, when he wrote his book.) Provided the complex space is normed, and is complete, which all finite-dimensional vector spaces are, the math works. No infinities are needed, which is good. --Tim May
Re: Quantum Probability and Decision Theory
On Monday, December 30, 2002, at 03:46 AM, Brent Meeker wrote: On 31-Dec-02, Hal Finney wrote: One correction, there are no known problems which take exponential time but which can be checked in polynomial time. If such a problem could be found it would prove that P != NP, one of the greatest unsolved problems in computability theory. What about Hamiltonian circuits or factoring an integer or roots of a Diophantine equation? Hal will probably answer. I initially had the same reaction you had (except only about Hamiltonian cycles and roots of Diophantines, but not factoring, as factoring is not known to be NP-complete). But upon rereading what Hal wrote, and what I had already drafted a nearly complete reply to, I saw that he was making the subtle point that there are "no known problems which take exponential time." All of the NP-complete problems (Hamiltonian cycle, Diophantine, etc.) currently only have exponential-time solutions. But there is no guarantee that a polynomial solution (which is not NP, that is, is not the result of a "guess" or an "oracle") will not be found sometime in the future. Proving that there "are" problems which can only be solved in exponential time but which can be checked in polynomial time is subtly different from saying that all problems in the class NP-complete admit no polynomial time solutions. (I'm trying to avoid using shorthand like P and NP and Exptime, as a lot of confusion enters when shorthand gets misinterpreted.) --Tim May
Many Worlds Version of Fermi Paradox
On Monday, December 30, 2002, at 01:18 PM, Jesse Mazer wrote: Hal Finney wrote: One correction, there are no known problems which take exponential time but which can be checked in polynomial time. If such a problem could be found it would prove that P != NP, one of the greatest unsolved problems in computability theory. Whoops, I've heard of the P=NP problem but I guess I was confused about what it meant. But there are some problems where candidate solutions can be checked much faster than new solutions can be generated, no? If you want to know whether a number can be factorized it's easy to check candidate factors, for example, although if the answer is that it cannot be factorized because the number is prime I guess there'd be no fast way to check if that answer is correct. Factoring is not known to be in NP (the so-called "NP-complete" class of problems...solve on in P time and you've solved them all!). The example I favor is the Hamiltonian cycle/circuit problem: find a path through a set of linked nodes (cities) which passes through each node once and only once. All of the known solutions to an arbitrary Hamiltonian cycle problem are exponential in time (in number of nodes). For example, for 5 cities there are at most 120 possible paths, so this is an easy one. But for 50 cities there are as many as 49!/2 possible paths (how many, exactly, depends on the links between the cities, with not every city having all possible links to other cities). For a mere 100 cities, the number of routes to consider is larger than the number of particles we believe to be in the universe. However, saying "known solutions" is not the same thing as "we have proved that it takes exponential time." For all we know, now, in 2002, there are solutions not requiring exponential time (in # of cities). This is also somewhat relevant to "theories of everything" since we might want to ask if somewhere in the set of "all possible universes" there exists one where time travel is possible and computing power increases without bound. If the answer is yes, that might suggest that any TOE based on "all possible computations" is too small to accomodate a really general notion of all possible universes. And this general line of reasoning leads to a Many Worlds Version of the Fermi Paradox: Why aren't they here? The reason I lean toward the "shut up and calculate" or "for all practical purposes" interpretation of quantum mechanics is embodied in the above argument. IF the MWI universe branchings are at all communicatable-with, that is, at least _some_ of those universes would have very, very large amounts of power, computer power, numbers of people, etc. And some of them, if it were possible, would have communicated with us, colonized us, visited us, etc. This is a variant of the Fermi Paradox raised to a very high power. My conclusion is that the worlds of the MWI are not much different from Lewis' "worlds with unicorns"--possibly extant, but unreachable, and hence, operationally, no different from a single universe model. (I don't believe, necessarily, in certain forms of the Copenhagen Interpretation, especially anything about signals propagating instantaneously, just the "quantum mechanics is about measurables" ground truth of what we see, what has never failed us, what the mathematics tells us and what is experimentally verified. Whether there "really are" (in the modal realism sense of Lewis) other worlds is neither here nor there. Naturally, I would be thrilled to see evidence, or to conclude myself from deeper principles, that other worlds have more than linguistic existence.) --Tim May
Many Worlds and Oracles
On Monday, December 30, 2002, at 11:57 AM, Jesse Mazer wrote: As I understood it, the basic idea here was to use the fact that history must work out consistently to get a machine that could solve problems much faster than a Turing machine. For example, for any problem that requires exponential time to reach a solution but for which possible solutions can be checked in polynomial time, you could have the machine pick a possible solution at random, then check to see if the solution actually works, then if it *doesn't* work it sends back a sort of override command that changes the original guess, which would create an inconsistency. Or just kills you and/or your world. This idea predates Max Tegmark by quite a while...I give Moravec the credit. My own version of an oracle was done for an article I sent out in 1994. Included below. RSA Broken By The Russians? Kolmogorov Cryptography System Possibly Cracked 1 Apr 1994 MOSCOW (AP) -- At a press conference held minutes ago in a crowded hall, Russian mathematicians announced that a breakthrough had been made nearly a decade ago in the arcane branch of mathematics known as "cryptography," the science of making messages that are unreadable to others. Leonid Vladwylski, Director of the prestigious Moscow Academy of Sciences, called the press conference yesterday, after rumors began circulating that noted Russian-American reporter John Markoff was in Russia to interview academicians at the previously secret city of Soviet cryptographers, Kryptogorodok. The existence of Kryptogorodok, sister city to Akademogorodok, Magnetogorsk, and to the rocket cities of Kazhakstan, had been shrouded in secrecy since its establishment in 1954 by Chief of Secret Police L. Beria. Its first scientific director, A. Kolmogorov, developed in 1960 what is called in the West "public key cryptography." The existence of Kryptogorodok was unknown to the West until 1991, when Stephen Wolfram disclosed its existence. American cryptographers initially scoffed at the rumors that the Russians had developed public-key cryptography as early as 1960, some 15 years prior to the first American discovery. After interviews last year at Kryptogorodok, noted American cryptographers Professor D. Denning and D. Bowdark admitted that it did seem to be confirmed. Professor Denning was quoted at the time saying that she did not think this meant the Russians could actually break the Kolmogorov system, known in the West as RSA, because she had spent more than a full weekend trying to do this and had not succeeded. "Believe me, RSA is still unbreakable," she said in her evaluation report. Russia's top mathematicians set out to break Kolmogorov's new coding system. This required them to determine that "P = NP" (see accompanying article). Details are to be published next month in the journal "Doklady.Krypto," but a few details are emerging. The Kolmogorov system is broken by computing the prime numbers which form what is called the modulus. This is done by randomly guessing the constituent primes and then detonating all of the stockpiled nuclear weapons in the former Soviet Union for each "wrong guess." In the Many Worlds Interpretation of quantum mechanics, invented in 1949 by Lev Landau (and later, independently by Everett and Wheeler in the U.S.), all possible outcomes of a quantum experiment are realized. As Academician Leonid Vladwylski explained, "In all the universes in which we guessed the wrong factors, we were destroyed completely. But since we are obviously here, talking to you at this press conference, in this universe we have an unbroken record of successfully factoring even the largest of imaginable numbers. Since we are so optimistic about this method, we say the computation runs in 'Nondeterministic Pollyanna Time.' Allow me to demonstrate..." [Press Conference will be continued if the experiment is a success.] MOSCOW (AP), ITAR-Tass, 1 April 1994 Appendix First, it was Stephen Wolfram's actual suggestion, a couple of years ago after the USSR imploded, that we try to recruit mathematicians and programmers from what he surmised must exist: a secret city of Soviet cryptographers. It probably exists. We did it at Los Alamos, they did it with their rocket scientists and others (Akademogorodok exists), so why not put their version of NSA a bit off the beaten track? Note that our own NSA is within a stone's throw of the Baltimore-Washington Parkway. I wouldn't be surprised to learn that their experts were ensconced somewhere in the Urals. I tried to acknowledge Steve with my comments. By the way, so far as I know, no word has come out on whether he was right in this speculation. (Maybe some of the Russians he does in fact have working at Wolfram are these folks? Naw...) Second, Kolmogorov did ba
Re: Quantum Probability and Decision Theory
On Monday, December 30, 2002, at 11:18 AM, Tim May wrote: On Monday, December 30, 2002, at 10:44 AM, Stephen Paul King wrote: QM comp seems to operate in the space of the Reals (R) and TM operates in the space of Integers (Z), is this correct? Any finite system, which of course all systems are, can have all of its quantum mechanics calculations done with finite-dimensional vector spaces. The "full-blown machinery" of an infinite-dimensional Hilbert space is nice to have, in the same way that Fourier analysis is more elegantly done with all possible frequencies even though no actual system (including the universe!) needs all frequencies. Lest there be no confusion, I meant that all actual systems can be computed with finite-dimensional vector spaces which have inner products. Or in Von Neumann's more precise language, "complete complex inner product spaces." (Since all Hilbert spaces with an infinite number of dimensions are isomorphic, this gives rise to just saying "Hilbert space" in the singular.) The point is that the arbitrary-dimension elegance of a full-blown Hilbert space is nice to have, especially for theorem-proving, but not essential. More speculatively, postulating that a quantum state in the real world (in a quantum computer, or atom cage, etc.) is "actually" a vector with an infinite degree of positional accuracy, is akin to saying that it computes with the reals, which touches on the Blum-Shub-Smale issue I talked about earlier this morning. As Hal says, the world is not actually Newtonian. And neither is it actually quantum-mechanical in the ideal, limiting, infinite-dimensional case. --Tim May
Re: Quantum Probability and Decision Theory
On Monday, December 30, 2002, at 10:44 AM, Stephen Paul King wrote: QM comp seems to operate in the space of the Reals (R) and TM operates in the space of Integers (Z), is this correct? Any finite system, which of course all systems are, can have all of its quantum mechanics calculations done with finite-dimensional vector spaces. The "full-blown machinery" of an infinite-dimensional Hilbert space is nice to have, in the same way that Fourier analysis is more elegantly done with all possible frequencies even though no actual system (including the universe!) needs all frequencies. We must additionally account for, at least, the "illusion" of time and concurrency of events. I don't see any problems with either. (Yes, I have read Huw Price's book.) --Tim May
Computing with reals instead of integers
ween the theory of
recursive functions (aka lambda calculus), cartesian closed categories,
and the effective topos of Hyland and others.
By the way, once we think in terms of the real numbers and points on
lines and planes not actually having any real existence, the
idealization of a manifold (e.g., a Riemannian spacetime manifold) as
being infinitely divisible becomes more and more farfetched.
(Now it may well be that spacetime is in fact an ideal manifold,
divisible and measurable at scales of 10^-35 m, or even at scales of
10^-100 m. But it will not be surprising at all to many of us if
spacetime is quantized, or foamy, or latticelike, at approximately
Planck-length scales. What those lattice points are "made of" is itself
a question, but the smoothness and continuity of spacetime is not
necessarily "real all the way down." Cf. the usual books and articles
by MTW ("Gravitation"), Smolin ("Three Roads..."), Rovelli,
Markopoulou, Crane, Baez, etc. for more on spin foams, lattice
structures, etc. Greg Egan's "Schild's Ladder" novel has a description
early on, a fictional description, of course.)
--Tim May
no quantum clones doesn't mean no for all intents and purposes clones
On Tuesday, December 24, 2002, at 11:02 AM, Stephen Paul King wrote: I just can't see any basis for invoking quantum mechanics and "no cloning" for why I am not you, or why I cannot plausibly experience being you, and vice versa, and so on. [SPK] I did state that my argument is "hand waving"! But, you seem to have missed this. ;-) ... Woah! Since when does Nature have to wait for Mankind to figure out anything? YOur argument here is so grossly anthropocentric that I hope you would re-think what you are saying here! I am not thinking in terms of technical or engineering limits but instead I am trying to get at the "in principle" notions of "what could Nature do?" If, as I wrote before, our minds are classical computational machines, we should have no problems in "knowing what it is like to be" any entity that had a mind that required less computational power than that available to our brains. We might not be able to know "what it is like to be a bat" but surely we could "know what it is like to be an ameoba"! Sorry, I misunderstood your chain of logic. I thought your paragraph from the earlier post said that you were attempting to explain why we _can't_ (as in "it is necessarily the case") simulate other minds or have first-person experiences of their minds. " The no cloning theoren of QM seems to have the "right flavor" to explain how it is that we can not have first person experience of each other's minds, whereas the UTM model seems to strongly imply that I should be able to know exactly what you are thinking." I read the "how it is that we can not have" as your claim that we know this to be the case. I see you are saying something close to what I am saying, "It may be the case" that minds cannot be simulated. And it may be the case, via some hand-wavy arguments, that this is "because" to do so would violate the no cloning theorem. But, even on this claim, I am intensely skeptical. I don't believe that any mind is critically dependent on a precise, perfect quantum state. Consider this thought experiment. Suppose the no cloning theorem does indeed mean that my mind in the state it is now in at this exact instant cannot be exactly duplicated. Well, would you settle for my mind as of a minute ago? A second ago? (And the usual chestnuts about whether the "myself" of _right now_ is the same person as a microsecond ago, an hour ago, etc.) I can imagine some variant of the usual epsilon-delta arguments of analysis to show that given any degree of closeness of states (possible worlds), there exists some time delay which gives a simulation and which still violates no theorems about cloned states. (I would guess the time for biological systems is on the order of what Max Tegmark and others have estimated for decoherence.) In other words, no quantum clones doesn't mean no for all intents and purposes clones. --Tim May
QM not (yet, at least) needed to explain why we can't experience other minds
On Monday, December 23, 2002, at 08:06 PM, Stephen Paul King wrote: Yes. I strongly suspect that "minds" are quantum mechanical. My arguement is at this point very hand waving, but it seems to me that if minds are purely classical when it would not be difficult for us to imagine, i.e. compute, what it is like to "be a bat" or any other classical mind. I see this as implied by the ideas involved in Turing Machines and other "Universal" classical computational systems. The no cloning theoren of QM seems to have the "right flavor" to explain how it is that we can not have first person experience of each other's minds, whereas the UTM model seems to strongly imply that I should be able to know exactly what you are thinking. In the words of Sherlock Holmes, this is a "the dog did not bark" scenario. I just can't see any basis for invoking quantum mechanics and "no cloning" for why I am not you, or why I cannot plausibly experience being you, and vice versa, and so on. Even if intelligence is purely classical (in terms of the physics), there are excellent reasons why there is no way today (given today's technology, today's interfaces, today's bandwidth) for me to "compute what it is to be a bat." Inasmuch as we cannot even build a machine which even remotely resembles a bat, or even an ant, the inability to simulate/understand/"be" a bat is not surprising. There is no mapping currently feasable between my internal states and a bat's. Even if we are made of relays or transistors. Saying that our inability to know what it is to be another person implies that some principle of QM is likely to be involved is, in my view, unsupported and unrealistic. It may well be that there are deep, QM-related reasons why Alice cannot emulate Bob, but we are probably a long way in _engineering_ terms from knowing that Alice can or cannot emulate Bob, or have a first person understanding of what a bat is, etc. Occam's Razor--don't multiply hypotheses needlessly. In other news, I am enjoying Barrett's book on quantum mechanics and minds. (Interesting to compare his views with Bub, Peres, Isham, and Wheeler.) Got a copy of Joyce's "Causal Decision Theory," to go along with the QM papers Bruno and Wei have been citing. Also read an interesting science fiction novel with some new twists on the Many Worlds Interpretation (esp. the DeWitt variant): "Finity," by John Barnes. A New Zealand astronomer/mathematician with some interesting ideas about "abductive reasoning" finds himself slipping between different realities. --Tim May
Mathematics and the Structure of Reality
somewhat like the Ocean - if an explorer worships the Ocean, then he will go off in any direction that Ocean seems to be leading ... Sorry, but this is a silly argument. Smolin and Rovelli may in fact be wrong in their theory of loop quantum gravity (and the closely related theories of spin foams, etc., along with Penrose, Susskind, Baez, Ashketar, and the whole gang), but it is almost certainly not for some simplistic reason that they were "ALGEBRAISTS." In fact, Penrose is a geometer's geometer. See, for example, the essays in his Festschrift. Now the geometry focus of Penrose does not prove _anything_ about either the internal consistency or the ultimate truth of some of his spin network and spinor models, nor about the truth or falsity of spin foams and so on. As for Lawvere and Mac Lane being "ALGEBRAISTS," I neither see your point nor its relevance. What Mac Lane may or may not be is open to debate...his work on homology theory tends to mark him as an algebraic topologist. And Grothendieck and Lawvere were looking into generalizations of the concept of a space--and they succeeded. (Personally, and speculatively, when the concept of a space is generalized so nicely, I think in terms of "this probably shows up in the physical world or its description someplace." If this ain't geometry affecting a physics outlook, what is?) Anyway, it's silly to argue along these lines. You ought to take a look at one of his recent books (co-authored when he was around 80): "Sheaves in Geometry and Logic: A First Introduction to Topos Theory." I'd call sheaves, presheaves, and locales some pretty deep geometrical/topological ideas, albeit at a level of abstraction that takes a lot of effort to master. What the structure of reality really is depends on a couple of important things: 1. What aspect we are looking at, whether the local causal structure of spacetime or the "explanation" of the particles and their masses, or even at some grossly different scale, such as fluid turbulence (still not understand, in many ways, and yet almost certainly not depending on theories of branes or strings or the Planck-scale structure of spacetime). 2. Scales and energies, whether the cosmological or the ultrasmall. 3. Our conceptual biases (if we only know geometry, we see things geometrically, and so on). One of the reasons I like studying math is to expand my conceptual toolbox, to increase the number of conceptual basis vectors I can use to build models with. --Tim May
Re: Applied vs. Theoretical
, did the Cartan-influenced differential forms approach to GR lead to new predictions that the classical, tensor-oriented approach did not? Probably few, if any, as most of the accessible predictions of GR were made a long time ago. But should students learning GR learn the methods for raising and lower indices in tensors or the more modern differential forms approach? The time saved, and the unity gained, may lead to new syntheses, such as in quantum gravity. Likewise, is the Hilbert space formulation of QM dramatically different in making predictions that the Schrodinger wave equation formulation? Working chemists still calculate Hamiltonians and wave equations--they don't need to think in terms of Hilbert space abstractions. (And in the area of observables, the great Von Neumann actually got it _wrong_ in his formulation, as Bell proved several decades later...) Geroch says this in his introduction: "In each area of mathematics (e.g., groups, topological spaces) there are available many definitions and constructions. It turns out, however, that there are a number of notions (e.g., that of a product) that occur naturally in various areas of mathematics, with only slight changes from one area to another. It is convenient to take advantage of this observation. Category theory can be described as that branch of mathematics in which one studies certain definitions in a broader context--without reference to the particular area to which the definition might be applied. It is the "mathematics of mathematics." "Although this subject takes some getting used to, it is, in my opinion, worth the effort. It provides a systematic framework that can help one to remember definitions in various areas of mathematics, to understand what many constructions mean and how they can be used, and even to invent useful definitions when needed." (p. 3) And apropos of one of the direct themes of this list, the chart on page 248 is a better chart of the categories which are of direct (known) relevance to modern physics than Max Tegmark's chart of what he thinks of as the branches of mathematics. (I don't mean this to sound snide...it's just a statement of my opinion. Further, Tegmark and others working on All Math Models need to get up to speed on this "mathematics of mathematics.") Division algebras like quaternions and octonions are not shallow in this sense; nor are the complex numbers, or linear operators on Hilbert space Anyway, I'm just giving one mathematician's intuitive reaction to these branches of math and their possible applicability in the TOE domain. They *may* be applicable but if so, only for setting the stage... and what the main actors will be, we don't have any idea... Sure, there's juicy stuff in the details of octonions. John Baez would agree with you. Getting down to making exact calculations is almost always necessary, and sometimes illuminating. But he also connects quaternions, octonions, etc. to n-categories and more generalized truths. Read his stuff for details--he writes more about both of these areas, various algebras and various categories, and their connections to physics, than anyone I know. Look, I'm happy that you looked at category theory and didn't find it to your taste. I had the opposite experience. Diversity is good. --Tim May
