Saturday, August 27, 2011

Semantic Foundations of Mathematics



In the late 80's I wrote a long essay (about 120 pages) on the "semantic foundations of mathematics."  I was quite sure at the time that the failed attempts at "founding" math (by the intuitionist school, formalist school, etc.) could be empirically resolved by resorting to ostension-based semantics based on the hypostasis of extension-based empirical data.  When I got into this idea, it was a time when semantic-based parallel processing was getting tossed around in the artificial intelligence world, so I figured they'd go hand in hand. 

The result of it was this, that a platonic-pythagorean world of numbers need not be postulated.  Those elusive sets of Cantor's infinities, Russell's paradox, Banach-Tarski's ball, the Axiom of Choice, etc. etc. need not be postulated either, i.e., in the universe of mathematical discourse.  The background ontology of the foundations of math gets nominalized to syntactically defined relationships and predication over semantically well-defined objects founded on an empirical lexicon.  You could say that the approach is a synthesis of intuitionism and formalism. 

Do sets exist?  Do numbers exist? Yes they do, according to this theory, as empirical extrapolations, in the semiotic world of language.  Mathematical grammar and universe of discourse make it possible to concoct objects such as the Klein Bottle, in the same way that synthetic a posteriori statements, a la Kant, are possible without any controversy whatsoever.  This  does not imply that Klein bottles exist as empirically ostensible objects.

As an aside, Quine's virtual theory of sets does away with sets altogether, reducing the axiomatic syntax of set theory to the predicate calculus, for example: 

x ∈ A is equal by definition to (∃x) Ax.  

The statement A ∩ B = ∅ is equal by definition to ~(∃x)[Ax ∧ Bx].  

And so on.


Friday, August 26, 2011

Planet Buddha

That which is reductionistically labeled 'animism' is in fact not incompatible with the quantum-theory based paradigm that everything is consciousness. Granted the latter view is epistemic Bishop-Berkeley-idealism, but I am not fond of its potentially Manichean implications: Quantum-field of potential of uncollapsed wave-function = good, material domain of collapsed particles = bad; 'bad' because it is "illusion," a deception of the senses. 

Yes, it is all illusion. But it is default illusion. Everything we experience by way of our five senses is extant as inherently systemic to Being; the Cosmos is the ground-level selection by default inherent to Being-as-such. So if we were to ask why motile beings have two eyes and trees look the way they do, or why there is anything at all, it's all moot; the answer is unobtainable, because everything is as it is, by default. And if we were to assent that the totality is infused with consciousness, that everything is consciousness, and take it one step further, that everything is conscious, we find ourselves living in a conscious, interpenetrating Mandala of which the phenomenal and noumenal are nondual. Mind is matter and matter is Mind. Everything is alive, everything is conscious. This is the stuff of meditation and practice, and not merely philosophical intellection.


Saturday, August 20, 2011

An Artistic Blog-Representation

It's like a Gestalt emanation of the Planet Buddha blog hitherto; it's got the Tsukumo-gami, the Alice in Wonderland motif, the Rabbit of 2011, and the card ... a little morphic-field condensation (and item on sale on Etsy), a nice melange of synchronicity.  Thanks Emi for sharing this link! 


Wednesday, August 10, 2011

R.O.C.K. in the U.S.A

In 1977 I befriended a guy called Frank in Tokyo (This is the same Frank in the blog article The Waiting Sentinel). He eventually became a multifaceted man of the world, but when I met him, the one thing that stood out about him, other than that we had resonated so well together, was that he was a wizard on the electric guitar.  He was a gentle guy, even when plastered on booze. He visited me in Palm Desert in '78.  

A buddy of mine at the time knew some guys who were musicians that lived out in a farm in Indio, so a bunch of us drove out there in a cloud of pot-smoke so Frank could jam with them.  The two guys at the farm (both teenagers) were on drums and electric guitar.  I think they might've been brothers, I don't know, but I had never met them before.  Their "studio" was wired for jamming at least. They played together a little at first to show what they could do. The drummer was actually quite serious, and quite good. The guitarist played simple hard rock riffs and looked more the part than sounded it. 

They hooked up Frank's guitar, Frank tuned it, and Frank said "Do you guys know 'Free'?" The two dudes looked at each other like, huh?  Frank was not from these neck of the woods, and his tastes of course were different.  Frank started playing, the volume was cranked high. The dudes played along, but they looked dismayed (they had the what the fug iz this glazed look).  Others in the room were getting nervous. Frank must've sensed it, so he suddenly switched to Robin Trower's "Day of the Eagle." 

