Russell Standish wrote:
>> Colin Hales wrote:
>> Hi Folks,
>> I have chewed this thread with great interest.
>> Our main gripe is the issue of emergent behaviour and the mathematical
>> treatment thereof? Yes? This is the area in which Wolfram claims to have
>> made progress. (I am still wading my way through his tome).
>> ***Isn’t the 'algorithmic revolution' really a final acceptance that
>> are behaviours in numbers that are simply inaccessible to "closed form"
>> mathematical formulae?  - That closed-form mathematics cannot traverse
>> complete landscape of the solution space in all contexts?
>If this were the case, the 'algorithmic revolution' is at least 200 years
old, as people have
>known at least this long that most integrals cannot be written in "closed
>Of course, from a practical point of view, it was so expensive to solve
mathematical problems numerically,
> that hardly anyone bothered until the advent of the electronic computer.
Since then, of course
> computational science has taken off like a rocket, and keeps the likes of
me employed. But this is
> hardly new news, or philosophically interesting.
>                                               Cheers

We have a small confusion here (probably caused by my own choice of words).
I’m not talking about the numerical solution to a given mathematical
formula. Agreed: Mundane++. Memories of laboriously iterating on my
calculator come to mind! :-)

What I’m saying (as you seem to agree) is that for some aspects of the
universe (including those aspects we identify as emergent phenomena) there
are no formulae possible in the first place. I think I find the answer to my
original question in the last 2 paragraphs of your paper section 2., as well in your ‘emergence’
thread para where I think you have ‘nailed it’:

………. “As to mathematics predicting emergent phenomena, I believe that the
answer is categorically no. Emergent phenomena are a result of a modelling
process - eg what a brain does, not an analytic process. Mathematics can be
used to describe the emergent phenomenon after it is discovered, but I don't
think the discovery process can really be called mathematics”…………..

Here is another possible confusion: ‘emergence’ as a descriptive artefact vs
‘emergence’ as real layered behaviour in a real system. The wording
initially looks as if you think emergence is not real. The emergence is real
(whatever we consider real is!). Example: There are at least 6 fundamental
layers of emergence from quantum froth to mind. The agreed view appears to
be that any formal mathematics of each layer stops at each layer whereas an
algorithmic approach generates/spans the layers, which are delineated by an
appropriately sensitised observer. Both styles of description seem
appropriate and able to coexist provided their character is understood.

I think we do have a ‘revolution’ and it is a revolution that will force us
to use a wolfamesque rule-based numerical combination descriptive/predictive
method to deal with emergence whether we like it or not because it’s the
only technique that can traverse the layers and there are a multitude of
problems where we need to do exactly that. It seems more than a passing fad.
I can foresee industry based on derivation of elaborate CA to fulfil useful
requirements that cannot be achieved otherwise. I can see scientific method
and Cog. Sci. in particular undergoing a transformation of a sort as a

Not really TOE stuff, so I’ll desist for now. I remain ever hopeful that one
day I’ll be able to understand Bruno…. :-)


Colin Hales

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