zabu of nΩd;10953163 said:
I just read this. A great overview of the capabilities and limits of logic, though the author may have cut a few too many corners in tailoring the article to a general audience.
After all those historical examples that drive home the point that "the ineffable" cannot be described, he somehow has no problem describing the ineffable as something that can be enclosed in a set.
I also found the argument from ordinal logic a bit suspect. It could do with more unpacking of the claim "After we have been through all the finite numbers (of which there is, of course, an infinity), there is a next number, ω, and then a next, ω+1, and so on, forever," -- especially after he goes on to say that "How far, exactly, the ordinals go is a vexed question both mathematically and philosophically." Also, how do you go from saying there's an infinite number of finite numbers to saying a language necessarily has a finite vocabulary?
One of the biggest problems with Aeon articles is that they make a lot of compelling points but don't develop them at length. Part of this is due to length, as well as the fact that Aeon isn't a peer-reviewed journal and so they don't need to provide the kind of substantive evidence required in most professional publications.
From what I understand - and it's a lot of set theory - he's saying that paradox forms a fundamental aspect of mathematics itself. Zermelo-Fraenkel Set Theory introduced the axiom of choice, which abolished some paradoxes; but the axiom of choice cannot be derived from set theory itself. Set theory is non-axiomatic in and of itself, and thus is theoretically expansive ad infinitum.
Language, at any given point in history, is finite. It is comprised of a totally measurable number of spoken words, utterances, gestures, etc. But it is always evolving and producing; so it is infinite in its potentiality, but finite in its actuality.
Set theory, and mathematics in general, introduce the problem of actual infinity.
zabu of nΩd;10953240 said:
Btw, I left a comment on your blog post "Toward the Nova: the Role of the Dialectic in Modern Thought". Great read.
Thanks man, that's a great comment. I love your suggestion that capitalism might operate, as you say, in a self-reflective manner. One of the most important attributes that scholars often identify in modernist art and culture is an apparent self-awareness and feedback energy through that indeed propels development. I think capitalism is, without a doubt, an effect (if not the driving impetus) of a historical epoch that is experience self-awareness for the first time - that is, experiencing itself as a historical subject, as expressed in Richard Powers's novel.
To be honest, I'm always wary of dialectics, but not because I find them deceitful or misleading. I read a lot of dialectical theory and am often fascinated by it. But dialecticians inevitably seem to pursue some form of
telos, and I believe that this precludes us from perceiving the dialectic as something less anthropomorphic. Let me try and explain quickly:
I think that the dialectic is the "self-refuting form of our relationship to reality" because I think, as a model, it closely approximates what occurs between our conscious perception and the external world. However, there's also a small problem in this reasoning because it separates human consciousness and constitution from reality itself; i.e. in this model, reality is something "out there," and human perception is something "in here." This is a misguided approach, because we - as humans - are part of the real. Our perceptions are material phenomena, not an effect generated apart from reality.
If we want to remain staunch materialists (as I do), then we must not sever perception from reality. Now we enter into the domain of process and flux; our internal perceptions, and our perceiving apparatuses, participate within the fluctuations of matter and energy that dictate all life and things. Now, this seems less dialectical, and indeed it sounds more like Deleuze's resistance to dialectics; but I don't think it means we can get rid of dialectics entirely. Rather, we have to realize that if perceptions are subordinate to the real - that is, if our perceptions are merely effects occurring within reality - then there is no reason to privilege our perspective on the dialectic itself. Instead, we have to entertain the possibility of what the dialectic becomes when we include our perceptions themselves as a part of reality. To paraphrase Hardt and Negri: "Reality is not dialectical, but our perceptions appear to be."
This is the point I'm interested in, and what I'm as of yet working on; but I feel as though there's something productive to say. The really exciting thing is that quite a lot of contemporary fiction that I'm interested in also expresses an interest in these ideas, and grapples with them (which gives me a lot to look at).
