Benjamin Rosenbaum |
Posted by Benjamin Rosenbaum at August 16, 2007 06:58 PMOh, I think level IV is clearly pretty far out from science, and in fact that's precisely what I'm nagging at here. If the only measure of falsifiability is "does it seem likely?", you're in trouble.
However, I'm not with you on computer programs. They spring from a few rules -- if you're lucky -- because we made them that way. It takes herculean standardization efforts to keep the number of rules in a living language small -- and that's with something that has to submit to the discipline of industrial use, not "anything you can imagine".
While a mathematical structure that says "F=MA, except in Montana on Tuesdays for purple spiky things which are near frogs" is still a mathematical structure. So is a mathematical structure composed of BB(1111) such uncorrelated rules.
Posted by Jackie M. at August 16, 2007 07:05 PMThe programming rules become complex because the compiler introduces another level or two between that and machine language. But in the end, it all boils down to a collection of very simple logical gates.
Posted by Benjamin Rosenbaum at August 17, 2007 05:56 AMMy impression is that's not even true in modern chip design -- look at the triumph of the godawful-complicated Intel chipset over the RISC chips.
Sure, in principle everything can be done with NAND gates, and if mathematical laws for universes must be computable (which sounds reasonable) then everything can be expressed as a Turing machine and a program. So if you ignore the program and just point to the Turing machine you can always say "ooo.. pretty! elegant and concise!"
But what I'm talking about is something like the Kolmogorov complexity of the program. A haiku and "War and Peace" are not the same complexity just because both can be expressed with 26 letters.
Posted by Jackie M. at August 17, 2007 07:14 AMSure. But Kolmogorov complexity still arises from a few fairly simple laws.
(furthermore, if you get enough of it, it cheerfully reduces back down to a few fairly simple statistical expectations.)
Posted by Jackie M. at August 17, 2007 07:42 AM(I'm not saying that studying physics isn't a leap of faith, mind--just that Occam's razor argues strongly against my getting too swept up in the mind-boggling existential implications of your extrapolation of Tegmark's fourth category of parallel multiversal cosmogenies.)
Posted by David Moles at August 17, 2007 08:46 AMdon't you think godawful sprawling Rube-Goldbergian messes full of tweaks, hacks, absurdities, special cases and dead ends would far outnumber elegant, simple universes following from a few simple laws?
That hypothesis seems to rest on one of two assumptions: either Rube Goldberg universes are common enough that this outweighs the improbability of sentience arising in them, or sentience is sufficiently more likely to arise in a Rube Goldberg universe that this outweighs the rarity of such universes.*
Either of those might be true, but neither seems particularly compelling to me, so...
...no?
*Or they're both true, and RG universes are both more likely and more conducive to sentience.
Posted by David Moles at August 17, 2007 09:13 AMFrom the Tegmark paper: although many if not most mathematical structures are likely to be dead and devoid of SASs, failing to provide the complexity, stability and predictability that SASs require, we of course expect to find with 100% probability that we inhabit a mathematical structure capable of supporting life.
(Emphasis added. And yes, if you know all the rules, any mathematical structure is predictable, but that is clearly not what he means.)
Posted by Matt Hulan at August 17, 2007 08:32 PMIt's funny:
Philosophers, physicists, and fiction writers are very similar in that they are all professional experts in the field of making shit up.
Good philosophers make arguments that are compelling, following the rhetoric of classical logic.
Good physicists make arguments that are compelling, following the rules of various mathematical systems.
Good fiction writers make stories that are compelling, based on observations of human beings and their interactions.
I suspect, and this is just a hunch (me being no good philosopher, physicist, or fiction writer), that fiction writers are the most useful to most people on a day-to-day basis, and their tools are the most subjective.
peace
Matt
Posted by Benjamin Rosenbaum at August 17, 2007 11:20 PMA program does not "arise" from a programming language the way elliptical orbits arise from Newton's three laws.
The fact that Windows Vista can be *expressed* in NAND gates -- or letters -- doesn't make it less complicated and arbitrary.
My reason for thinking that there are way more complicated and arbitrary universes which are complex, stable, and predicatble enough to support SASs than simple and elegant ones goes like this:
For each simple and elegant universe, add one single random arbitrary rule that is not compressible with the other rules. So, let's say, the universe Newton thought we lived in could support SASs, right? Do we agree on that?
And so could that universe, except that there's one arbitrary place that gravity doesn't work except on Tuesdays. That could still support SASs, right? If they go that particular place on a Wednesday, they might be confused, but they are still sentient. Right?
Note, I'm positing here that that really is the simplest explanation of this universe: Newton's laws plus one funny place. The funny place isn't a pointer secretly revealing an even more fundamental and elegant set of laws. The laws are right, except that one place.
With me?
OK, now add another such place. Or another silly rule.
It seems clear we can add a whole lot of such rules before we begin to degrade the ability of an SAS to function. Sure, if 50% of events are wholly random, consciousness may break down. But universes that approximate a few elegant laws -- but in which those laws are actually mere rules of thumb with a host of exceptions -- are going to be perfectly amenable to SASs.
Since for *each* elegantly expressible universe there seems to me to be a *huge* number of potential exceptional-kludges you can add without disallowing SASs, each combination of which constitutes a new mathematical model, it seems to me that the "Aristotelian" universes -- in which our laws are approximations -- far outnumber the "Platonic" universes -- in which the elegant laws are the true nature of reality.
It's true that the whole overarching system -- the level 4 multiverse itself -- is simple and elegant. Just like a Turing machine is simple and elegant. But the vast majority of *programs* which you can run on a Turing machine are anything but.