Applied vs. Theoretical
On Sunday, December 1, 2002, at 10:00 AM, Osher Doctorow wrote: 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. As I hope I had made clear in some of my earlier posts on this, mostly this past summer, I'm not making any grandiose claims for category theory and topos theory as being the sine qua non for understanding the nature of reality. Rather, they are things I heard about a decade or so ago and didn't look into at the time; now that I have, I am finding them fascinating. Some engineering/programming efforts already make good use of the notions [see next paragraph] and some quantum cosmologists believe topos theory is the best framework for "partial truths." The lambda calculus is identical in form to cartesian closed categories, program refinement forms a Heyting lattice and algebra, much work on the fundamentals of computation by Dana Scott, Solovay, Martin Hyland, and others is centered around this area, etc. As is so often the case, the mathematical physicist John Baez has done a fine job of introducing the subject to physicists and providing some motivation. Here's one of his articles: http://math.ucr.edu/home/baez/topos.html As for the mix of concrete and abstract, I studied plenty of abstract stuff on set theory, topology, analysis, and of course in physics. But I also did a lot of applied physics and engineering during my career at Intel. Believe me, I would have been in deep trouble had I proposed that we look into applications of Tychonoff's Theorem when we having problems with our dynamic RAMs and CCDs losing occasional stored bits in what were called "soft errors." But knowing a lot of abstractions helped me in countless ways. And now that I am free to pursue what I wish (have been since 1986), studying math that has some points of contact with ontology, physics, even AI, is what I am enjoying. I should be receiving Peter Johnstone's massive 2-volume set, "Sketches of an Elephant: A Topos Theory Compendium," in the next few days. And ya gotta crawl before ya can walk. I'm only recently gaining a good appreciation of S4, the logic system closely related to time and causality. Had I not learned S4 vs. S5, more computability theory than I used to know, a lot of stuff about lattices, quantum logic, and category theory, I surely would not be able to make sense of _any_ of what Bruno talks about! 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]. I have experience in both of the areas you talk about. Now I'm not saying this is why you should believe what I write, but at least my background spans both the *applied* (in college, working in a Josephson junction lab on superconductivity, and at Intel, working on microchips, and with some startup companies I've been working with for the past decade or so) and the *theoretical* (math, physics, computer science, logic, topos theory, etc.). Few things thrill me more than taking something which seems to be as abstract as unworldly as anything imaginable and applying it in the real world. (P.S. Could I encourage you to not include the full text of the messages you are replying to?) --Tim May
Funding AI
A slight sidetrack from pure Everything topics... On Saturday, November 30, 2002, at 06:44 PM, Ben Goertzel wrote: (stuff about physics which we are partly in agreement about, mostly not in agreement about...no point in arguing it further right now) Well, that depends perhaps on what you mean by "new physics," I think. Right now our physics is basically stumped by most complex systems. We resort to -- computer simulations -- crude "phenomenological" models Except I'll add that I don't agree physics is stumped by most complex systems. Physics doesn't try to explain messy and grungy situations, nor should it. Turbulence is a special case, and I expect progress will be made, especially using math (which is why Navier-Stokes issues are on the same list with other math problems for the prize money). ... (For example, some friends of mine are doing interesting work on using systems of several million machine agents to data mine large amounts of financial data. It seems likely that this kind of work on machine learning, pattern extraction, support vector machines, and a plethora of other "AI tools" will have major effects on the world of economics and forecasting. And on creating financial derivatives (synthetics) which are alien to human thinkers/investors.) Yeah, financial forecasting with AI does not require Artificial General Intelligence (AGI) in any sense, it is a classic domain-specific narrow-AI application. Whereas, coming up with new physics will require a significant degree of general intelligence, I believe. In this sense, physics theorizing is certainly a much harder problem than financial prediction-- it's hard to argue with that!! I tend not to even consider that kind of narrow-AI work "AI" -- I just think of it as computer science. But I have to remind myself periodically that the mainstream of academia does consider this AI, and considers AGI work to be a foolish and faraway dream... Funding is the key issue. Someday I'll write a thing for this list about successes vs. failures in terms of auto-funding each successive stage of a complex technological path. In a nutshell, the electronics/computer industry was essentially self-funding for the past 50 years, with the products of 1962, for example, paying for the work that led to the 1965 products. Same thing with aviation. By contrast, space development and controlled fusion have not been. We "know" that there exists a reasonable combination of ignition temperature-containment time--cost that lies several orders of magnitude away in Temp-time-power-cost space, but getting there is like crossing the Gobi desert without any watering holes or fuel stops on the way. The difference between "island colonization" models, akin to colonizing the fertile U.S. heartland (automobiles, aviation, electronics, etc.), versus "desert travel" models, akin to funding the first commercial fusion reactor or building the first space colony, is crucial. It is unlikely that the "path to AI" will be successful if there are not numerous intermediate successes and ways to make a _lot_ of money. My tip to all AI workers is to look for those things. (This is more than just banal advice about "try to make money," I am hoping. I have seen too many tech enthusiasts clamoring for "moon shots" to fund what they think is needed...)) The ""AGI" may come from the distant great-great grandchild of financial AI systems. --Tim May "Dogs can't conceive of a group of cats without an alpha cat." --David Honig, on the Cypherpunks list, 2001-11
Alien science
e quantum-domain-natural minds as mildly hilarious... Probably so. But at any given point in time the best we can do is to do the best that we can. We of course cannot just wait for the machines... Whether there "are" branes and strings and spins and suchlike at the Planck scale is unknown to me, but physicists seem to be making progress in acting as if such things have some meaning. The universe is not a rubber sheet, either, but it can help to think of gravity with the rubber sheet model (though an Arcturan squid creature might use a Flozzleblet to picture gravity, and an AI might use something entirely different). It is true that taking the "at hand, all around us" experience we have with physical objects and with the logic of physical objects is problematic at the quantum level. The fact that small things do not behave the way rocks and spoons behave, being either here or not here, having some speed which can be measured, etc., is why quantum mechanics is so weird to newcomers and others. So, yes, the physical world is not really made of rubber sheets or strings or little blue balls called electrons. But the fact that reality is so weird is not, in my opinion, an argument of any kind that we should not try to make some sense of it with the best arsenal of tools and concepts we can gather. Humans may or may not arrive at a workable TOE before the advent of AI's with quantum-level sensors and actuators. Following this advent, however, the progress of fundamental physics will be unimaginably fast, and will move in humanly-unimaginable directions. Will mathematics be central to this new physics? Maybe. But not our mathematics. I disagree fairly strongly on this point. I think our mathematics is what is most lasting, albeit the mathematical ideas change names and new ideas become more important. And I expect the mathematics the AIs develop, or that alien cultures may already have, will look like a "coordinate transformation" on our space of mathematical basis vectors. Their categories may be slightly different, but the underlying structure will be similar. (If natural transformations are what "slide" one category and its morphisms into other categories and morphisms, 2-categories, 3-categories, and n-categories in general are the tools for looking at how these natural transformations slide around.) Anyway, this was part of why I decided to start thinking about AI rather than fundamental physics ;-> AI remains interesting, but I think new views of physics will be coming from AIs long after other important things come out of AI. Just my opinion. (For example, some friends of mine are doing interesting work on using systems of several million machine agents to data mine large amounts of financial data. It seems likely that this kind of work on machine learning, pattern extraction, support vector machines, and a plethora of other "AI tools" will have major effects on the world of economics and forecasting. And on creating financial derivatives (synthetics) which are alien to human thinkers/investors.) I think Greg Egan's fiction is great, but I also think Diaspora is badly flawed futorology, because his uploaded minds never get tremendously more intelligent than humans. I don't think that's a very realistic prognostication, though it makes for easier storytelling. I totally agree. His characters were all recognizably human. Where were the entities with the equivalent of a truly alien intelligence, or with an IQ of 1000? (Not of course in the sense of thinking 5-7 times faster than the average bright person on this list, but of having many times the difference in conceptualizing power than an Einstein or a Wolfram has over a 100 IQ drone.) Where were the "Jupiter-sized brains" so beloved of the Extropians and Transhumanists? Vinge would say, apropos of your "easier storytelling" point, that such minds are on the other side of some flavor of Singularity, with little to say except to say that there are "Entities" out there, brooding and thinking their deep alien thoughts like some kind of unseen Lovecraft monster. --Tim May
Re: The universe consists of patterns of arrangement of 0's and 1's?
hink 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
Re: Is emergence real or just in models?
On Wednesday, November 27, 2002, at 11:42 PM, Eric Hawthorne wrote: I'm in the camp that thinks that emergent systems are real phenomena, and that eventually, objective criteria would be able to be established that would allow us to say definitively whether an emerged system existed in some time and place in the universe. I see "emergent properties" in very simple systems. I'll get to my main example in a few minutes. Why do higher-level systems emerge in our universe? Is there something about some systems that allows the system and its constituent parts to out-compete alternative configurations of matter and energy? Competition and differential reproduction is important, but the example I'll give here involves neither. OK, the example. Go. Black and white stones, with rules for moves that can be written on a small index card. Similar to a cellular automaton, though not as general. And yet from simple rules on a simple grid, emergent properties: * "thickness" (a measure of strength or weakness, depending) * "influence" (ability to influence direction of evolution) * a host of other emergent behaviors named by the main countries playing Go (Anyone who has played Go has seen the "reality" of thickness, influence, gote, sente, and other "properties." They were not obvious from first principles in the simple, CA-like rules of Go, but they emerge very quickly. Granted, the very concept of "influence" is partly shaped by human (or predator) notions of what influence means, but it seems clear to me that the ontology of Go (and by extension, other CAs) involves higher-order emergent behavior descriptions.) The moral is that even very simple CA-like systems have behaviors with "apparently" higher-level behaviors, aka emergent behaviors. --Tim
Re: Algorithmic Revolution?
long time, does not mean the universe is in any meaningful sense itself a cellular automaton. (And, pace Gleason's Theorem and the aforementioned Kochen-Specker Theorem, and the work of Bell, I am suspicious for other reasons that a purely local theory of the universe, one based on CA-like iterations, can be consistent with quantum mechanics. "No local hidden variables" and all that.) My own approach has been to regard emergence as the repositioning of the observer of a system such that the mathematical descriptions you have been using fall over/cease to be relevant. The idea that the math can seamlessly transcend an observers scope is, I concluded, simply meaningless as the math is defined by the observers scope. The prejudices of our position as observers are therefore automatically destined to be embedded in our descriptors of things. If this is the case then one cannot overlook the use of computers or the AIT approach if you need to study, understand and replicate real-world phenomena (in particular, MIND) that transcend the boundaries of emergence. Will the historians look back on our obsession with closed form math and see it as the machinations of mathematical youth? Para *** above is the clincher and I have been unable to distil a definitive stance from all the writings. Clues anyone? There are many phenomena which have no closed-form, simple description. That watch on the beach clearly is not going to have some master differential equation describing it. And e. coli is not likely to have some simple theory behind it: it emerged/evolved as the result of many, many interactions with other e. coli, with a complex environment of chemicals and proteins, and the resulting code is "packed" with a lot of stuff. All of these things are interesting from an information theory point of view, an AIT point of view, and other mathematical and philosophical points of view. But the evidence is slim that these things have anything at all to do with what's going on 30 orders of magnitude away in space, time, and energy, down where perhaps spin foams are bubbling with instantons, where perhaps wormholes are opening and closing, where perhaps Kalabi-Yau topological structures are vibrating or whatever it is they do. Fascinating stuff, to be sure. --Tim May "He who fights with monsters might take care lest he thereby become a monster. And if you gaze for long into an abyss, the abyss gazes also into you." -- Nietzsche
Re: Algorithmic Revolution?
On Tuesday, November 19, 2002, at 05:12 PM, [EMAIL PROTECTED] wrote: I would take all of TCMs own citations and turn them around in my favor. I would classify all the following as occurring under the heading "algorithmic revolution" (not the greatest moniker I admit.. a provisional one for me right now) - advent of cyberspace, web, email, browsers etcetera - advent of mass software - PC revolution - microsoft & intel from zero to billion dollar companies in short decades - quantum computing is on the way - fractals. could not be discovered without algorithms. a new metaphor for not only nature but all reality. - complexity theory. again, not possible before the algorithmic metaphor and mass computational capabilities - simulation, "in silico science" - moore's law - photorealistic rendering - (relational) databases - mass economic shift into information technology as driving force.. "bits versus atoms".. (negroponte) - video games - etcetera!! I agree that these are all huge changes. I interpreted your "algorithmic revolution," in the context of this list and the Kevin Kelly article and the Wolfram brouhaha, to be about a revolution in terms of thinking of the universe (or multiverse) as being primarily computational. My point is that the verdict on the Zuse/Fredkin/Wheeler/Lloyd/Wolfram/Tegmark/Schmidhuber/etc. views of reality is still way, way out. I stand by this point. If by "algorithmic revolution" you meant that computers are increasingly important, then of course I agree. ... ahem!!! what is the relevance to a TOE??? well historically it is clear our perception of reality is based on our favorite metaphor of the times. in recent ages it was (a) the clock, "clockwork universe", (b) the steam engine. and now it is (c) computer/algorithm/information. clearly it is no coincidence whatsoever that new TOEs are essentially algorithmic. its the human race's latest-and-greatest metaphor for reality. Yes, and our past experience in going through all of these metaphors or "mathematical fictions" has made many of us wary of saying things like "The universe is like a hologram" or "The universe is about connection" or even "The universe is a gigantic Game of Life." The issue of an ontology being a metaphor is an interesting one. I currently have no view of any particular metaphor for what the universe "is." It may have computational aspects, and mathematics (superset of computer science, of course) may be woven throughout the structure of reality. It may even have "holographic" or "clockwork" or "cellular"-like aspects. But aspects are not the same thing as equivalence. Greg Egan makes a good point in "Diaspora" about the limitation of the mathematical models we sometimes use as metaphors for reality. A mass falling through a borehole through the Earth acts exactly as if it's a mass on a spring tethered at one end. Same precise equation of motion. Yet a spring is not at all what the Earth is, and confusing the mathematical model with reality is dangerous. Of course, at this point we have much, much less reason to speculate that the universe "is" a cellular automaton of some sort. scientists have been slow to adopt to this shift, and I would argue they are still underutilizing simulation to some extent. science & physics is still yet to be influenced fully by the algorithmic revolution. one striking example I think will happen-- I believe billion dollar particle accelerators may be downgraded in importance in favor of extremely effective simulations. The reason experiments are still done is because they are the real proof of the pudding about what the universe really _is_. (besides-- does anyone fully realize how much software plays already such a crucial, foremost role in existing accelerators??) Not sure what you mean. I did some coding of Monte Carlo simulations in my physics days, and I hired some of the coders from SLAC to work on some of the stuff we were doing at Intel. Software is used to design the accelerators, the detectors, the experiments, etc. As with the rest of the world, computers and software are undeniably important. I'm not doubting the importance of computers. Nor the importance of clocks and wristwatches. But just as we know "the universe as a clockwork mechanism" was not the whole picture, I think "the universe as a computer" is not, without a lot more evidence, very compelling. To me, at least. (I am interested in being convinced otherwise. And I have my own interests. Today I ordered the Peter Johnstone 2-volume set "Sketches of an Elephant: A Topos Theory Compendium." Not that I am saying "the universe is a topos.") --Tim May
Algorithmic Revolution?
On Monday, November 18, 2002, at 10:15 PM, [EMAIL PROTECTED] wrote:
as just noted by TCM, kevin kelly on a computational/algorithmic TOE,
wolfram, wheeler, etcetera, from current issue of wired.
http://www.wired.com/wired/archive/10.12/holytech.html
I would say we are all in the midst of some kind
of "algorithmic revolution" that is sweeping across
culture, industry, & scientific fields etc. .. more
on that theme here
I just don't see any such sign of a revolution. No more so than 10
years ago, 20 years ago. Yes, computers are now more powerful. Problems
tend to grow faster in size than computers do, however, and often
having 100x the power only yields a slight improvement in accuracy, not
qualitative leaps or breakthroughs. (Paralleling, no pun intended, the
spacing of the Mersenne primes, where it's taking longer and longer to
brute force find the next one, even with dramatically more computer
power. Or the accelerator energy gap, where 10 times the accelerator
energy doesn't produce much more new physics.)
There are aspects of computers that are always touching on cultural
issues. In the 60s and 70s there was much hype about "general systems
theory" and modeling (a la Bertanlanffy, Arrow, others). Some social
scientists expected a revolution. In the 1980s it was chaos theory, and
fractals, with books on how financial markets are chaotic, how art is
fractal, how civilization lives at the boundary between order and
chaos, and so on. Trendy, and probably implicated somewhere in the
Sokal hoax ("Transgressing the Boundaries," the quantum
mechanics/litcrit/hermeneutics put-on). Not much of lasting value came
out of it, insofar as the revolutions outside of the narrow fields
directly involved are concerned.
In recent years it's been stuff about string theory, to some extent.
The Brian Greene book, "The Elegant Universe," became a best-seller,
even if probably fewer than one out of a hundred buyers got past the
first 20 pages. I don't think many of the coffee table book buyers are
expecting many revolutions outside of physics qua physics.
And of course Wolfram's book is a big seller. I won't comment, except
that I see no particularly strong evidence that he has changed the way
science is done, or will be done. Others have written harsher reviews.
I admire him for his dedication, but I think he missed the boat by not
working with others and working on specific problems.
(Tegmark works on lots of cosmology and observational astronomy
problems, with his Everything paper as just one small facet, almost a
hobby. Working on that theory full-time might make him a frequent
contributor to this list, but would probably not be good either for his
career or for getting any kind of progress or confirmation (!).)
My belief is that basic mathematics is much more important than
computer use, in terms of understanding the cosmos and the nature of
reality.
--Tim May
Re: The number 8. A TOE?
