Regarding Godel, there is no doubt that his incompleteness theorems are among the most astounding and puzzling results in formal logic, but the fact that they're so baffling on an intuitive level tends to compel people to impart a quasi-mystical bent on the theorems themselves. This leads non-specialists to frequently invoke the theorems whenever vague notions of "infinity" or "absolution" are brought up, regardless of the context. However, Godel's theorems are only applicable to very specific formal systems of logic that are capable of generating Peano arithmetic. Again, to extrapolate from a highly abstract and symbolic domain of discourse to something like social systems or physical phenomena subject to countless variables just doesn't work. This isn't to say that we shouldn't keep someone like Godel in mind when thinking about those issues, we just need to be careful about precisely how we cull things from his highly formalized thought.
Ah, I see. Thanks for clarifying. It seems to me as though "Infinity" itself presents a pretty big problem for mathematicians/philosophers in general. There's a book that I really want called
The Infinite by A.W. Moore that apparently is a pretty good summary/exposition of the different interpretations of infinity.
The idea of a Planck length, Planck's constant, etc. is indeed fascinating. Somewhat related, take a look at
Graham's number. It's an integer so large that a decimal representation in the observable universe is impossible, assuming each digit occupies a single Planck volume. In spite of the fact that it's
really fucking huge, we actually know the last 500 digits of it. There's something to be said there about the philosophical underpinnings of things in mathematics where we know a certain property of an object, but we can't necessarily "flesh out" the entire object, so to speak (this comes up a lot when you study infinite series and matters of convergence/divergence).
That's awesome; thanks for the link! I love reading about stuff like this. Granted, for me, since I very little direct experience with these sorts of subjects, my interest fails to go much deeper than a superficial fascination with the ideas (bordering, I'm sure, on the "quasi-mystical" thing with Godel, although I take considerable pains to avoid mystifying these concepts). Still, I really think a new complementary structure, or process, between philosophy and science/mathematics is on the horizon, and I'm trying to familiarize myself with the vocabulary so as to be a part of this movement. If Badiou and his successors (Meillassoux, Brassier, Thacker, etc.) are ushering in a rigorous new philosophical process (as I believe they are), then mathematics and science are going to inseparable from that process!
I only have a passing familiarity with set theory. Every mathematician needs to know the basics of axiomatic set theory since virtually every field of mathematics is explained in set-theoretic terms; but when it comes to odd-ball non-standard sets of axioms, I'm clueless. ZFC does resolve Russell's paradox; I'm pretty sure this isn't contested by a significant number of people. The crux of the resolution lies in treating every object as a
set itself, and not saying anything about "elements" that themselves aren't sets, which is the downfall of naive set theory. In spite of ZFC's popular acceptance, I know that its use is far from unanimous. There are a wide range of different axiomatic systems with varying results, and one can't forget about imminently important unsolved problems like the
Continuum hypothesis, which has been proven to be undecidable (cannot be proved or disproved) in ZFC, thanks to the work of Godel.
Ah, okay. I've read about the Continuum hypothesis before as well. With regards to Russell's Paradox: I've always understood it as treating merely the abstract nature of Cantorian set theory, but I was never certain as to how one makes the jump from "naive" set theory to the more complex formula of Ernst Zermelo. From what I understand, it involves the axiom schema of restricted comprehension; but how did Zermelo and Fraenkel arrive at this axiom? Was it purely chosen for its practical applicability and the resolution it offered for Russell's Paradox? If so, that seems quite arbitrary; or, is there some strong mathematical basis for deciding to restrict set theory to actual things/objects that effectually impose their own limits, so as to satisfy certain predicates?
I hope I didn't entirely botch that explanation.
My apologies if the reasoning behind this is rather simple; I'm just a complete ignoramus when it comes to this stuff.
I really need to get around to Meillassoux given the things you've been saying about him. Sentences like "there is no reason to assume that external things truly exist in the way that consciousness perceives them" certainly appeal to me, and his attack on the principle of sufficient reason seems very well-founded; but I'm very skeptical about his proposition of "knowable" noumena and extra-sensory items. It's probably just from my ignorance of his work, but I simply can't help but be skeptical of anything that treats certain actions and traits of our being such as "consciousness", "sentience", and the like as things emanating from a different, "extra-sensory" source in contrast to more base stimuli responses.
I should clarify that, because I'm not sure if Meillassoux means that we can know things
in-themselves in the sense that I phrased it. I believe that he claims we can know at least some
conditions of things; most explicitly, he contends that we can think the world without having to think it as given to/for consciousness.
His major points of contestation involve the philosophies of Locke, Berkeley, and especially Kant, all of whom posited essentially that consciousness amounts to sensory perception and that those objects we perceive as existing exist, basically, for-us, or for our consciousness. This creates that ever-pernicious gap between things and subjective consciousness. Berkeley follows on Locke's thesis of human experience as sensory perception, and concludes that if things only exist because we perceive them, then there must be something that exists separate from humanity perceiving the world in order for it to also exist (that separate being is, for the Catholic Berkeley, God). Kant followed in a similar tradition, particularly challenging David Hume's extreme skepticism, and asserted that, as observers, we can't know the object in-itself, or the noumena that exists behind/beneath phenomena; but all consciousness does share the effect of being conscious, and since collective consciousnesses observe the same objects in a similar way, there is some sense that our perceptions adhere to a certain ideal form, or come very close.
Meillassoux wants to avoid the problem of perception, which he calls the "correlationist circle". He believes that we can arrive at a theory of things-in-themselves even if we can't intuitively know them as they are. For Meillassoux, the big examples of his methodology are the world before human consciousness ever developed (which he calls the "ancestral"), and the "becoming-nothing" of consciousness, or mortality. Meillassoux claims that these concepts, as objects of thought, necessarily entail the absence of consciousness as such; ancestrality pre-exists consciousness, and mortality entails the ceasing-to-be of consciousness. Are we then to think that, since consciousness cannot perceive these objects, they do not exist
as such, but only exist as pre-givens?
Meillassoux basically argues that the correlationist circle results in a logical paradox. Thinking the possibility of ancestrality (the world existing prior to the appearance of consciousness) or the possibility of mortality (the ceasing-to-be of consciousness) means that we must think of these things as
absolute possibilities. If we try and conceive of the existence of ancestrality, or mortality, purely as a correlation of thought (i.e. that these things only exist through our perception of them) then we preclude the possibility of even thinking of ancestrality or mortality in the first place, since they would require our consciousness in order to exist. Correlationism wants to de-absolutize facticity; but in doing so, the capacity-to-be-other of consciousness becomes unthinkable.
Meillassoux contends that, in fact, when we think of our own mortality or ancestrality, we are subverting the correlationist circle, although we've been taught to believe in correlationism since Berkeley, Kant, and onwards. Meillassoux challenges the supposed "Copernican" revolution of Kant, claiming instead that the true Copernican revolution was in the Galileo event and the subsequent flourish of empiricism and the natural sciences. In this light, Kantianism and post-Kantianism become what Meillassoux calls a "Ptolemaic revenge."