So the irony I'm pointing to is that, following Tegmark's logic, if we live in a *multiverse* -- level 4 --- that is the epitome of Platonic elegance, then there's a huge probability that we live in a particular observable *universe* -- not multiverse -- the program, not the idea of programming -- which is irreduceable to a cogitable number of accurate natural laws.
Posted by Benjamin Rosenbaum at August 17, 2007 11:28 PMBy the way, this is neither an argument against studying physics, nor an argument against Occam's razor. It is an argument that Tegmark's paper does not strengthen the case for what he's calling a Platonic, vs. Aristotelian, approach to studying physics -- and therefore, by implication, an argument that we should regard science as useful, rather than finally true in the Weinbergian "any day now we will understand it all!" sense, and regard Occam's razor as an excellent tool for our convenience, not a Revelation From Above Of The Nature Of Things.
Posted by Dan Percival at August 18, 2007 12:10 AMIt's true that the whole overarching system -- the level 4 multiverse itself -- is simple and elegant. Just like a Turing machine is simple and elegant. But the vast majority of *programs* which you can run on a Turing machine are anything but.
I admit that I have not yet RTFPDF. However, I refer you back to Schmidhuber's argument about the computation of all computable universes that came up back a while ago in this venue. (In the context of "everything is true", I think -- if you can't find it, I seem to remember that it's linked somewhere from Schmidhuber's home page.) He makes a persuasive argument that if you run all possible programs (universes) in a manner that allocates resources evenly between them, the programs with lower complexity (incorporating both program length and running time) will progress faster than higher-complexity programs. Correspondingly, from the point of view within a running program, simple programs that describe your observable state are more likely than complex ones -- a formalization of Occam's Razor.
Posted by Ted at August 20, 2007 09:56 AMI've only skimmed the PDF and so I can't really address it in detail, but I do have a comment regarding your thought experiment of adding a single anomaly to an otherwise consistent universe. I think that such a universe could not be described by what Tegmark calls a mathematical structure. The condition "gravity doesn't work at position X on Tuesdays" cannot be represented as a mathematical axiom of the sort that the rest of the system is built on.
You can't define a formal system as "Peano's axioms, except that four plus three equals six." That anomaly would contradicts all the other axioms; you'd essentially need an infinite number of rules to reconcile it with the theorems that arise from the other axioms. As another example, there's no single term you can add to one of Maxwell's equations and get a specific, localized anomaly in the phenomena of electromagnetism. If you add a term, you will change the nature of electromagnetism everywhere, all the time.
The level 4 multiverse isn't the Turing machine in your analogy; our universe is the Turing machine, and the level 4 multiverse is some broader set of machines. You're essentially saying that you could define the rules for a Rosenbaum machine that runs all programs that a regular Turing machine runs, except that it replaces the ASCII string 'BENJAMIN ROSENBAUM' with 'benjamin rosenbaum' when executing DOS. I don't believe that's possible.
Posted by Benjamin Rosenbaum at August 20, 2007 10:18 AMTed, I think there's a circularity in that argument, in that you seem to be saying "only elegant, simple mathematical structures are actually mathematical structures." That may be what Tegmark's saying, but if so the level IV multiverse becomes an even more spectacular leap of religious faith, and ultimately much less interesting.
It's true that adding some simple deviation from natural laws, such as "gravity doesnt work over there", while easy to express in English, might require a hugely complex muddle of rules to formalize as a mathematical system. Indeed, it would have to be a total rewrite of the rules to somehow account for locality. But it's not *inexpressible* as a formal mathematical system. Ptolemaic physics, after all, had just such a "special point" in it -- the center of the earth, which was the center of the universe -- and it was certainly formalizable.
Let's say the universe *really is* like that -- that there's one or more special points where things work differently. (If you say that the universe cannot be like that because then it would not be a reflection of mathematical laws, then you have already decided so completely for the Platonic worldview that the conversation is over.) Let's say we had enough time and data to figure out that it did work like that. Would we despair of designing laws?
Well, probably we would have a law that looked like "law A except in the following points which have special laws, look them up", which seems to me to be adequately formal to be a "mathematical structure" (if you recall that that original question is "why these laws and not others?"). In some sense, having such an exception is no different than saying "there's a stong force, here are its laws, and then over here we have electromagnetism, and over here gravity." But if we did want to have a single "unified theory" which would account in one set of equations for the above described weirdness, it could be done, it would just be crazily complex (like, worst case, a separate term for each point in the universe).
When I said "take a Newtonian worldview and add one simple change", I mean "simple" from the perspective of an SAS living in the model, not "simple" from the point of view of mathematical formalization.
I think what it really comes down to is that you either have a Platonic or an Aristotelian intuition. If your intuition is Platonic, these examples are going to look like perverse, nonsensical, sophistic fooling around. If you have an Aristotelian intuition, the notion of the world actually being a simple discoverable mathematical structure looks like absurd, crazy hubris.
Posted by Benjamin Rosenbaum at August 20, 2007 10:21 AMA Rosenbaum machine, in your example, is a Turing machine plus an extra finite state machine attached to it with a small program that says "running dos? then replace..."
You will then say "no, no, that's still a Turing machine in principle, you have to change something *fundamental*". I'm saying I don't. Adding a stupid kludge to an elegant model is just as possible a variation as designing a new elegant model. If you think it's not, you are essentially asserting that the universe must be pretty.