On Monday, November 18, 2002, at 07:12 AM, Marchal Bruno wrote: Hi, I hope you have not missed Ian Steward's paper on the number 8, considered as a TOE in the last new scientist. It mentions a paper by John Baez on the octonions. The octonions seems to be a key ingredient for the quantization of general relativity. http://math.ucr.edu/home/baez/Octonions/ I am too buzy now to make comments but it seems *very* interesting, if not convincing. I happened to see Stewart's article at a news stand. He writes good general math books, so he was able to do a good job explaining octonions and hinting at why they may be important. (I was struck by the point that the sequence "1, 2, 4, 8" is the only sequence satisfying certain properties--the only "scalars, vectors, quaternions, octonions" there can be--and that the sequence "3, 4, 6, 10," just 2 higher than the first sequence, is closely related to allowable solutions in some superstring theories, and that these facts are related.) Ironically, in the Bogdananov/Sokal controversy being discussed in sci.physics.research, the topic of articles in "New Scientist" came up a week or so ago. Baez said he no longer reads "Scientific American," "New Scientist," and similar popular magazines because of their watered-down, sensationalized, dumbed-down, breathless hype. Someone (maybe Baez) said that cover stories in "New Scientist" are a good place to look for what _not_ to take seriously! I have to wonder what Baez thinks of being quoted in this latest cover story! I actually enjoy the speculative cover stories in "New Scientist." I take them with a grain of salt, especially as every few weeks there's a new article about a new theory of everything, a new theory of how the universe arises out of nothingness or out of some sort of dream state. (Perhaps like some of the theories people here on this list have!) The articles, especially those by Marcus Chown, are wildly speculative hints at what may be aspects of reality...at least this is how I treat them. And what appears to be just idle speculation sometimes is linked with things I know to be important (a cover story on the work of Greg Chaitin comes to mind...anyone not familiar with Chaitin's work would probably think the article was hype, but it contained hints and nuggets which might inspire some folks unfamiliar with his work to take a closer look.). --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Good summary of Bogdanov controversy
A good summary of the Bogdanov controversy is in the New York Times today. URL is http://www.nytimes.com/2002/11/09/arts/09PHYS.html Some of the folks we like to quote here are quoted in the article, including Lee Smolin, John Baez, Carlo Rovelli, etc. Also, the latest "Wired" print issue has a fairly good survey article by Kevin Kelly about theories of the universe as a cellular automaton. Konrad Zuse gets prominent mention, along with Ed Fredkin. I didn't read the article closely, so I didn't notice if either Tegmark or Schmidhuber were mentioned. The usual stuff about CA rules, Wolfram's book, etc. Things have been quiet here on the Everything list. I haven't been commenting on my own reading, which is from books such "Physics Meets Philosophy at the Planck Scale" and "Entanglement." Isham's collection of essays on QM should arrive momentarily at my house. My interest continues to be in topos theory, modal logic, and quantum logic. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Modal Realism vs. MWI
On Friday, October 4, 2002, at 09:13 AM, Bruno Marchal wrote:
> At 9:36 -0700 1/10/2002, Tim May wrote:
>
>> MWI looks, then, like just another variant of "modal realism." To
>> wit, there IS a universe in which unicorns exist, and another in
>> which Germany won the Second World War, but these universes are
>> forever and completely out of touch with us.
>
> Not quite due to possible interferences. We do have empirical evidences
> for those "worlds" imo. (if only the two slits + Bell or better GHZ)
While I find Deutsch fairly persuasive, the verdict is of course not
yet in whether MWI is the correct interpretation. The double slit
results had a "traditional" wave mechanics interpretation 75 years ago
("wave-particle duality"), and this remains a viable interpretation
even today. (I'm not talking about popularity, either on this list or
in the overall community, just "technical viability.")
However, I take your point that full Lewis-Stalnaker-D. Lewis modal
realism is "more disjoint" than the "less disjoint" (initial
interference of branching worlds) MWI. In terms of topology, one might
say full modal realism is the discrete (perhaps Zariski) topology,
while MWI has more notions of closeness, overlap, etc. (I think this
could be worked out, but I haven't.)
Certainly after a time interval where decoherence occurs, the
interaction between macroscopically different worlds is essentially
zero.
So, I will amend my earlier statement to read: "After the very early,
entangled period, MWI looks, then, like just another variant of "modal
realism." To wit, there IS a universe in which unicorns exist, and
another in which Germany won the Second World War, but these universes
are forever and completely out of touch with us."
And since the time of entanglement/coherence is small for most systems,
most worlds in MWI are as "far apart" as modal realism worlds are.
(Digression: I wonder what kind of work has been done on _evolution_ in
topology, e.g., the transition of systems from "overlapping open sets"
to the "discrete" topology? Looks like nucleation and growth out of a
continuous medium, or formation of tree structures, perhaps.)
>
> A very natural generalisation (!). Just replace the hom Sets by hom
> Categories.
> In which you can again replace the hom sets by hom categories
> What is intriguing is the existence of coherence conditions making
> those
> constructions apparently very genuine for many stuff from quantum
> field theories.
Baez (IIRC) has an anecdote about talking with a noted quantum field
theorist at a conference. The theorist was highly skeptical of
"generalized abstract nonsense" (i.e., category theory). Baez told him
about some of the developments and the theorist went off to sleep on
it. The next morning he buttonholed Baez and said "Braided monoidal
categories are really cool" (I'm paraphrasing from memory).
>
> I have used the smullyan trees for the G and Co. theorem provers. The
> tableaux
> structure reflects in some way the Kripke structure. Posets appears
> with
> S4-like modal logic.
> You should study Gentzen presentation of logic which are naturally
> related
> to categories. An indigest but brilliant introduction to many
> (intuitionnist)
> logics is the North-Holland logic book by Szabo: Algebra of proofs.
> To bad he miss the braided monoidal categories ... For a categorician,
> knots
> theory is a branch of logic.
I haven't gotten to knots yet, except for a look a few years ago at the
Vaughan Jones stuff on classifications of knots (more related to string
theory, which I did a little bit of reading on).
Gentzen is referred to, of course, in the books on logic I'm reading,
but I'm still absorbing the more basic stuff.
>> "Possible worlds," something I only encountered in any form (besides
>> Borges, Everett, parallel universes sorts of references) in the past
>> several years, is my real touchstone.
>>
>> And, more mundanely, I think it applies to cryptography and money. I
>> had a meeting/party at my house a few weeks ago with about 50 people
>> in attendance (gulp!). We had a series of very short presentations. I
>> gave a very rushed 10-minute introduction to intuitionistic logic,
>> mainly focused on my "time as a poset, a lattice" example, citing the
>> natural way in which "not-not A" is not necessarily the same as A. If
>> the past of an event is A, then not-A is its future. But the
>> not-future is larger than the original past, as "incomparable" (in
>> the poset/trichotomy sense) events influence the future. Or, put in
Re: Many Fermis Interpretation Paradox -- So why aren't they here?
On Tuesday, October 1, 2002, at 06:37 AM, Bruno Marchal wrote:
> At 12:26 -0700 30/09/2002, Tim May wrote:
>> If the alternate universes implied by the mainstream MWI (as opposed
>> to variants like consistent histories) are "actual" in some sense,
>> with even the slightest chance of communication between universes,
>> then why have we not seen solid evidence of such communication?
>
>
> I am not sure I understand why you oppose the "mainstream MWI" and the
> consistent histories (although many does that, I don't know why).
> In all case, if QM is right (independently of any interpretation),
> parallel
> histories or parallel universes cannot communicate, they can only
> interfere(*). The same happens with comp. Probability measures are
> global
> and depends on the whole collection of relative computational
> histories, but
> this does not allow the transfer of one bit from one computation to
> another.
I of course was not claiming such communication (or travel, whatever)
would be easy. Just doing a thought experiment settting some very rough
bounds on how impossible the communication or travel would be.
One of the conclusions of "How come they're not here?" is that, in
fact, such communication or travel is essentially impossible (else
they'd _be_ here).
MWI looks, then, like just another variant of "modal realism." To wit,
there IS a universe in which unicorns exist, and another in which
Germany won the Second World War, but these universes are forever and
completely out of touch with us.
>
> BTW, Tim, I am discovering n-categories. Quite interesting. John Baez
> has written good papers on that, like his categorification paper.
> Have you read those stuff. Could be useful for the search of coherence
> condition in "many world/observer" realities ...
I've been reading Baez for a while. An excellent teacher. I hear he's
working on a book on n-categories. And Baez and my namesake, J. Peter
May--unrelated to me, are leading a consortium to research n-categories
more deeply. I confess that I have only vague ideas what they
aresort of generalizations of natural transformations, I sense.
(I'm still studying categories at a more basic level, having "jumped
ahead" to other areas, as is my wont.)
His "From Categories to Feynman Diagrams" (co-authored with James
Dolan) and several of his related papers are good introductions.
Chris Isham is also very good on drawing the connections between
conventional quantum mechanics (i.e., stuff in the lab, not necessarily
quantum gravity or quantum cosmology) and category/topos theory. (In
particular, the collapse of the wave function and measurement looks
like a subobject classifier, or, put another way, the usual transition
from "neither true nor false" in a Heyting algebra to the "one or the
other" we _always_ see once there is any chance to
observe/measure/decide. That is, Heyting --> Boolean is what the
mystery of QM centers around.
(I am intrigued to find that Jeffrey Bub, in his "Interpreting the
Quantum World," 1997, makes central use of possible worlds, lattices,
and such. While he does not explicitly mention Heyting algebras, the
connection is close, and is implicit in the math. Had I encountered
this approach when I was studying QM, I might have pursued it as a
career. Instead, I was bored out of my mind solving partial
differential equations for wave functions inside boxes. Ugh.)
I'm reading Graham Priest's "An Introduction to Non-Classical Logic,"
2001, which covers various modal logics, conditional logics,
intuitionist logic, many-valued logics, and more ("first degree
entailment," "relevant logic," etc.).
The tableaux approach is new to me. They look like the trees of
Smullyan, and hence like semilattices. (I'm also reading Davey and
Priestley's "Introduction to Lattices and Order," along with parts of
Birkhoff's classic, and the lattice/poset approach continues to appeal
to me greatly. It's a vantage point which makes all of this
heretofore-boring-to-me logic stuff look terribly interesting. I'm
viewing most programs/trees/refinements/tableaux as branching worlds,
as possible worlds (a la Kripke), to be further branched or discarded.
Hence my focus on MWI and "Everything" remains more on the mathematics.
(I just ordered my own copy of Goldblatt's "Mathematics of Modality.")
"Possible worlds," something I only encountered in any form (besides
Borges, Everett, parallel universes sorts of references) in the past
several years, is my real touchstone.
And, more mundanely, I think it applies to cryptography and money. I
had a meeting/party at my house a few weeks ago with about
Many Fermis Interpretation Paradox -- So why aren't they here?
If the alternate universes implied by the mainstream MWI (as opposed to
variants like consistent histories) are "actual" in some sense, with
even the slightest chance of communication between universes, then why
have we not seen solid evidence of such communication?
Amongst the universes, many ("many" is a huge number, obviously) of
them will be way ahead of us. Some will have had galactic civilizations
for a billion years. Some will be versions of Earth except that the
Egyptians pioneered electronics and hence the world is a few thousand
years "ahead" of our world...even assuming time is commensurate with
ours.
And so on. You can all imagine the rich possibilities.
If these universes are even remotely able to affect each other, through
perhaps enormously advanced technology, then the vast number of such
possible worlds would suggest that at least some of them have figured
out how to do so.
And yet they aren't here. No visitors from alternate universes. No
signals sent in, a la Benford's "Timescape."
Perhaps we don't know how to listen. Perhaps there are so many possible
universes to potentially visit that we just haven't been gotten to yet.
Perhaps in a multiverse of so many possibilities, ours is just not an
interesting destination. Maybe there's a kind of MWI censorship going
on: since we are still debating the validity of MWI, we obviously are
in a universe where MWI has not been proved through such a visit.
(There are many divergent series here, making even crude estimates
difficult and probably worthless.)
Hmmm
--Tim May Prime, resident of Earth Prime
Re: MWI of relativistic QM
On Wednesday, September 25, 2002, at 10:09 AM, Wei Dai wrote: > On Tue, Sep 24, 2002 at 03:20:54PM +0200, Bruno Marchal wrote: >> I mentioned Deutsch for his account of time in term of parallel >> universes. >> I don't remember if Deutsch deduced this explicitly from relativity. >> (I lend his book so I cannot verify now). >> I was just doing the following caricatural reasoning: >> General Relativity (GR): gravitation = space-time curvature >> Quantum mechanics (QM): forces should be quantized (and unified >> through >> symmetry/broken-symmetry) >> Now GR + QM gives: space-time itself should be quantized. A MWI view >> of this >> doesn't give many minkowski worlds, but something more like a >> discrete minkowski multiverse. > > Is there a paper or book that describes this discrete minkowski > multiverse > in more detail? Several of the papers by Rafael Sorkin, Carlo Rovelli, Chris Isham, Fotini Markopoulou, John Baez, and others discuss "causal sets" as a model of spacetime. For example, picking just one of them, arXiv:gr-qc/9910005 v 1 2 October 1999 C.J. Isham, J. Butterfield, "Some Possible Roles for Topos Theory in Quantum Theory and Quantum Gravity." Here's one small part to provide some of the flavor: "By a causal set we mean a partially-ordered set P whose elements represent spacetime points in a discrete, non-continuum model, in which p <= q, with p, q elements of P, means that q lies in the causal future of p. "The set P is a natural base category for the presheaf of Hilbert spaces in which" (etc.) I talked about these issues in my article several weeks about time as a lattice of partially-ordered events. Now, whether time and space are "really" continuous or discrete (at some very small scale, presumably near the Planck scale) is not terribly important for this analysis. Just as both QM and relativity are usually involved with events (measurements, clocks, light flashes, etc.), and just as much of the traditional "causal analysis" of everyday events (a la Pearl) is of discrete, chunked events, the causal set model is very generally applicable. And again I have no choice but to recommend Lee Smolin's "Three Roads to Quantum Gravity" as a good introduction to the ideas of the authors named above. --Tim May
Re: Tegmark's TOE & Cantor's Absolute Infinity
ps the multiverse has variants of all of of the axioms of these branches, isn't terribly useful except as a stimulating idea (hence this list, of course). Naturally Tegmark is not claiming his idea is _the_ theory, so stimulation is presumably one of his goals. In this he has succeeded. --Tim (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks --Tim May "Dogs can't conceive of a group of cats without an alpha cat." --David Honig, on the Cypherpunks list, 2001-11
New special issue of "Scientific American" on "Cosmology"
I was disappointed in the thin, banal issue of SciAm on "Time," but now there's a new special issue devoted to "The Once and Future Cosmos." It's very good, filled with excellent illustrations of models of cosmology, key experiments, and what we currently know about the structure of the cosmos. Some of the graphics are so good that I am tempted to buy a second copy just so I can clip the graphics and put them up in places where I can ponder them. For this list, there's a little bit of discussion of the Rees/Linde/Vilenkin/Barrow/Smolin/Tegmark sort of model where new universes are created with slightly different laws of physics. This is mentioned in a fanciful figure showing such hypothetical pocket universes being formed: "Multiple universes are continuously being born, according to some cosmologists. Each universe is shown here as an expanding bubble branching off from its parent universe. The changes in color represent shifts in the laws of physics from one universe to another." [Figure on page 85] By the way, I'm reading Smolin's "Life of the Cosmos," where he makes a good case for his thesis that the cosmos we are in is one strongly rigged (basic laws, parameters) so as to have a very high production rate of black holes. The idea is similar to the Boostrum sort of Bayesian argument about assuming the world we find ourselves in is "typical." Smolin argues that if a universe creates few or no black holes (for example, it expands and collapses in a Planck time, or has no aggregations of matter worth mentioning, and so on), it leaves no (or just one) children. A universe which creates, say, 10^18 black holes, leaves 10^18 children, some of which may leave even more children, and so on. Thus, there's a fitness landscape with the x and y axes (in a simple diagram) being some set of parameters which affect black hole formation and the vertical or z axis being the number of black holes (and hence universes, a la Linde, others) created. While there is no "competition amongst the universes" directly, that we know of, differential reproduction is enough, in general, to produce evolutionary effects. After some number of "bounces or black hole formations with slightly different laws of physics" the result will be that nearly all universes in some multiverse are ones where black hole formation is commonplace. Smolin argues that we will be able to measure the number of black holes, from small (stellar masses) to larger (recently found, it is believed, 1000-sun mass black holes, to massive galactic core black holes, and, he thinks, reach the conclusion that we are in a universe which is "selected" for maximum ease of black hole formation. Maybe. By the way, it may also be the case that universes in which intelligence is possible are dominant. My notion, refecting some of the fantastical fiction of Steven Baxter and Greg Egan, is that advanced civilizations will be able to _make_ black holes, possibly in vastly greater abundance than astrophysical sources can. (Yeah, of much smaller mass. And presumably of very short duration. The weirdness of spacetime and the insides of black holes are enigmas, but it is speculated by some that even a microscopic black hole could have a "full cosmos" inside. We don't, at least, have any compelling evidence that a galactic core black hole with 10,000 solar masses is going to produce a richer or more complete "cosmos inside" than a microscopic black hole made up of a few grams or less of our matter.) Using the same kind of reasoning Smolin uses, a civilization which has special accelerators cranking out 10^30 black holes per second, say, is going to "outbreed" and thus be more represented, than universes where only astrophysical processes are making only a relatively paltry 10^18 or 10^25 black holes in the age of that universe. (And, even more speculatively, the difference is not great between the number of black holes a single civilization can make and the number a million civilizations can make, so it's more significant that a universe supports AT LEAST SOME advanced intelligence/civilization than that it supports UBIQUITOUS intelligence/civilization. But this reasoning is speculative and needs a lot more thought...maybe. Someone could probably write a nice Baxterian story with this theme.) --Tim May, Occupied America "They that give up essential liberty to obtain a little temporary safety deserve neither liberty nor safety." -- Benjamin Franklin, 1759.
Re: MWI of relativistic QM
On Friday, September 20, 2002, at 10:03 AM, Wei Dai wrote:
> On Thu, Sep 05, 2002 at 12:08:39PM +0200, Bruno Marchal wrote:
>> This comes from the fact that MWI is explained most of the time
>> in the context of non relativistic QM (which assumes time and space).
>> But this problem disappear once you take into account the
>> space time structure of relativistic QM, where roughly speaking
>> moment of time are handled by "parallel" universes (see Deutsch FOR).
>
> I got Deutsch's book, but it doesn't mention relativistic QM at all.
> Can
> you elaborate on what the MWI of relativistic QM is, or point me to
> another paper or book, or give me a page number in FOR that deals with
> this?
This topic dovetails (no pun intended) on several points I've made as
well, so I'll add some comments.
* Deutsch's "Fabric of Reality" is a slender book, with only the first
few chapters really making his main point (about how the single- and
double-slit experiments already "proved" the MWI interpretation a
century ago, had we known what to look for, and that quantum computers
make the point as well). I don't recall whether he says much about
relativistic vs. nonrelativistic QM, but I'll take your word that he
says nothing. His focus is on the quantum aspects, not cosmology or
relativity or a unified theory, so this is not too surprising.
* Much more is said in a book I have recommended a couple of times
here: Lee Smolin's "Three Roads to Quantum Gravity." Also, his earlier
book, "The Life of the Cosmos."
* The idea is this:
-- conventional ("classical") QM assumes Newtonian space and time,
i.e., a universal coordinate system
-- conventional ("classical") relativity (SR and GR) assumes a
non-Newtonian, non-constant space and time, via Lorentz transforms on
a Minkowski spacetime, but it has no quantization a la QM
-- in other words, two very different spacetimes. This is sometimes
characterized as the "very small" (quantum effects) vs. the "very
large" (astrophysics), and experiments at most ranges don't produce
contradictions, as gravity effects are miniscule at the usual quantum
levels and quantum effects are miniscule at cosmological or
astrophysical scales. However, understanding black holes will almost
certainly require a unification of these two theories or outlooks. And
of course a coherent, unified theory ought not to have two radically
different views of spacetime.
