On Tue, Jun 07, 2005 at 05:57:17PM +0200, Bruno Marchal wrote:
> Le 07-juin-05, ? 12:28, Russell Standish a ?crit :
> >On Tue, Jun 07, 2005 at 10:37:10AM +0200, Bruno Marchal wrote:
> >>
> >>OK. it seems to me that (equation 14 at
> >>http://parallel.hpc.unsw.edu.au/rks/docs/occam/node4.html  )
> >>
> >>?
> >>
> >
> >In LaTeX, this equation is
> >
> >\frac {d\psi}{d t}={\cal H}(\psi)
> >
> >It supposes time, but not space (TIME postulate). Moreover, it
> >supposes continuous time,
> Yes but that is a lot of assumptions. Why a linear time capable of 
> being represented by the very special line with the usual topology of 
> the reals? I can imagine many topology on the reals.

I thought the definition of the reals defined its topology? Perhaps
you're using topology somewhat differently.

It is a good question as to why time should have a topological
dimension of 1, and I admit to not having a good answer to that. All I
can say is that computationalism also introduces a time with
topological dimension of 1. Tegmark has some arguments as to why the
topological dimension is 1, and not any other number, but these are
not overly persuasive (the argument depends on properties of 2nd order
PDEs, which already carries a greater amount of baggage)

I only assume continuity to make contact with standard QM. It is an
arbitrary assumption in my opinion, and should rightly be viewed with

> >but I do suggest in the paper how it might
> >be generalised to other possible timescales.
> yes but if you pretend to derive your equation, I don't understand what 
> you mean by generalizing your conclusion (if only by: I have not derive 
> it and it remains some work to do).

Of course there remains some work to be done.

> >Perhaps it also supposes
> >continuity in time for \psi, although this probably flows from
> >assuming continuity of time.
> Why should a function be continuous just because it is defined on a 
> topological space (which is what I assume you are saying when you say 
> continuity of time).

Because then it wouldn't be an evolution. For a state \psi(t') to
depend on the state \psi(t), there must be a corresponding limit
\psi(t')->\psi(t) as t'->t. Of course if t were not continuous, then
this condition is no longer necessary.

> >I do not think time is necessarily
> >continuous - I think it is interesting to explore alternative QMs
> >without this assumption.
> Sure. But again how to talk on derivation then. I mean if someone 
> pretend to derive B from A, then if someone else derive something m?ore 
> general than B from A, it is a critic of the assertion that B has been 
> derived from A. If from facts I can derive the murderer is among John 
> and Charles, I am not so interested in  knowing the derivation can be 
> generalized into leading that the murderer is among John, Charles, Lee, 
> Bruno and Nicole!

To use your analogy, the situation is more like:

Assuming the murderer is John, Charles, Fred or Diana, I have shown
that the murderer must be one of John or Charles. However, if we also
consider Lee, Bruno and Nicole ... there is more work to be done.

The assumption of J,C,F or D corresponds by analogy to the continuity

L,B and N are not continuous - does that sound right? :)

> >
> >The question is whether this is the most general evolution equation
> >for continuous time, or whether there is some more general
> >equation.
> Absolutely.

I think it is, for the reason above. I willing to stand corrected
should that not be the case.

> >Remember, we do have already that \psi is a member of a
> >Hilbert space, so we can write things like:
> OK, but you assume Set theory (that by itself is huge in our context). 
> I show only that you have a preHilbetian space (why should "cauchy 
> sequence of vectors converges).

Yes I do assume set theory. That is stated.

I prove Cauchy sequences converge in the paper. It requires the use of
Kolmogorov axiom no. 6. Hence the space of states is a Hilbert space.

> >
> >\psi(t')-\psi(t) = ...
> >
> >What do you mean by derivability notion for H, and topological notion?
> Topological notion are needed for talking of continuity (a continuous 
> function is just a function from topological space into a topological 
> space such that the inverse image of open set is an open set).. You 
> "assume" the familiar topology of reals, complex number, etc.

Ah yes, you're talking about topological spaces. I just did a quick
refresher course on these using Wikipedia. Indeed, when I make the
arbitrary continuity assumption of time, it is an assumption that time
is a subset of the reals, which has the usual metric and topology
defined. When you say you can imagine many topologies on the reals,
what you are really saying is that you can imagine sets isomorphic to
the reals (1-1 corrspendence), that have many different topologies, eg
the trivial or the discrete topology for instance, or anything defined
by an arbitrary metric. For instance the set of descriptions which is
isomorphic to the reals might more naturally have a Hamming metric
between pairs, which would induce a different topology from that
defined by the reals.

The usual topology also applies for probability, which is defined by
Kolmogorov's axioms as a map X:->R. This induces a topology on the
space of states \psi. It does more than that, of course.

I point to generalisation of the evolution equation by means of the
theory of time scales. I don't know that much about the theory, but I
think it does assume ordering of elements of time as a minimum. It may
also assume a metric - which would in turn induce a
topology. Unfortunately the relevant book is not in the library, and
very little appears on the internet.

> Derivability is a stronger requirement (although some algebraist would 
> introduce many nuances). Someday I will show you make also assumption 
> on consciousness, but that is more subtle, and then all physicist if 
> not almost scientist are doing them when they pretend to solve the 
> consciousness problem like Dennett, or when they put it under the rug 
> (a little bit like Lee in his last posts, I would say).

Please, I would be most interested - both in derivability and consciousness.

> Look Russell, as I said I appreciate your attempt, it is just that, as 
> Hal and Paddy mentionned, there remains quite a lot of work to make it 
> thoroughly communicable. You should really put more clearly your 
> assumptions. You assume a vast part of mathematics, and I would say of 
> physics, mainly with your time postulate and your equation. 

True. The point of the work is that it forces into the open what needs
to be assumed in order for standard QM to work as it does. Things like
set theory, real numbers, Kolmogorov probability axioms and so
on. Some of these things are undoubtedly definitional (one could use a
different set of concepts, but still end up with an isomorphic
theory), whilst others relate to the nature of consiousness. I'm
trying to pack all of the consciousness assumptions into TIME and

> Compare 
> your work with those I have mentionned (I will give the reference for 
> those you don't have yet). 

A reference for Henry would be great. I have the paper by Hardy.

> Don't compare it to quickly  to mine where 
> the assumptions are made still at a much more basic (logical and 
> arithmetical) level. I assume less than Peano arithmetic. I know I 
> could seem a little bit presomptuous, but nuance would make the post 
> more long and more boring. Hope you don't mind, (actually I would be 
> glad someone criticize the most severely possible my work),
> Bruno
> http://iridia.ulb.ac.be/~marchal/


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