Putting the last nails in the coffin lid... ---------------------------------------------------------------------- Matthew P Wiener <firstname.lastname@example.org> wrote: > >>>>>>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.
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, 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.
The model can be varied to take into account time dependence and whatever else. 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. 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.
>>>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*. That was the basis for assuming iterating a single operator. 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.
>>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.
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.