Philosophy and NKS

June 29, 2010

Transients matter

Filed under: finite nature — jasonca @ 1:28 am

I am currently at the NKS 2010 summer program at the University of Vermont – a great experience as always, with lots of bright enthusiastic people. A topic of continual discussion is breaking our understanding of computation out of its “one countable infinity” frame. And not in the continuum infinity direction, but downward into finitude.

A system that bubbles around for a little bit and then resolves into just a few simple periodic structures, or straight lines running down the page, we call class 2 and dismiss as generally uninteresting. For short transients below any level of aggregation we can imagine mattering, this quick assessment may be merited. But extended indefinitely it collides with our actual experience – it is another way that one infinity over directs our thinking.

A bubbling computational system wanders around Vienna for a few dozen years and writes a few score symphonies. Then it goes remarkably stable. Ah, just a transient – class 2 behavior – uncomplex and therefore uninteresting, right? OK, how about a transient between a speck and a photon bath of non interacting light streaming about without interacting, eventually, that only took oh I don’t know, 100 trillion years to resolve?

If a transient is at least as long and interesting as you are, its temporal finitude is no reason to dismiss it – and if it is a material universe through all its immensities of space and time, its supposedly being spanned by your one imaginary countable infinity as a reason to dismiss it as uninteresting, is simply laughable.

Long enough and complicated enough transients matter. We can’t dismiss them, and if our notions of computation do not take them seriously, then our notions of computation are flat wrong and need to be revised.

Mathew Szudzik is fond of citing the Kalmar elementary or primitive recursive functions (PRF for short) as families sufficiently involved and powerful to capture practically everything we mean by computation, without crossing the threshold of universality, because they don’t have that one countable infinity. In programming terms, they have “Do” loops but not “While” loops. But how much can that matter, operationally and physically, in a transient actuality?

You have some practical computer and you know its operations speed. You want a “While” loop construct, but can only program with the PRFs. OK, so you do a little math and figure out an upper bound for the heat death of the universe in seconds, divide by your cycle speed, underprofile to pretend you can get off way more steps in your little routine than you ever actually will, and set the top of your “Do” iterator to that integer. Then put a “Throw” command inside the “Do” instead of a condition on the “While” you wanted to write.

It isn’t a While. It will provably halt. It can’t be universal. But then, no actual While running in your hardware in the actual universe will get any farther. So what operational difference can there be between them?

December 15, 2008

The computational capacity of the universe

Filed under: NKS Midwest — Tags: — jasonca @ 9:15 pm

In the afternoon session of NKS Midwest 2008, we had several remote talks by video conference. The first was by Seth Llyod, speaking on his famous calculation “the computational capacity of the universe”. He gave a nice background to the idea in thermodynamics of entropy and how it tied in with Shannon information, and then the stages he went through in coming up with it. Originally he had the question of the “ultimate laptop”, how fast could one possible get a single kilogram of matter in a one liter volume to compute, subject only to the limits of QM, arriving at the figure of 10^51 ops/second on 10^31 bits. The basic idea is then to extend this calculation to the knowable universe, and derive from it an upper bound on the logical operations (or, distinctions possible, perhaps more accurately) in the history of the universe.

When explaining entropy he got an audience question of typical epistemological bent, claiming that information is a relation between a system and us, not something about the system. He shrugged this off with the statement that entropy is info we don’t know about the system and info is info we do know about the system, but the total info is what is2nd-law non-decreasing. He was on the other hand careful to distinguish mere information processing in the sense of logical operations from computation in the sense of universality. Someone makes the claim that the universe is capable of universal computation and presents as evidence our own practical computers; they wouldn’t be able to do it if the universe cannot. There is a potential objection from cardinality, that universality needs an unbounded store, or to add memory as needed. This relates to the cosmology question whether the universe is infinitely extendable.

I thought he could have done a better job with this, as the point of his calculation is to show the universe to date is only capable of a finite number of operations, yet everything we see was able to result from it, including all of our practical computers and their actual flexibility. To me this shows that infinite cardinality is not actually required for everything we know, empirically, as either practical computation or physical complexity. Our mathematical idealizations in computation theory make use of a single countable infinity, that the real world cases are not seen to require in practice. One can say this shows our computers or the universe are not universal, or one can say the mathematical definition of universality misses somewhat. I prefer the latter, not being wedded to that abstraction.

He next explained the QM origins of the processing speed limit, both the Heisenberg limit on the time to go from one distinguishable state to another related to an energy spread, and the Margolis limit related to the absolute energy. QC has a tendency to operate at the Margolis limit, which is the most one could expect, QM being posited of course. This gives around 10^123 ops since the big bang, maximum. The maximum memory calculation on the other hand comes from a volume calculation and the amount of information that can reside on the boundary of a given volume (evolution inside said boundary being deterministic etc). The finite volume comes with the qualifier “knowable” applied to universe – take a light cone from us into the past as far as the big bang, then forward from that region at light speed. Note that in principle this means a larger present region could access more memory than “here”, and a future one likewise more than “now”.

One then asks, what is the max entropy of this much energy (bounded below by the critical density, since the universe is seen to expand) in that much volume, and one gets 10^92 bits, which he noted are about the amount stored in the black body radiation (most of it, in other words). He speculated that maybe this figure can be pushed higher when dark energy is allowed for, but that was frankly a bit hand-wavy. He wanted to point out these numbers seem to be similar to the universe age over the planck time to various powers (3/2 for 10^92 and 2 for 10^123, one hopes, since the former is about 5×10^60 planck times), but it isn’t really exact.

Overall it was a fun talk, on an argument I had read before. He was clear and the history of physics bits on Maxwell, Boltzman, Gibbs, Planck, Heisenberg, Shannon, and Margolis were interesting.

Here is what I like most about this sort of argument, from my own philosophic perspective. So often we are presented with QM indeterminacy or continuous value cardinality as reasons for expecting hypercomputation or a universe that exceeds all finite grasp, but in fact the theory itself has a quite different operational tendency, rigorously limiting the operational distinctions possible and requiring “a distinction” to have a clear physical meaning. Said clear physical meanings always having an actual “spannedness” or positive measure in both time and energy terms. In a walnutshell, if QM is true then the universe is rigorously finite in information-theoretic terms.

But two provisos have to be allowed to that statement, for the partisans of actual infinities (I think e.g. of Max Tegmark). One, Llyod is speaking of a knowable (in principle) region of space-time and not making claims about inaccessible infinities beyond it, pro or con. And two, he nowhere addressed the many-worlds interpretation or any possible “contemporary orthogonals” it might posit. So one might allow a “knowable” between “the” and “universe” in the last sentence of the previous paragraph.

November 21, 2008

Invariants, native primitives, and computation

Filed under: NKS Midwest, reduction & emergence — Tags: — jasonca @ 7:45 pm

 

Paul-Jean LeTourneau of Wolfram Research, a former student of Stuart Kaufman, gave a fine pure NKS talk in the afternoon of the first day.  I want to discuss a few of the issues it raises because they seem to me to go beyond the specific case he was analysing. 