Posted by Ted at August 21, 2007 01:49 AMI suspect that neither of us have the mathematical background to discuss this properly, but we can try to proceed with analogies.
That said, what do you mean by "formalizable"? How can you be sure that the Ptolemaic system was formalizable? As I see it, just because something is easily expressible in English doesn't mean it's capable of being expressed as a formal mathematical system. For example, I can easily express, in English, the idea of a machine that can solve the halting problem; I can even talk about its properties in some detail. But that doesn't mean that such a machine can exist.
Similarly, I can talk about a universe where all numbers behave like ours except that 3+4=6, but that doesn't mean it can exist either. Have you ever read Carl Sagan's novel Contact? The final revelation of the novel is (spoiler alert) that there's a message hidden in pi; it's described as the signature of the artist, a message to the inhabitants of the universe from the creator. Note that this wasn't a matter of carefully measuring the curvature of space or anything; it was accessible by pure computation. I found it a suitably wondrous ending when I first read the novel, but much later I read some online discussions about the ending; one person said it was exactly as easy to encode a message in the value of pi as it was to encode a message in the value of 1, and there was some discussion about whether even God could do that.
(Or, as the problem has been stated elsewhere, how much choice did God have in making the universe?)
So, do you think that Tegmark's Level 4 multiverse includes an aleph-one number of universes which each have a different value of pi? Or different values of of 1? If not, would you say that the reason such universes are excluded is an aesthetic one?
Posted by Lucy Dee at August 21, 2007 07:18 AMI just chose the last option (radio button) because it was funny....!
I just wanted to email and say thanks for your comment on my blog...
You had a lot of insightful things to say.
I hope you'll come back and share more of your thoughts.
I wanted to return the favor and contribute a comment, but I think the
subject matter was a bit over my head and out of my league...anyway, thanks again!
Posted by David Moles at August 21, 2007 11:30 AMThe thing that got to me about Contact and π, several years later -- I think the movie π kind of takes this tack, too -- is that in an infinite random sequence, there's a non-zero probability of finding any message, including all false messages. (But I suppose it's harder to dramatize some discovery that would be actually interesting to mathematicians than it is to sketch a circle in dot-matrix on fanfold.)
Posted by Ted at August 21, 2007 07:00 PMThat point was actually raised in the novel, but the protagonist pointed out that you could calculate the likelihood of finding a specified sequence, and thus judge whether a perceived message was significant.
Posted by Matt Hulan at August 21, 2007 10:35 PMORLY?
peace
Matt
Posted by Benjamin Rosenbaum at August 22, 2007 09:28 AMTed, you're probably right that this is over both of our heads mathematically.
I'm not sure what "a universe with a different value of pi" would mean. If you mean, a universe where the ratio of the circumference of a circle to its diameter is a number other than 3.1415..., I don't know, but my bet is that that is trivial in terms of designing an alternate geometry. For one thing, in Lineland (that simpler cousin of Flatland), pi in this sense is undefined as there are no circles. For another, some theories in physics, I hear, assume that only taxonomy and relationships are fundamental and that geometry is an emergent effect thereof; it does not stretch the imagination too much, then, to suppose that if the taxonomy and relations were arranged differently, the epiphenomemon of geometry would not emerge at all, so that you would have a mathematical system without geometry. If such a system would support an SAS (and I can't even begin to formulate a real argument for or against this), then you would have universes without geometry, so definitely no pi-as-ratio.
If you mean a universe with a different value of pi, the number, or of one, the number, my gut feeling is that this is nonsensical. "One" is not an empirical fact about the universe which we observe; nor is it a law governing such a universe. It is just a label in a language game. The language game may be useful for describing something real, or not useful. It may have certain internal properties like consistency or completeness. But it can't be empirically proven wrong.
It's interesting to speculate about, say, how strong gravity is. You can say "hey, maybe the gravitational force falls off at long distances." Then you can go test this. Once you have an answer, or a current best guess, you can imagine universes where the answer was different. In doing this you are imagining a counterfactual scenario in which you found different evidence. You would then come up with math which would account for the evidence you found. The findings are primary; the math is secondary. You don't say "we reject these findings because the math doesn't work."
You therefore can dream up a universe in which things work differently, and then find the simplest math to describe that universe. Those are, by definition, its physical laws. And I think the import of Tegmark's paper is that the Level IV multiverse contains, at the minimum, any universe we might in fact be living in: not only ones with pretty laws.
This notion of counterfactual imagining of different results, however, makes no sense when you are speculating about *mathematics itself*. You can say "hey! what if there were no odd numbers? Imagine a universe which is just like ours, except that all the numbers are doubled, so that the integers consist of 0, 2, 4... Hey, maybe we even live in such a universe! Maybe every time we encounter one thing, it is actually two things, because there really is no 1 and 2 is the smallest integer!"
Well, that's fine, except where do you go from there? The doubled-integer universe looks exactly like ours. You've actually just renamed the integers. There's nothing observable here. "One has a different value, like, say, 2" is meaningless -- certainly untestable, and possibly pure nonsense.
(On the other hand, if you look at the chart of Tegmark's heirarchy of mathematical models and systems, you'll note that there are mathematical models simpler than the integers. If an SAS could arise in such a mathematical system -- which I'm betting against! -- then it would be a "universe" and it would "have no numbers" not in the sense that numbers are observable things, but in the sense that nothing in that universe would require the use of numbers to explain it. Again, this seems to me absurdly implausible. I bet any mathematical system complicated enough to have an SAS has numbers. But YMMV.)
