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 there >> are behaviours in numbers that are simply inaccessible to "closed form" >> mathematical formulae? - That closed-form mathematics cannot traverse the >> 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 form". > >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

## Advertising

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. http://life.csu.edu.au/ci/vol09/standi09/, 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 result. Not really TOE stuff, so I’ll desist for now. I remain ever hopeful that one day I’ll be able to understand Bruno…. :-) Cheers, Colin Hales