His rule system is ECA 146, and the point he noticed in analysing it is that this rule differs from the well-known additive rule 90 only in cases of adjacent blacks within the pattern.  Which in turn are produced in rule 90 (or 146, necessarily) by runs of white of even length.  Therefore, in any region in which there are neither adjacent black cells nor even runs of white, the evolution must be identical to rule 90.  Then the idea is to track these things through various evolutions of rule 146.  And he finds persistent structures, which themselves meander about on the background, can annihilate in pairs, etc.  These are present but not obvious in the rule 146 evolution, but comparing what rule 146 and rule 90 do from the same initial lets them pop out.

What is of more general interest about this?  One question is whether these localized structures might be manipulated to get the rule to perform meaningful computations, which would be a step toward solving the “class 3 problem” in the affirmative.  The class 3 problem being the question, are there class 3 random-looking rules that are universal?  Wolfram’s principle of computational equivalence (PCE) predicts there are, but this has not been proven.  All the simple rule universality proofs to date have exploited localized structures and their predictability in constructing analogs to practical computer components.  Class 3s might have been thought to lack the necessary local stability to support programming, though PCE conjectures otherwise.  This may be of broader information-theoretic interest – class 3s are thought of as “maximum entropy” systems, and appear to be computationally irreducible.  But universality is a strictly stronger attribute than irreducibility, leaving the class 3 question – are many instances of apparent complexity reducible but not universal, or are most universal?  So this is a pure NKS issue broader than the rule system itself.

But there are others, tangential to the core concerns of NKS, perhaps, but not to philosophy and issues of reductionism.  Notice these patterns refer to invariants of the evolution of rule 146, but to invariants that exceed the scope of its native primitive states.  The case of runs of even length is a particularly fine instance of that, not being a single pattern but a whole class of them.  (Those are not, however, strictly preserved by the evolution – instead they always give rise to a black-pair particle at some point, which then is preserved, up to collisions with like particles etc).

Next notice that regions of the rule seem to behave with the same additivity as rule 90, where information always passes through unchanged, at maximum (we say, “light”) speed.  While the regions marked by non-rule-90 behaving subpatterns move more slowly, and interact in a non-information-preserving way (e.g. a particle collision leading to both ceasing, has many possible time-inverse pre-images, etc).  As though two rules were operating on the same lattice, one having the information transfer properties we associate with light, the other having the information transfer properties we associate with matter (at least, macroscopically).  But it is a single underlying rule – there are simply many possible subpatterns that behave the first way because the deviations from additivity in the rule cannot arise without special subpatterns being present.  To me this is a fine example of emergence.

Notice further that in principle the two regimes are reducible to a single underlying rule (146), but to understand its internal complex behavior it may actually be superior to decompose into a simpler rule followed “some of the time” (additive rule 90) and to analyse the behavior of the “emergent” particles (black cell pairs and their even-white-run generators) “empirically”.  Why?  Because the reducibility of the rule-90 portion of the behavior can be exploited fully, if it is separated off from the non-additive remainder.

The point is to notice that these relationships (among levels of analysis, apparent particles, reduction, simplification, “factoring” of laws, etc) can arise in a purely formal system, for entirely analytical reasons.  They are not facts about physics.  They are practical realities of formalism and analysis itself.

November 7, 2008

Computability of real valued relations

Filed under: NKS Midwest, real numbers — jasonca @ 10:57 pm

Gilles Dowek (Ecole Polytechnique and INRIA, France) gave a nice talk trying to formalize the requirements for keeping real numbers real, in my terms.  More specifically, he was interested in the restrictions required on functions or relations that take reals to reals, so that they remain formally computable — with various distinguished levels of the latter.  He laid it all out in relation to the more familiar discrete valued cases and was quite clear.

First notice that when we are discussing functions, aka deterministic or single valued relations, computability in maps from reals to reals is exactly equivalent to continuity of the mapping.  In that case we have the typical uses of analysis, and theorems telling us that we can approximate as close as we need to with rationals, and they will converge, etc.

So next he relaxes single valued and asks, first, which of the relations of reals to reals are fully decidable.  The answer is not much, only relations that are either empty or full (the only open sets that don’t care whether you distinguish members or not). 

So relax the map category to semi-decidable, meaning that the relation is either continuous as you shrink a ball around a source point or it becomes undefined at that point.  This gives a stable concept of semi-decidable but not a very useful one, because even the identity relation is not semi-decidable (since the domain is an open set) — which I see as a fine formulation of the objection computational “constructivists” have to reals in general.

So he wants a looser concept, analogous to being effectively enumerable for discrete sets.  That is equivalent to semi-decidable in the discrete case but not with reals, where it is strictly weaker.  Effectively enumerable in the discrete case means you start from the image set, and since it is countable (as is the domain, of course), you can just impose some enumeration scheme on the image, and equate any points in the domain that map to that image-point, identifying them by their image-index, in effect.

To extend this notion to the reals, what is required is that up to some intervening index (perhaps countable), the mapping be continuous.  You can regard the extra index as a random variable or some other sort of unknown, but once it is fixed then the map must become continuous.  All deviations from deterministic continuity are ascribed to an extra variable.   This then allows a stable identity relation – it is simply a projection operator that ignores the extra hidden index.

What I liked about this formalism is it fits the use we actually make of non-determinism in real valued models (or at least machine precision valued ones, not to beg too many questions).  We interpose some probability measure or we generate random samples and branch the relation-behavior on them.  I wasn’t so keen on the name he wanted to give it, however — since both domain and range are non-denumerable, “effectively enumerable” seemed a stretch to describe the real-number analogue.   He means it to apply to the relation, of course, but it is still too confusing a name.  Constructable or model-able are perhaps the right idea, but no better as terms.  Call it EE for now.

Then his programmatic statement is that we should restrict ourselves when using non-deterministic relations of reals to only such maps as are EE, that is, such that there exists some random variable that if fixed makes the outcome fully computable, because the projection of the relation to the plane of that specific value of said random variable, is fully continuous.  He notes that if the relation is a true function, this collapses to the usual definition, leaving all deterministic-model experiments or tests unaffected, etc.

It was a point-set topology sort of talk, and quite clear if you have a background in that sort of analysis.  I found it useful for getting more precise about an intuition I’ve had for some time, and something stressed by Gregory Chaitin, that real numbers (without any continuity requirements) can have pathological properties in an information-theoretic context.  There are any number of constructions in which one encodes entire possible universes in single real numbers — Chaitin pointed out one due to Borel later in the conference, that lay behind his own Omega.  Well, when we tried to say what we meant by a real number we were thinking of the limit of a rational series, not entire histories of possible universes.  The definition evidently misses, if it means the latter when it is trying to mean only the former.  Dawek’s definitional work helps us pin down where we can safely use reals without a qualm, and where they imply a formal non-computability too extreme for rationalism.

I should also say that I spoke with Dawek repeatedly throughout the conference and found him a talented and interesting guy, with a good take on everything going on etc.  He asked the last question of the conference, left unanswered by the panel for want of time.  (There was a discussion panel of luminaries on the last day, I’ll cover it later).  He wanted to know each operative’s take on whether there could be experimental tests for the computability of the universe, and for discreteness as well.  (With some preface comments wondering if everything looks like a nail now that we know what computation is and consider it our finest new hammer – sensible question).

Internal time in dynamic graphs

Filed under: NKS Midwest — jasonca @ 10:04 pm

Tommaso Bolognesi is working on dynamic trivalent networks, along the lines suggested by Wolfram in the NKS book.  He has presented on them in the past, and in this talk he first reviewed how they work, then got to his recent questions.  Basically he distinguishes between a global time for the model as viewed from outside, from the sense of time “experienced” (or perhaps more properly, undergone) at a specific location within one of his graphs. 