The Turing Machine is a mathematical notion. It isn't something you can necessarily construct in reality. (If our universe is finite, to begin with, there is no room for a Turing machine). To say "I can easily express, in English, the idea of a machine that can solve the halting problem; I can even talk about its properties in some detail. But that doesn't mean that such a machine can exist" is, in a sense, a category error. It is mathematically impossible for a machine to solve the halting problem for the same reason that it is mathematically impossible for a machine to show that 1=2 -- because you're talking about a machine doing something which makes no sense.
But noone says that you can't build an actual, physical machine that can compute the halting state of any other actual, physical machine. (I expect that, for instance, in a unvierse finite in space and infinite in time, you CAN build such a machine).
If I say "anything I could potentially observe, no matter how screwy, is a possible universe, and must have a place in the level IV multiverse," I mean anything I could potentially observe. I could observe cows flying, beans flying up in one particular location, or the world of Hell Is The Absence of God (its observable details, mind you, not the assertions of truth of its narrator).
I cannot observe that 1=2, or that a Turing machine solves the halting problem, not because these are some kind of forbidden fruit, worlds from which we are banned by an angel with a fiery sword who says "you may not pass", but rather because they don't mean anything. They are self-contradictory things.
You are saying "perhaps the Ptolemaic universe is not formalizable". I don't buy this. I have never heard any suggestion that there is anything self-contradictory, in the sense that 1=2 or that a Turing machine solves the halting problem, about the Ptolemaic universe, and I don't see how there could be anything self-contradictory about it.
Let us say there is a set of sensory data, S, for which the simplest explanation is a Ptolemaic universe. For instance, we fly away from the earth, observe the sun and planets orbiting it, go "clunk" against the heavenly sphere, etc.
What I'm saying is that there's *some* set of rules and axioms that would then account for it. If you had observed such a set of data S, you would build a model to account for it. If the model turned out to be self-contradictory, you would not change S -- you would change the model. I don't think there is ANY set of sensory data for which NO mathematical model can be constructed -- because there is no sensory data which *obliges* us to believe self-contradictory things like 1=2.
Can you think of a counterexample?
Hi Lucy!
Posted by Ted at August 22, 2007 09:45 PMYou're using two different approaches with regards to ascertaining what's possible. You say it's mathematically impossible for a machine to solve the halting problem (which I'm relieved to hear you say), and I agree. But then you talk about a set of sensory data that describes a Ptolemaic universe. I don't think that imagining a set of sensory data is a good indicator of what is possible.
For example, let's consider the halting problem again. I can imagine a set of sensory data for which the simplest explanation is a program that solves the halting problem (roughly, "I run the Haltcast program over any set of source code and input data, and it tells me whether it halts or not, and it's always right"). It is not at all obvious that this set of sensory data involves a mathematical contradiction; someone could spend their entire life as a computer programmer and not suspect that this idea is impossible if they hadn't taken the relevant coursework.
Now let's consider the value of pi. I can likewise imagine a set of sensory data that involves me computing the sum of the appropriate infinite series of fractions, and I come up with a sequence of digits that encodes the message "kilroy was here" beginning at the 1,000,000th decimal place. This is a mathematical impossibility; there is one and only one sequence of digits that can result from performing that computation. It's not immediately obvious what that sequence is, but any set of sensory data that involves a different sequence of digits is mathematically impossible.
Determining whether a given set of sensory data involves a mathematical impossibility is extraordinarily difficult. You and I have no way of knowing whether a universe where gravity doesn't work at position X is possible or not. But Tegmark seems to be positing a close correspondence between the physical universe and abstract mathematics, and even though one can come up with lots of alternate mathematical systems, mathematics is still highly constrained: there's no set of axioms which will cause the computed value of pi to contain a specific message at a specific decimal place. Just because you can imagine a particular set of sensory data, it doesn't mean that there exist a set of axioms that will generate it. Which is why I don't think Tegmark's multiverse includes what you describe as universes with tweaks, hacks, absurdities, and special cases.
Posted by Benjamin Rosenbaum at August 23, 2007 08:54 AMI don't think you are taking seriously what I mean by "sensory data for which the simplest possible explanation is X".
The simplest possible explanation cannot be something self-contradictory; a self-contradictory statement is not an explanation, any more than a random series of grunts is an explanation.
You cannot imagine sensory data in which Haltcast solves the halting problem, for two reasons: 1) you cannot imagine running Haltcast over "all" programs, because that would take an infinite amount of time; you can only running Haltcast over "a whole bunch" of programs, which would not prove anything; and 2) if you imagine running Haltcast over itself and it returning "yes, I would halt", then the simplest explanation is not that you have solved the halting problem, but that Haltcast has a bug.
You cannot actually imagine, in detail, working through a known mathematical proof and coming to a different conclusion without making a mistake. You can imagine imagining it, in a fuzzy lyrical way, maybe, but that's not what I mean by "sensory data for which the simplest possible explanation is X".
Similarly, you cannot imagine computing "kilroy was here" in pi any more than you can imagine writing down "1 + 1 =" and then writing down "3" and being right. That is, of course it's easy to imagine if you simply redefine 3 (or make up an arbitrary encoding system for which the digits of pi do encode "kilroy was here"); but that is just imagining yourself cheating.
This is because math is a game. Imagining the same game, played the same way, yielding different results, is not imagining something empirically wrong; it is imagining nonsense.
Math does not say anything about the physical universe unless it does, if you see what I mean. You cannot look at math and say "ah, now I know that such-and-such an event in the physical universe is impossible." You can only look at math and say "ah, now I realize that if such-and-such an event in the physical universe occurs, this would be the wrong math to describe it."
