## Saturday, September 17, 2016

(some fairly out-there technical/cosmic musings/speculations …)

This post is best read while listening to 1:39:55 and onwards of https://www.youtube.com/watch?v=GzBslbMJdRU

I have been skimming (not yet carefully reading) some bits and pieces of the radically imaginative physics theories of Matti Pitkanen.   Among other innovations, he founds his physics theories on p-adic analysis rather than conventional differential and integral calculus.   Pitkanen has been looking at p-adic physics for  a long time; but in recent years various applications of p-adic math to physics have gotten more mainstream, with a large number of researchers jumping into the fray.

Inspired by a vague inkling of some of Pitkanen’s ideas, this afternoon I started thinking about the possibility of developing a notion of p-adic uncertainty.   Among other things, I have in mind Knuth and Skilling’s elegant derivation of probability theory from basic axioms regarding lattices and orders, and my sketchy ideas about how to extend their derivation to yield Youssef-style complex-valued probabilities.

Now, one of Knuth and Skilling’s initial assumptions is the existence of a valuation mapping elements of a lattice of events into real numbers.   So, it becomes natural to wonder – what happens if one replaces this assumption with that of a valuation mapping elements of a lattice of events into p-adic numbers?

Perhaps some variant of their symmetry arguments follows through … it seems at least plausible that it could, since the p-adic numbers also have field structure, and also have continuity.

If so, then one would obtain a concept of p-adic probability, with an elegant foundation in symmetries.

Extending this argument to complex numbers, one would obtain a p-adic analogue of Youseff-ian complex probability.

One key difference between p-adic numbers and real numbers is ultrametricity – p-adic numbers obey a strong triangle inequality of the form

d(x,z) <= max{ d(x,y), d(y,z) }

Conceptually, ultrametricity can be modeled via drawing a tree structure with the elements of the ultrametric space at the leaves.  The distance between x and y then corresponds to the number of levels one has to go up in the tree, to form a path between x and y.

If one arranges the elements of one’s event lattice in a hierarchy, one can naturally define an ultrametric distance between the lattice elements using this hierarchy.   Intuitively, it seems that p-adic probability might provide a way of quantifying “size” of lattice elements that correlates with this sort of hierarchical distance.

Viewed in this way, and making all sorts of thinly-substantiated conceptual leaps, one is tempted to think about ordinary probability as heterarchical probability, and p-adic probability as hierarchical probability.  Or in the complex case: heterarchical vs. hierarchical complex probabilities.

It’s not immediately obvious why physics would make use of hierarchical rather than (or along with?) heterarchical complex probabilities.   But with a bit of funky lateral thinking, one can imagine why this might be so.

For instance, it seems to be the case that if one looks at the distribution of distances among sparse vectors in very high dimensional spaces, ultrametricity generally holds to within a close degree of approximation.   This suggests that if one embeds a set of sparse high-dimensional vectors into a lower-dimensional metric space, one may end up doing some serious injustice to the metric structure.   On the other hand, if one embeds the same set of sparse high-dimensional vectors into an ultrametric space, one may preserve the distance relations more closely.   But any ultrametric structure one imposes on finite datasets, if it’s going to have reasonable mathematical properties, is going to be equivalent to the p-adic numbers.

So, suppose the set of events in our universe is viewed as a sparse set of events drawn from a much higher-dimensional space --- then projected into some sort of smaller space to form our universe.   It follows then that, to preserve something of the metric structure that our universe has when embedded in the original higher-dimensional space, we want to model our universe as having ultrametric structure.  But if we also want our universe to have some nice reasonable symmetries, then we end up with a p-adic structure on our universe, rather than a traditional real metric structure.

And finally – I mean, since we’re already way out on a latticework of extremely flimsy and weird-looking ambiguously-dimensional limbs, we may as well go all-out, right? – this would appear to bring us back to my modest proposal that our universe can be viewed as embedded in some broader eurycosm.   If our universe is embedded in some “wider world” or “eurycosm” which is viewed as very high dimensional (perhaps as an approximation of some sort of nondimensional structure), then one would appear to have the beginning of an argument as to how a p-adic foundation for physics would emerge.

It’s also worth noting that a finite non-dimensional structure can be turned into a high-dimensional structure via tensorial linearization – so that, to the extent we can describe a eurycosmic order via Boolean descriptors, we can also describe it via very high-dimensional vectors.    So we have a path from any logical description of a eurycosm, to a picture of our universe as a sparse set of high-dimensional vectors, to a picture of our universe as an ultrametric low-dimensional embedding of these vectors, to a p-adic foundation for physics…

And that, dear friends, is some hi-fi fuckin’ sci-fi !!!!

(I mean ... conspiracy theories about the Rothschilds or the Reptilians or whatever just can’t compete, in my not so humble opinion…)

The question, a la Max Born, is if it’s crazy enough to be (after appropriate fleshing out and tweaking, yadda yadda) pointing in the vague direction of some sort of useful truth…

(“Truth?  What is truth?”)

--> This was a fun post to write – it was written on a flight from the Milken conference in Singapore, where I served on a panel about the future of AI in Asia, back home to Hong Kong.  After blowing much of the Milken audience’s minds with videos of OpenCog-controlled Hanson robots and rather obvious observations about the imminent obsolescence of humanity and the potentials of nanotech, femtotech and superintelligence … I needed to plunge a bit into deeper questions.   Mathematics gives us intriguing, amazing hints at aspects of superhuman realms; though of course superhuman minds are likely to create new cognitive disciplines far beyond our concept of mathematics….
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#### 1 comment:

Stephen Paul King said...

:-) Nice! Now I wonder if Kevin H. Knuth's other ideas can fit in too?
http://arxiv.org/abs/1504.06686
http://arxiv.org/abs/1411.2163