* Einstein attempted to merge the two, but failed. Beginning in the
1970s, with the work of Ashtekar, Witten, Rovelli, Crane, Susskind,
Baez, and many others, progress was made toward unifying the models.
The quantum gravity program, as pursued by the several different
schools (strings and branes, spin foams, twistors, etc.), is to unify
these two fundamentally different outlooks. As of now, this hasn't
happened.
* Personally, I think there is much of interest in the "discrete at
Planck scales" relational approach.
--Tim May
Good article in "American Scientist" on cosmology and cosmic background variations
I took a quick look at a newstand copy of "American Scientist," the current issue. A good article on variations in the cosmic background and how this might be able to give some indications about very early "forks" taken in the evolution of the universe we are in. In other news, am reading Graham Priest's "An Introduction to Non-Classical Logic," 2001. A good survey of various kinds of modal logic, multi-valued logic, intuitionistic logic, etc. I also found an interesting book by Robert Goldblatt, "Mathematics of Modality," 1993, which contains a paper "Diodorean Modality in Minkowski Spacetime." He points out that Arthur Prior, in books from the late 60s, early 70s, demonstrated that the lattice of partially-ordered events in Minkowski spacetime corresponds to a modal logic system called "S4.2." (Sidenote: Bruno uses these names for axiom systems more comfortably than I can at this point. Just citing a name for some system is not very convincing to me, without having the background to know what the names imply.) The point is that apparently my hunch about time being viewed as a poset, which I wrote about several weeks ago, is already known to people like Goldblatt and Prior. This remains my focus. --Tim is a Democrat, as he is always looking for a handout" --Unknown Usenet Poster
Time-varying sets and modal logic
On Monday, September 9, 2002, at 01:39 AM, Bruno Marchal wrote: > > In one little sentence: modal logic is a tool for refining truth > by making it relative to context, situations, etc. Those last > notions are in general captured by some abstract mathematical > spaces, like set + binary (accessibility) relations with Kripke, > quasi topological space with Scott and Montague, etc. Or, seen naturally all around us in the world, with time-varying sets. A time-varying set, informally, is one whose set of members varies with time. (Time is just about the most important kind of _context_ mentioned above by Bruno.) The set of nations in the U.N. varies with time, the set of air molecules in a room varies with time, the set of descendants of a person varies with time, and so on. The logic and algebra associated with such variable sets are Heyting logic and Heyting algebra, not the more commonly studied Boolean logic and Boolean algebra. I outlined this in some earlier posts. (And there are synonyms for Heyting: intuitionistic logic, Brouwerian lattices, forms of modal logic, etc.) The connection between time-varying sets and time-varying logic is of course straightforward. Propositions within a logical system can be translated into set inclusion relationships. The connection with branching forks of a universe, where different forks are BY DEFINITION contradictory (and hence are not analyzable with Boolean logic), is clear...to me at least. Two outcomes of the flip of a coin, for example, form a fork which is part of a poset. The outcomes, H or T, do not obey the usual law of trichotomy, hence the set is a poset. I outlined this in earlier posts as well. --Tim May
Re: MWI, Copenhagen, Randomness
On Thursday, September 5, 2002, at 09:34 AM, Jesse Mazer wrote: > But even if one understands that conscious observers are not necessary > to "collapse the wave function," Tim's questions do not go away. One > could always imagine that the box in the Schroedinger's cat experiment > was made of some super-material that blocked interaction between the > inside and the outside so effectively that decoherence was completely > eliminated, so from the outside the cat would have to be treated as > being in a macroscopic superposition until the box was opened, even > though the cat (or a video camera inside the box) would remember > having been in a single definite state all along. In fact, all formulations I have seen of the SC experiment are exactly as you describe: a sealed box, with the necessary condition of no information from inside the box, whether meows of the cat or portholes cut in the side of the box or video signals coming out. Hence the "mixed state" (as it described) holds up to the time the box is opened. Whether or not a camera is inside recording the process. Whether or not other humans are also inside the box, wearing gas masks perhaps, observing the process. Whether or not that box is inside an even larger box, a la the Chinese boxes thought experiment. > >> > One could arrange a thought experiment involving literally >> > a series of boxes within boxes, each being opened at, say, >> > one minute intervals after the cyanide was released or not >> > released. One set of observers sees the cat either alive >> > or dead at the end of the canonical one hour period. But >> > they are sealed inside a box. After one minute, their box >> > is opened, and the observers in the next-larger box then >> > see the "collapse of the wave function at the 61-minute >> > point." After another minute, their box is opened and a >> > new set of observer sees "the collapse of the wave >> > function at the 62-minute point." >> >> > And so on. (I don't know if I'm just reinventing a thought >> > experiment someone developed many decades ago...it seems >> > like a natural idea.) > > Yes, this is similar to the "Wigner's friend" thought-experiment. The > physics dictionary entry on Schrodinger's cat at > http://physics.about.com/library/dict/bldefschrdingerscat.htm > describes it briefly: > > "Wigner's friend is a variation of the Schrdinger's cat paradox in > which a friend of the physicist Eugene Wigner is the first to look > inside the vessel. The friend will find a live or dead cat. However, > if Professor Wigner has both the vessel with the cat and the friend in > the closed room, the state of the mind of the friend (happy if there > is a live cat but sad if there is a dead cat) cannot be determined in > Bohr's interpretation of quantum mechanics until the professor has > looked into the room although the friend has already looked at the > cat. These paradoxes indicate the absurdity of the overstated roles of > measurement and observation in Bohr's interpretation of quantum > mechanics." Thanks for the reference. It matches my own thought experiment. (Which, aside from showing some "overstated roles" for human observation, also shows that this whole business of "the wave function being defined everywhere and then suddenly vanishing" is a deeply flawed notion, as we've just shown many such points of potential "collapse." Fortunately, nothing in the "shut up and calculate" practical side of QM depends on "collapse of the wave function," so it has mainly been a side show.) But this thread is stimulating me to refresh my memory of QM and to study it more deeply. (I have a bunch of the recent Zeilinger papers on delayed-choice and double-slit experiments, but haven't had a chance to read them except by skimming.) --Tim May
MWI, Copenhage, Randomness
On Wednesday, September 4, 2002, at 02:44 PM, Hal Finney wrote: > Tim May wrote: > >> In weaker forms of the MWI, where it's the early state of the Big Bang >> (for example) which are splitting off into N universes, De Witt and >> others have speculated (as early as around 1970) that we may >> _possibly_ >> see some evidence consistent with the EWG interpretation but NOT >> consistent with other interpretations. > > I'm not familiar with the details of this. But I know that much of > the impetus for increased acceptance of MWI models comes from the > cosmologists. It was in DeWitt's article, "Quantum mechanics and reality," Physics Today, September 1970, reprinted in the collection "The Many-Worlds Interpretation of Quantum Mechanics," edited by Bryc DeWitt and Neill Graham, 1973. "Moreover a decision between the two interpretations may ultimately be made on grounds other than direct laboratory experimentation. For example, in the very early moments of the universe, during the cosmological "Big Bang," the universal wave function may have possessed an overall coherence as yet unimpaired by condensation into non-interfering branches. Such initial coherence may have testable implications for cosmology." (p. 165 of the reprint volume). (Glad to see my memory hasn't failed me. DeWitt's article made a big splash when it first got wide notice with that 1970 article. Around that time, "Physics Today" was where we found many wild things. A beautiful cover painting of a black hole, the first such graphic I'd seen...perhaps it's scanned and on the Web someplace, as it was a seminal image, from January 1970, if I remember correctly. And another cover from around that era was of O'Neill's proposal for L-5 colonies and powersats.) >> >> What's the problem here? I find it utterly plausible that we would be >> in a universe where matter exists, where stars exist, where entropy >> gradients exist, etc., and NOT in a universe where the physical >> constants or structure of the universe makes life more difficult or >> impossible (or where the densities and entropy gradients mean that >> evolution of complex structures might take 100 billion years, or more, >> instead of the billion or so years it apparently took). > > The problem is more formal, that if we abandon measurement as a special > feature of the physics, there is no longer an axiom that says that > probability is proportional to amplitude squared. I'm not an expert on this. Jeffrey Bub, in "Interpreting the Quantum World," 1997, cites several classes of resolutions of the "measurement problem." He calls them the "For all practical purposes" (FAPP) model, after Bell, the "change the linear dynamics" model, and the "modify the orthodox Dirac-von Neumann interpretation principle." From what I can tell, the Copenhagen interpretation is already a mixed state, so to speak, of bits and pieces of Bohr's and Heisenberg's interpretations. By the way, issues of observers and measurements are obviously fraught with "Chinese boxes" types of problems. In the Schrodinger's Cat pedantic example, if the "cat alive or cat dead" measurement is made at the end of one hour by opening the sealed box, what if a video camera had been also sealed inside the box, and had seen the cat breathe in the cyanide gas at 10 minutes into the experiment? Does this imply the "wave function collapsed" at the time of the measurement by the human observers, at the one hour point, or at the time the video camera unambiguously recorded the cat's death? One could arrange a thought experiment involving literally a series of boxes within boxes, each being opened at, say, one minute intervals after the cyanide was released or not released. One set of observers sees the cat either alive or dead at the end of the canonical one hour period. But they are sealed inside a box. After one minute, their box is opened, and the observers in the next-larger box then see the "collapse of the wave function at the 61-minute point." After another minute, their box is opened and a new set of observer sees "the collapse of the wave function at the 62-minute point." And so on. (I don't know if I'm just reinventing a thought experiment someone developed many decades ago...it seems like a natural idea.) Seen this way, the "collapse of the wave function" in the Schrodinger's Cat thought experiment is seen as a problem of knowledge, not something quasi-mystical about an instantaneous collapse of some psi-squared function. (More interesting are the delayed choice experime
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
On Wednesday, September 4, 2002, at 10:08 AM, Hal Finney wrote: > I think on this list we should be willing to seriously consider the > many-worlds interpretation (MWI) of quantum mechanics as the ontology > for > our universe. I remain agnostic on the MWI or EWG interpretation. While I don't strongly believe that the MWI is "reality" (cough cough), I agree with Hal that it's a plausible ontology. Further, I take more seriously than many the "plurality of worlds" ontology of the late philosopher David Lewis. (The guy who argues that we should not give special linguistic treatment to "our" world and should give equal standing to "the world in which World War II was won by Germany," for example. Lewis is sometimes caricaturized by capsule summaries of the sort "David Lewis believes unicorns really do exist," but what Lewis is claiming is fully consistent with modal logic and possible worlds semantics.) > There are a few objections which I am aware of which have been raised > against the MWI. The first is its lack of parsimony in terms of > creating a vast number of universes. We gain some simplification in > the QM formalism but at this seemingly huge expense. The second is its > untestability, although some people have claimed otherwise. The latter is the more important. If, for example, the plurality of worlds are out of communication with each other, forever and always, then it means nothing to assert that they "actually" exist. In weaker forms of the MWI, where it's the early state of the Big Bang (for example) which are splitting off into N universes, De Witt and others have speculated (as early as around 1970) that we may _possibly_ see some evidence consistent with the EWG interpretation but NOT consistent with other interpretations. > And the > third is that it retains what we might call the problem of measure, > that is, explaining why we seem to occupy branches with a high measure > or amplitude, without just adding that as an extra assumption. What's the problem here? I find it utterly plausible that we would be in a universe where matter exists, where stars exist, where entropy gradients exist, etc., and NOT in a universe where the physical constants or structure of the universe makes life more difficult or impossible (or where the densities and entropy gradients mean that evolution of complex structures might take 100 billion years, or more, instead of the billion or so years it apparently took). > > The point is, all of these objections apply equally to the more > ambitious multiverse models we consider here. Our multiverse is even > more profligate than the MWI; it is if anything less observable; and > the problem of measure is at least as acute. I certainly agree with this! Tegmark's and Schmidhuber's and Egan's "all mathematics, all programs" models form supersets of the conventional MWI. > By the metrics we typically use for > universe complexity, basically the number of axioms or the size of a > program to specify the universe, the MWI is in fact simpler and > therefore > more probable than the traditional interpretation. This I'm not convinced of at all. I don't find the Copenhagen (aka "Shut up and calculate") Interpretation requires any more axioms. So long as we don't try to understand what is "really" happening, it's a very simple system. > Quantum randomness does not exist in the MWI. It is an illusion > caused by > the same effect which Bruno Marchal describes in his thought > experiments, > where an observer who is about to enter a duplication device has > multiple > possible futures, which he treats as random. If Schmidhuber would > adopt > this model for the physics of our universe it would improve the quality > of his predictions. And, putting in a plug for modal/topos logic, the essence of nearly every interpretation, whether MWI or Copenhagen or even Newtonian, is that observers at time t are faced with unknowable and branching futures. (In classical systems, these arise from limited amounts of information available to observers and, importantly, in limited positional information. Even a perfectly classical billiard ball example is unpredictable beyond a few seconds or maybe tens of seconds, because the positions and sets of forces (turbulence in the air currents around the balls, even gravitational and static electricity effects, etc.) are only known to, say, 20 decimal places (if that, of course). Because the "actual" positions, masses, sphericities, static charges, etc. are perhaps defined by 40-digit or even 200-digit numbers, the Laplacian dream of a suffiicently powerful mind being able to know the future is dashed. Unpredictability, or randomness, arises even in a fully classical real world. --Tim May "As my father told me long ago, the objective is not to convince someone with your arguments but to provide the arguments with which he later convinces himself." -- David Friedman
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
cs and even math to explore ideas about the nature of our reality, the anthropic principle, and the colonization of cyberspaces. It also turns out that Egan has been doing some Java and Mathematica programming for some of the spin foam papers by Baez and others. This doesn't mean that any particular idea he explores, whether in "Distress" or "Diaspora" or "Schild's Ladder," is "right," just that Egan is obvious technically competent to write about these ideas. What fired me up about "Distress" in particular was the several-page synopsis of the "All Topologies Model." For some reason, this got my juices flowing. (The rest of the novel just sort of plodded along to a fairly predictable conclusion.) I hope this explains why I don't look to Egan's fictional character for actual theories, just stimulation. --Tim May
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
On Tuesday, September 3, 2002, at 02:21 PM, scerir wrote: > Tim May: > I don't have a comprehensive theory of time, > but I am very fond of "causal time." > > Sometimes we read papers saying there is now > experimental evidence that quantum phenomena > are "a-causal" or "non-causal" or "out-of-time". > > See, in example, these recent papers > http://arxiv.org/abs/quant-ph/0110124 > http://arxiv.org/abs/quant-ph/0201036 > > Now, can lattices capture also those important > features? I haven't read the papers, just the abstracts. I could wait to comment for a few days or weeks until I've had a chance to absorb the papers, if ever, or comment now. First, it looks like these events are the usual "entangled states," which can be spacelike (the usual example of particles separated by light years). Second, for such spacelike intervals, they are outside each others' light cones in the extreme cases, so it would not be expected for any partial ordering to exist. Third, my own idiosyncratic view is to look at entangled particles as a single system, regardless of separation. Fourth, as to the mechanics of lattices: the essence of a partially-ordered set (poset) is that it does not require trichotomy, where either a is less than b, a is greater than b, or a is equal to b. In a chain, a linear form of a lattice, trichotomy holds. So, the integers obey trichotomy, as one integer is either less than, greater than, or equal to any other integer. Orders which obey trichotomy are said to be well-ordered. But not all sets are well-ordered. If the ordering relation is set inclusion, then a series of sets need not obey trichotomy. Some sets may be disjoint, with one neither including the other, being included by the other, or equal. In terms of causality, not even getting involved with speed of light issues and light cones, it is quite possible to say "event A neither caused event B nor was caused by event B nor is the same as event B." That is, the events A and B are incommensurate, or disjoint...they fail trichotomy. Clearly, most events all around us are such examples of incommensurate. They form posets. What a lattice does is to formalize the notions of order and to say there is only one edge between two events, and nothing in between (no other nodes in between). If two events are separated by many instants of time, many other events, then the lattice is made up of the smallest identifiable events. The events look like a lattice. (As I said, the Web has many nice pictures. No point in my spending 20 minutes drawing an ASCII lattice here, having it reproduced poorly, when entering "lattice poset" into Google will turn up nice pictures.) So, I would say from reading the abstracts that the Bell example just fits the ecample of a poset, where two events, which may or may not be entangled, are spacelike to each other. (This is the essence of the usual "instantaneous action" of EPR/delayed choice experiments.) --Tim
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
time has dominated for most of the past century (time lines, time as a river, the flow of time, Riemannian manifolds, etc.). In particular, whether space and time "really" are discrete at the Planck scale or "continuous all the way down" (to 10^-35 cm, 10^-50 cm, 10^-100 cm, etc.), is not what I am thinking about right now. It may not even matter, except for the unification of QM and gravity, the main reason for trying to resolve this issue. Rather, I am more interested in the issue of ordered sets and lattices, posets especially, for the study of time and causality at human scales. (Without going into details, here, there are issues relating to the nature of belief and trust which have to do with these causal orders and possible worlds. I've hinted at some of these points in other posts here...someday I'll solidify enough of the swirling ideas to summarize them.) I agree strongly with Lee Smolin that topos theory (and related ideas, tools) is not just the right logic for dealing with quantum cosmology, it is also the right logic for dealing with a huge number of other things. --Tim May
Re: Time as a Lattice of Partially-Ordered Causal Events or Moments
On Monday, September 2, 2002, at 09:22 PM, Osher Doctorow wrote: > From: Osher Doctorow [EMAIL PROTECTED], Mon. Sept. 2, 2002 9:29PM > > It is good to hear from a lattice theorist and algebraist, although I > myself > prefer continuity and connectedness (Analysis - real, complex, > functional, > nonsmooth, and their outgrowths probability-statistics and > differential and > integral and integrodifferential equations; and Geometry). > Hopefully, we > can live together in peace, although Smolin and Ashtekar have been > obtaining > results from their approaches which emphasize discreteness (in my > opinion > built in to their theories) and so there will probably be quite a > battle in > this respect at least intellectually. I'm not set one way or the other about discreteness, especially as the level of quantization is at Planck length scales, presumably. That is, 10^-34 cm or so. Maybe even smaller. And the Planck time is on the order of 10^-43 second. One reason discrete space and time isn't ipso facto absurd is that we really have no good reason to believe that smooth manifolds are any more plausible. We have no evidence at all that either space or time is infinitely divisible, infinitely smooth. In fact, such infinities have begun to seem stranger to me than some form of loops or lattice points at small enough scales. Why, we should ask, is the continuum abstraction any more plausible than discrete sets? Because the sand on a beach looks "smooth"? (Until one looks closer.) Because grains of sand have little pieces of quartz which are smooth? (Until one looks closer.) But, more importantly, the causal set (or causal lattice) way of looking at things applies at vastly larger scales, having nothing whatsoever to do with the ultimate granularity or smoothness of space and time. That is, a set of events, occurrences, collisions, clock ticks, etc. forms a causal lattice. This is true at the scale of microcircuits as well as in human affairs (though there we get the usual "interpretational" issues of causality, discussed by Judea Pearl at length in his book "Causality"). You say you prefer continuity and connectednessthis all depends on the topology one chooses. In the microcircuit case, the natural topology of circuit elements and conductors and clock ticks gives us our lattice points. In other examples, set containment gives us a natural poset, without "points." (In fact, of course mathematics can be done with open sets, or closed sets for that matter, as the "atoms" of the universe, with no reference to points, and certainly not to Hausdorff spaces similar to the real number continuum.) The really interesting things, for me, are the points of intersection between logic and geometry. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Time as a Lattice of Partially-Ordered Causal Events or Moments