If you can imagine *making an observation*, then science must be broad enough to encompass the possibility that you *will in fact make that observation*. You cannot imagine making an observation that 1=2, or of solving the Halting Problem, because these are not *observations* -- they are incorrect (because self-contradictory) *conclusions* about observations. Other than being complex, imagining a computer program seeming to solve the Halting Problem leading you to conclude that Turing machines can solve the Halting Problem (and therefore that all Cretans really are liars and that the set of all sets that do not contain themselves contains itself, since these problems are all isomorphic), is not any different than seeing conjoined twins and concluding that 1=2.
Observations do not prove anything about math, other than, perhaps, that you are using the wrong math. When we observe cosmic microwave background radiation that suggests that the universe is flat, when we thought it was round, we do not say "oh, the math must be wrong, flat must really mean round." No, the math is fine; we were using the wrong math to describe physical reality. Roundness has not changed any; it just turns out we are living in flatness.
If one believes otherwise -- if one thinks that just by looking at the math, without the previous assumption that this particular math happens to fit this universe, one can prove that certain facts of physical reality cannot be true (because, presumably, it has been revealed to one which math is the special privileged math) then one is living on the island of Hayy Ibn Yaqzan.
Now, it may be that I am misreading Tegmark in a sense, and that he really means some subset of *all possible* mathematical models, so that it is not merely things which are self-contradictory which are excluded, but some other set of things, so that if I posit a particular exception such as "on Tuesdays in Idaho", etc., that may turn out to be excluded by the constraints on his definition of "mathematical model". I doubt it, but it doesn't matter.
Here's why. Consider this equation:
F = MA
I assume you accept that this is a formalizable equation. :-) But note that it has some "externals", if you will. Take "mass". Mass is not derived from other things in Newtonian mechanics. It is simply a quality that things have. You measure it, and plug it into the equations. It is, if you will, arbitrary.
Now consider this:
F = MA + K(d^(1/2))
Where K is an arbitrary constant and d is the distance of the center of mass from a "special point" at the center of the universe (so that the force on an object is equal to mass times acceleration plus a constant times the square root of the distance from the center of the universe).
The "special point",like mass, is an external -- it is axiomatic. It does not arise from the theory any more than mass does. The second equation happens to be wrong. But it seems absurd to me to say that it is "not physics". There is nothing self-contradictory about it. A contemporary of Newton could have proposed this equation, and it would have been a coherent, falsifiable, wrong hypothesis.
Note that if K is very, very small, Newton's contemporaries would probably not have been able to make the determination for lack of appropriate data (especially if they were far from the center of the universe).
Now consider this equation:
F = MA + K(d^(1/2)) + K2(d2^(1/3))
Here the above equation is adjusted by K2 (a constant) times the cube root of d2, which is the distance to another special point, this one hovering somewhere near Jupiter.
This equation, surely, is equally admissable as a mathematical model.
Now consider adding a fourth root term, a fifth root term, and so on, so that the equation becomes an arbitrarily long polynomial relating an arbitrary number of special points in space to the fundamental laws of nature.
We could be living in such a universe, especially if the "extra term" effects are very small.
Occam's razor certainly suggests that we are not. But there is no Occam's razor, at least on a local-universe level, in Tegmark's model. Tegmark's multiverse, by including all non-self-contradictory, admissible mathematical models, contains an infinite number of universes in which each coherent simple equation of the form F = MA is adjusted by an arbitrary number of wacky outlier terms such as the above.
(If for some reason you think adding such terms to F=MA would break the relationship with the other laws, rendering the whole self-contradictory -- though I can't see why, since the other laws would be derived from this first law anyway -- you can imagine that instead of that, the gravitational constant G is a function of the distance to an arbitrary number of special points. Slightly weaker, but makes the point just as well).
Occam's razor is the foundation of science. You can use Occam's razor as an Aristotelian pragmatist ("since we are never going to know the true laws, we might as well choose the simplest ones that work") or as a religious Platonist ("I feel deeply that the universe must be elegant").
But you can't claim that Occam's razor is true and that you can prove it, or its likelihood, and that thus we are definitely living in a Platonic universe. This is, one way or another, just an updated version of medieval proofs of the existance of God, resting on the same "it must be so because I cannot bear to imagine otherwise".
Posted by Ted at August 23, 2007 10:41 AMThe simplest possible explanation cannot be something self-contradictory
True, but my point is, it is not at all obvious whether a given explanation involves a contradiction.
What is the billionth digit in the decimal expansion of pi? A mathematician may propose a formula for rapidly calculating any given digit, and come up with a prediction for the billionth digit. Someone else may make a different prediction. Can you tell whose prediction involves a mathematical contradiction? Not easily.
You cannot actually imagine, in detail, working through a known mathematical proof and coming to a different conclusion without making a mistake.
But if it's not a known mathematical proof, it is not at all clear whether you've made a mistake. That's why mathematics is hard. Proving the correctness of theorems is nontrivial.
Likewise, it may be that you cannot actually imagine, in detail, a set of sensory data where F = MA + K(d^(1/2)) without making a mistake. The mistake would certainly not be readily apparent, but neither is the mistake involved in predicting the wrong billionth digit of pi.
I obviously don't know for a certainty that this is the case. But someone could argue that a different billionth digit of pi is a much smaller departure from our universe than a point in space where gravity doesn't work. And someone could likewise deny that this would necessarily be self-contradictory by using your argument "the other laws would be derived from this first law anyway."
