image: kevindooley

There’s a nicely done article in the recent issue of Skeptic that would be of interest to those with an interest in all things foundational. Sensibly titled “Whatever Happened to Stephen Wolfram?”, it poses a question that’s been in the back of my own neural net for a while.

After all, seven years ago Wolfram’s long-awaited work, *A New Kind of Science*, was hailed by some as the Next Big Thing, nothing less than a Principia-style hinge-point in intellectual history. Appropriately enough for such a revelation, its arrival was attended by scandal: the author went outside the mainstream by self-publishing, dispensing with the usual peer-review and journal process altogether. He had spent over a decade secreted away on his researches; when the curtain was drawn aside he simultaneously produced his opus and pronounced its insights revolutionary. In something that looks like cellular Hegelianism, the 1,200 page tome tackles everything from air turbulence to evolution to the existence of free will.

Needless to say, it wasn’t the way science is usually conducted. But then, the title told us that.

The difference between Wolfram and some math crank with a Theory of Everything is that he was already known to be the Macarthur genius prodigy who created Mathematica. His contribution to complexity theory is unquestioned. (Full disclosure: Wolfram is a member of FQXi, but has nothing to do with this blog.) So is NKS the real thing?

Reviews ranged from the awed to the generally positive to the tentative to the outright hostile. Proponents of NKS continue to attend the yearly Wolfram Science Conference, spend the warm months at the NKS Summer School, and post their thoughts here. Critics take Wolfram to task for his personal style, and – more substantively – the question of whether NKS is actually science, or wholly new.

But whether you feel this is Newton reborn or Chaos revisited, no one can accuse NKS of not shooting high. Among a good many other things, it promises an end to our fatuous enthusiasm for mathematical equations as the optimal method for modeling nature, their clunky crudity to be replaced by the “mining” of abstract computational systems. Why hammer at calculus-based descriptions of the end result when we could be empirically testing various cellular outcomes in search of the actual Rule?

I’m surprised, actually, that there isn’t more discussion of NKS here at FQX (that would be AOK). My sense is that even those intrigued by it shy away from is its ontological implications – which is the actual focus of the article in Skeptic. But that’s just where I find NKS most interesting. If any nontrivial computational system is indeed a universal computer, and nature is filled with – or identical to – stacks and stacks of such systems, then what exists is, in a fundamental sense, an infinitely large computer (or possibly a computation). In its largest sense, NKS implies that we are living in, and are ourselves part of, some kind of titanic cellular automaton, chugging through its endless iterations.

The mind begins adventuring with such an idea, though perhaps down some familiar paths. Would confirmation of NKS in nature (as opposed to only in abstract systems) hint that we are in a synthetic reality? At what point do the simulated consciousnesses in a VR do enough research to conclude that their cosmos looks suspiciously like an algorithm? Or, leaving such thoughts aside, if even the vast complexity contained in the evolution of a single cell emerges from nontrivial but still fairly simple rules, might the NKS people be onto the design instructions for building the living cosmos?



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I tend to think that this NKS sort of logic is crucial. The paper I submitted to this essay involves a recursive sequence of quaternions or quivers thereof, based on a branching pattern. This pattern is induced by a tesselation of AdS spacetime. This tessellation is the root space of the E_8 group. The interesting thing about the E_8 group is that the root space is the group! In a hyperbolic setting this induces a fractal geometry. Since a symmetry of the roots are deterimined by the the golden mean this induces a nested Fibonacci sequence as seen in the attachment.

This branching pattern in the the correspondence between the AdS tessellation and field theory means this induces a renormalization of fields. The fractal-like structure, which I relate more in my paper to the Klein quartic system, seems to be a potential way of renormalizing the quantum gravity problem.

Lawrence B. Crowell

attachments: auliflower_Fracta.jpg

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The New about NKS is that we run an automaton over and over with different initial conditions and see what statistics we get as a way to generate probability distributions --- instead of using traditional classical and quantum ways to generate probability distributions relatively directly, with statistics essentially a feature of experimental data that we use to constrain parameters of a class of models.

NKS is too much a model building approach, whereas Old Kinds of Science at their best maintain more of a balance between building specific models and formulating empirical constraints between phenomenological observables as inverse problems for parametrized classes of models.

NKS as it stands, certainly in the book, doesn't have an adequate account for characteristically quantum mechanical phenomena.

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Chaos theory, fractals and their extension into hypotheses of emergent complexity are admittedly not well formulated into a coherent whole. This is also a sort of empirical theoretical study, where dynamical equations are iterated and solutions are observed output. The study also involves running a whole set of initial and boundary conditions and comparing those to a set of dynamical outcomes. Traditionally problems were solved by knowing with an element of certainty how any set of initial conditions were mapped into a future dynamical state. With chaos theory there are Lyapunov exponents which are measures of how this traditional structure breaksdown.

Yesterday I was watching the slushy snow as it accumulated on the outside of a large window slide down to the bottom and form this rather stunning fractal at the bottom. If you step outside the door you almost instantly stumble all over NKS types of systems: the shape of clouds, branching patterns of trees, the winding of a river, and if you look inside the branching patters of blood vessels, nerves in brains, the knitting patterns of kerotin in bird feathers and the examples are huge. The problem with NKS is that in order to avoid stubbing our toes on these complexities we temd to ignore them.

The relationship to quantum physics is a bit odd. A chaotic system has a fractal geometry of paths (or paths on Poincare sections), which when quantized cut off at a scale where a change in momentum approaches hbar/Delta x. Zurek demonstrated how the chaotic motion of the Saturnian moon hyperion due to quantum fluctuations which are amplified by Lyapunov exponential behavior have changed the rotational position of the moon with in a 1/2 century time period. So there is plenty of territory to explore here.

Lawrence B. Crowell

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"there is plenty of territory to explore here."

I think that's what it is all about.

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Exploration; Our powers of observation radiating outward, with the occasional clicking together of insight like some gravitational collapse. Then we wonder when the entire picture will come together and everything fits together so neatly...

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I tend to think honestly that fundamentally everything is actually linear, but due to how things couple together or how linear regions of space are patched together, and so forth, that one arrives at nonlinearity. General relativity is based on a linear group system over a linear basis of 1-forms and so forth.

If science has any prospect of arriving at some semi-complete understanding of the universe, at least up to the limits of what is knowable, then the structure of the universe should conform in some approximate way to how our minds are able to abstract things about it.

Even with NKS and chaos and the rest the nonlinearity is due to some iteration of a function. The nonlnearity or breakdown of predictability comes from an exponential growth of trunctation errors or coarse graininess in how we simulate things. Nature in fact "knows" what it is doing.

Lawrence B. Crowell

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I'm thinking it's the other way, that we are linear, so we see reality as linear. Not that major aspects of reality are not effectively linear, but that is emergent. Think of the linear functions as nodes in the network, yet they are essentially bottlenecks of input from and output to the larger context. The ultimate linearity is the Big Bang universe, but now multiple universes are being proposed. As well as it is filled with objects that formed from matter and energy drawn into them and are shedding and radiating it back out, often to be combined into other forms/nodes. Sort of like the trunk of a tree is a bottle neck between the leaves and the roots, or a bolt of lightning is between the clouds and ground. Are quantum particles inviolate, or is their fuzziness do to emerging from and fading back into the field?

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Lawrence,

Please email me offline at buck.field@fieldoperative.com.

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Stephen Wolfram thank you for WolframAlpha!

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I found the NKS

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Nice NKS find "Anonymous"!

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!Thanks anonymous too, but I should point out that it is easy to spot the real TOE!

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