A year and a half ago I relayed the content of an intense, heavily symbolic dream with obvious pregnancy of urgently-needed deciphering to a colleague. Several months afterwards I learnt of its startling prescience and its clever depiction of a complicated situation involving five people's emotional stances towards each other. This colleague was in a position to realise the gravity of some of my dream but declined to inform me of its literally life-threatening meaning because of his loyalty to a mutual friend. He did so despite the fact that we had both previously collaborated in ensuring that this mutual friend didn't finally succeed at one of his many suicide attempts. I didn't know at this point that the colleague had deliberately wiped the entire contents of my external drive holding years of work and memories at the behest of the mutual friend who was in the grip of crystal meth- and opiate- induced psychosis. The colleague's response to my dream-retelling was, "I'm glad you've reached a singularity." I only knew the meaning of the term "singularity" from my knowledge of physics and mathematics so found the reply baffling - I just immediately sensed pretentiousness and misapplication of a scientific term to the human/psychological/societal/cultural sphere a la Alan Sokal, especially since the colleague is an English major and cultural theorist. Yeah, I'd deck him if I ever saw him again. Gratifyingly I've since read a dissertation by him and was surprised to learn that he can't write and fundamentally misunderstands some basic concepts in the fields of philosophy and law, both of which I'm trained in. The mutual friend had lead me to think he might be a genius. The mutual friend has also told me things that would destroy the colleague's academic career on revelation. But I'm nice.
That was a bitchy little rant. Anyway.
Today I simultaneously, synchronistically stumbled on the concept of "singularity" in two different contexts: the metaphysical and the technological/futurist. This was after a period of re-evaluation of my opinion about the validity of the technical use of the designator "infinity" in mathematics.
My approach to the metaphysics of mathematics is a bit unusual. I think it could be described as "Kantian". I definitely have to elaborate because I don't think there have ever been many philosophers that have actually ever come close to understanding the thoughts of Kant that are relevant here, including Kant himself. I don't care how revered they are/were. Kant's writing itself feels like he was aware that he was touching the boundaries of human comprehension, and that he knew humans are strictly confined to a certain type of rationality that honestly reaches out to truth, but in a particular way that prohibits access to objective/universal/ultimate truth. But this isn't relativism. The first thing to try to understand is that empiricism vs rationalism/idealism is a false dichotomy. I think Hegel got this. Space-time, the physical universe and consciousness are both real and human constructs, as if human consciousness and perception interact with the objective field in a way that structures this field in a particular way for us, but just for us. And at the same time, because we can't exceed the strictures of reason and perception, which are in turn real, it's worthless to speculate about objective reality; we have our own true reality. Again, don't confuse this with relativism. I think Aldous Huxley got this. I definitely can't put it into words. The second thing is that I think even Kant himself underestimated how much of our thinking is synthetic a priori. I really don't think people can intuit even the most simple arithmetic or even small natural numbers. I've written about this before. When I was a kid I was amazed that maths, which proceeds from whole reams of assumptions that are unprovable but intuitively commanding, could proceed in abstract obliviousness to anything outside itself and yet matched up with the physical world, and that some mathematical proofs would just lie around for centuries before some genius applied it to something in nature. Maths is never wrong! So obviously empiricism is out of the question. Strict Platonism was always more attractive but I was still in the darkness of lack of imagination exhibited by those who counter that there is no way the abstract can interact with the physical. Kant finally gave me more of a concrete way of elaborating what I thought was code written into the Universe of which our physical brains are a part, and thus share the same code. Learning maths never felt like learning, it felt like remembering, like inheriting our evolutionary birthright sitting in the depths of our brains all along. But now I'm a Platonist in a Kantian way. Mathematics is written in the sky, but in our sky. And that's where exploration of the ontology of maths ends. I can imagine things like Dialetheism being valid, but not in our Universe.
So I've always had a problem with infinity and asymptotes. They just can't be grasped by human reason, and of course not experienced. And yet they're absolutely essential concepts for a lot of maths, and a lot of that maths in turn is verified by the existence of parallels in the natural world that are beyond any probability of mere coincidence of the abstract with the physical. Infinity isn't something you can even just be agnostic about precisely because it exceeds comprehension. I think Levinas rambled something about this. You simply can't have an opinion about infinity. Full stop.
Then there's Goedel's Theorem.
A word on recursion, which applies both to Goedel and proponents of technological singularity theory. We need to use it but be aware that it's epistemologically problematic.
"Singularity" in metaphysics is the "Omega Point". It's basically an unprovable, spiritual idea that is quite attractive. It's teleological, but I don't personally have a problem with that. The concept of infinity don't present a problem here.
Infinity causes problems for believers in the Technological Singularity. By definition an asymptotic curve cannot reach - let alone progress - past its asymptote. Therefore it's ridiculous to speak of reaching a singularity, or the future beyond a singularity. Time (the x-axis) will not slow down or stop or something when computers' processing power goes into hyper-growth (the y-axis).
A side note on Technological Singularity and Artificial Intelligence: believers mustn't have much understanding of neurology and the nature of human intelligence. The brain is the most complex known system in the Universe. Its functioning involves chemical, electrical and magnetic transmission. It's a chaotic system. There are all kinds of crazy things about it I won't go into. We know next to nothing about it, and by definition, being constrained by the consciousness it produces/facilitates, will never be able to understand it in totality. Computers crash because they're made by fallible humans and rely on insurmountably fallible systems of logic. They'll get faster than us but will always be limited by our limitations.
One last thing: Stephen Hawking makes all kinds of inferential mistakes from his mathematics. Don't take the current theories on event horizons of black holes seriously.