We are both of us eventually going to have to read Badiou...after we better understand Cantor...
Cantor was the first to formulate what later came to be known as the continuum hypothesis or CH: there exists no set whose power is greater than that of the naturals and less than that of the reals (or equivalently, the cardinality of the reals is exactly aleph-one, rather than just at least aleph-one).
Some intellects are more playful than others.
e.g. the Parthenon and the Leon Cathedral. It is such an excellent contrast. So rich! So full of meaning. A building is worth a million words.
Ah yes, I know that Badiou is a math man. As far as Cantor goes, it's probably an inexhaustible mine, but I do understand this:
I'm quite fascinated by the Cantor/Kronecker clash. I sympathize with both, to the degree that I understand either.
For me, it's amusing/annoying that T doesn't see that one/unity is important exactly because it is indeed the model (and essence?) of all simplicity. Also the unary number system is important in the same way. Just as the black painting, however boring in some ways, forces the question of what painting/art is upon the viewer. Same with that famous urinal. Art representing art. Well, I want to look at the root(s) of number. 1 and " "?
I was bored and my inner consumerist got the best of me today and I picked up a book "Badiou and Theology". I think I may even get around to reading it rather than just adding it to my extensive half-read library (I'm trying to call myself out on this so please excuse my self-berating). It's written by a theologist who wants to take some of Badiou's thoughts and copt them for the Christian cause. The author (Depoortere) also offers it as an introduction to Badiou's thought.
Badiou's central thesis: "mathematics is ontology" is based almost entirely upon Cantors work.
Love the analogy between number theory and minimalist art.
It's possible to imagine some more practical Greek mathematician scoffing at Pythagorous in much the same way that the uninitiated scoff at modern art.
Pythagorous a minimalist? Well maybe not...seems that there was a lot of mysticism grafted onto his number theory but maybe that was just for the entertainment of those who were not allowed behind the curtain.
Everyone was religious in the past, North.
Was he just saying what Galileo said, that math was the grammar of nature?
"Just" saying? And only 2000 years beforehand, without the benefit of telescopes or anything we would recognize as hard data?
Do you know that Pythagoras was the first to call himself a 'philosopher'? There were 'wise men' before him but he was the first to create a synthetic rational philosophy based on the combination of reason, mathematics and observation, which he called a 'philosophy'. I still reckon you can make the case that the main divergence between Western and Asian culture commenced with Pythagoras, and he is the main reason the West went on to develop a truly scientific culture, as opposed to a purely religious-mystical one. (Huge claim, I know - but I dare any historian of science to differ with it!)
But the reason these ancients seem 'mystical' might simply be, because 'the mystical' is actually a much deeper, or bigger, view of reality than ordinary people (the hoi polloi - that's us!) have. (I don't think there was anything wrong with your tone incidentally, I am just waxing poetic here.)
That's my view. Of course we can never speak with certainy about that which is "above" us. So we can only understand what we have become.. Now, this is not a denial of the virtue of modesty. But it's a good point I think. We can infer that someone is beyond us, as I infer Euler is way the F beyond me, but we cannot know it, for to know it would be to know exactly what we are claiming is beyond us. (But surely you are using hoi polloi for us with a sense of humor.)
---------- Post added 04-18-2010 at 09:12 PM ----------
One of the themes I am working on is the 'transformation of perception'. When we look, what are the eyes through which we see? As you keep saying, rightly, we are seeing through a mesh of concepts, perceptual constructs and the like. (In Sanskrit, vijnana, or conceptual constructions.)
As Blake said:- "if the doors of perception were cleansed, we would see everything as it is: infinite" (one of the founding phrases of the psychedelic era, and the source for the name of the band, The Doors.) Now I think the pythagorean vision is transformational in this sense. It requires its adherents to 'see differently'. Certainly his is a spiritual/religious vision, and pythagoras a deeply religious figure, but again diifferently to the sense that this is understood in today's world. It is the fact that he is kind of theosophical, but also with his scientific leanings, which make him so fascinating. Archimedes was one of his lineage, and was similarlly a kind of 'mathematical sage' (and genius!).
Anyway, I don't really have a point here, I will do some more reading of the Pythagorean Sourcebook tonight and maybe come up with some more passages. I am just expressing General Enthusiasm.:bigsmile:
Well written, J. Yes, that's my view too. The word "infinite" is tricky, but I think if it is understood as the negation of the finite, it's perfect. Hence my signature. And the high I get from "Reason" does inspire me to use religious myth metaphorically, so I certainly understand the connection. Archimedes is great! I like the Blake line interpreted this way: as the dissolution of dualisms. And of course love is a big part of his system, but one can only say so much about feelings. I think if we foolosophers look to the forms, the feelings will take care of themselves. :flowers: