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From: Russell Standish <R.Standish.domain.name.hidden>

Date: Mon, 15 Nov 1999 11:59:23 +1100 (EST)

*>
*

*> Russell Standish wrote:
*

*> >
*

*> > My much hyped paper is now available for review and criticism
*

*> > (hopefully constructive). The URLs are
*

*> > http://parallel.hpc.unsw.edu.au/rks/docs/ps/occam.ps.gz or
*

*> > http://parallel.hpc.unsw.edu.au/rks/docs/occam/ depending on whether
*

*> > you like your papers in postscript or HTML.
*

*> >
*

*>
*

*>
*

*> > Abstract:
*

*> >
*

*> > In this paper, I show why in an ensemble theory of the universe, we
*

*> > should be inhabiting one of the elements of that ensemble with least
*

*> > information content that satisfies the anthropic principle. This
*

*> > explains the effectiveness of aesthetic principles such as Occam's
*

*> > razor in predicting usefulness of scientific theories. I also show,
*

*> > with a couple of reasonable assumptions about the phenomenon of
*

*> > consciousness, that quantum mechanics is the most general linear
*

*> > theory satisfying the anthropic principle.
*

*> >
*

*> >
*

*> > Introduction
*

*> >
*

*> > Wigner[8] once remarked on ``the unreasonable effectiveness of
*

*> > mathematics'', encapsulating in one phrase the mystery of why the
*

*> > scientific enterprise is so successful. There is an aesthetic
*

*> > principle at large, whereby scientific theories are chosen according
*

*> > to their beauty, or simplicity. These then must be tested by
*

*> > experiment -- the surprising thing is that the aesthetic quality of a
*

*> > theory is often a good predictor of that theory's explanatory and
*

*> > predictive power.
*

*>
*

*> I would go so far as to say that it is a good predictor of a theory's
*

*> validity, or truth. Explanatory or predictive power is not dependent
*

*> on simplicity. Or, perhaps, one might say that the explanatory power
*

This is quite true. However, Newton's theory of gravitation predicts

far more than the Ptolomeic tellurcentric model (because it decsribes

Jupiter's moons as well eg), and General Relativity is better that

Newtonian gravity in precisely the same reason. In each case, we have

a simpler theory providing better predictive power than the more

complex, even if they both had same quantitative utility on the

orginal dataset (the solar system). Maybe I lavished the explanatory

power bit too much. Clearly, a simpler theory has better explanatory

power over a more complex one when both agree with the observations,

as you say.

Now as for the theory's validity or truth - just exactly what do you

mean by that? This _is_ a trick question, so answer carefully :)

*> of a theory is enhanced by its simplicity, for the obvious reason that
*

*> you explain the same thing with fewer principles, so the
*

*> explained/explaining ratio goes up (am I making sense?)
*

*>
*

*>
*

*> > This situation is summed up by William de Ockham
*

*> > ``Entities should not be multiplied unnecessarily'' known as Ockam's
*

*> > Razor.
*

*> >
*

*> > We start our search into an explanation of this mystery with the
*

*> > anthropic principle[1]. This is normally cast into either a weak form
*

*> > (that physical reality must be consistent with our existence as
*

*> > conscious, self-aware entities) or a strong form (that physical
*

*> > reality is the way it is because of our existence as conscious,
*

*> > self-aware entities).
*

*>
*

*> I don't see the need to introduce the SAP at all, here.
*

*>
*

*>
*

*> > The anthropic principle is remarkable in that it
*

*> > generates significant constraints on the form of the universe[1,5].
*

*> > The two main explanations for this are the Divine Creator explanation
*

*> > (the universe was created deliberately by God to have properties
*

*> > sufficient to support intelligent life), or the Ensemble
*

*> > explanation[5] (that there is a set, or ensemble, of different
*

*> > universes, differing in details such as physical parameters, constants
*

*> > and even laws, however, we are only aware of such universes that are
*

*> > consistent with our existence). In the Ensemble explanation, the
*

*> > strong and weak formulations of the anthropic principle are
*

*> > equivalent.
*

*>
*

*> I disagree with this last sentence, although it is a minor point.
*

*> I tend to interpret the SAP as being more or less equivalent to
*

*> the Divine Creator argument (or at least what Tipler calls the
*

*> "first cause" argument), and therefore in contradiction to any
*

*> ensemble explanation.
*

This is curious. I've never heard of the SAP being identified with the

God hypothesis before. For me, the strength of the AP (shifting now to

a continuous scale) is related to how much predictive power the AP

actually has (its surprise factor - as it were). If the universe was

going to be conducive to intelligent life for other reasons, than all

the WAP will do is confirm that fact. The stronger the AP, the more

there will be anomalies (or contingencies) in the universe

unexplainable by any other means than the AP. The WAP needs no further

explanation, the SAP does, and traditionally, the two arguments I

mention are the explanantions given.

*>
*

*>
*

*> >
*

*> > Tegmark introduces an ensemble theory based on the idea that every
*

*> > self-consistent mathematical structure be accorded the ontological
*

*> > status of physical existence. He then goes on to categorize
*

*> > mathematical structures that have been discovered thus far (by
*

*> > humans), and argues that this set should be largely universal, in that
*

*> > all self-aware entities should be able to uncover at least the most
*

*> > basic of these mathematical structures, and that it is unlikely we
*

*> > have overlooked any equally basic mathematical structures.
*

*>
*

*> Frankly, I don't see where this last statement is relevant. If I
*

*> remember correctly, Tegmark discussed this in order to argue that
*

*> in his anthropic arguments for why we find ourselves in a 3+1
*

*> dimensional universe, etc., etc., that he probably hadn't missed
*

*> any structures of importance.
*

*>
*

*> >
*

*> > An alternative ensemble approach is that of Schmidhuber's[4] -- the
*

*> > ``Great Programmer''. This states that all possible programs of a
*

*> > universal turing machine have physical existence. Some of these
*

*> > programs have will contain self-aware substructures -- these are the
*

*> > programs deemed interesting by the anthropic principle. Note that
*

*> > there is no need for the UTM to actually exist,
*

*>
*

*> You don't justify this statement at all.
*

Shouldn't need to - Schmidhuber does it for me.

*>
*

*> > nor is there any need
*

*> > to specify which UTM is to be used -- a program that is meaningful on
*

*> > UTM1can be executed on UTM2 by prepending it with another program that
*

*> > describes UTM1 in terms of UTM2's instructions, then executing the
*

*> > individual program. Since the set of all programs (infinite length
*

*> > bitstrings) is isomorphic to the set of whole numbers N, an enumeration
*

*> > of N is sufficient to generate the ensemble that contains our universe.
*

*>
*

*> Huh?
*

Isn't this obvious? What part didn't you understand?

*>
*

*> > The information content of this complete set is precisely zero, as no
*

*> > bits are specified. This has been called the ``zero information
*

*> > principle''.
*

*> >
*

*> > In this paper, we adopt the Schmidhuber ensemble as containing all
*

*> > possible descriptions of all possible universes, whilst remaining
*

*> > agnostic on the issue of whether this is all there is.
*

*>
*

*> That's confusing, I don't know what you mean. In one phrase you say
*

*> "all possible universes", and in the next you say you're not sure that
*

*> "this is all there is". What else could there be besides all possible
*

*> universes?
*

The Schmidhuber ensemble is some well defined entity. The Tegmark

ensemble is less well defined (as Bruno points out endlessly). The "all

possible universes" is completely undefined. Schmidhuber and Tegmark

have two differing definitions of what AUH means. I remain agnostic

as to whether these two different schemes exhaust the possibilties.

*>
*

*> > Each
*

*> > self-consistent mathematical structure (member of the Tegmark
*

*> > ensemble) is completely described by a finite set of symbols, and a
*

*> > finite set of axioms encoded in those symbols, and a set of rules
*

*> > (logic) describing how one mathematical statement may be converted
*

*> > into another. These axioms may be encoded as a bitstring, and the
*

*> > rules encoded as a program of a UTM that enumerates all possible
*

*> > theorems derived from the axioms, so each member of the Tegmark
*

*> > ensemble may be mapped onto a Schmidhuber one. The Tegmark ensemble
*

*> > must be contained within the Schmidhuber one.
*

*>
*

*> No, no, no, this is the same error that you've persisted in making
*

*> for some time now. Just because two sets have the same number of
*

*> members doesn't mean one set is contained within the other set.
*

*> Your statement here is just like saying "I count here two apples,
*

*> and here three oranges, so each member of the apple ensemble may
*

*> be mapped onto the orange one. The apple ensemble must be contained
*

*> within the orange one."
*

That is not the argument. The identity relationship I describe is an

isomorphism relationship between members of Schmidhuber's ensemble and

members of Tegmark's. That is the meaning of a true embedding. Apples

and Oranges are clearly not isomorphic (unless you're almost

completely blind :).

*>
*

*>
*

*> >
*

*> > An alternative connection between the two ensembles is that the
*

*> > Schmidhuber ensemble is a self-consistent mathematical structure, and
*

*> > is therefore an element of the Tegmark one.
*

*>
*

*> This, I'll buy.
*

*>
*

*> > However, all this implies
*

*> > is that one element of the ensemble may in fact generate the complete
*

*> > ensemble again, a point made by Schmidhuber in that the ``Great
*

*> > Programmer'' exists many times, over and over in a recursive manner
*

*> > within his ensemble. This is now clearly true also of the Tegmark
*

*> > ensemble.
*

*> >
*

*>
*

*>
*

*>
*

*>
*

*>
*

*> --
*

*> Chris Maloney
*

*> http://www.chrismaloney.com
*

*>
*

*> "Donuts are so sweet and tasty."
*

*> -- Homer Simpson
*

*>
*

*>
*

----------------------------------------------------------------------------

Dr. Russell Standish Director

High Performance Computing Support Unit,

University of NSW Phone 9385 6967

Sydney 2052 Fax 9385 6965

Australia R.Standish.domain.name.hidden

Room 2075, Red Centre http://parallel.hpc.unsw.edu.au/rks

----------------------------------------------------------------------------

Received on Sun Nov 14 1999 - 17:02:22 PST

Date: Mon, 15 Nov 1999 11:59:23 +1100 (EST)

This is quite true. However, Newton's theory of gravitation predicts

far more than the Ptolomeic tellurcentric model (because it decsribes

Jupiter's moons as well eg), and General Relativity is better that

Newtonian gravity in precisely the same reason. In each case, we have

a simpler theory providing better predictive power than the more

complex, even if they both had same quantitative utility on the

orginal dataset (the solar system). Maybe I lavished the explanatory

power bit too much. Clearly, a simpler theory has better explanatory

power over a more complex one when both agree with the observations,

as you say.

Now as for the theory's validity or truth - just exactly what do you

mean by that? This _is_ a trick question, so answer carefully :)

This is curious. I've never heard of the SAP being identified with the

God hypothesis before. For me, the strength of the AP (shifting now to

a continuous scale) is related to how much predictive power the AP

actually has (its surprise factor - as it were). If the universe was

going to be conducive to intelligent life for other reasons, than all

the WAP will do is confirm that fact. The stronger the AP, the more

there will be anomalies (or contingencies) in the universe

unexplainable by any other means than the AP. The WAP needs no further

explanation, the SAP does, and traditionally, the two arguments I

mention are the explanantions given.

Shouldn't need to - Schmidhuber does it for me.

Isn't this obvious? What part didn't you understand?

The Schmidhuber ensemble is some well defined entity. The Tegmark

ensemble is less well defined (as Bruno points out endlessly). The "all

possible universes" is completely undefined. Schmidhuber and Tegmark

have two differing definitions of what AUH means. I remain agnostic

as to whether these two different schemes exhaust the possibilties.

That is not the argument. The identity relationship I describe is an

isomorphism relationship between members of Schmidhuber's ensemble and

members of Tegmark's. That is the meaning of a true embedding. Apples

and Oranges are clearly not isomorphic (unless you're almost

completely blind :).

----------------------------------------------------------------------------

Dr. Russell Standish Director

High Performance Computing Support Unit,

University of NSW Phone 9385 6967

Sydney 2052 Fax 9385 6965

Australia R.Standish.domain.name.hidden

Room 2075, Red Centre http://parallel.hpc.unsw.edu.au/rks

----------------------------------------------------------------------------

Received on Sun Nov 14 1999 - 17:02:22 PST

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