On Saturday, August 31, 2002, at 11:31 PM, Brent Meeker wrote: > Time is a construct we invented to describe things. Most > basically we use it to describe our sequence of experiences > and memories. We feel hot and cold, but we needed to > quantify hot and cold and give them operational definitions > in order make definite predictions about them. So we > invented temperature and thermometers. For mechanics we > needed a quantified, operational definition of duration - > so we invented time and clocks. > > Besides psychological time,there are at least three > different possible definitions of time used in physics What > they all have in common is that they assign numbers to > different physical states, i.e. they index different states > into some order so that this sequence of states can be > compared to that sequence of states. I don't have a comprehensive theory of time, but I am very fond of "causal time." Picture events as a series of points in a lattice (a graph, but with the properties I talked about a while back in a post on partially-ordered sets). Basically, a lattice of events where there is at most one edge connecting two points. (There are formal properties of lattices, which the Web will produce many good definitions and pictures of.) Lattices capture some important properties of time: * Invariance under Lorentzian transformations...any events A and B where B is in the future light cone of A and A is in the past light cone of B, will be invariantly ordered to all observers. * The modal logic nature of time. Multiple "futures" are possible, but once they have happened, honest observers will agree about what happened. (Echoing the transformation of a Heyting algebra of possibilities into the Boolean algebra of actuals...this sounds like it parallels quantum theory, and Chris Isham and others think so.) * Personally, I believe the arrow of time comes from more than statistical mechanics. (I believe it comes from the nature of subobject classifiers and the transformation Heyting --> Boolean.) * I am indebted to the books and papers of Lee Smolin, Fotini Markopoulou, Louis Crane, Chris Isham, and several others (Rovelli, Baez, etc.) for this interpretation. None of us knows at this time if time is actually a lattice at Planck- or shorter-time-scale intervals. But discretized at even the normal scales of events (roughly the order of seconds for human-scale events, picoseconds or less for particle physics-scale events), the lattice-algebraic model has much to offer. * I don't see any conflict with Huw Price, Julian Barbour, and others (haven't read Zeh yet), though I don't subscribe to all of their idiosyncratic views. --Tim May
Time
The September issue of "Scientific American" is usually/always devoted to some special theme. This issue is ostensibly devoted to "Time" and problems associated with it. Articles include some physics articles, some perception/psychology articles, and one or two on clocks and timepieces. Sad to say, "Sci Am" has fallen far from its once lofty perch. Flipping through the issue at a boostore, I found the first _half_ of the thin magazine devoted to advertising, general news, and a special 20-plus-page insert devoted to Italy and its industries, blah blah. Once the articles started, they were of course no longer the meaty, detailed dozen or so solid articles. (Used to be the special September issues were thicker than usual!) The articles were short, filled with colorful graphics (but with less content than the SciAm graphics of the 1950s-recent), but carried little information. The articles may be of use in introducing people to notions like "block time," but the entire idea is covered in just a few paragraphs. Not much to go on. Paul Davies does one of the physics articles on time...nothing in his article not covered in much more detail in the books by Huw Price, Julian Barbour, Kip Thorne, and others. I didn't buy the issue. Meanwhile, my study of lattice and order continues. I'll say more in the future (if it exists, that is). --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: Yetter's "Functorial Knot Theory" and the Mind/Body Problem
as the one about the "reversability of time" arguements, raised initially by Wei Dai and then argued by Hal Finney. Writing the "Professor Ludwig" piece, in which an 1860 Prof. Ludwig (Boltzmann, obviously) predicts that the simplest time-reversed pocket of the universe means telescopes will likely see nothing but chaos outside the local region, helped me to clarify my thinking on anthropic arguments. And motivated me to finish reading Huw Price's book. This is the real blessing of mailing lists like this one! I may now be motivated to understand the kinds of logic you discuss if only to try to refute you! (no offense intended) --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: Entropy, Time's Arrow, and Urns
(A minor typo is corrected) On Sunday, August 18, 2002, at 01:00 PM, Tim May wrote: > In Sequence One, the two urns are filled with stones of mixed color at > the start of the film. As the main transfers stones, the number of > black and white stones in each of the urns fluctuates, but there are > never, in this particular film, any excursions outside the ratio 450 of > one color to 550 of the other. There are only 500 stones in each urn, so what I meant was, for example, 275 of one color, 225 of the other, adding up to 500 total in either of the two urns. --Tim
Entropy, Time's Arrow, and Urns
Hal has brought up Huw Price's book, "Time's Arrow and Arhimedes' Point," and especially the thermodynamic/entropy arguments related to recurrence a la Poincare, Boltzmann, and others. A point Price makes several times is th "..though it needs to be borne in mind that not everyone had a clear grasp of the fact that the low-entropy past is itself in need of explanation." (p 37) "In sum, the puzzle is not about how the universe reaches a state of high entropy, but about how it comes to be starting from a low one." (p 40) "For all their intrinsic interest, then, the new methods of nonlinear dynamics do not throw new light on the asymmetry of thermodynamics. Writers who suggest otherwise have failed to grasp the real puzzle of thermodynamics--Why is entropy low in the past?--and to see that no symmetric theory could possibly yield the kind of conclusions they claim to draw." (p 44) "There is no separate problem as to why entropy in branch systems always increases towards the future, in other words: only the big problem was in the bottle in the first place." (p 45) And so on. Price repeatedly bases many arguments on his dissatisfaction with assuming a state of high order. (To be fair, his book is an interesting romp through many theories of time asymmetry, touching on Feynman-Wheeler absorber theories, delayed choice experiments in quantum mechanics, psychology, etc. Not a bad place to get exposed to a lot of the current issues. Just don't take his particular "crotchet" too seriously, is my advice.) Frankly, I don't worry how the beer got in the bottle (one of his example, about gas expanding out of a beer bottle...Price worries that the analysts of time are not asking proper questions about how the beer came to be in the bottle in the first place...most of us, dullards that we are, assume that a bottling company _put_ the beer in the bottle!. I'm not being flip. It's an observed fact of our universe, and likely derivable from anthropic arguments, that there's a lot of "free energy" around: a lot of unfused hydrogen, a lot of gravitational potential energy, a lot of stored chemical energy, etc. How this came to be from "first principles" from an initial singularity is of course unknown at this time. Time for a digression. The classic urn experiment, with Price's objections. And let me throw in something several members of this list will likely appreciate: a bet on the outcomes (a la Bayesian reasoning, a la market processes, a la Robin Hanson's idea futures, a la probabalistic definitions of the truth). Imagine two urns. Imagine, say, 500 black stones and 500 white stones. A person is reaching inside one urn, removing a stone, transferring it to the other urn, picking up a stone "at random" (a regrettably loaded term, but one which will hopefully become clearer...imagine that the man is blind and cannot possibly see the color of the stone he is picking up). A group of people is show two filmed sequences: In Sequence One, the two urns are filled with stones of mixed color at the start of the film. As the main transfers stones, the number of black and white stones in each of the urns fluctuates, but there are never, in this particular film, any excursions outside the ratio 450 of one color to 550 of the other. In Sequence Two, the the film begins rolling with one urn filled with white stones and the other urn filled with black stones. The man reaches in, takes a white stone, transfers it to the other urn. He reaches in, takes a black stone, transfers it to the first urn. As the film progresses, the two urns eventually reach a state where each has about 250 white stones and about 250 black stones. The group is told that one of the films is presented in correct chronological order while the other is presented in reverse chronological order. The group is told that bets will be taken on which is which. Oddsmakers are standing by. A terminal linked to the Idea Futures Market is available. (The Barbourian jumps up and yells "There is no time! All events happen at the same time!" The organizer says "Fine, but I'm still taking bets. The Barbourian sits down.) Which way would you bet? And what do think oddsmakers would make odds? Not surprisingly, nearly everyone will bet that Sequence Two was shown in correct chronological order and that Sequence One, if one of the two sequences was shown in reverse chronological order, must have been the one that was reversed. (The Quibbler points out that Sequence One could easily be shown in chronological order, just either a long time after the mixing started or starting with an initially mixed set. "Sure," the organizer points out, "but I told you one was in correct order, one was reversed, so place your bets.") Now the urn example is one that does not use the "molecular chaos" that Huw Price is so critical of in "gas mixing" examples, arguing that "molecular chaos" is assuming the conclusion. Here w
Re: Doomsday-like argument in cosmology
On Saturday, August 17, 2002, at 11:37 PM, Hal Finney wrote: > Now you might say, so what, the whole idea that we formed in this way > was so absurd that no one would ever take it seriously anyway. But the > authors of this paper seem to be saying that if you assume that there is > a positive cosmological constant (as the cosmological evidence seems to > show), eventually we will get into this de Sitter state, and based on > some assumptions (which I didn't follow) we really should see Poincare > recurrences. Then by the anthropic principle we should be > overwhelmingly > likely to be living in one. OK, let us assume for the sake of argument that we should be overwhelmingly likely to be living in one of these "time-reversed cycles" (which I distinguish from "bounces" back to a Big Bang state, the more common view of cycles). By the same Bayesian reasoning, it is overwhelmingly likely that any observer would find himself in a TRC in which other parts of the universe eventually visible to him (with telescopes) are "incompletely reversed." Let me give a scenario to make the point clearer. It is 1860. Telescopes exist, but are still crude. The Milky Way is only known to be a nebula, a swirl of stars. The existence of galaxies other than our own is unknown. Professor Ludwig calls together several of us friends (perhaps on the Vienna version of the Everything List) and outlines his theory. "We are very probably in a recurrence phase of the Universe, where a worn-out, gaseous phase of the Universe has randomly arranged us into this low-entropy, highly-ordered state we find ourselves in today. It took a very long time for this to happen, perhaps 1,000,000,000,000,000 million years, but here we are." (Reactions of his audience not presented here...maybe in the novel some distant version of me will write.) "All that we see around us, our Sun, the planets, even the gas balls we call stars, were formed thusly out of a random rearrangement of gas molecules. My young mathematician friend in Paris, Msr. Poincare, says this sort of recurrence is inevitable in any sufficiently rich phase space." "Now, if this is correct, it is overwhelmingly likely that of all of the time-reversed cycles, or TRCs, the TRC we find ourselves in will have only reversed time (or created low entropy structures) in our particular region of the Universe. In a hugely greater amount of time, even more regions of the Universe we will be soon be able to observe would be subject to this reversal, but the times involved are even more hideously enormous than the very long times needed to create our own TRC pocket in which we find ourselves." "So, overwhelmingly, observers who draw the conclusions I have reached will find themselves in a Universe where only a region sufficient to have "built" them and their supporting civilization will have the low entropy order of a TRC." "Thus, gentlemen, by a principle I call "falsifiability," I predict that when the new telescopes being built now in Paris and London become operational, we will see nothing around our region of the Universe except gas and disorder." And, of course, within his remaining lifetime Professor Ludwig was astonished to learn that distant galaxies looking very much like nearby galaxies existed, that if a Poincare recurrence had in fact happened, it must have happened encompassing truly vast swathes of the Universe...in fact, the entire visible Universe, reaching out ten billion light years in all directions. The unlikelihood that an observer (affected causally only by events within a few light years of his home planet) would find himself in one of the comparatively-rare TRCs which affected such a big chunk of the Universe convinced Professor Ludwig that his theory was wrong, that the new ideas just being proposed of an initial singularity, weird as that might be, better explained the visible Universe. --Tim May (who also thinks the difficulty of time-reversing things like ripples in a pond, radiation in general, and all sorts of other things makes the Poincare recurrence a useful topological dynamics idea, but one of utterly no cosmological significance)
Recurrence in the universe
On Saturday, August 17, 2002, at 01:57 PM, Hal Finney wrote: > > After an extremely long interval, we may get a Poincare recurrence. > (Actually, I'm not sure this is the right term for this; I think a > Poincare recurrence is a more general thermodynamic effect. But I will > use the phrase here to specifically talk about a low-entropy fluctuation > out of a high-energy equilibrium state.) The gas will randomly happen > to > move back into a low-energy state, perhaps even the same state we > started > with, all the molecules in one corner. At that point we once again get > dissipation, structures, the passage of time, and the possibility of > life. > This cycle can and will repeat indefinitely. As usual, I am intensely skeptical. From the magnitudes of the calculations, not from a "gut feel." The gedankenexperiment of all the molecules in a box being found in one small volume/corner of the box is a classic textbook calculation. I haven't done the calcs in a long, long time, but it's fairly clear that even a mole-magnitude quantity of molecules might "easily" take some incredibly huge amount of time, something like 10^500 years, to "randomly" end up in a volume 1% as large as the box. (It might just as easily be 10^2000 years or more6 x10^23 molecules bouncing around is hard to find in one region of the phase space.) How long before 10 moles find their way to a small part of the phase space? Or galactic-cloud-sized quantities? And biological structures with chemical reactions driven by concentration gradients on lipid layers...whew. I guarantee that the "time for gas molecule-type recurrence" is something like 10^10^10^10^10^10.10 years. Now I realize that "infinity" is a much larger number than "10^10^10^10^.10 years," and so a Cantorian correspondence might suggest that "it could happen." But in any finite chunk of time, no matter how large, my hunch is that the "divergence" issues utterly dominate. That is, in the first 100 billion years of the universe's life after the stars all burn out (say, 200 billion years from now), there still will not be a single instance anywhere in the universe where a mole of hydrogen in some reasonable volume (a thousand cubic kilometers, maybe) has "fluctuated" into an ordered state where the hydrogen is at much higher concentration. (This doesn't preclude bounces which reset to very dense states.) > That is, if we really assume that somehow this gas in the corner evolved > life which then died out in the heat death of the universe, then the > most likely path back into the corner is to evolve life backwards. > We would see the formless void of space begin to cluster together to > form > structure. That structure would include the pattern of dead life-forms. > These life-forms would come to life, and they would live their lives > backwards. They would grow young and be un-born. Each generation > would be replaced by its ancestors. I don't claim to understand the physics of time asymmetry (despite the books I have read, including starting Huw Price's book recently, based on recommendations here), but this extension of billiard ball gedankenexperiments to "living lives backwards" is just too bizarre. So many chemical reactions, so many biological "objects," so many issues of functional causality (as but one example, the heart pumping blood, enabling cell growth, etc.). To hypothesis that "if we wait long enough, cells will randomly get smaller, will ungrow, causing reverse fluid flow to then cause the heart to beat backwards" (as but one of a vast number of examples...and this effect has to happen across all organisms in all places, else the Universe has not really "unaged." If a mole quantity of hydrogen may take "10^10^10^...10" years just to get to a "low entropy state, but not necessarily the same structure as before," then how long well, it ain't something I'll lose sleep over. > I believe it follows, then, that if we are living in such a Poincare > recurrence, it is overwhelmingly likely that the universe did not really > go all the way back to the Big Bang. Rather, our past is an illusion. > Time ran backwards far enough to form us; but among those recurrences > where we formed, the overwhelming majority of them don't have time go > back much farther than that. (My son is reading the Price book now and > says that this idea goes back to Boltzmann, that our past is false and > the universe no older than us, if our experience are explained by such > a recurrence.) Nietzsche had similar ideas of the "eternal recurrence," circa 1870. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: modal logic and possible worlds
On Saturday, August 17, 2002, at 08:06 PM, George Levy wrote: > The arbitrariness of "my," "your" or anybody's own mind point to the > need for the relativistic approach which I have been advocating. The > frame of reference here is the logical system residing in the > observer's mind. It may not be the type of formal system which has > been discussed in the list. There may be a need to develop some kind of > "fuzzy" logical system for human mental processes corresponding to the > formal systems already in existence. As far as I know Fuzzy Logic has > not been developped to the same extent as the branches of logic that > have been discussed in the list. Well, count me as skeptical that the hype about "fuzzy set theory" and "fuzzy logic" has ever, or will ever, live up to some of the claims made by Bart Kosko, Lofti Zadeh, and others. Most of what passes for fuzzy logic just looks like ordinary Bayesian probability. Here's a comment from Saunders Mac Lane in his book "Mathematics: Form and Function," 1986: "Not all outside influences are really fruitful. For example, one engineer came up with the notion of a _fuzzy_ set--a set X where a statement x elementof X of membership may be neither true nor false but lies somewhere in between, say between 0 and 1. It was hoped that this ingenious notion would lead to all sorts of fruitful applications, to fuzzy automata, fuzzy decision theory and elsewhere. However, as yet most of the intended applications turn out to be just extensive exercises, not actually applicable; there has been a spate of such exercises." (. pp 439-40). While maybe Mac Lane is a little too snippily dismissive, here we are more than 15 years later and what do we have? Fuzzy rice cookers which look like nothing more than rice cookers with various algorithms Newton could have calculated, fuzzy-logic elevators which are simply implementing similar acceleration algorithms, and not much else. Certainly fuzzy logic has not been significantly in the foundations of mathematics. Logicians have not been using fuzzy sets and fuzzy logic in any significant way, judging by the books and articles I've seen. I agree that formal logic is not easily applied to minds. Logicians would agree. A mind is weighing large numbers of inputs, far beyond what would normally fill an entire page of First Order Logic equationssurvival has made the ability to reason with uncertainty (a better core concept that calling it "fuzzy logic," in my opinion) a survival trait. Those minds which can find solutions in the midst of noise and uncertainty tend to reproduce more than those minds which are paralyzed or too slow in reaching survival-enhancing conclusions. What we have talked about here in this sub-thread on _modal logic and possible worlds_ is an idealization of logic, just a snapshot or facet of things, in much the same way a "line" or a "plane" is a facet of the world around us (and understandable at some level by birds and reptiles even). --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: modal logic and possible worlds
On Tuesday, August 13, 2002, at 08:47 PM, Wei Dai wrote: >> Seen this way, category and topos theory are worth studying for their >> own sake. I don't think it is likely that "every conceivable universe >> with consistent laws of mathematics has actual existence" (to nutshell >> my understanding of Tegmark's theory) is actually true (whatever that >> means). Nor do I take Schmidhuber's "all running programs" notion very >> seriously. Interesting ideas to play with, and to use some tools on. > > Well why don't you take these ideas seriously? Lack of even the slightest piece of evidence for "all possible mathematical universes actually exist" and/or "the all runnable computer programs.' I also don't believe there are gods or other supernatural beings, for the same reason. If and when I see an experiment that points to there being other universes which have tangible existence, then I'll start to believe. --Tim May "That the said Constitution shall never be construed to authorize Congress to infringe the just liberty of the press or the rights of conscience; or to prevent the people of the United States who are peaceable citizens from keeping their own arms." --Samuel Adams
Re: modal logic and possible worlds
On Tuesday, August 13, 2002, at 06:16 PM, Wei Dai wrote:
> On Tue, Aug 13, 2002 at 10:08:50AM -0700, Tim May wrote:
>> * Because toposes are essentially mathematical universes in which
>> various bits and pieces of mathematics can be assumed. A topos in which
>> Euclid's Fifth Postulate is true, and many in which it is not. A topos
>> where all functions are differentiable. A topos in which the Axiom of
>> Choice is assumed--and ones where it is not assumed. In other words, as
>> all of the major thinkers have realized over the past 30 years, topos
>> theory is the natural theory of possible worlds.