Global time simply means the model steps themselves – at each of those one local subgraph is rewritten according to a definite list of rules with a preferred order of application.  The update specifies the location of the next “active site” or the next updated node.  He tracks where this occurs by node number and notices that those rules that produce instrinsic randomness tend to revisit all nodes “fairly”, in an average or statistical sense.  While globally the network may show exponential growth, for example, if the spacings out of revisits keep pace, the size of the graph at each (global) step on which a given node is updated, appears to grow linearly.  Similarly for a rule system that showed square root growth globally – again the “internal” sense of time is of linear growth.  It is a feature of fair revisitation that local updates on a graph growing at any speed appear as linear growth in a frame that regards each self-update as a clock-tick, and ignores all other updates.

He also discussed constructing the causal network of site updates, by tracking the faces shared by a given node-update.  I suggested linking those backward through time at the update time-slices given by the “effected” node, to get an internal sense of locality, as well.

September 24, 2008

Finite does not mean small

Filed under: knowledge — Tags: — jasonca @ 9:16 pm

Cardinality issues block and confuse discussions of NKS and the significance of universality.
Some QM fans stump for more than Turing computation as physically real based on the notion that
access to continuum cardinalities can break the limited fetters of countable computables.
Others stick to Turing computables but make more than practical use of its own countable
infinity and span more than anyone will ever span, and pretend things are the same or have
been reduced to one another, sub species infinitum. We even see men educated in computational
complexity theory speak as though anything exponential corresponds to uncountables while
anything polynomial corresponds to countably infinities. Against all these loose associations,
it is necessary to insist forcefully that finite does not mean small. Even finite computation
exceeds all realizable grasp. Against cosmologists dreaming of towers of continuum
infinities and microscopists strident for infinitessimal distinction, both as the supposed
origin of limitations on knowledge or uncertainty in the external world, I must insist that
even denying all such claims a purely finite, discrete, and computable universe has nothing
simple about it. The operative cause of a limit on exhaustive knowledge is not the
hypothetical presence of infinite cardinals of any description, but follows simply and directly
from the term “universe”, and our existing minimal knowledge of its scope and complexity, and
is there already even if every such infinity is denied. People need to give up the equation
between “finite” and “simple”; it is a mere mistake. And if this is established it is
already enough to show that no appeal to experienced limits of knowledge can count as evidence
of any kind, for a real existence for any such hypothetical infinities.

I present a mesh to cover the practical universe and allow for its possible laws and regularities. The
smallest spatial distinction we think can matter for physical behaviors is the planck length.
But I allow for underlying generators below that scale, 100 orders of magnitude smaller. We
experience 3 large scale spatial dimensions, but some theories employ more; I’ll allow for
10,000. We differentiate a zoo of a few score particles, themselves understood as quanta of
fields, on that space. I’ll allow a million, no need to skimp. The smallest unit of time
we think likely to have physical meaning is the planck time, but that’s too long. Slice the
time domain in a manner unique at every single location as specified above, into time units
100 orders of magnitude shorter than the planck time. We can see tens of billions of light
years in any direction, but extend this outward 1000 fold, and allow for every location from
which a light signal might enter a forward position on our light cone 800,000 billion years
in the future, as the spatial extent we care about. Now let us consider every possible field
value for every possible hypothetical field quanta at each such location – quanta to the power
of the number of locations. Those are states. Now let us consider their transitions one
after another, not as compressed by some definite law, but the pure power set, any state can
go to any other state, as a purely formal and one-off transition rule. Allow this transition
to be multivalued and indeterministic, such that the same exact prior state can go to
literally any other distinct subsequent, or otherwise put, multiple by the number of time
instances at each location as though they are all independent. Any regularity actually seen
is a strict compression on this possibility space. Now add elaborate running commentary on
all events as they happen, in 3000 billion billion languages, surrounding the physical text.
All independent and voluminous, billions of times larger than all human thought to date, about
each infinitessimal instance. Don’t worry about where or how the last is instantiated, let
it float above reality in a platonic mathematical realm.

I am still measure zero in the integers. I can fit pure occasionalism in that mesh. I can
fit any degree of apparent indeterminism you can imagine. I can fit all possible physical
theories, true approximate or completely false. But it is all discrete and computable and
moreover, finite. All the realized computations of all the physically realized intelligences
in the history of the slice of the universe observable by us and all of our descendents or
successors for hundred of thousands of billions of years, along with all their aides or
computational devices, cannot begin to span that possibility space – but it is strictly
finite. So, what is it I am supposed to detect operationally, that I can’t fit into a theory
within that mesh, or above it? Notice, I didn’t even posit determinism let alone locality or
the truth of any given theory. It is enough if I can characterize a state by millions of values
at each of an astronomical number of locations. If supposedly I can’t, then no operational
theory is possible period. If any operational theory is possible, it will be strictly less
fine or exhaustive than the thought-experiment mesh given, and strictly more determined or
restrictive, as to transitions that actually occur according to that theory. I further note
that the mesh given is already completely intractable computationally, not because it is
formally noncomputable or has halting problems, let alone because of higher order infinities.
No, it is intractable already, for all finite intelligences and anything they will ever know,
without a single countable infinity. Naturally this does not preclude the possibility of
tractable, even more finite models. But it does show that intractability arises for reasons
of pure scale, within finitude.

Finite simply doesn’t mean small, nor simple.

September 2, 2008

Against simulation

Filed under: Against skepticism, Epistemology — jasonca @ 9:23 pm

I want to trash the idea that we are living in a computer simulation. I will specifically examine Bostrom’s argument that either advanced civilizations don’t run simulations, or most civilizations go extinct, or we are living in a simulation. I will show that anyone who believes his argument is forced to believe in the flying spaghetti monster as well, and in any other item of superstitious nonsense that anyone wishes to impose upon the credulous. He reprises Pascal’s wager, misuses the notion of a Bayesian prior, and falls into cardinality pitfalls as old as Zeno. In passing I will slander Hanson’s more limited claim that at least it is not impossible that we are living in a simulation, explain a few philosophy background items for the Matrix, and defend instead a robust form of the Kantian transcendental deduction – we are living in the universe, which is actual and not an illusion; even illusions live in the actual universe.

First the form of Bostrom’s argument. He claims that one of the following is true, he does not decide which -

(B1) the human species is very likely to go extinct before reaching a “posthuman” stage;

(B2) any posthuman civilization is extremely unlikely to run a significant number of simulations of their evolutionary history;

(B3) we are almost certainly living in a computer simulation.

The form he desires for the conclusion of this supposedly necessary triparition is, “It follows that the belief that there is a significant chance that we will one day become posthumans who run ancestor – simulations is false, unless we are currently living in a simulation.”

Notice, on the surface he does not claim to argue independently that any of the three is more plausible than the others, let alone that all three hold. But in fact, he really comes down for B1 or B3, offering the alternatives “we are in a sim or we are all going to die”, and his money is on the “sim” answer.

