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Re: More on 0-1 Laws (was Re: 0-1 Laws
Posted:
Jun 15, 1997 11:11 AM
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In article <5npp7o$tlo$1@news.fas.harvard.edu>, kubo@abel (Tal Kubo) writes:
>Putting the last nails in the coffin lid...
Yawn. The lid of your blustered credibility?
>>>>>>>It can be modelled as a Markov chain M on consistent extensions
>>>>>>>of the alpha theory A. You argue, in effect, that iterates of M
>>>>>>>converge to the identity. This is *impossible* for any nontrivial
>>>>>>>chain.
>>>I *assume* nonlinearity. [...]
>>Yes, but your argument doesn't apply whatsoever, and has absolutely no
>>features that connect with what you claim to have proven impossible.
>I'm not claiming an impossibility proof, just an implausibility argument.
Uh, I see the word "impossible" up there. Emphasized even. Feel free to
confuse me all you want, it really doesn't matter.
>To fix ideas, model your argument as iterating some operator M on certain
>probability distributions (of "TM mind beliefs", say). Your "alpha=omega"
>assertion implies that M, which can be thought of as a nonlinear Markov
>chain,
What the heck is a nonlinear Markov chain, and why can M be thought of as
one? Really, you're jumping too fast. Markov trees? Markov fields? I
have seen them a little, but I don't recall any results of the sort you
seem to be claiming. They are very distinct from Markov chains.
> has essentially trivial dynamics: starting at a distribution
>concentrated on one type of TM-mind, we stay there. That is, for alpha to
>equal omega, it must also equal beta, gamma, delta and so on.
No! The beta, gamma, etc are almost surely drifted away from alpha. The
omega limit is an idealized claim about this limit, and the most that
consideration of it can do is render the notion of TM-minds evolving in
the real world to beyond PM seem peculiar.
>The model can be varied to take into account time dependence and whatever
>else.
If so, you can't toss around the word Markov whatsoever as part of your
arguments.
> For instance, the "alpha=omega" assertion was later qualified to a
>claim about consistency strength (i.e. an "observable" depending on the
>state, rather than the state itself). You have quibbled about some
>of these details (not that details of your model are available in any
>case), but they change absolutely nothing as far as the plausibility
>of an "alpha=omega" statement. The basic problem remains:
> --> comprehensive "alpha=omega" statements don't come for free.
You are misunderstanding the alpha=omega claim here, though.
> --> They
> result from conservation laws, or in this case an infinite system
> of conservation laws. Conservation laws for complicated dynamics
> are RARE, not ubiquitous as your argument presumes.
No, there is no conservation going on. Complicated dynamics with fixed
points around which stuff circulates, or some attractors of some sort,
are *not* rare. That is a better analogy.
>>>>When identifying something as "Markov", you have further questions.
>>>>Do you have temporal independence and lack of memory? Not here.
>>>You assume nothing about the initial conditions (since we can't
>>>identify what Physical Mathematics is) or the state of the world,
>>Eh? I make assumptions, which is not the same as identification.
>The assumptions other than TM population state can be assumed
>*constant*.
I do not assume that. In fact, I think it is false, but I haven't thought
about it.
> That was the basis for assuming iterating a single
>operator.
No! Taking an average over something messy and having it equal the start
is not the same as
> Presumably if your argument works for TM-minds evolving
>in veldt it also works for evolution in forest and desert -- but
>the environmental change over time in any of these is much less than
>the difference between them, so we can assume a constant environment.
Fine by me.
>>>and in particular we can start the argument again from the state
>>>resulting from just a brief bit of evolution.
>>Meaning what?
>The environment is the same, and TM "beliefs" are presumably encoded
>in the distribution of TM-mind types. So instead of asking what
>happens in evolving from a single starting theory (i.e., an atomic
>distribution), ask what happens starting from a more diffuse distribution
>of TM's that results from evolution of that starting theory for a while.
Eh? Do I have to explain about the birds and the bees to you? The exact
single starting theory doesn't matter that much to my argument. In fact,
the argument I've given goes through (at the same level of credibility)
with a family of starting alphas, pretty much word for word. They mate
and intermix and mutate and select out. At the beginning, and throughout
time.
>Hence my claim that, heuristically, "it's not much of a stretch to model
>your proposed evolution of theories as iterating a fixed operator on
>populations of TM-minds". This is all in the absence of an actual
>evaluable argument, to indicate the problem about conservation laws.
Since conservation is not an issue, you are blustered again.
--
-Matthew P Wiener (weemba@sagi.wistar.upenn.edu) If Apple owned
NBC, they would sue Nike for comedy-interface copyright violation.
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