But I don't think that you can adjust the mathematical structures shown in Tegmark's Figure 8 to produce universes with any desired value of pi. By the same token, neither can you produce universes with versions of F=MA that contain arbitrary wacky outlier terms. (Tegmark's level 2 multiverse would be the closest you could come.) Alternate mathematical structures, and their corresponding universes, are possible, but any changes you make will be bottom-up, so to speak; if you try to make a pinpoint change at the top, you will find yourself in a contradiction. Physical universes are constrained in the same way that mathematical structures are.
Posted by Benjamin Rosenbaum at August 23, 2007 11:04 AMWow, we just seem to have profoundly different worldviews here. I feel like at this point we should go dig up an expert who actually *does* have the math and science background to adjudicate this thread.
This seems to be the core issue: "Physical universes are constrained in the same way that mathematical structures are."
That strikes me as a statement coming straight from the island of Hayy Ibn Yaqzan. You are just *asserting* that the universe corresponds to nice math.
*Models* of physical universes are constrained in the same way mathematical structures are -- because the models ARE mathematical structures.
It is possible that I cannot imagine, in detail, a set of sensory data where F = MA + K(d^(1/2)) -- but, as I implied above, only if the set of laws implied by that statement is self-contradictory. But there's nothing about the external physical things, like adding the notion of distance from a special point in space, which can make it self-contradictory.
It is certainly possible that the particular form of the equation I chose somewhow couldn't be plugged into the others to give a self-consistent set of physical laws. But that would be an error in my math, and it would be beside the point. It would just be a bug in my example.
I cannot see any coherent argument for the idea that NO set of physical laws can self-consistently make reference to locations. Why, pray tell, is a specific distance in space -- like Planck's constant -- appropriate to bind into a physical theory, but a specific point in space would not be? Newtonian and post-Newtonian physics axiomatically assume a homogenous spatial universe with no special point... but you can't elevate that axiom into a general feature of all possible physical universes.
Of course, you are right, all changes are bottom up. Yes, if I propose a *specific mathematical structure* and make a pinpoint change at the top, like "gravity doesnt work Tuesdays in Idaho", I am probably wrong about that prediction.
But if I suppose a specific prediction and say "the specific framework is left as an exercise to the reader", then, either the prediction itself *already contains a self-contradiction without reference to the proposed mathematical structure* (like "solving the halting problem" or "finding that the set of all sets which do not contain themselves contains itself"), or else such a framework can be found -- it may simply be arbitrarily complex.
When you say "to produce universes with any desired value of pi", do you mean "in which the ratio of a circle's circumference and radius is any given number"? Because that seems to me like it would be either impossible, or trivial, and my bet is on trivial. But it's also irrelevant to the conversation, because pi is a *fact of mathematics*, not a *fact of reality*.
Reality comes first. We do not know what occurs in reality. We made math up.
No possible event in reality is self-contradictory. Only an *explanation* of the event can be self-contradictory. Events happen; then we use math to put them in a framework.
I suppose, though I think it may be nonsense, but I can imagine that there are possible (ie the specification of them is not intrinsically self-contradictory) universes which are "unmodellable", in the sense that no mathematical structure can be created to predict or derive the events which occur in them, even if you had an infinite amount of time and space in which to describe the mathematical structure. This seems impossible, but maybe it even follows from Gödel.
But that would not prove anything about our universe; just put limits on our ability to describe it.
Posted by Ted at August 23, 2007 07:54 PMI feel like at this point we should go dig up an expert who actually *does* have the math and science background to adjudicate this thread.
I imagine that experts would likewise disagree.
When you say "to produce universes with any desired value of pi", do you mean "in which the ratio of a circle's circumference and radius is any given number"?
When I say "pi" in this discussion, you can take me to mean the sum of an appropriate infinite series of fractions.
We made math up.
Yet we cannot make math do whatever we want. Why should a physical universe be less constrained?
there's nothing about the external physical things, like adding the notion of distance from a special point in space, which can make it self-contradictory.
You seem awfully confident. As another example, consider the two statements "Fermat's theorem is true" and "Fermat's theorem is false." One of these statements implies a contradiction. Is it readily apparent which one?
I cannot see any coherent argument for the idea that NO set of physical laws can self-consistently make reference to locations. Why, pray tell, is a specific distance in space -- like Planck's constant -- appropriate to bind into a physical theory, but a specific point in space would not be? Newtonian and post-Newtonian physics axiomatically assume a homogenous spatial universe with no special point... but you can't elevate that axiom into a general feature of all possible physical universes.
As an aside, Plank's constant is not a distance, it's a unit of action. But I'm not arguing that it's impossible for a physical theory to incorporate a specific point in space. I'm arguing that you cannot incorporate a point in space into a physical theory and restrict the implications arbitrarily.
It seems to me that, if you did have a theory which incorporated a specific point in space, virtually everything in that universe would be changed, not just gravity but electromagnetism and chemistry and everything else. The various laws of our universe are all very tightly bound; to expect that you could change one but not the others is to demand a contradiction. (It's like saying, "I want pi to have this value, but everything else to remain the same; the axioms necessary to make that happen are left as an exercise to the reader.") And so this point in space would not constitute an "exceptional kludge"; it would be as fundamental an aspect of that universe as, say, the fact that our universe has both space and time as dimensions. And you wouldn't, I hope, say that time is a Rube-Goldbergian kludge in our universe.
Posted by Benjamin Rosenbaum at August 26, 2007 12:04 AMWell, at bottom it's an aesthetic issue.