The idea is either civilizations generally die out, are uninterested in simulation, or we are in one already. The first B1 is meant to be his catastrophe-mongers “out” and a possible resolution of the Fermi paradox (why aren’t the super smart aliens already here? — they are all dead, they routinely destroy themselves as soon as they have the tech to do so). While he is personally quite given to apocalyptic fantasies of this sort, the “very likely” in the phrasing is a give-away that he regards this is less plausible than B3. Some, sure, nearly all, not so much.

The second he actually regards as extremely implausible. On an ordinary understanding of human interests, if we can we probably would, and B2 requires an “extremely” modifying unlikely, as even a tiny chance that a society with the means would also add the desire is sufficient to generate astronomical numbers of sims. B3 is meant to be plausible given a denial of B1. Why? He counts imaginary instances and decides reality is whatever he can imagine more instances, of.

The primary fallacy lies there, and it can be seen by simply imagining a wilder premise and then willfully alleging a bit more cardinality. The flying spaghetti monster universes are much more numerous than the simulations, because the FSM makes a real number of entirely similar universes for each infinitessimal droplet of sauce globbed onto each one of his continuous tentacles. These differ only in what the FSM thinks about each one of them, and they are arranged without partial order of any kind. The sims, on the other hand, are formally computable and therefore denumerable and therefore a set of measure zero in the infinite sea of FSM universes. My infinity is bigger than your infinity. Na nah nana na.

This shows the absurdity of counting imaginaries without prior qualification as to how plausible they are individually. Worse, it shows that we have here just another species of Pascal’s wager, in which a highly implausible premise is said to be supported by a supposedly stupendous but entirely alleged consequence, which is supposed to multiply away any initial implausibility and result in a practical certainly of… of whatever. The argument in no way depends on the content of the allegation – simulation, deity created universe, dreams of Vishnu, droplets of FSM sauce. It is merely a free standing machine for imposing stupendous fantasies on the credulous, entirely agnostic as to the content to be imposed.

As for B1, well we are all going to die. Sorry, welcome to finitude. Doesn’t mean the civilization will, but we aren’t the civilization, we are mortal flies of summer. Theologically speaking it may be a more interesting question whether we stay dead, or whether the Xerox corporation has a miraculous future ahead of it, post human or not, simulated or lifesized. But one can be completely agnostic about that eventual possibility without seeing any evidence for simulation-hood in the actuality around us.

Hanson, a much lesser light, breathlessly insists instead that at least the simulation hypothesis is not impossible. This truly isn’t saying very much. According to the medievals, the position one ends up in denying only the impossible but regarding anything else as equally possible is called Occasionalism. Anything possible can happen and it is entirely up to God, the occasionalists claimed. They denied any natural necessity as limitations on God’s transcendent freedom. Logic and math could be true in all possible worlds, but everything else could change the next instant as the programmer-simulator changed his own rules. This anticipated Hume’s skeptical denial of causality by at least five centuries. The modern forms add nothing to the thesis; I’ll take my al-Ghazali straight, thank you very much.

The immediate philosophical background of these ideas and their recent popularization in the Matrix is the brain in a vat chesnut of sophmore skepticism, and its more illustrious predecessor, Descartes’ consideration in the Meditations whether there is anything he can be certain about. Descartes posits an evil genius deliberately intent on deceiving him about everything, and decides that his understanding of his own existence would survive the procedure. This evil genius has metastasized in the modern forms. As an hypothesis it actually has gnostic roots (caught between a world they despised and a supposedly just omnipotence, they required an unjust near-omnipotence in the way to square their circle), though come to that an occasionalist God would fit Descartes’ idea to a tee. Descartes himself, on the other hand, is quite sure that no intentional deceiver of that sort would deserve the more honorable title.

Why is the evil genius evil? Why is the brain-in-a-vat manager managing brains in vats? Why are the alien simulators studiously absent from their simulations? And why do their make believe realities feature so little of the harps and music and peaceful contemplation of which imaginations of paradise abound, and so much of the decrepitude and banal horror of actual history? No, they don’t need batteries – no civilization capable of such things would have material exploitative motives. One might allow for a set of small measure to discover things; beyond that the sim managers presumably actually have preferences and are fully capable of realizing them. Naturally you should therefore pray to them for a benign attitude toward your own trials and tribulations. The flying spaghetti monster appreciates a fine sauce.

The original Matrix at least kept up the dualism between a technological mask over a theological world and a theological mask over a technological world. The sequels couldn’t maintain it and collapsed into their own action movie absurdities, but the ability in principle to maintain the dualism is a better guide to the argument’s actual tendency. Which is superstition for technopagans.

Against those signs of contemporary intellectual flacidity, a Kant argued on essentially Cartesian grounds that we could arrive at an at least minimalist piece of information about the actual world beyond our interior of experience – that it exists and is real and structures the possibilities of experience. We need not go so far as to accept his characterization of space and time as necessary forms of intuition to see the soundness of this point. Even illusions happen in some real world, and “universe” refers to that ultimate true ground, and not to any given intervening layer of fluff.

Every robust intelligence starts from the actuality of the world, from feet firmly planted on the ground, not airy fantasies between one or another set of ears. The impulse to look everywhere for fantastic possibility instead is a desire for fiction. If you don’t like the truth, make something up. It is a power-fantasy, but shows remarkably little sensitivity to the question what any intelligent being would want to do with power of that sort. The morals of such pictures are a farce, from the gnostics on. But it is enough to notice that the entire subject is a proper subset of historical theology.

August 28, 2008

Against Many-Worlds

Filed under: Epistemology, Many worlds — Tags: , — jasonca @ 7:00 pm

I’ve written in the past about a few of my objections to the many-worlds interpretation of quantum mechanics, for example on the NKS forum, the thread “do many-worlds believers buy lottery tickets?” I want to discuss here my favorite single objection to many-worlds (MW for short) and be specific about where I think it lands, and what it tells us.

The objection is founded on a classic paradox in probability theory called the St. Petersburg paradox. I’ll give the details below. The idea of the paradox is to show that the expectation of a probabilistic process is not the relevant practical guide to behavior toward it, when the sample size is left unspecified. More generally, operations research and optimal stopping problems are full of cases in which the sample size is a critical decision variable, with direct operational consequences for how one should deal with an uncertain outcome.

This is relevant because many worlds advocates frequently claim that their position is operationally neutral – that one would behave the same way if MW is true as if it is false. They essentially argue that the expectation is the right rational guide either way, and that this does not change across views. But the expectation is not a proper rational guide, as examples like St. Petersburg show. MW in effect posits a different value for the sample size, and this has operational consequences if actually believed. Since it does not appear to have such effects on the empirical behavior of MW advocates, one is left with two alternatives – they do not fully understand the consequences of their theory (unsurprising, and charitable), or they do not subjectively believe their own theory (also plausible), and instead advocate it in only a debator’s pose or provocative sense.

First the details of the St. Petersburg paradox. I give the traditional form first and will then reduce it to its essentials. One player offers the outcome of a game to a field of bidders – the highest bidder gets to take the “paid” position in the game, in return for paying the offering player whatever he bid. The paid position earns a chance outcome, determined by repeatedly flipping a fair coin. The payoff for a first flip of “heads” is one cent, and it doubles for each subsequent flip of “heads”, until the first “tails” outcome occurs. With the first “tails” the game ceases and the balance so far is paid to the second, purchasing player. Thus for example if the first flip is tails, player 2 gets nothing, if the first is heads and the second tails, player 2 gets 1 cent, if the first 2 flips are heads and the third tails, player 2 gets 2 cents, for 3 heads he gets 2^2 = 4 cents, and so on.