>
> How does this compare to the situation in classical logic, where you can
> have theories (and corresponding models) that assume Euclid's Fifth
> Postulate as an axiom and theories that don't?
Because such a dichotomy ("and theories that don't") means the logic is
ipso facto modal. The very form tells us that a modal (and hence
intuitionist) assumption is at work: "If it were the case that the
parallel postulate were valid, then..." and "Suppose the parallel
postulate is not true, then..."
If the Fifth Postulate is independent of the others, then within the
framework of the other postulates one may have one "branch" where the
Fifth holds (Euclidean Geometry) and another branch where it doesn't
hold (all of the various non-Euclidean geometries).
Now this turns out to be a not very important example, as various
geometries with various geodesics on curved surfaces are sort of
mundane. And the details were mostly worked out a hundred years ago,
starting with Gauss, Bolyai, Lobachevsky, Riemann, and continuing to
Levi-Cevita, Ricci, and Cartan. The fact that by the mid-19th century we
could _see_ clear examples of geometries which did not "obey" the
parallel postulate, e.g., triangles drawn largely enough on a sphere,
great circles, figures drawn on saddle surfaces and trumpet surfaces,
etc., meant that most people didn't think much about the modal aspects.
But they are certainly there.
(I believe it's possible to cast differential geometry, including the
parallel postulate or its negation, in topos terms. Anders Kock has done
this with what he calls "synthetic differential geometry," but I haven't
read his papers (circa 1970-80), so i don't know if he discusses the
parallel postulate explicitly.)
Both category theory and topos theory have been used to prove some
important theorems (e.g., the Weyl Conjecture about a certain form of
the Riemann zeta function, and the Cohen "forcing" proof of the
independence of the Continuum Hypothesis from the Zermelo-Frenkel
logical system), but it is misleading to think that either will give
"different results" from conventional mathematics. It is not as if
Fermat's Last Theorem is true in conventional logic or in conventional
set theory but false in intuitionist logic or category theory.
I'm going to have to slow down in my writing. You ask a lot of short
questions, but these short questions need long answers. Or, perhaps, I
feel the need to make a lot of explanations of terminology and
motivations. I'll have to tune the length of my responses to the length
of your questions, I think!
--Tim May
(.sig for Everything list background)
Corralitos, CA. Born in 1951. Retired from Intel in 1986.
Current main interest: category and topos theory, math, quantum reality,
cosmology.
Background: physics, Intel, crypto, Cypherpunks
Re: modal logic and possible worlds
g programs" notion very seriously. Interesting ideas to play with, and to use some tools on. Strangely, then, I view these notions as places to apply the math I'm learning to. And I'm thinking small, in terms of simple systems. A paper I have mentioned a couple of times is directly in line with this approach: Fotini Markopoulou's "The internal description of a causal set: What the universe looks like from the inside." Here's the paper number and abstract: gr-qc/9811053 From: Fotini Markopoulou <[EMAIL PROTECTED]> Date (v1): Tue, 17 Nov 1998 19:28:10 GMT (41kb) Date (revised v2): Thu, 18 Nov 1999 17:32:41 GMT (42kb) The internal description of a causal set: What the universe looks like from the inside Authors: Fotini Markopoulou Comments: Version to appear in Comm.Math.Phys. (minor modifications). 37 pages, several eps figures Journal-ref: Commun.Math.Phys. 211 (2000) 559-583 We describe an algebraic way to code the causal information of a discrete spacetime. The causal set C is transformed to a description in terms of the causal pasts of the events in C. This is done by an evolving set, a functor which to each event of C assigns its causal past. Evolving sets obey a Heyting algebra which is characterised by a non-standard notion of complement. Conclusions about the causal structure of the causal set can be drawn by calculating the complement of the evolving set. A causal quantum theory can be based on the quantum version of evolving sets, which we briefly discuss. --end of excerpt-- Take a look at these papers (hers, the Guts paper, the various Baez, Smolin, Crane, etc. papers). All free. Some are introductions. All have a fair amount to say about the nature of reality. The stuff on causal sets (lattices and posets) is of direct relevance to several areas of modern physics. The relevance to MWI and Tegmark-style meta-branchings seems clear to me. As far as the math of nonstandard logic goes, I think the most interesting application within our lifetimes will come with AI. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: modal logic and possible worlds
On Monday, August 12, 2002, at 11:18 PM, Wei Dai wrote: > Tim, I'm afraid I still don't understand you. > > On Mon, Aug 12, 2002 at 06:00:26PM -0700, Tim May wrote: >> It is possible that WWIII will happen before the end of this year. In >> one possible world, A, many things are one way...burned, melted, >> destroyed, etc. In another possible world, B, things are dramatically >> different. > > Ok, but what about my point that you can state this by explicit > quantification over possible worlds rather than using modal operators? > I.e., "There exist a world accessible from this one where WWIII happens > before the end of this year." instead of "It is possible that WWIII will > happen before the end of this year."? That is indeed saying just the same thing (though the language is slightly different). The important part of modal logic is not in the "accessible from this one" or "it is possible" language. Rather, the "forking paths" (a la Borges) picture that is described by posets and lattices. > >> There can be no implication from one world to the other. That is, we >> can't say "A implies B" or "B implies A." > > What does that have to do with my question? Anyway A and B are supposed > to > be worlds here, not propositions, so of course you can't say "A implies > B". I don't know what point you're trying to make here. Worlds _are_ propositions. And the "causal operator" (time) is the same as implication. With some important caveats that I can't easily explain without drawing a picture. In conventional logic, implication is fully-contained or defined from some event A (or perhaps some combination of events A, B, C, etc., all causally contributing to a later event). There are two interesting cases to consider where implication does not follow so easily from A: 1. Possible worlds. The event A forks down two (or more) possible paths. A future where war occurs, a future where war does not. A future where Fermat's Last Theorem is proved to be true. A future where it is not. A future of heads, a future of tails. 2. Quantum mechanics. Schrodinger's cat. (It was Einstein and Podolsky's belief that classical logic must apply that led to their belief that there _must_ be some other cause, some hidden variable, that makes the outcome follow classical logic. Bohm, too. But we know from Bell's Theorem and the Kochen-Specker "no-go" theorems that, basically, these hidden variables are not extant.) (By the way, the book "Interpreting the Quantum World," by Jeffrey Bub, has an interesting section on how modal logic applies to QM.) Bruno is much more of a logician than I am, but the various terms of logic, lattices, and set theory are analogous (probably a very efficient category theory metaview, but I don't yet know it). 1 is True 0 is False lattice infimum or Boolean meet, ^ , is conjunction (AND) lattice supremum or Boolean join, v , is disjunction (OR) lattice or Boolean orthocomplement is negation (NOT) (Understanding this is not essential to my arguments here...I just wanted to make the point that there are mappings between the languages of logic, set theory, and lattices. In a deep sense, they are all the same thing. Definitions do matter, of course, but e-mail is not a great place to lay out long lists of definitions!) > >> This branching future is exactly what I was talking about a week or so >> ago in terms of "partially ordered sets." If the order relationship is >> "occurs before or at the same time as," which is equivalent to "less >> than or equal to," A and B cannot be linearly ordered. In fact, since >> both A and B are completely different states, neither can be said to be >> a predecessor or parent of the other. In fact, A and B are not >> comparable. > > I'm with you so far in this paragraph. > >> We cannot say "A or not-A." > > Now I'm lost again. Again A is a world not a proposition so what > would "A > or not-A" mean even if A and B are comparable? The two forks in the road are given the same truth value weighting in this "possible worlds" approach. We have _assumed_ A in this fork I described, so "not-A" is certainly not necessarily the other path. In fact, the meaningful interpretation of "not-A" in the complement sense is "that which precedes A," that is, the events leading up to A in this world. I realize this sounds confusing. Draw a picture. Just have three points in it, arranged in a triangle: A B \ / X Time is in the upward direction. The points/events/states X, A, B form a poset. One arrow
Re: modal logic and possible worlds
On Tuesday, August 13, 2002, at 10:08 AM, Tim May wrote: > This graph, this set of vertices and edges, is a "per-ordered" set. > More than just a set, any category with the property that between any > two objects "p" and "q" there is AT MOST one arrow "p --> q" is said to > be "pre-ordered." I meant to type "pre-ordered" in the first line above. I don't normally worry overmuch about minor typos, especially when I used the correct spelling right after the typo, but I wouldn't want anyone thinking there's some kind of "per-ordered" set! --Tim May
Re: modal logic and possible worlds
as) event C.
Example: (short version--you know the drill by now): If A contains B and
B contains C, then A contains C.
Discussion: These are all simple points to make. Obvious even. But they
tell us some important things about the ontological structure of many
familiar things. I encourage anyone not familiar with these ideas to
think about the points and think about how many things around us are
pre-ordered.
If a pre-order has an additional property we call it a partial-order:
3. Anitsymmetric: Whenever pRq and qRp, then p = q.
Example: If p implies q and q implies p, then p and q are the same
thing. (Equality, isomorphism, identity.)
Example: If p is LTE q and q is LTE p, then p = q.
Example: the time example is left as an exercise!
Example: ditto for set containment.
A set with a partial-order is called a "poset." These feature
prominently in all sorts of areas. For our purposes, posets are
essentially what _time_ is all about.
In addition, we can define things like "meets" and "joins" and the
result is a _lattice_, studied extensively by Dedekind, Von Neumann, and
Garrett Birkhoff. Lattices look exactly like lattices, or trellises. Two
vertices have at most one link (arrow, R, etc.) between them, though
many links may point to any particular vertex.
In this view, it doesn't really matter (at this level) whether the
vertices are the outcomes of a coin toss or entire worlds.
This was the sense in which I was using "WWIII happens this year" or
"WWIII doesn't happen this year" for my MWI-type example.
The essential point is that the natural logic of such posets is not
necessarily Boolean. There are several names for this "not necessarily
Boolean" aspect, depending on the interest of the researcher or writer:
* He may call it "non-Aristotelean logic," as even Aristotle was said to
have realized that a statement like "The fleet at Carthage will either
be sunk tomorrow or it won't be" is not always meaningful, and that
attempting to force future or time-varying truth into the Stoic model of
"A or not-A" is not the most useful thing to do.
* He may call it "Intuitionist" or "Constructivist," asking that
mathematical proofs be _constructive_ in nature rather than using proof
by contradiction. ("Assume the proposition not to be true, then we see
that,...then, and this is a contradiction, therefore the proposition
must be true.") This turns out to be fairly important when proofs use
the so-called "Axiom of Choice." (Which is equivalent to many other
axioms and theorems.) Some important results of the past 40 years have
come about by challenging the role of the Axiom of Choice. (BTW, as an
aside which may be of interest to some list members, John Nash used the
Axiom of Choice to prove that certain solutions to multi-party protocols
must exist, but he did not give a constructive proof of what those
solutions are.)
* He may call it a "Heyting algebra," as opposed to a Boolean algebra.
I've discussed Heyting algebras and logic here in the past, and I refer
readers to the Web for many articles of varying levels of assumed
background.
* He may call it "possible worlds semantics," after the work by the
logician Saul Kripke on the logic implicit in possible worlds.
* And most generally of all, at this time, he may call it topos theory.
Does it relate specifically to the speculations of Max Tegmark, Greg
Egan, and others on "all mathematics" and "all topologies" models?
I can't say for sure, but it's the direction *I* am taking. As I said, I
think small. I can't "reason about" entire worlds and draw meaningful
conclusions. I _think_ this kind of thinking about posets, lattices, and
toposes is the right way to think about systems with varying choices,
even varying mathematics (*).
* Because toposes are essentially mathematical universes in which
various bits and pieces of mathematics can be assumed. A topos in which
Euclid's Fifth Postulate is true, and many in which it is not. A topos
where all functions are differentiable. A topos in which the Axiom of
Choice is assumed--and ones where it is not assumed. In other words, as
all of the major thinkers have realized over the past 30 years, topos
theory is the natural theory of possible worlds.
So, I think small. I think about flips of coins, about simple lattices
and simple posets.
These are not the Universe, let alone the Multiverse, but it seems clear
to me we cannot reason about the entire Universe or Multiverse unless we
can reason about very simple sub-parts of it.
In any case, it's my particular interest at this time.
I hope this helps clarify things a bit.
--Tim May
(.sig for Everything list background)
Corralitos, CA. Born in 1951. Retired from Intel in 1986.
Current main interest: category and topos theory, math, quantum reality,
cosmology.
Background: physics, Intel, crypto, Cypherpunks
Re: modal logic and possible worlds
On Monday, August 12, 2002, at 12:07 PM, Wei Dai wrote: > According to possible world semantics, "it's necessary that P" means > that > P is true in all worlds accessible from this one. Different modal logics > correspond to different restrictions on the accessibility relation. > Before > the invention of possible world semantics, people argued about which > modal > logic is the correct one, but now philosophers realize that different > notions of accessibility (and the corresponding notions of modality) are > useful at different times, so there is no single correct modal logic. > > That's my one paragraph summary of possible world semantics. Please > correct me if I'm wrong, or read these articles if you're not familiar > with this topic: > > http://www.xrefer.com/entry.jsp?xrefid=552831 > http://www.xrefer.com/entry.jsp?xrefid=553229 > > My questions is, why not just quantify over the possible worlds and > refer > to the accessibility relation directly? This way you can talk about > multiple accessibility relations simultaneously, and you don't have to > introduce new logical symbols (i.e. the box and the diamond). Is > modality just a syntactic shorthand now? Modal logic is a lot more than syntactic shorthand. Consider this example, phrased in MWI terms. It is possible that WWIII will happen before the end of this year. In one possible world, A, many things are one way...burned, melted, destroyed, etc. In another possible world, B, things are dramatically different. There can be no implication from one world to the other. That is, we can't say "A implies B" or "B implies A." This branching future is exactly what I was talking about a week or so ago in terms of "partially ordered sets." If the order relationship is "occurs before or at the same time as," which is equivalent to "less than or equal to," A and B cannot be linearly ordered. In fact, since both A and B are completely different states, neither can be said to be a predecessor or parent of the other. In fact, A and B are not comparable. We cannot say "A or not-A." We have thus left the world of classical logic and are in the world of non-classical, or intuitionistic, or Heyting logic. Posets are not just a different syntactic shorthand from linearly-ordered sets. Branching worlds, aka possible worlds, aka MWI (when QM is involved) is a more accurate way of talking about time and successions of events than is attempting to force time into a strait-jacket of linearly-ordered sets (chains). Besides the topos work of Saul Kripke, Vaughan Pratt at Stanford has written a lot on concurrency, lattices, and posets. Lee Smolin's book "Three Roads to Quantum Gravity" is very good at explaining how this relates to cosmology. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Time, causality, posets
Everything folks, Here's a posting I made last night to another list, a list of folks who meet to discuss math. I had been telling them about nonstandard logic, notably Intuitionist or Brouwer/Heyting logic, and the natural logic of toposes. This post below expands on a few points we had been talking about at our session in Palo Alto a few days ago. (By the way, if anyone is local to the Bay Area and wants to try one of the evening gatherings, let me know. We just started meeting and it's too soon to know how it'll go in the future. The basic idea is to have an informal group similar to the "Assembler Multitudes" nanotechnology discussion group that Ted Kaehler ran in the early 90s. I enjoyed that group immensely and was disappointed to see it fade out. With all of the new excitment in math, and with links to the cosmology and Everything universes, it seems to be a good time to try something again. We had six people at our gathering a few days ago.) The background for this article is not given here, so I'll make a very few points now: * conventional logic (Aristotelian, Boolean) uses "law of the excluded middle": A or Not-A, something is or is not, the complement of an open set is a closed set. The complement of a complement of a set is the set. * alternative or nonstandard logics exist, and turn out to be quite natural...when looked at properly. * one of these is the logic pursued by Brouwer early in the 20th century: Intuitionism (which is not mysticism, by the way). Brouwer argued that only constructible entities have meaning, that abstractions about infinite sets or things like the axiom of choice are misleading. His student Heyting formalized the axioms of Intuitionist logic. Marshall Stone proved in the 1930s that the set of operations on open subsets of a set (think of blobs drawn on a page, or time intervals, etc.) forms a Heyting algebra, that is, that the natural logic for these open sets is not Boolean logic, but Heyting logic. * lattices are sets of node and links between the nodes which satisfy certain properties, such as that any two nodes have a "meet" and "join." Events in time are a good example of a lattice. * partially ordered sets (posets) are those with some relationship (such as "less than or equal to" or "preceeds or happens at the same time" or "is contained in or equals") such that certain properties of comparison exist. Posetss are less ordered than the integers, for example, which are fully-ordered. An example is containment and inclusion of open sets (or intervals on the line). (The Web has a lot of good definitions, complete with diagrams and drawings, of these ideas. For example, MathWorld has this article on posets: http://mathworld.wolfram.com/PartiallyOrderedSet.html.) * To relate this to the Everything list, sort of, imagine the lattice of events in "our" universe. It forms a poset, basically. What about possible "branch points" where other universes form (as in MWI)? What about the overall notion of "possible worlds"? (Branching, fictional, AI planning, plurality of worlds a la David Lewis, etc.). * Fotini Markopoulou has been looking at causal sets and the nature of time. Her articles are available at the xxx.lanl.gov arXive site. Here's the article: From: Tim May <[EMAIL PROTECTED]> Date: Sat Aug 03, 2002 10:57:18 PM US/Pacific To: x Subject: Time, causality, posets, Heyting Second, while watching a fairly silly movie called "Signs" today, I was thinking about the issue of "when is a negation of a negation of something not the same as that something. That is, "not not A !=! A" or "not not A NEQ A" or A' ' NEQ A. (Lots of symbologies exist, and our keyboards and screens can't easily handle the most common ones.) An example Mac Lane gives in "Form and Function" is this: Consider the real number line. Consider the topology of open sets (or intervals). Suppose that we define an open set (or interval) U which is the open set of all of the positive reals _EXCEPT_ the number 1. Then "Not U" or "Complement of U" would be the set of all negative reals. (1 would not be in this complement because any actual number is of course a _closed_ set (endpoints and all that stuff from the definition of open and closed intervals. (Drawing a picture on a blackboard would help!) So far, no surprises. However, the negation or complement of his open subset (the negative reals) is the open subset of all of the positive reals. So Not Not U is bigger than U. This phenomenon of Not Not something being larger than something is common in Heyting algebras. Think about time. Think of a "lattice" of events combining in various causal ways to product events, which then combine with other events, a
Plurality of Worlds
I'm reading David Lewis' book, "Plurality of Worlds," 1986.
Lewis argues that not only are all possible worlds possible, but that
they actually exist. He does not claim they exist in any form we can
visit, or communicate with, but that the most economical philosophical
position to take is that they do in fact actually exist.
Lewis is no dummy. (Actually, Lewis _was_ no dummy: he died last year.