The dudes' faces lit up. They couldn't believe what they were hearing. It was perfect. Frank went with it into the entire song. The drummer was grinning from ear to ear with delight as he drummed away. The other guitarist couldn't follow, he put down his guitar and listened with glassy bloodshot eyes. One song after the other, hard rock, blues-riffs, etc., Frank blew them away. The session lasted about 2 hours.  Everyone in the room cheered.  Calls were made.  The dudes invited all the locals, they came out of the woodwork. 

Suddenly there was a huge party in the middle of a barren desert. Everybody was getting wasted, one way or another.  Everyone was trying to get Frank's attention. "Hey Frank!" "Hey Frank!" He was a local hero. They crowded around poor Frank, stoned, dazed, and confused. Joints and bongs and pipes and bottles got passed his way. And there he was, a metropolitan city-boy from Tokyo, surrounded by all the local desert rats in the desolate desert, suddenly a hero and a celebrity; we partied on late into the night. As usual.


Tuesday, August 9, 2011

Indra's Net


A little philosophical tidbit about reified nonlocality: According to D. Bohm's nonlocal hidden-variables theory (sounds almost contradictory), all particles in the universe at all times can exchange instantaneous info with each other.  It's like Indra's Net, but only superficially.  According to classical mechanics, the instantaneous exchange of particle-information would violate special relativity.

So something called the "transactional" theory has been proposed, which postulates that statistical-probabilistic quantum histories are temporally bidirectional, i.e., they go backward and forward in time.  But ironically, the transactional approach is a reversion to another kind of Newtonian spatiotemporal absolutism. 

The many-worlds, the transactional solution, and so on, in my opinion are inadvertently like the Schrödinger’s Cat argument; they are reductio ad absurdum. They reduce to absurdity, viz., that epistemological realism, materialism, and classical mechanical paradigms cannot account for nonlocality. The missing ingredient is the conscious agent, i.e., consciousness. 

So the special and general theories are at the apex of classical mechanics, because they don’t require the postulation of consciousness to account for why they work, like doing applied math; but when it comes to doing the foundations of math, for example, consciousness in all eventuality elbows its way in, because pure math is a castle in the sky; math is sustained by consciousness, by Nous, by Mind. That’s why math has no inherent bottom, no inherent hypostasis; that’s why Cantor inadvertently conjured the axiom of choice, that’s why the old Hilbert programme failed, that’s why Russell and Whitehead’s proof that 1+1 = 2 failed, that’s why Gödel was able to concoct the incompleteness proof. And so on.  Those Pythagoreans had one horn of the dilemma right. 

Applied math empirically works, the special and general theories of relativity empirically work. With utmost precision. But the empyrean cannot be ontologically extrapolated from the empirical; the empirical is a continuum with infinitely fuzzy borders; the borders fuzz out because there's a complex dynamic at work, beyond the empirical per se.  There's no bootstrapping into Being; everything simply is, by default, like space; but that's not all there is to it.  Quantum theories have to do with the foundations of the empirical world, and as such, like mathematics, the empirical alone cannot account for its own inherent foundation; without the conscious agent, everything becomes absurd, like the probabilistically dead and not-dead cat, the infinite proliferation of probabilistic trajectories of universes, particles traveling instantaneously backward in time to exchange information, and so on. 

As an aside, von Neumann’s establishment of the mathematical foundations of quantum mechanics led to his theoretical conclusion that the collapse of the wave-function cannot be accounted for without the conscious agent. Ironically, by taking his argument one step further, mathematics and physical apparatuses alone can’t account for quantum mechanical conundrums. 

The foundation is both matter and Nous, the foundation is the phenomenal and Mind. Mind is all-pervasive, embraces all, and like Indra’s Net, everything is simultaneously, hologram-fractally interfused, interpenetrated. I'm not arguing about the ontological priority of Nous, however. I’ve heard of the “downward causation” argument, and this is another kind of covert theology, and implicit theogony. We don’t need this, it’s yet another kind of Cartesian Manicheanism. I view Mind and matter as non-dual and nonlocal; Mind and matter are interwoven fabrics of reality.

When Mind and matter are viewed in terms of nonlocality, their all-pervasiveness makes us appreciate everything as precious. Everything participates in amazing infinitude, and it’s all there, inherent in absolutely everything, because everything is Mind, and Mind is everything. There is no room for scientific or ideological hubris in this world. The only proper attitude is that of gratitude. Gratitude for being alive, of being aware, nonlocally aware, to witness the splendor of the universe and everything therein, as the domain of our consciousness, and as our very own Mind.