The expectation for player 2 is the sum of the infinite series of possible payoffs, which is simply equal to the number of terms times half a cent, which formally diverges. In other words the expectation is infinity, and the formal “value” of the game to player 2 is therefore infinity, and if we think a rational rule to adopt toward a probabilistic payoff is to value it at its expectation, then at the bidding stage, a group of supposedly rational players should be willing to outbid any finite bid so far. Notice, however, that all of the payoff is concentrated in the tail of the distribution of outcomes, as the product of an exponentially vanishing probability with an exponentially increasing payout. For any given bid, the probability of a positive result for player 2 can be computed and e.g. for $100, it is less than 1 out of 8000.

Economists typically use the St. Petersburg paradox to argue for adjustments to expectation-centric rules of rational dealing with risk, such as a strictly concave utility function for increments to wealth, that they want anyway for other reasons. But such adjustments don’t really get to the root of the matter. One can simply modify the payoff terms so that the payout rate grows faster than the utility function declines, after some sufficiently lengthy initial period, to insulate player 1 from any likely effect from this change.

Another frequent objection is to the “actual infinity” in the paradox, insisting that player 1 has to actually be able to pay in the astronomically unlikely event of one of the later payoffs. This can deal with the infinite tail aspect and require its truncation. However, if one then makes player 1 a sufficiently large insurance company, that really could pay a terrific amount, the probability of actually paying can still be driven below any practical limit, without recourse to actual infinities. While trivial bids might still be entertained in lottery-ticket fashion, the full expectation won’t be (with sane people using real money and bidding competitively, I mean), especially if it is stipulated that the game will be played only once. Rational men discount very low probability events, below their expectation. It only makes sense to value them at their full expectation if one is implicitly or explicitly assuming that the trial can be repeated, thus eventually reaching the expectation in a multiple-trial limit.

So just imposing a utility function with fall-off or just truncating the series will not eliminate the paradox. (The latter comes closer to doing so than the former, to be sure). The root of the matter is that the utility of improbable events is lower than the product of their value and their probability, when the sample size imposed on us is radically smaller than required to make the event even plausibly likely, in our own lifetimes. In similar fashion, rational men do not worry about getting struck by lightning or meteors; superstitious men might. It is in fact a characteristic sign of a robust practical rationality, that events below some threshold of total possibility over all the trials that will ever be made of them, are dropped outright from consideration.

Now, if MW is true, this would be irrational rather than rational. All the low probability events would occur somewhere in the multiverse that one’s consciousness might traverse and experience. Effectively MW posits the reality of a space of independent trials of every probabilistic outcome, that is unbounded, raising every unlikely outcome to the same practical status as certain ones. Certainly a MW-er may distinguish cases that occur in all possible worlds from those that branch across them, and he may practically wish to overweight those that have better outcomes in a space of larger probability measure. But he will still bid higher in the St. Petersburg game. He should buy lottery tickets, as I put it elsewhere – or at a minimum, lottery tickets that have rolled over to positive expected value (or have been engineered to have one from the start). Non-MWers, on the other hand, willingly “write” such risks – who rationally cares about the magnitude of a downside that is 100 billion to 1 against, to ever happen?

Peirce defined belief as the willingness to stake much upon a proposition. By that definition, I submit that MWers do not believe their own theory. In fact it has all the marks of a theory-saving patch, held to avoid some set of imagined possible criticisms of a valuable physical theory. I think it is superfluous for that – QM can stand on its own feet without it. I consider a more honest assessment of QM to be, that it simply underspecifies the world, or does not attempt to tell us what will actually happen, or that the determinism of wave function evolution is a conventional holdover from classical Laplacean physics. We can take this position without accepting any of the outright silliness of subjectivist interpretations of QM.

The principle noticed in this criticism, stated positively, is that the sample size posited by a theory is just as important a parameter as its expectation. Positing sample sizes that do not really occur will lead to errors in prediction and implication just as real as getting a magnetic moment wrong. And the actual, instantiated universe is sample size one – its first syllable says as much. When we refer to larger sample sizes, we are always talking about proper subsets that really are formal duplicates of each other, or we are talking about unknowns whose variation we are trying to characterize. Attempts to replace actualization with a probability measure will work if and only if the actual sample size truly involved suffices for the two to be practically close substitutes, and this is emphatically not guaranteed in general. Wherever the actual sample size is small, a probability measure view will underspecify the world and not fully characterize its practical reality, for us. And the deviation between the two widens, as the absolute probability of the events involved, falls.

August 7, 2008

Against fully describing the universe

Filed under: knowledge — jasonca @ 8:25 pm

One task that philosophy can perform for scientific researchers is to criticize the logical structure of their arguments and to explore the possibility spaces they may be missing, wrapped up as they generally are in details of the content of their theories, and self – limiting as they often are, in the candidate propostions they entertain to fufill different explanatory roles within them. Philosophy takes a broader and longer view of such things, and can entertain more speculative possibilities than are generally considered in such theorizing.

Hector Zenil had an interaction with Seth Lloyd on quantum computation that he attempts to summarize on his blog. Without imputing Hector’s characterizations to Lloyd himself, I propose to criticize the argument Hector describes. Whether it is a fair characterization of Lloyd is Hector’s affair. The subject being discussed is quantum computation as a model of the universe, and the argument as Hector presents it is as follows -

“His chain of reasoning is basically as follows : a and b imply c

a) the universe is completely describable by quantum mechanics
b) standard quantum computing completely captures quantum mechanics
c) therefore the universe is a quantum computer.”

I will call the claim “a and b together imply c” the overall claim or LZ (read, “big LZ”, for Lloyd according to Zenil), and retain Hector’s labeling of the subcomponents or separate propositions, just using capitals to distinguish them from surrounding text.

The very first thing to notice is that LZ is a directional claim, and not simply an independent statement of the conclusion C. C might hold while LZ is false. Or C might be unsupported while LZ is false, or C might be false while LZ is false. Secondary claims not in evidence that independently support C or claim to, are therefore out of the scope of LZ and of consideration of it. I need not show that C is false or dubitable to refute or render dubitable the claim LZ. I will accordingly direct none of my criticism at C as a substantive claim. That leaves 3 ways in which LZ can fail – A may be false or unsupported; B may be false or unsupported; or A and B combined may not imply or support C.

While Lloyd himself may be chiefly concerned with establishing B, specifically the claim that the Turing computable version of QC can fully describe any QM system, I will direct none of my criticism at that claim. I consider it largely sound, but my reasons for doing so are tangential to LZ or to Lloyd. That intermural debate between different camps of QC operatives, the Turing complete and more than Turing claims for QC, are not relevant to my criticism. My sympathies in that dispute are with the Turing complete version of QC.

Instead my first criticism is directed at A, which I take Lloyd to consider a piece of unremarkable allegiance to a highly successful theory and nothing more, or exactly the sort of faith in theories that have withstood rigorous tests and led to new discoveries that we all honor and support etc. But I do not see the claim A that way, as stated. It isn’t a statement about QM as a theory and its great virtues compared to other theories, it is a statement about the universe and about something called *complete description*. And I deny that QM is a complete description of the universe, or even wants to be. I deny further that the universe is completely describable, full stop. I do not deny that the concept “complete description” can be well formed and can have a real domain of application – there are things that can be completely describable, and even that have been completely described (the times table up to 10 e.g.). But the universe is not one of them, and it isn’t going to be. I will elaborate more on all of this below, but first my second main criticism.

