Just to make sure we're on the right track, yesterday I reread a book called Incompleteness, on Gödel and his theorems. We've discussed this book in the past, but one of Goldstein's main objectives is to correct the common view that the theorems forever seal us in a closed world of subjectivity, cut off from reality.
She compares this misconstrual to the common misunderstanding of Einstein's theory of relativity, which is actually a theory of absoluteness. Yes, motion is relative, but to an absolute: the speed of light. Nevertheless, many vulgarians think it implies that "everything is relative" or something.
Likewise, a common postmodern interpretation of the Theorems maintains that
the very notion of the objectively true is a socially constructed myth. Our knowing minds are not embedded in truth. Rather the entire notion of truth is embedded in our minds.... Epistemology is nothing more than the sociology of power (Goldstein).
And we all know where this leads: to Foucault and other metacosmic perverts. Could Gödel actually be an oddfather of the left? Not bloody likely. Yes he was crazy, but not that crazy.
Rather, as Goldstein explains,
Gödel's theorems don't demonstrate the limits of the human mind, but rather, the limits of computational models of the human mind (basically, models that reduce all thinking to rule following).
Along these lines, I also re-skimmed Robert Rosen's Life Itself for additional hints at where this series of posts might be headed. Lots of important stuff in there, but I'll try to limit myself to the most relevant passages:
The celebrated Incompleteness Theorem of Gödel effectively demolished the formalist program. Basically, he showed that, no matter how one tries to formalize a particular part of mathematics, syntactic truth in the formalization does not coincide with (is narrower than) the set of truths about numbers.
In short, semantics cannot in principle be reduced to syntax, or meaning to order. This is big. No, bigger than big: infinite. For "no finite set of numerical qualities"
exhausts the set of of all numerical qualities. There is always a purely semantic residue that cannot be accommodated by that syntactical scheme.
Or Woo Hoo!, depending on how you look at it, because it means that meaning always persists despite our efforts to contain it via form. Which goes precisely to our larger point -- or intuition -- that esoterism can never be reduced to the exoteric formulations of dogma.
In one sense this is blandingly obvious, but in another it is anything but, being that for the great majority of people, esoterism is by no means obvious. It may be obvious to you and me, but we're weird. Ofttimes the conventionally religious person makes even less sense to us than does the babbling village atheist. At least the latter isn't an affront to our hindbrain, even if he's a hindrance to our front brain.
Biology may look "soft" to a physicist, but don't confuse complexity with softness. Rather, as Rosen points out, a material system is not necessarily -- in fact, usually not -- a simple system but a complex system, because the latter has more qualities than can be accommodated by mere syntax.
For example, this is why climate science is such a travesty. Economics too, at least the kind that pretends to leap over Hayek's wall and know the unknowable.
Now, meta-questions about this or that discipline aren't usually contained in the field itself -- for example, physics isn't equipped to answer metaphysical questions, and to the extent that it tries, it merely beclowns itself (e.g., via materialism, positivism, scientism, or any other approach that tries to magically explain the higher via the lower).
Gödel, in the words of Goldstein, was primarily interested in shedding metalight on the metalevel of reality. But I would say it was more of a.... not light per se, but a cleaning of the window, so to speak. The theorems themselves aren't the light, but they do allow us to transcend the relative darkness of mere syntactical formulations, if you're following me, and it is this higher metalight that is of interest to us.
But ironically, Gödel himself was limited and hemmed in by his own Platonism. On the one hand he was "committed to the possibility, pace the positivists, beyond our experiences to describe the world 'out yonder.'"
But he limited this yonder to "a reality of pure abstraction, of universal and necessary truths," as opposed to plugging into the living God -- or let's just say to the Cosmic Person, who can never be reduced to any form of syntax, not even to his own name! While he is the logocentric source of speech, he also shatters speech, for the Singer transcends the song.
We're running out of time, and there is much more to come, but let's just stipulate at this point that the Principle that ultimately explains why semantics cannot be reduced to syntax is the Person.