My very usage above and here of the present tense, as in "Lewis argues
that," is a kind of illustration of the weaker form of the plurality of
worlds thesis. Namely, it is natural to take a subjunctive ("had it been
the case") or possible world and treat it as a real world. For example,
arguing in the present tense by treating even a dead author as if he
were amongst us: "Aristotle tells us..."
This is recommended reading for Everything subscribers. Lewis has _some_
familiarity with the MWI, and cites Niven's 1968 story, "All the Myriad
Ways." But Lewis is not depending on a physics interpretation for his
thesis, although the physical (MWI or Tegmark/Schmidhuber/etc.) theories
of mulitiple realities would fit in as a subset of Lewis' plurality of
worlds.
The strong form of the thesis, that all possible (no violations of
logic) worlds have actual existence is dubbed "modal realism." A weaker,
more common-sensical form is called "ersatz modal realism." This is form
in which we can temporarily instantiate possible worlds, as, for
example, "In a world where Microsoft had never existed, the software
industry probably would have"
I'm deliberately not choosing whether to believe the strong form, though
it seems natural that everyone would believe the weak form. What's more
useful at this point is to learn the methods of reasoning these analytic
philosophers use, at least in this one world of counterfactuals and
possible worlds. (As you may recall, I am very interested in the links
between possible worlds and toposes, notably that the natural logic of a
possible worlds model is Heyting logic. A colleague/fellow writer of
Lewis's is Saul Kripke, who did interesting work in the 1960s showing
the connections between Intuitionism and possible worlds semantics.)
The weaker form of modal reality is used by nearly every person to
describe the possible worlds of the future: "Tomorrow it may rain." "I
can see a world in which peace exists." And so on for millions of
examples. We also use possible worlds semantics when discussing
alternative theories of how things happened: "Maybe the way it worked
was like this..." and "Had the Second World War not happened, the atom
bomb wouldn't have been developed."
Alas, David Lewis uses some math in his books, especially in an earlier
one I also have called "Counterfactuals," but the math is somewhat
lacking in its generality. (Reading "Counterfactuals," 1973, all I could
think was "This man is reinventing parts of point set topology! Give him
a copy of Kelley's "General Topology" and let him use the accepted
jargon for his inventions.")
It seems to to me that there are several communities (worlds) of writers
and researchers in this general area of "multiple realities" and they
have only tangential and fleeting communcation with each other:
* the "traditional" world of MWI, Everettistas, consistent histories, De
Witt, Hartle, etc., with newcomers like Tegmark and Schmidhuber (adding
models of computation)
* the philosophical world of Kripke, Lewis, Montague, and others, with a
focus on possible worlds semantics, reasoning in different worlds,
ontology, etc. This world somewhat intersects the above world through
the medium of fiction. Novels are examples of possible worlds, and "What
if" novels are a staple of science fiction. Larry Niven, Phil Dick, Rudy
Rucker, Greg Egan, and many others have even written thoughtfully about
what MWI means. Their novels discuss possible worlds and are in fact
themselves possible worlds.
* the world of topos theory and study of synthetic realities derived
from propositions, with the work of Lawvere, Johnstone, Kock, and
others. This world intersects the possible worlds semantics world
through the work of Kripke, as I said. It also is beginning to intersect
the MWI world through the work of Chris Isham -- cf. that streaming
video presentation I mentioned, at the URL:
http://www.newton.cam.ac.uk/webseminars/hartle60/1-isham/
Now it is my current interest to unify these three worlds, at least in
my own mind.
I do recommend taking a look at David Lewis's work. It's a bit off the
beaten track for most MWI thinkers, but it clearly deals with the same
general ideas. And it offers new language and new tools.
--Tim May
(.sig for Everything list background)
Corralitos, CA. Born in 1951. Retired from Intel in 1986.
Current main interest: category and topos theory, math, quantum reality,
cosmology.
Background: physics, Intel, crypto, Cypherpunks
Re: Causality
nd, which is what they are. Most "interesting things" involve temperature differences, energy inputs, life surrounded by nonlife, and such. I don't claim to know the nature of time, or what produces the "arrow of time." It's not a resolved question. Several of the "physics of information" conference papers, especially ones by W. Zurek and Charles Bennett, are useful. (But, before I tackle the issue at a deeper level, I'm trying to learn the language of what I think temporal evolution is: time's arrow being one of the arrows of category theory, hence my interest, or one additional reason for it.) > We can look at a > microscopic part of such a system and see fluctuations which appear to > be describable in causal terms. For example, a temporary void forms > randomly, causing matter at the edges to move towards the center. > Or in reverse we say that matter moved away from the center, causing a > void to form. Both are equally valid ways of describing the situation, > indicating that there is no true causality. Physicists would say that a lot of these explanations of random phenomena in terms of "voids causing matter to move..." are just abuses of language. Comparable to financial market babblers saying things like "And then at 2 pm the market sensed an oversold situation and shifted its buying to tech stocks, causing a late rally in the NASDAQ." Nonsense, mostly. (Rest of Hal's post, no time to think about and respond to tonight.) --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Causality
On Friday, July 12, 2002, at 09:47 AM, Bruno Marchal wrote: > OK I will try to read Joyce's book asap. In general I am quite skeptic > about the use of the notion of "causality". I have also no understanding > of your posts in which you argue about a relationship between the search > of a TOE and decision theory. More so than in a lot of areas, we here on this list are sometimes like orthogonal vectors in some Hilbert space...not having read the same books, not having a good understanding of the language of others, etc. I'm not exactly sure what you mean specifically by "causality," but in my worldview it has central importance: -- causes precede effects -- the structure of spacetime is more causal than it is geometrical (Smolin's point that most of what we mean when we talk about the geometry of spacetime is about the causal structure of spacetime, especially the orientations of light cones) I haven't read Joyce's book either, but I have read (some of) Pearl's book, "Causality," On the issues of mind-body, first person vs. third person, etc., I have no particular views. I've never thought there was any fundamental dichotomy between mind and body: our brains and sense organs are part of the package. But I have no particular philosophical or cognitive special competence in this area, so I won't participate in the debate. Maybe later I will turn my attention to it. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Pointers to places in vast spaces
On Wednesday, July 10, 2002, at 07:24 AM, Stephen Paul King wrote: > > I can't seem to get the idea out of my head that information can not > just refer to information itself but merely can encode the "address" of > where and when it can be found - this is how I think Goedelization > works. This is quite correct in many important respects. Here's an example. Consider the space of all strings (DNA, RNA) which make up living things, from bacteria to reptiles to humans, and perhaps to other organisms, past, present, and future. Even assume that this space contains strings for beings which are "possible" (would be living if they were instantiated, made, grown) but which have never existed in the past and don't exist at present. I usually draw this on a blackboard or sheet of paper as a two-dimensional plot, with the x- and y-axes not explicitly labeled. If it helps, consider the x-axis to be something like body size in cc, the y-axis to be something like total number of neurons, and so on. Clearly these axes are just extreme simplifications. But what one can reasonably see is that in this space there are places where the single-celled organisms live, "islands" for the reptiles and birds, islands for the mammals, and some region where homo sapiens is found. Now humans have something like 4 billion base pairs in the genome. I don't recall what the conversion is from ATCG sorts of base pairs to bytes, but it's within a small factor, so something like 4 GB or 32 Gbits represents the human genome. Fits on a handful of CD-ROMs, uncompressed. But this is not the full story. This 32 Gbit sequence is effectively a _pointer_ into a space of 2^ (32 Gbits) points, the space of all strings of the same length as the human genome. (And the space of all living things is even larger, as it includes all strings of our length, plus all orderings of shorter strings. If we include beings with even longer strings, it gets much bigger, of course.) Where living things in this space can be found depends on many other things, including the environment around the living thing (e.g., a lizard in a desert which only eats wheat or rice cannot live, and did not ever evolve). In Bennett-type terms, the strings of living things have great logical depth. They evolved, changed, got more "complex" (in the logical depth sense) as the phenotypes competed, lived, reproduced, etc. In a sense, the genome is a _pointer_ to a particular address in that vast space of all possible living things. (Just as the library call number of "War and Peace" is much shorter than the actual text of "War and Peace.") (Completely aside: We even have some ideas about the topology and geometry of "life space": we know something about what "nearness" means, through single-point mutations and their effects of organism viability, and we are learning what rearrangements and insertions of string sequences may mean.) --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: Some books on category and topos theory
On Tuesday, July 9, 2002, at 11:08 AM, Bruno Marchal wrote: > > Me too. Now, I feel almost like you about ... knot theory. > And this fit well with your cat-enthusiasm, for knot theory is > a reservoir of beautiful and TOE-relevant categories > (the monoidal one). I've just > ordered Yetter's book: functorial(*) knot theory. It is the number 24 > of Kauffman series on Knots and Everything (sic) at World > Scient. Publ Co. A series which could be a royal series for this > list ... > May I recommand the n° 1, by Louis Kauffman himself: knots and physics? > A must for the (quantum) toes, and (I speculate now) the comp toe too! I've looked at some of the knot series books, but have put them off for now. A good book to prepare for these books is Colin Adams, "The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots," 1994. Whether knots are the key to physics, I can't say. Certainly there are suggestive notions that particles might be some kind of knots in spacetime (of some dimensionality)...a lot of people have played with knots, loops, kinks, and braids for the past century. One thing that Tegmark got right, I think, is the notion that a lot of branches of mathematics and a lot of mathematical structures probably go into making up the nature of reality. This is at first glance dramatically apposite the ideas of Fredkin, Toffoli, Wheeler1970, and Wolfram on the generation of reality from simple, local rules. Wolfram has made a claim in interviews, and perhaps somewhere in his new book, that he thinks the Universe may be generated by a 6-line Mathematica program! However, while I am deeply skeptical that a 6-line Mathematica program underlies all of reality, enormous complexity, including conceptual complexity, can emerge from very simple rules. A very simple example of this is the game of Go. From extremely simple rules played with two types of stones on a 19 x 19 grid we get "emergent concepts" which exist in a very real sense. For example, a cluster of stones may have "strength" or "influence." Groups of stones develop properties which individual stones don't have. Abstraction hierarchies abound. The Japanese have hundreds of names for these emergent, higher-order structures and concepts. All out of what is essentially a cellular automata. So even if our universe is a program running as a screen saver on some weird alien's PC, all sorts of complexity can emerge. Getting down to earth, most of this complexity is best seen as mathematics, I think. I expect to take a closer look at knots after I get more math under my belt. > Now I know the z logics really should have "tensorial semantics", > sort of many related (glued) von neumann type of logics (which are > themselves atlases of boolean logics). > But where (in Zs logic) those damned tensorial categories come from??? > Knots gives hints!!! This would explain the geometrical appearance > of realities. > > Bruno > > (*) For the other: "functorial" really means categorial. Functors are > the morphisms between categories. The first chapter of Yetter's book > is an intro to category theory, the second one, on Knot theory, ... Exciting stuff. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Fwd: Which universe are we in?
Apologies. I accidentally sent this last night to another mailing list. Here it is. --Tim Begin forwarded message: > > On Monday, July 8, 2002, at 07:43 PM, Stephen Paul King wrote: > >> Dear Tim, >> >> Are you tacitly assuming some kind of communication between >> observers >> when you make the claim of a "convergence"? Adsent said communications, >> could we show that the convergence would still obtain? Have you ever >> seen >> any discussion of the notion of cyclic or periodic gossiping in Comp >> Sci? >> >> > > No, I was arguing that while the future may be multi-worlded, > everything we know about science (evidence, archaeology, > measurements, ...) points to a _single_ past. > > For example, a single past world line for me, for you, for Hal, for > Chaucer, for Einstein. > > Now we may not know what this world line is very accurately, but as we > look at more closely, e.g., by examining the photographs someone may > have taken, or their diaries, or whatever, the more we home in on what > that world line was. We never look closely and see two or three or N > different histories, we just see a higher fidelity view of what we must > assume is the One True Past. > > I don't doubt that Hal gets the sense that many potential Hals could > have resulted in the current Hal...an interesting notion. But > everything does in fact point to a One True Past which various > measurements get closer and closer to, and which no measurements > contradict. > > This is what I meant by "convergence." Homing in, getting closer, > sharpening the image, filling in the details. > > As for "tacitly assuming some kind of communication between observers," > I am _explicitly_ saying that observers get together and compare > notes...and they find no contradictions, if they are honest observers. > > Hal may have meant something different, perhaps. > > > --Tim May > > --Tim May > (.sig for Everything list background) > Corralitos, CA. Born in 1951. Retired from Intel in 1986. > Current main interest: category and topos theory, math, quantum > reality, cosmology. > Background: physics, Intel, crypto, Cypherpunks > >
Re: Some books on category and topos theory
s excited about a new area in more than a decade. I expect I'll be doing something in this area for at least the _next_ decade. My apologies if this explanation of enthusiasm is too personal for you the reader, but I think enthusiasm is a good thing. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: Which universe are we in?
On Monday, July 8, 2002, at 03:40 PM, Hal Finney wrote: > Future uncertainty is familiar to us, but one of the things that the > many universe model introduces is past uncertainty. There is a sense > in which the past is not unique and determined. My mental state is > consistent with many macroscopically distinct pasts. I'm not convinced that this is so. Sure, there are many views of past events, of history, faulty memories, changing memories, etc. However, the "single past" model is quite well-supported by science and a kind of "convergence" of knowledge: -- we may not have complete knowledge of the past, but experience points to the fact that the more different observers learn about the past, the more they will (if they are honest) agree on what that past was. -- archaeology is a good example: more and more bits and pieces add together to "converge" to a unitary past, not to multiple, diverse pasts -- this is analogous with measurements in QM: honest observers will report the same measurement > My brain and my mind hold only a certain amount of information. > Vastly more information than that has existed in my past light cone, the > history of the universe which has led up to me. My brain is therefore > very probably consistent with a great many past histories, each of which > will lead to a brain, a mind and a mental state which is > indistinguishable > from that which I am now experiencing. From my first-person > perspective, > the past is indeterminate in much the same way as the future is, > although > to a lesser degree. I agree that many possible causal pasts lead up to what you are. The placement of grains of sand on a beach in Greece is not going to significant affect who you are right now, so this is just one of a vast multitude of possible causal pasts which will not affect your currrent mental state. But this does not mean these possible pasts have equal "actuality." For example, two different observers may have carefully photographed the patch of beach where the possible variations occurred. The more accurate their observations or photographs are, the more closely they will agree on what that past was (again, assuming honest observers). Nothing in science points to the "many actual pasts" possibility, even though I acknowlege your point that "many _possible_ pasts" would lead to a indistinguishable equal mental state for you or me. In other words, science points to a single actual past. There is, so far, no evidence for multiple actual paths. (And in the consistent histories picture, we should not be surprised. We find ourselves in whichever universe we are in, and we will see one actual trajectory through space-time.) > --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Which universe are we in?
I want to give a solid example of a time-varying set and how it relates to possible worlds (and even to MWI). Consider that cat being chased by the dog. Is the cat in the world in which he will escape the dog or is in the world in which the dog catches him? One answer is "That lies in the future. We don't know yet." Another is: "All knowledge is Bayesian. Based on his running in the past, I'll lay odds of 5 to 2 that he'll escape the dog." A MWI-flavored version is: "There are many worlds in which he escapes, many where he doesn't. We'll only see one of the possible worlds." And then there's the strictly Boolean, determinist point of view: "The cat is in one of the two possible worlds you describe. A sufficiently powerful being or computer able to calculate all of the factors, including the wind speeds, the slipperiness of the stairs, and so on, knows which world we are in." I think the last point is actually the most naive of all of the views, as it simply "punts" the question and asserts that some omiscient, omnicomputing entity knows the future...or, regardless of any such being, that the future is determined. (This gets into free will issues, obviously.) In the other approaches, the Boolean "cat is in one world or in other world" is replaced by a time-varying set: -- "We don't yet know which world the cat is in, or which world we are in along with the cat, but in a few minutes we'll know for sure." (And everyone will agree on this...there will be no disagreement amongst honest observers as to whether got away or got caught by the dog.) I give this example to show that we don't need quantum weirdness to show how useful/important time-varying sets are, and how the logic of reality can be "non-Boolean becoming Boolean," how the time morphism (passage of time) results in assignment of an event to one of N possible worlds. In other words, the naturalness of Heyting logic instead of Boolean logic. In other words, topos logic. (Aside: I don't claim that more and more powerful computes and analysis tools don't help us to either determine which universe we are in--by making predictions of stock movements, or weather, or wars, or the escape of that cat--or even by helping us to make our own changes which change the future. My ideas are not firm on this, but I think computational and cognitive power relates to how far forward in time this "knowability" extends. In the case of a billiard table, knowability may extend a few seconds into the future, for accuracy within some range. In the case of planetary motions, many millennia, for some accuracy. In the case of the cat? My point about the "omniscient" model being a "punt" is that it simply defines omniscience as being enough to have complete knowledge. There is no evidence that such omniscience is possible, not even with all the computer power in the universe.) --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Fwd: being inside a universe
I got a bounce (" - The following addresses had permanent fatal
errors -
"|flist everything-list"
(expanded from: <[EMAIL PROTECTED]>)
so I'm trying to send this a second time:
Begin forwarded message:
> From: Tim May <[EMAIL PROTECTED]>
> Date: Mon Jul 08, 2002 12:17:27 PM US/Pacific
> To: [EMAIL PROTECTED]
> Subject: Re: being inside a universe
>
>
> On Monday, July 8, 2002, at 10:39 AM, Bruno Marchal wrote:
>> Mmh... Most people here have a good understanding of the "many"-idea,
>> by which
>> I mean have realised that the idea of a unique universe is far more
>> speculative
>> that O universe or many universe. I will not insist.
>> Perhaps you could read the very interesting paper by Louis Crane,
>> which is quite
>> convincing on the importance of category theory in frames of quantum
>> gravity+
>> observers, and which use cleverly Everett relative states. He even
>> concludes that
>> his proposal can be seen as an attempt to fuse the many worlds with
>> general
>> relativity. It is a paper by Crane entitled "categorical physics". He
>> gives
>> the TEX source somewhere on the net. (If you don't find it I can
>> eventually
>> send you a photocopy).
>
> Thanks, but I can no doubt find it at UC Santa Cruz. I have most of his
> later papers already printed out, from the xxx.lanl.gov archive site,
> and his papers will refer to his earlier papers that may not be on the
> arXive site.
>
> Crane is one of the Sultans of Spin, as Egan dubs them, along with
> Rovelli, Sen, Ashtekar, Smolin, Baez, Markopoulou, Susskind, etc.
>
> I have to admit that I'm more prosaic in my approach...the quantum
> gravity and MWI stuff is interesting to think about, to speculate
> about, but my focus is a bit closer. And I'm still learning this new
> language (category, topos theory).
>
>
>> One day I will talk you about the work of Yetter on
>> models for non commutative linear logic. Yetter is at a relevent
>> crossroad of
>> logic and physics imo. (Then I guess you heard about Crane, Kauffman
>> Yetter
>> papers?)
>
> I have the Crane-Yetter paper " On the Classical Limit of the Balanced
> State Sum," but I haven't read it yet. From glancing at it, it didn't
> seem to be cosmic. I'll look at it more closely.
>
> Is it related to the noncommutative geometry work of A. Connes?