My second criticism is directed at LZ proper, or the claim that A and B even if both true, together imply C. I detect an additional unstated minor premise in this deduction, which I will formulate below and label M. I consider that unstated minor M to itself be false. I also detect an equivocation in the statement C itself – the predication O is P can mean for the copula to be understood in one of two distinct senses. Either the predicate P can be truthfully predicated of the object O, whatever else might also be predicated of O (“accidental copula” or C1, “in addition to being blue and round and crowded on Sundays, the universe also has quantum computation going on somewhere, sometime, and can therefore truthfully be said to “do” quantum computation”) or the predicate P is meant to exhaust O essentially (“being in the strong sense” or C2, “the universe *is* a quantum computer”). I will claim that the second reading of C aka C2 is unsupported by the preceding argument, and the unstated minor premise cannot sustain it. But I believe C2 is the intended reading of C. I will argue, moreover, that even C1 does not strictly follow from A and B jointly, although it is made plausible by them.

In the end, therefore, I will claim that LZ is false, whatever the case may be with C in either sense; that C1 is plausible but does not forcefully follow from A and B jointly; that the unstated minor M is false; and that independently A is also false. Without in any way denigrating QM’s status as our most successful physical theory to date, by far, nor positing any revision to it as a theory. B is untouched, C is left unsupported by LZ but is not affirmed or denied absolutely. Such in preview are my claims. It remains to explain my reasons for each claim, including the formulation of M. In order, those will be (1) what M is and why it is false; (2) why A and B jointly fail to imply more than C1 and that only suggestively, and above all (3) why A is independently false.

What minor premise is required to think C logically follows from A and B jointly?

(M) If A is “completely describable” by B, and C “completely captures” B, then A *is* C.

With suitable equivocation on “complete”, and on what is meant by “capture” and “describe”, this is a claim about the *transitivity* of *models*. And I claim it is in general false. Translations may exist from some system into another that are truth preserving but not reversible. As a first example, a translation may contain many to one mappings. B may lump together some cases that A distinguishes, and C may do the same to B, leaving C underspecific and many-valued in its “inverse image” claims about A. But it might be objected that this would apply to “capture” and “describe”, but not to either modified by “complete” – a complete description might mean no many to one mappings are allowed.

This will not help, however. I need only pick some other aspect of the original A to distort every so slightly in B – say, the number of logical operations required in B to arrive at the modeled result in A. Then there could be behaviors in A that are in principle calcuable with the resources of the universe within the time to its heat-death, that are not calculable in B within that resource limit within that time limit. Such behaviors would then have the attribute “physically realizable, in principle” in A, but not in B. To avoid inverse asymmetries of that kind, not only would one require the complete description to reach the same outcomes, but to do so in the same steps, memory, and speed. If to avoid any “miss” or slipperiness between levels of that sort, one requires the complete description to be instantiated in the same manner as what it describes, then description of the universe becomes impossible. The universe is a set of size one – that is what its first syllable means. An instantiated full copy of it is a contradiction in terms.

In other words, descriptions are always reductions or they are not descriptions, but copies of the original. But there are no actual copies of the universe, and reductions are always many to one maps. And many to one maps are not in general reversible. The unstated minor wants to read “description” as an equality sign. But description as an operator is truth-extracting, but not in general fully preservative. A description cannot say things about what it describes that are untrue, without failing as a (faithful) description. But it can readily leave things out that the thing described, contains. The terms of A attempt to avoid this possibility by saying “complete description”. But if complete is meant in a strong enough sense, a complete description is a contradiction in terms. Description means primarily discourse intended to evoke a mental image of an experienced reality, and secondarily a reduction of anything to its salient characteristics. A description of something is a predication structure – a set of true statements – or occasionally a diagram or map, preserving some formal, relational “bones” of the intended thing. An exactly similar physical instance of the original thing is not a description but a copy. No description, as distinct from a copy, is complete in the sense of invertible. For example, “this is not a body of true statements, but a physical instance” is predictable of the original but not of its description.

Leaving that model-terminological issue aside, consider instead a chain of emulations between universal computational systems, having the structure M claims between three of them, UA, UB, and UC. Then M claims “if UA can be emulated by UB, and UB and be emulated by UC, then UA *is* UC”. This is equivalent to the claim that there is only one computationally universal system. And it is false. As there is one concept of computational unversality (at least for Turing computation), a fair substitute inference M* would claim only that UA can be emulated by UC, and this would be true. Where “emulated by” allows for the usual issues of requiring a translation function and the potential slow down it may involve. Worse for LZ, M* follows regardless of whether the systems involved are the universe, or quantum anything. It would follow if the universe is Turing computable and the UC at the end of the emulation chain were rule 110. But nobody thinks this shows that the universe *is* rule 110, nor that the only alternative is to deny that the universe is Turing computable. In other words, M* applies to too much, and M doesn’t apply at all, to anything. It is just false.

Next I turn to (2), the issue of the equivocation in the sense of the copula in C. In ordinary speech, we sometimes say “X is A” where we intend merely to predicate A of X accidentally – the moon is spherical. Here it is enough that the statement be true (even roughly true, the approximate sense involved in all predication being understood), and no claim is being advanced that the essence of the moon is its spherical shape, nor are any other predicates being denied to it. At best, a few implicit constrasting predicates are being denied by implication – the moon isn’t triangular or completely flat, since those predicates contradict the true one. Call C1 the reading of C that understands the copula in this sense. C1 claims that whatever else it happens to be, the universe also has the distinct characteristics of a quantum computer. Given the encompassing definition of “universe”, this reduces to the statement that quantum computation is physically realized, anywhere and at any time. For if quantum computation is ever and anywhere physically realized, C1 holds. What else is supposed to be realizing it? Unless QC were nothing but a mathematical abstraction, C1 would hold, independent of LZ. I submit this shows that the stronger claim, C2, is in fact intended by C.

But first, leave aside the truth and inherent plausibility of even just C1, and ask whether it actually follows from A and B jointly. I submit that it does not. For B could hold even if QC *were* a mathematical abstraction, and never realized anywhere in the actual universe. QC is a theory and a description. Call QC beyond a few bits, worked up to the level of a working universal computer, UQC (U for universal). And call QC not as a theory but actually instantiated in the history of the universe with physical components, IQC (I for instantiated). Call the joint case of instantiated and universal QC, IUQC. Assume that behaviors actually occur in the universe sufficiently complex to require universality to model them (this is a more heroic assumption than it may seem, given finite nature possibilities and cardinality issues with universality, but let that slide). Then I claim that B could hold even if IUQC did not. There could be IQC, as QM provides sufficient basis for it. And there could be UQC in the realm of mathematical abstraction. But IUQC might fail to appear – if, say, all the instances of universal computation the universe actually happened to instantiate used unentangled components, and all the instances of true QC physically realized were simple enough to fall below the threshold of universality. If this does *not* occur, we will know it does not occur because of empirical evidence from actual instances of IUQC. Not as a supposedly logical consequence of B, assuming B is true. Logical consequences of possibilities of theories like B do not tell us what actually happened in the instantiated history of the universe. Observation does. Personally I think C1 is true. But I don’t think C1 follows logically from B. And it won’t be theoretical attributes of QC as a formal theory that convince me C1 is true, it will be an actual, empirical instance of IUQC. Hence my statement that C1 is plausible given A and B (or even if not given A, see below), but does not follow from them with logical necessity.