If you can look at a universe whose natural laws demand that you incorporate reference to 12,453 special points unevenly distributed throughout it, each with a different relevance to it, as profoundly elegant, then, by all means, more power to you. It is certainly true -- indeed, tautological -- that each of them is as fundamental an aspect of that universe as time or space. None is an afterthought, or a minor footnote, it is true; they are all tightly bound into the very fabric of a tightly coupled set of laws. On a theological level, I think that the generosity of accepting a complex set of laws as every bit as natural and beautiful as a simple one displays an admirable flexibilty, generosity and humility.
However, it's not exactly standard in physics. The aesthetic that usually prevails in science is one of desiring to minimize the number of bits in which the laws of nature can be described. Hence the discomfort with "fine-tuned constants", magic numbers, and separate unrelated fundamental forces, and the jubilation when forces can be unified and some previously axiomatic elements of theory can be shown to be logical consequences of other elements.
General relativity and string theory, if I have my history of science right, both were motivated not, initially, by data that previous models failed to explain (there was no experiment analogous to Michaelson-Morley for special relativity, or Jupiter's moons for Newtonian mechanics), but rather by the sense that the theory could be made more elegant -- in a sense not too far from saying it could be potentially expressed in a smaller number of bits -- by drastically changing its foundations.
Setting aside the issue of whether I can give a specific example of what might constitute an added complexity in the laws of nature which passes the self-consistency test (values of pi, special points in space, or whatever), do you think there's necessarily a maximum number of bits in which each of the mathematical models of Tegmark's Level IV multiverse can be expressed? Does the notion of "mathematical model" itself bound the complexity of the set of laws? Or can "mathematical models" be arbitrarily complex (in terms of the minimum number of bits in which they can be accurately represented)?
Or is there any reason to believe that the subset of possible mathematical models in which an SAS can survive as an observer is so bounded? Setting aside my specific guesses at how, isn't it likely that many complex sets of laws can function as close enough approximations of simple ones to generate a locally stable enivronment somewhere for an SAS to begin (if not finish) the task of experiencing and observing?
And, if the set of laws can be of any given specific complexity, then since Tegmark's Level IV multiverse contains all possible mathematical models, that means that for any finite number of bits N, there are infinitely more universes whose laws require more than N bits to describe, than universes whose laws require less than N bits to describe.
You are welcome to see the more complex ones as just as elegant as the simple ones. You are welcome to reject my characterization of complexity in natural laws as messy, or full of "hacks". It's a polemical characterization, to be sure; each unit of complexity is no more a "hack" than the notion of time is in our laws.
However, you had still better despair of coming up with a simple set of laws to describe our particular universe. If you do come upon a simple (in total number of bits, setting aside qualitative terms like "elegance") set of laws that seem to describe everything you've seem so far, the odds are overwhelmingly that your data so far has just been insufficient to expose the universe's true complexity.
Thus -- lacking a reason to put a bound on the possibly complexity of SAS-containing mathematical models -- in any given universe inside Tegmark's Level IV multiverse, the expectation that Occam's razor will lead you to the truth is tantamount to hoping for a miracle.
Posted by Benjamin Rosenbaum at August 26, 2007 12:06 AMto begin (if not finish) the task of experiencing and observing?
(I mean "not finish" compiling the true set of laws.)
Posted by Ted at August 28, 2007 11:36 AMyou had still better despair of coming up with a simple set of laws to describe our particular universe.
I had still better despair? Okay, I'll get right on that, although I'm not even sure what argument you're trying to make anymore.
I think you're mistaken about the history of general relativity and string theory, but that's neither here nor there. It's not at all clear to me that we can usefully talk about the number of bits in which a mathematical structure can be expressed, let alone that a more elegant theory requires fewer bits. Using examples from Tegmark's Figure 8: how many bits does it take to express the idea of integers? How many bits does it take to express Abelian fields? Does the sum or product of those bits equal the number of bits needed to express rational numbers? Does this mean that rational numbers are less elegant than integers?
Are you saying that Tegmark's multiverse contains far more universes that contain rational numbers than ones that don't? I initially understood you to be trying to make a different argument, but perhaps I misunderstood you.
the odds are overwhelmingly that your data so far has just been insufficient to expose the universe's true complexity.
Again, I'm impressed by your confidence, although nothing you've said has caused me to share it. What is your basis for estimating these odds?
Posted by Benjamin Rosenbaum at August 28, 2007 05:29 PMI trust you will forgive the rhetorical flourishes. "You had better" doesn't really mean you, personally; I mean "following Tegmark's logic to its ultimate conclusion leads one to".
The argument I am trying to make is "assuming -- as Tegmark does -- a multiverse of all possible mathematical models, those of higher complexity will be more common, and thus, according to Tegmark's probabilistic reasoning about universes, more 'likely', than those of lower complexity; hence, while Tegmark's multiverse is simple/elegant as a whole, it leads to a high likelihood that any given observer is situated in a complex/inelegant universe."
This does indeed rest on three assumptions, which you have every right to question:
1) mathematical models differ in their complexity
2) it is sensible to talk about "some things being more likely than others" in terms of probabilities in an infinite set, and to reason from this to conclusions about the nature of our universe
3) complex things are less elegant than simple ones.
All of these assumptions are open to debate. I am somewhat dubious about #2 myself, and aware that #3 is on some level pure prejudice.
However, these are not my assumptions. They are Tegmark's assumptions.
I am not making claims about the universe. I am making claims about where following Tegmark's assumptions leads. Perhaps that is where we got off track.