>
>
>> About linguistic use of Kripke, most of them, imo, use technical
>> approach
>> of language to explain problems ... away. Beware sophisticated tricks
>> for
>> putting interesting (but hard) problems under the rug.
>
> My main interest in Kripke is his discussion of "possible worlds,"
> which is a kind of superset of the MWI/Tegmark view. (Supersets can be
> nebulous, and I am not claiming the MWI/Tegmark stuff is some trivial
> subset of a larger theory.)
>
> We constantly make plans and think about futures in terms of these
> possible worlds. David Lewis gives an example in one of his many papers
> on possible worlds (plurality of worlds). A cat being chased by a dog,
> for example. The cat imagines one possible world in which he has gotten
> safely away, another possible world in which the dog's jaws have gotten
> him. It seems likely that reasoning about possible worlds is much more
> innate than, say, reasoning using formal syllogistic logic!
>
> I believe, in fact, that the conventional semantic networks of AI,
> exemplified by some large knowledge engineering efforts like Doug
> Lenat's CYC project, may need to be scrapped in favor of networks
> embodying the morphisms, functors, and functors of functors of category
> theory. This is not _directly_ linked to MWI and Tegmark, of course,
> but it has some partial links.
>
>>
>>> Personally, I'm not (yet) "taking seriously" either the David Lewis
>>> "plurality of worlds" or Max Tegmark "everything" or Greg Egan "all
>>> topologies model" ideas. At least not yet. I need to learn a lot more
>>> of the language first.
>>
>> I am more problem driven, and even "mind-body" problem driven. I gave
>> an
>> argument in this list (argument on which my phd thesis is based) that
>> IF we are machine then physics is utimately reducible to machine's
>> psychology. The laws
>> of physics emerge from some collection of sharable "dreams" by
>> machines, where
>> a dream is basically a computation seen from first person point of
>> view.
>
The relevance of category and topos theory
ies.) > Does it help understand or formalize the notion of "all possible > universes"? I know in logic there is the concept of a categorical theory > meaning all models of the theory are isomorphic. Does that have anything > to do with category theory? Yes, there are deep and important connections. Models form a category. The book I mentioned by Paul Taylor, "Practical Foundations of Mathematics," is very good on these issues. In my view, category theory (and topos theory) represents the "modern" way of looking at a lot of seemingly unrelated areas. As we know, as Hal Finney and several of us used to discuss about ten years ago on the Extropians list, Chaitin's formulation of algorithmic information theory gives us a much more understandable and comprehensible proof of Godel's Theorem than Godel himself gave! (For the best explanation of this, and why this is so, either see Greg Chaitin's own papers and books or the wonderful summaries by Rudy Rucker in his "Mind Tools" book. These modern viewpoints are much more comprehensible than the classics. Which is not surprising. Shoulders of giants and all that. The same is true of category theory. It's a relentlessly modern approach to seeing the similarities that pervade mathematics and physics. Whether it answers the question about whether lots of other universes exist is doubtful...I'm not convinced we'll know the answer to that question in the year 3000. But it's a powerful and elegant approach, with perhaps a slightly misleading name, and it looks to me to be the right language for talking about the world around us and possibly the worlds we cannot directly see. I agree with Lee Smolin that topos logic is not just the logic of cosmology, but also the logic of our everyday world of limited information, bounded rationality, Bayesian decision making, and information horizons. Even if this is not useful for answering questions about "the Everything theory," because we may need to wait 600 or 6000 years for experimental tests to become feasible, I believe this outlook will be of great utility in many areas. I'll keep you all posted! --Tim May
Re: Some books on category and topos theory
On Friday, July 5, 2002, at 01:16 PM, Tim May wrote: > The category and topos theory books I actually _own_ (bought through > Amazon) are: > > Oops! I left out one of the most important and accessible of the books I have and recommend: * McLarty, Colin, "Elementary Categories, Elementary Toposes," 1992. An intermediate-level, moderate-length book. Covers a lot of interesting material. Here's what Baez says: "3) John Baez, Topos theory in a nutshell, http://math.ucr.edu/home/baez/topos.html and then try the books I recommended in "week68", along with this one: 4) Colin McLarty, Elementary Categories, Elementary Toposes, Oxford University Press, Oxford, 1992. which I only learned about later, when McLarty sent me a copy. I wish I'd known about it much sooner: it's very nice! It starts with a great tour of category theory, and then it covers a lot of topos theory, ending with a bit on various special topics like the "effective topos", which is a kind of mathematical universe where only effectively describable things exist - roughly speaking. " (end of Baez comments) By the way, the Web is a great resource for finding online books. Barr and Wells, who Bruno referred to, have put an updated version of their book "Toposes, Triples and Theories" online in PDF form. Search for it in the usual way. --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Some books on category and topos theory
On Friday, July 5, 2002, at 10:54 AM, Bruno Marchal wrote: > But, perhaps more importantly at this stage I must recall the book > "Mathematics of Modality" by Robert Goldblatt. It contains fundamental > papers on which my "quantum" derivation relies. I mentionned it a lot > some time ago. > And now that I speak about Goldblatt, because of Tim May who dares > to refer to algebra, category and topos! I want mention that Goldblatt > did wrote an excellent introduction to Toposes: "Topoi". (One of the big > problem in topos theory is which plural chose for the word "topos". > There > are two schools: topoi (like Goldblatt), and toposes (like Bar and > Wells). :) > > Goldblatt book on topoi has been heavily attacked by "pure categorically > minded algebraist like Johnstone for exemple, because there is a remnant > smell of set theory in topoi. That is true, but that really help for an > introduction. So, if you want to be introduced to the topos theory, > Goldblatt Topoi, North Holland editor 19?(I will look at home) is > perhaps the one. Yes, this is an excellent book. It has more of an expositional style than many books on category and topos theory. It's out of print and Amazon has been looking for months for a used copy for me. (Amazon can search for books which become available. I also have them searching for a copy of Mac Lane and Moerdijk's book on sheaves, logic, and toposes, also out of print.) Fortunately, I live near UC Santa Cruz, which has an excellent science library. The category and topos theory books I actually _own_ (bought through Amazon) are: * Cameron, Peter, "Sets, Logic and Categories, 1998. An undergraduate level primer on these topics. One chapter on categories. (By the way, most modern algebra books, e.g., Lang's "Algebra," Fraleigh, Dummitt and Foote, etc. have introductory chapters on category theory, as this is the "language" of modern abstract algebra.) * Lawvere, F. William, Schanuel, Stephen H., "Conceptual Mathematics: A first introduction to categories," 1997. This is a fantastic introduction to categorical thinking. The authors are pioneers in topos theory, but the presentation is suitable for any bright person. There is not much on applications, and certainly no mention of quantum mechanics a la Isham, Markopoulou, etc. But the conceptual ideas are profound. (And this should be read before tackling the "formalistic" presentations in other books.) * Pierce, Benjamin, "Basic Category Theory for Computer Scientists," 1991. A thin (80 pages) book which outlines the basics. Includes material on compilers, the "Effective" topos of Hyland and others, cartesian closed categories, etc. * Mac Lane, Saunders, "Categories for the Working Mathematician, Second Edition," 1971, 1998. Wow. A dense book by the co-founder of category theory. As someone said, reading along at 10% comprehension is better than reading other books at full comprehension. I find the book sort of dry, on historical and conceptual motivations, but Mac Lane has written many longer expositions in MAA collections of reminiscences...I just wish mathematicians would do more of what John Baez in his papers: show the reader the motivations. (Much of mathematical writing came out of the tradition of "lecture notes." In fact, the leading publisher of mathematics, Springer-Verlag, calls their series "Lecture Notes," or, more recently, "Graduate Texts." Brilliant mathematicians like Emil Artin and Emmy Noether would have their lectures on algebra transcribed by grad students or post-docs, like Van der Waerden, who would then republish the notes as "Moderne Algebra," the first of the "groups-rings-fields" modern algebra books. Which is why one of E. Artin's students, Lang, writes so many dry books! These books are often very short on pictures or diagrams, very short on segues and motivations. It's as if all of what a good teacher would do in class, with drawings on blackboards, with historical asides, with mentions of how material ties in with material already covered, with mention of open research problems and unexplored territory...it's as if all this material is just left out of these texts. Too bad.) * Lambek, J., Scott, P.J., "Introduction to higher order categorical logic," 1986. Way too advanced for me at this point. So no comments on content. But it's useful to glance at topics so as to get some idea of where things are going (part of the issue of motivation I raised above). * Taylor, Paul, "Practical Foundations of Mathematics," 1999. Another advanced book, covering logic, recursive function theory, cartesian closed categories, and a lot of the second half I can't comment on. A wo
Re: being inside a universe
human (or machine) observers. Isham makes an excellent point about time-varying sets, echoed by Smolin. In a nutshell, while the logic of a quantum universe (or cosmological universe, perhaps) may follow a Heyting logic where "the cat is neither alive nor dead," once _any_ observation or measurement, whether a machine or a written note or a memory or whatever, then the logic is Boolean, as we "are used to." Now obviously we're all familiar with this as the basic "measurement collapses the wave function" model, so there is at first glance nothing new here (you skeptics out there are right to be skeptical). However, the topos-theoretical point of view, in which topos logic (Heyting) is used instead of Boolean logic, seems to me to make the "interpretation" problem (Copenhagen vs. MWI vs. Cramer vs. ...) largely go away. The "naive realism" view is that whether we can see the cat or not, it "must" be "really" either alive or dead. The Heyting/Isham/Smolin/etc. point of view is that speculating about whether the cat is alive or dead is as meaningless as speculating about what the "actual number of cats living at this moment in Andromeda" is, given that that place is outside our light cone (our causal past) and that the earliest we could even conceivably answer that question is two million years from now. Smolin covers this territory convincingly, for me, in his "Three Roads to Quantum Gravity." In fact, the elimination of the absolute view is refreshing. Take, for example, the very model of past and future light cones. We are familiar with the conventional world line of, say, me or you. Our world line moves from out past to our future in this Minkowski (or some variant) space-time. This is the point of view of the "outside, omniscient, sees all events and objects in all parts of space-time" point of view. The God viewpoint. This very point of view encourages (some) people to think in deterministic terms. "The future" and all that (emphasis on "the"). One thing reading a lot of science fiction has done for me is to disabuse me of any notion of "the" future. Instead, sheafs of possible futures. Locally determistic, and past-deterministic (pace the point about Heyting-->Boolean), but various possible worlds of various futures are unknowable to observers in the real universe. (Smolin makes the case that the universe is everything there is, that it is pointless to speak of external observers who can see the entire structure of space-time. The links between this viewpoint and other areas are fascinating.) We are finite beings in an effectively finite, though very large and of effectively unlimited potential complexity, universe. The logic of time-varying sets (essentially topos logic) is the natural way to describe such systems. Locally, and in most everyday situations, Boolean logic works very well in physical situations (all honest observers will agree on any observation)...just as Euclidean geometry works very well in most situations, just as other theories work very well in most situations. I'm amazed at how well humans can understand reality. As I said, lots of people are way ahead of me in understanding the math. Seeing how once obscure parts of mathematics turn out to be very useful for Theories of Everything, I'm more convinced than ever that essentially all branches of mathematics are somehow "built in" to the structure of reality. (And this is one reason I'm skeptical of models that reality is just a cellular automaton running local rulesets on some computer. I have a hard time conceiving of how so much interesting mathematics would exist with simple local CA rules. But I could be wrong. :-) ) --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Logical depth
On Thursday, July 4, 2002, at 09:26 AM, Hal Finney wrote:
> Brent Meeker writes:
>> What's Wolfram's critereon for "randomness"? how it looks? It would
>> seem
>> that he's using something different than Chaitin if we thinks a simple
>> system can produce random output. Chaitin *defines* random as an
>> output
>> that can't be reproduced by a system simpler than the output itself.
>
> That's a good point. It's curious that Chaitin used the primes as an
> example of mathematical randomness when they must have inherently low
> algorithmic complexity. A handful of relatively short axioms define the
> integers and the operations of addition, multiplication, etc., and this
> is sufficient to produce the "random" spacing of the primes.
However, the "dual" of "descriptional complexity" is "logical depth":
the running time of a program to generate some string or set from some
initial conditions or axioms. An apparently complex system, one which
apparently has no description shorter than itself, may arise from a
short description run through a machine or process a number of times.
Logical depth is the term coined by Charles Bennett, of course.
(Arguably, the universe is such a high logical depth system.)
I say "apparently complex" rather than "complex" to admit this very
possibility: that an object with seemingly no shorter description of
itself _than_ itself ("random" in Kolmogorov-Chaitin sense) actually has
a much shorter description.
Examples of this abound. An apparently random string may be a very
simple string run through an encryption process. An apparently random
string may have a very short description once someone spots the pattern.
A DNA string coding for some organism which has evolved over billions of
generations has "packed" a lot into a fixed length string. Each
twiddling of the string through mutation and crossover, followed by each
test of reproductive fitness, is effectively a computation. The "spring"
of this string is "compressed" with more and more energy. The DNA string
for an organism has a lot of "energy" or "cleverness" (viewed in
different ways). The string length remains the same, more or less, and
naive calculations of entropy (which are always simplistic) give
essentially the same measure. But the logical depth is different, and
that matters a lot.
We can say "the set of primes is not random, but has high logical
depth." It will take a certain amount of energy to generate/compute the
primes up to some number.
I think Chaitin is making a point about the set of primes being a
readily-visualizable example of something which has a seemingly random
structure but with high logical depth. Which is what a lot of cellular
automata, e.g., LIFE, have been showing us.
Simple initial conditions + Long running times ---> Complex structures
with no simpler descriptions than themselves
(unless we know the initial conditions and algorithms, in which case we
can do the computation ourselves)
Aside: Trying to compute the initial conditions from a final
configuration is difficult. This was dealt with in some of Wolfram's
earlier papers, collected together in his World Scientific book
"Cellular Automata."
--Tim May
"Stupidity is not a sin, the victim can't help being stupid. But
stupidity is the only universal crime; the sentence is death, there is
no appeal, and execution is carried out automatically and without pity."
--Robert A. Heinlein
Egan's "All Topologies Model"--an excerpt
of perfectly balanced chaos which space-time would become if so much energy was poured into it that literally everything became equally possible. Everything and its opposite; the net result was that nothing happened at all. But some local fluctuation had disturbed the balance in such a way as to give rise to the Big Bang. From that tiny accident, our universe had burst into existence. Once that had happened, the original "infinitely hot," infinitely even-handed mixture of topologies had been forced to become ever more biased, because "temperature" and "energy" now had a meaning and in an expanding, cooling universe, most of the "hot" old symmetries would have been as unstable as molten metal thrown into a lake. And when they'd cooled, the shapes into which they'd frozen had just happened to favor topologies close to a certain ten-dimensional total space one which gave rise to particles like quarks and electrons, and forces like gravity and electromagnetism. By this logic, the only correct way to sum over all the topologies was to incorporate the fact that our universe had by chance emerged from pre-space in a certain way. Details of the broken symmetry had to be fed into the equations "by hand" because there was no reason why they couldn't have been utterly different. And if the physics resulting from this accident seemed improbably conducive to the formation of stars, planets, and life . . . then this universe was just one of a vast number which had frozen out of pre-space, each with a different set of particles and forces. If every possible set had been tried, it was hardly surprising that at least one of them had turned out to be favorable to life. 85 It was the old anthropic principle, the fudge which had saved a thousand cosmologies. And I had no real argument with it even if all the other universes were destined to be forever hypothetical. --end excerpt-- --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: being inside a universe
quick search confirmed the details. Pearl's book is one that would profit immensely from a "collision" with other areas, especially the causal set work of Isham, Markopoulou, etc. (As in a lot of areas, the "invisible colleges" (communities of researchers) are often operating in micro-universeses of discourse, referencing from the same set of papers, using the same basic ideas. For example, the only real point of commonality between the universes of [Pearl-Dempster-Shafer-cognitive-AI] and [Smolin-Markopoulou-Isham-physics] seems to be the term "causality" and varius references to Saul Kripke (who is better known in linguistic circles!). > >> [SNIP] What's fascinating is that a topos is a kind of "micro >> universe.' Not in a physical sense, a la Egan or Tegmark, but in the >> sense of generating a consistent reality. More on this later [SNIP]. > > > In the sense of (brouwerian-like) mental construction? > I mean in the sense that the history of modern science seems to me to be a succession of "throwing out the "centered" object," throwing out a world centered around the Sun, or centered around God, or centered around even Newtonian physics. A good example is throwing out Euclidean geometry. Or, rather, truly understanding that Euclidean geometry is just one of _many_ geometries. Euclidiean geometry, Riemannian geometry, Newtonian physics, Lorentzian transforms in a Minkowski space (aka Einsteinian physics), etc. are just special kinds of universes. Are they "real"? (This gets into Tegmark territory, about "actual" (whatever that means!) physical universes with different mathematics and then, obviously, different physical laws. Egan, in "Distress," calls this the "all topology model.") Personally, I'm a partial Platonist (these things in mathematics have some kind of existence) and a partial formalist (we are pushing symbols around on paper and in our minds). I increasingly view the constructivist (Intuitionist/Brouwerian) point of view as being consistent with what we see in the world and what we can model or simulate on computers. There are aspects to the world which are Newtonian, which are Boolean, which use the discrete topology, which use only continuous functions, which are beyond Boolean, which are compact (in the compact space sense), which are non-compact, which are Banach, Hilbert, Fock, etc. spaces, and so on. While I think the Universe is remarkably understandable, I don't think it makes much sense to talk about what "the" topology or laws of mathematics of the Universe is/are. (I apologize if this is not clear...this is just a glimpse.) There is much, much more to say on all of these topics. --Tim --Tim May (.sig for Everything list background) Corralitos, CA. Born in 1951. Retired from Intel in 1986. Current main interest: category and topos theory, math, quantum reality, cosmology. Background: physics, Intel, crypto, Cypherpunks
Re: being inside a universe
or near/inside event horizons, perhaps). Perhaps more strangely, the conventional Boolean algebra and logic get superceded by time-varying sets where the law of the excluded middle (A or not-A, not-not-A is A) is replaced by a richer logical system: Heyting algebra and logic. I'll get into this stuff more in future posts. In particular, Isham has a topos perspective on "consistent histories" (MWI) which is quite interesting. A streaming video lecture on "Quantum theory and reality" is available at http://www.newton.cam.ac.uk/webseminars/hartle60/1-isham/ This is not easy going, but watching it a couple of times may get across some of the ideas. And he and his main collaborator, Butterfield, have written several papers. My last comment will be that I am not really a Tegmarkian. Frankly, I thought Greg Egan treated the same ideas better than Tegmark did. In "Distress" we find the "all topologies model," yet another overloading of the acronym ATM. (AOL, acronym overload.) "Distress" was published in hardback in June 1997. Tegmark's TOE preprint appears in April 1997. So roughly simultaneous publication. Anyway, Tegmark is a professional physicist, and has done much good work on conventional cosmology, so I'm not dissing him. More on this later. --Tim May -- Timothy C. May [EMAIL PROTECTED]Corralitos, California Political: Co-founder Cypherpunks/crypto anarchy/Cyphernomicon Technical: physics/soft errors/Smalltalk/Squeak/ML/agents/games/Go Personal: b.1951/UCSB/Intel '74-'86/retired/investor/motorcycles/guns Recent interests: category theory, toposes, algebraic topology
