We are all aware of the many ways in which ideas and our ideas about ideas influence the propagation, acceptance and adoption of ways of thinking -- especially those ways of thinking that impact ways of doing. Personally, i never cease to marvel at scope of this. For example, it was very clear that the imaginations of those thinkers in the 20's and 30's who were crafting the agenda for physics for the next hundred years, those imaginations and imaginers were captivated and compelled by images that were startlingly removed from the human scale. The large and the small best measured in distance from the human scale by counting orders of magnitude held sway in the imaginations of those who were framing the physical theories most of us think as intellectual ediface. What is the human-scale frame for general relativity? What is the human-scale framing of quantum mechanics? It's only in the past couple of decades that things like GPS (note the 'global' in that moniker) or quantum cryptographic protocols have put these theories into strands that weave into the fabric of daily life for most folks.
At the opposite end of the spectrum we have notions of computation whose verisimilitude, the very life-likeness of which is a stumbling block for adoption. There are essentially two successful compositional models of computation: the lambda calculus and the pi-calculus. These are really representatives of classes of models, so forgive me for employing a little metonomy. At their root, both make an common ontological commitment: both rest on the idea that being is only witnessed through doing. We can only classify programs in terms of how they behave. Beyond this fundamental commitment, however, there is a marked difference in world-view and outlook. The functional paradigm does not support the idea of autonomous composition. In the functional world there is always a 'head' term in a composition, and the head enjoys an essentially governing role in the computation.
By contrast, the pi-calculus supports a notion of autonomous composition. Terms in parallel composition are essentially peers. They freely mix and may interact/compute/evaluate/reduce in a non-deterministic (ungoverned) manner. i submit that this is much, much more life-like. This is a model of computation that much more closely aligns with my personal understanding of the physical world. Moreover, it is a model much more closely aligned with the western ideal of human society. It's proto-market, in the sense that it's the sort of conditions necessary for markets to emerge -- in a western viewpoint.
In my experience, however, this model hits a real stumbling block in its adoption in various human communities. Part of this -- i will concede -- is the commitment to doing as the only witness for being. Many, many people keep asking "where's the result? where's the data" and it takes them a long time to get to a bridging notion like, "ah, the result is a channel to a process that behaves like the data i was looking for." But, part of this is the deep-seated inability to understand a notion of composition that supports autonomous execution with the possibility of mutual engagement. i need only point out that this idea escaped every model of computation until the '70's. That's 3K recorded years of computing (yes, Euclid and Pythagoras were computing) and thinking about computing before we stumble on the obvious idea that computation is a form of interaction. With the exception of Chemistry, none of the dominant mathematical apparati supporting physical theories supports this kind of composition.
And then we wonder why a society of peers is so far from our grasp. As a parent i have learned a lot from watching my children engage with toys. Many of my cherished ideas about nature and nurture were trashed merely from careful observation of how my 5 children engaged their playthings. There's a way in which mathematical theories are just that: toys. They're reduced, idealized versions of situations and phenomena. Observing our models of computations can be like a parent watching children play. The kinds of 'toys' we make and the kind of games we play say a lot about who we are as a society and a species. There is very little mystery about the struggle to reach a society of peers. Our proclivities are writ large -- like brand names on the packaging and very form -- of our toys. Actually, in a funny way, the situation is so profoundly skewed in one direction that the fact that we ever stumbled on such a radical model as computation as interaction is cause for hope. If phylogeny recaptulates ontogeny, maybe the picture of children learning even from playing with the same old toys is a picture of a deeper process that holds true for all of us.
Functional Equations VII: The p-Norms
5 days ago