I’ve been writing a bit about infinity, so I thought it might be good to take a step aside and look at this some more.
Imagine a line. Now remove the middle third. You have two shorter lines with an equal-sized space between them. Now remove the middle thirds of the two lines you have left. Keep going!
You are creating something like a Cantor set. It was discovered by the brilliant mathematician Georg Cantor in the 1880s. Cantor got into a lot of trouble for his thoughts on infinity. But his discoveries laid the foundations for set theory, Gödel’s Incompleteness Theorem, and Alan Turing’s thinking on Artificial Intelligence.
If you think about it, the Cantor set contains an infinite number of points. Yet it also contains an infinite number of no-points! It appears to contain two different infinities. Does this make it weirdly larger than an infinity of points alone?
Talk about holding infinity in the palm of your hand. A two-dimensional version is known as Cantor dust: infinite dust, and infinite no-dust. If you make a three-dimensional version, you will produce something like a Menger sponge, a fractal object with infinity spaces and infinity points. You can’t squeeze a Menger sponge. But there’s something there all the same.
The strange stranger I referred to in the last posting is like the Menger sponge. Somehow, we have discovered infinity on this side of phenomena.
Who or what is a strange stranger? The category includes, but is not limited to, “animals,” “nonhmans,” and “humans.” In The Ecological Thought I refrain from using the word “animals” (unless in quotation marks). “Nonhumans” strictly refers to the set of those entities who are not Homo sapiens.
Now behold this Menger-sponge-like strange stranger, Astrophyton darwinium:
Alain Badiou refers to his Lacanian “set theory” as “pre-Cantorian.” (See Kenneth Reinhard’s essay in The Neighbor.) Now I’m not convinced you can actually have pre-Cantorian set theory—this would be like having pre-Newtonian gravitational theory (strike one against Badiou!). But you can have a non-Cantorian set theory. This has to do with whether or not you accept Cantor’s Continuum Hypothesis, a project that ended up driving him insane. The Continuum Hypothesis states that there is no set whose size is strictly between the set of integers (1, 2, 3…) and the set of real numbers (rational numbers—integers and fractions—plus irrational ones like pi). As far as I know (I’m no mathematician) the issue is open right now. I’d like to know more about this, and I’d like to know why Badiou and Lacan appear hostile to Cantor.
Intuitively, I find Cantor’s view of infinity (nay, infinities) very satisfying. Since I am by no means a mathematician I can’t explain this properly. Still, I believe that the kind of infinity to which Lévinas refers when he writes of the other (autrui)—my strange stranger—is not “beyond” this side of reality, if by “beyond” we mean an outside. An outside would imply an inside—and this would imply a metaphysical system. Inside–outside distinctions are the basic ingredients of metaphysics.
I find the idea of an ontologically incomplete Universe where there is no neat holistic nesting of parts in wholes very satisfying, though at present I lack the precise language in which to articulate this idea.
Rigorous materialism must take seriously the seemingly theological idea that infinity is on this side of reality. I believe that work on infinity will counteract the Heideggerian tendency in ecological discourse. Since I hold that we cannot avoid a form of fascism unless we circumvent Heidegger, I also believe that this work is of the utmost political significance.
Burying our heads in the vulgar materialist sand, or the utilitarian environmentalist sand, won’t do.
In general, we humanities scholars need some remedial math and science lessons. I’m dismayed that I have nothing but vague intuition to go on in suspecting Kenneth Reinhard’s essay (noted above) of Badiou hagiography—mostly the preponderance of “According to Badiou”s in it.
I would love it if a kind Romanticist would help me. Paging Arkady Plotnitsky…