The intended meaning of C, however, is C2, strong *is*, essence or exhaustion. And while a claim of C2 might be rendered more plausible by A and B, it does not follow from them logically. Again it would be possible that QC models QM but is a mathematical abstraction that captures QM, without actually being the real, external, instantiated process that the universe follows – even if A and B were true in a suitably strong sense of “completely”. For it is one thing to say that QC models either QM or physical reality, and another to say that QC is physical reality. It might be, but this would be an independent claim and does not follow from QC modeling physical reality. Any more than rule 110 is my laptop because it can emulate any computation performed by my laptop, albeit more slowly.

In addition, C2 can fail because its stated premises fail. Leaving aside M as already adequately addressed above, we arrive at a still more basic problem. The primary premise A is also false. The universe is not fully describable by QM. And this, for multiple reasons – because QM does not even try to fully describe even the systems and relations to which it properly applies (NotA1), because QM is not all of physics (NotA2), because all of physics does not exhaust description of the universe (NotA3), and because the universe is not fully describable, full stop (NotA4).

Let us start with NotA1. Quantum mechanics is a deliberately underspecific theory. It describes the evolution of a probability amplitude, and only predicts that the absolute square of this quantity will approximate the sample average of classes of events it regards as equivalent. Strictly speaking, it can’t make contact with empirical reality (meant in the precise, exact sense of that compound term – not in a loose sense) in any other manner.

Anything is a reality, in the scholastic definition of that term, if its truth and content do not depend on a specific observor or that observor’s internal state. Loosely, realities are not matters of opinion. Anything is empirical, in the early modern sense of that term as appropriated by empiricists, if it is given directly in sense experience. An empirical reality, therefore, must be given directly to sense experience yet remain invariant across experiencers. Single observations of events of a given class described by quantum mechanics do not have this character, and QM does not claim that they do (some observors see an electron go that-a-way, and some see an electron go this-a-way). Sample averages of events described by quantum mechanics and regarded by it as equivalent to one another, do have this character. And QM deliberately restricts itself to making claims about such averages.

The universe, on the other hand, does not consist of probabilistic claims about classes of events regarded as equivalent under some theory. Definite things happen. The universe has a sample size of one, as its first syllable emphasizes. If one adopts a many-worlds interpretation of QM and regards all possible outcomes as real, one still has not explained the specific experience of oneself as an observor, which is part of the universe. Stipulate that there are a plethora of instances of oneself experiencing every possible outcome of each successive quantum event. QM does not describe which of these “you” will subjectively experience. At most, the voluntary addition to QM of a many-worlds interpretation (which is not itself a part of QM) may wave hands about the orthogonality of each of the possible experiences QM suggests could have occurred. Leaving aside the problem of probability measures across possible worlds in a universe of sample size one, QM itself is not even attempting to fully describe any selection, objective subjective or illusory, of world-lines through that branching possibility space.

Now, a doctrinaire many-worlds advocate may wish to deny the reality of anything left unspecified by QM. But he is abusing the term “reality” in doing so. It is exactly the actual historical observations shared by all inhabitants of any of his hypothetical orthogonal slices, that fit the scholastic definition of “real” – invariance over existing observors, each fully capable of talking to each other and communicating their experiences. While it is exactly his hypothetical alternate realities to which neither he nor anyone he talks to has immediate experiential access, that fail to match the definition of “empirical”. But let us leave aside the complexities of many-worlds interpretations of QM and their claims of exhaustive determinism; it would take us too far afield.

The more basic point must be seen that all descriptions as reductions are sets of true statements about a described, but that such sets are not the original being described. It is one thing to provide a map and another to provide a fully instantiated copy of the original. Descriptions as such achieve their reliability, accuracy, and usefulness by lumping some things together and making statements about the lumps that are true. QM does not attempt to describe the space-time trajectories of every electron since the begining of the universe, but instead, precisely, lumps together lots of disparate space-time events as involving a simplifying unity called “electrons”. The usefulness of a description lies in its generalities, but generalities precisely underdescribe single instances. In scholastic language, theories are about reality, while history is an actuality.

Next, NotA2, QM is not the whole of physics. There is no consistent theory of quantum gravity, but gravity is physically real. Perhaps there will someday be a consistent theory of quantum gravity, or perhaps conventional gravity theory will be found to be incorrect and will be replaced by some superior (but independent) theory. But A is a claim about QM as it stands, and QM as it stands does not even attempt to describe the gravitational behavior of bodies. QM is therefore not an exhaustive description of even physical reality.

Next, NotA3, physics and physical reality, defined as the subset of consistent relations discoverable about the material universe, do not exhaust the universe. The second syllable of “universe” refers to truth as such, and truth as such is not limited to the material world. While cosmologists these days use the term quite loosely, its proper meaning encompasses all truth of whatever character. Mathematical abstraction describes truths which are not described by physical law, but are no less true for that. Mathematical truth is more general than any specific possible world, but is smaller than and properly contained by the concept “universe”.

For that matter, there are truths about conventional categorizations, specifics of actuality, and of individual instances, that physical science even in the most general sense does not even attempt to speak about, let alone claims to fully describe, let alone actually captures fully. “Julius Caesar was a Roman politician assassinated in the senate in the year 44 BC”, is not a piece of physics, but it is a truth about the universe. It differs from physical law in the other direction, so to speak, than the mathematical truth case, above. It is more particular than the laws of physics, and refers to a single definite actuality. But it is part of the universe, just the same. The point is, physical reality lies in a specific place in terms of its generality-relations with the universe as a whole. A map isn’t the territory, nor mapping in general, nor geometry, nor logic, nor the history of the territory, nor its specific configuration at a specific instant in time, etc.

One might imagine reducing each of the categories involved in that statement to underlying physical relations, along the lines of the positivist program of trying to reduce every statement to either a statement about empirical facts or to nonsense, but this is actually hopeless. Truth is not confined to one level of description, and you could not derive many readily accessible “overlying” truths from underlyings buried deep enough below them, within the lifetime of the universe, using physically realizable amounts of computational resources. We can’t solve N body problems analytically, and quantum field theory for single large molecules is already intractable. Minimum energy configurations e.g. in protein folding are NP complete, generalized Ising models are known to be formally undecidable by reduction to known tiling problems, etc.

The idea that reduction can solve all overlying problems “in principle” is computationally naive. Overlying relations may in fact be vastly simpler than the reduction and may already encompass all the computationally important substitutions, when a process is reducible enough to actually be solved. And an accurate reduction may simply land in a morass of intractable unsolvability. The idea that ultimate and simple answers to everything lie at the bottom is simply false. It cannot withstand rational scrutiny. Since computational intractability will arise in any attempt to reformulate all problems in “lowest level” terms, QM will not succeed in exhaustively describing the universe, and other higher level descriptions will frequently succeed where a full QM treatment would be utterly hopeless from the start. One might hypothesize that in all cases of useful higher order descriptions, those descriptions are ultimately compatible with QM. But that needn’t mean they are derivable from QM in the lifetime of the universe, and in fact the higher level descriptions could be logically independent of the microlevel (the Taj Mahal is made out of bricks; this does not make the artistic character of that building a property deducible from the nature of bricks) and yet still hold for every actuality anyone will ever encounter. Enough, infatuation with the bottom level is simply that, a piece of boasting self importance from certain physicists, and not a dictate of reason.