Yes, all mathematical systems can be represented in bits. Here is Tegmark, p. 13:
Despite the plethora of mathematical structures
with intimidating names like orbifolds and Killing
elds, a striking underlying unity that has emerged in the last century: all mathematical structures are just special cases of one and the same thing: so-called formal systems. A formal system consists of abstract symbols and rules for manipulating them, specifying how new strings of symbols referred to as theorems can be derived from given ones referred to as axioms. This historical development represented a form of econstructionism, since it stripped away all eaning and interpretation that had traditionally been given to mathematical structures and distilled out only the abstract relations capturing their very essence. As a result, computers can now prove theorems about geometry without having any physical intuition whatsoever about what space is like.
Posted by Dan Percival at August 29, 2007 01:55 AMHi, Ben... checking back in, I really think that the Schmidhuber article (which, I note, Tegmark does cite) would really put a different perspective on these quotes of yours:
do you think there's necessarily a maximum number of bits in which each of the mathematical models of Tegmark's Level IV multiverse can be expressed?
[...]
Or is there any reason to believe that the subset of possible mathematical models in which an SAS can survive as an observer is so bounded?
[...]
since Tegmark's Level IV multiverse contains all possible mathematical models, that means that for any finite number of bits N, there are infinitely more universes whose laws require more than N bits to describe, than universes whose laws require less than N bits to describe.The paper I'm thinking of is Algorithmic Theories of Everything:
The probability distribution P from which the history of our universe is sampled represents a theory of everything or TOE. We assume P is formally describable. Since most (uncountably many) distributions are not, this imposes a strong inductive bias. We show that P(x) is small for any universe x lacking a short description...You really don't have to impose strict upper bounds -- instead, you draw confidence limits on the probabilities. Tegmark alludes to this in his segment about observing our Level I multiverse, when he says:
Hot and cold spots in CMB maps have a characteristic size that depends on the curvature of space, and the observed spots appear too large to be consistent with the previously popular "open universe" model. However, the average spot size randomly varies slightly from one Hubble volume to another, so it is important to be statistically rigorous. When cosmologists say that the open universe model is ruled out at 99.9% confidence, they really mean that if the open universe model were true, then fewer than one out of every thousand Hubble volumes would show CMB spots as large as those we observe -- therefore the entire model with all its infinitely many Hubble volumes is ruled out, even though we have of course only mapped the CMB in our own particular Hubble volume. The lesson to learn from this example is that multiverse theories can be tested and falsified, but only if they predict what the ensemble of parallel universes is and specify a probability distribution (or more generally what mathematicians call a measure) over it.I'm not saying that you have to be convinced by Schmidhuber's argument, but considering that it resolves the objections you're raising pretty neatly within the author's (noted) assumption that the universe may be formalized, you might want to read it over.
Posted by Dan Percival at August 29, 2007 01:57 AM(or, rather, "the author's (noted) assumption that the multiverse may be formalized.")
Posted by Benjamin Rosenbaum at August 29, 2007 07:06 AMThanks, Dan. I will check it out. That sounds like he does respond to my objections.
Though it may take a little while as I am kind of swamped.
I would also like to note that, glancing back, I realize that I have been, somewhat confusedly, arguing two things. Ted's original objection was to my imposing specific arbitrary end conditions and saying "the math is left as an exercise to the reader":
I do have a comment regarding your thought experiment of adding a single anomaly to an otherwise consistent universe. I think that such a universe could not be described by what Tegmark calls a mathematical structure.
Your quote from Schmidhuber suggests that Ted is right about this, and I am wrong: most (uncountably many) probability distributions are not formally describable.
(This positions the leap of faith earlier -- rather than hoping that our universe is a simple one within a complex multiverse with many overcomplicated ones, if only simple universes are formalizable, that means Tegmark's multiverse is then a leap of faith -- or "strong inductive bias" if you will -- that all extant universes happen to be formalizable. My bias is in the other direction, but if it's true that assuming formalizability imposes a strict bound on complexity, the irony disappears, and Tegmark is just advancing a Platonist hypothesis with Platonist consequences, rather than -- to my misplaced glee -- a Platonist hypothesis with Artistotelian local consequences.)
My second argument, entangled with the first, was that even if most "not on Tuesdays" systems would not be formalizable, enough would be formalizable to yield high (or infinite, even) average complexity. I have no idea if this is true, but I haven't heard a good argument that it is not true, and if Ted is willing to concede that we can build a "special point" in some regard or other into the laws of the universe, for instance, then it's not clear why we can't have an arbitrary number of such special points and thus arbitrarily complex laws.
I read you as reporting that Schmidhuber argues that I am right about this and that you can have arbitrarily complex formalizable laws (and that "complex" is meaningful in this context), but that more complex ones are increasingly unlikely, yielding a high probability of simplicity. This is fascinating, and I will have to read his paper to find out why.
Posted by Dan Percival at August 30, 2007 12:07 AMBriefly:
Schmidhuber's caveat in the abstract there is that infinitely many probability distributions over the space of all computable universes are not formally describable. Any universe with a finite history is, at worst case, a formal description of itself, but it may not be possible to formally describe the likelihood of all possible universes -- this is why I posted my correction that it is at the level of the multiverse that Schmidhuber lays his bias.
As near as I can tell, Schmidhuber would have no problem with your description of infinitely many possible universes with special-case crazinesses. The case he's making is that the likelihoods of universes are not all equal and that the prior probability of any particular universe being one of the pathological plethora is small compared to it being a computationally-simpler one.