But I am still not done with the ways in which A is false. I claim NotA4, that the universe is not fully describable, full stop. First as to mathematical truth, you aren’t going to capture all of it in one model, see Godel. Second as to historical actuality, stipulate that you have a perfect theory of all physical truth, and can moreover guess the boundary conditions of history. You still won’t be able to work out what happens inside – the consequence of your own theory – with the computational resources of only a subset of the universe. Sure, in principle you might be able to “run faster” that many subsets of the universe, by clever substitutions say, avoiding the need to mimic the universe’s own, full, computational capacity. But you are still a flyspeck on a galaxy – beyond the lightcone of everything that has ever interacted with you, or will in the next 60 billion years, you have no idea what physical immensities remain. Stipulate that you can guess averages and tendencies, and that every law you heroically generalize beyond the range of your empirical experience holds up perfectly. Completely describing the universe still requires you to give an elaborate aesthetic commentary on the exquisite electromagnetic symphonies of Hordomozart the Twenththird, of an obscure planet around an unseen star buried in the heart of an unsensed galactic supercluster 400 gazillion parsecs east-south-east of the great attractor, 46 billion years from now.

Actuality is measure epsilon in possibility-space. The most a good theory can give you is a map of generalities that will hold across wide swathes of space and time. You will not find the specific actuality that actually obtains by staring at the possibility-space such a map describes forever. At some point you have to go look at the actual world. And when you do, you will find yourself at the bottom of a well looking out through a straw at an immensity. Honesty and rationality start with an elementary humility in the face of this inescapable fact. This does not mean you can’t know things; you can. It does mean you can’t know everything.

And it means the actual evolution of the universe, actual history, is continually and meaningfully accomplishing something and adding “more”. Even if the universe follows definite laws (deterministic ones or objectively stochastic ones, either way) and even if you can guess those laws; even if the Church-Turing thesis is true and the universe is formally within the space of the computables, and even if the physical universe happens to be finite. None of those things implies “small” or “simple” or “solved”. Even a finite, computable, deterministic or QC universe will be bigger than every attempt to describe it ever achieved by all finite rational beings combined. It is vastly bigger than we are and just as complicated. Knowledge is not about getting all of it inside our heads – our heads are not big enough and would split wide open – but about getting outside of our heads and out into the midst of it, discovering definite true things about it that are helpful, for us.

July 29, 2008

Tolerance and knowledge

Filed under: Against skepticism, knowledge — jasonca @ 2:59 pm

Although it really has nothing much to do with NKS, whenever discussing skepticism the moral argument for it comes up. I don’t find those convincing, and I think I should explain why.

Part of the attraction of arguments from epistemic weakness comes from a set of moral claims commonly advanced for them, or against the imagined contrary position of epistemic dogmatism. I don’t consider those common moral claims remotely sound, and their association with epistemic weakness is too loose to bind the two together. Roughly, people think it is intolerant to claim to know, sometimes about specific issues and sometimes anything at all, and more tolerant or nicer somehow to refrain from such claims. As though knowledge were inherently oppressive and ignorance, freedom.

At bottom I think this is a post hoc fallacy, a loose association arising from flip diagnoses of what was wrong with chosen bete noirs. So and so believed he knew something and look how awful he was, ergo… Ergo not very much. Your favorite bete noir probably also thought that 2 and 2 are 4, and never 5 nor 3. This won’t suffice to dispense with the addition table and make arithematic voluntary. Evil men had noses. This doesn’t make noses the root of all evil. Believing you know things is as normal as having a nose.

For that matter, I can comb history and find any number of convinced skeptics who were personally as nasty as you please, or even as intellectually intolerant on principle. Al Ghazali will argue from the impotence of human knowledge that philosophy should be illegal and the books of the philosophers burned. You won’t find any skeptical argument in Hume that he didn’t anticipate by centuries. But in his cultural context, it was the theologians that were the skeptics and philosophers who believed in the possibility of (human) knowledge. As this context makes clear, you need a reason to challenge entrenched convention, and if human thought cannot supply one you are left to the mercies of convention. Convention can reign without making any epistemic claims; it suffices to destroy all possible opposition.

There is a more basic problem with the idea that tolerance requires epistemic weakness. It misunderstands the reason for tolerance, and because of it will narrow its proper domain of application. The real reason for tolerance is that error is not a crime, but instead the natural state of mankind. Tolerance is tolerance for known error, or it doesn’t deserve the name. Imagine some Amish-like sect that wants to opt out of half of the modern world, and believes demonstrable falsehoods, but keeps to itself and bothers no one. What is the tolerant attitude toward such a group?

People can think what they like. You can’t require truth of them as a moral matter, because it is rarely available at all, for one, but also because truth can only be grasped internally, as a freely offered gift. You can’t make someone else think something, and it is a category error to try. Minds are free, and individual. All you can do it offer a truth (or a notion or thought), for someone else to take or leave. In the classic formulation of the age of religious wars in Europe, a conversion obtained by duress simply isn’t a conversion. Yes men err, and sometimes their errors issue in actions that are crimes. But no, you cannot eliminate the possibility of crime by first eliminating error. You couldn’t eliminate error even if you had full possession of the truth (which you don’t, to be sure). Persecution isn’t made any better if the doctrine for which you persecute is rational – it remains persecution. (The historian Ignaz Goldhizer made this point in a convincing Islamic context, incidentally).

Human beings are falliable and they are mortal. They have short lives full of personal cares, trials, and difficulties, whose incidence and urgency are peculiar to each individual. They are born in utter ignorance and dependent on their immediate fellows for most of their categories and systems of thought. They grope for knowledge because they need it in practical ways, they attain bits and pieces of it in scattershot fashion, with more found by all combined than possessed by any specific subset among them. Most knowledge stays distributed, particular, and operational – not centrally organized, general, or theoretical.

You can’t require conformity to some grand theoretical system of men in general without violence to half of them. Equally you can’t deny them the possibility of knowledge without maiming them all; humility for export isn’t a virtue. Real tolerance is a patient acceptance of these facts, a charitable and kindly view of our mutual difficulties. We offer one another such knowledge as we have found, and recipients freely take it or leave it, after judging for themselves what use it may have in their own lives. If instead you try to force everyone to acknowledge that they don’t know anything, one you are wrong because they do know all sorts of things, and two you are exactly requiring the sort of grand theoretical conformity you are pretending to be against. You end up making disagreement with your epistemological claims some sort of crime. In this case, that disagreement isn’t even an error, let alone any crime.

So at bottom, my objection to arguments in favor of epistemic weakness on the basis of its supposed tendency to further moral tolerance is that it has no such tendency, and that it misses the point of true tolerance. Which isn’t restricted to a response to ignorance. It isn’t (just) the ignorant who require tolerance, it extends to people who are flat wrong, but innocently so. The moral requirement to practice tolerance is not limited to people unsure of themselves, but extends to people who are correct and know it. The real principle of tolerance is simply that error is not a crime.

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