Thanks for these references. I just went on a quick google of Clifford. Glanced through his textbook on classical mechanics (google books). He starts off with the concept of a center of mass. He looks like another of those guys with a great mind.
I just did a google books on Badiou and google came up with page 34:
``It is here that Dedekind's `ordinal' orientation comes into effect: function f, via the mediation of the concept of the chain, is that which defines N as the space of a total order. The first `point' of this order is obviously 1. For philosophical reasons (compare 1.17) Dedekind prefers a denotation beginning with 1 to one beginning with 0; `1' denotes the first link of a chain, whereas zero is `cardinal' in its very being; it marks lack, the class of all empty extensions''
[BTW Dedekind, Essays on the Theory of Numbers, is in the Dover collection. It's very accessible and at once mathematical...]
Now isn't that above quote interesting? Halmos argued 0 was essential to all ordered sets and built his axiom of infinity on the idea of 0.
So then, looking up sec 1.17 listen to this:
``On the infinite. Dedekind with admirable profunity, _begiins with the infinite_ , which he determines with a celebrated postive property; `A system S is said to be infinite when it is similar to a proper part of itself.' And he undertakes immediately to `prove' that such an infinite system exists. The finite will be determined only subsequently, and it will be the finite that is the negation of the infinite (in which regard Dedekind's numberical dialectic has something of the Hegelian about it). Frege, on the other hand, begins with finite, with natural whole numbers, of which the infinite will be the `prolongation' or recollection in the concept.
On zero. Dedekind abbors the void and its mark, and says so quite explicity: `[We] intend here for certain reasons wholly to exclude the empty system which contains no elements at all.' Whereas Frege makes the statement `zero is a number' the rock of his whole ediface.''
Whatever else Badiou is, he is a serious philosopher in my book. [And like everybody on my shelves, I don't have to own him all the way down. I can steal what I want and leave what I don't.]
I hope people see some of the issues at stake in the above.
The first one that popped into my mind is the interesting ordinal series (Null, 0, 1, ..., Inf). You see that Null functions like a bracket with Infinity at the other end. This problematic idea crops up over and over in number theory analysis because there is something wrong with the basic theorems on limits, the summation of an infinite series, convergence, etc.
The other interesting issue that popped next was that Dedekind and Frege seem like the progenator-followers of the whole constructionist v. formalist, started by Kronecker v. Cantor.
Then the last and most interesting to me, problem is the relationship between this set-number theory discussion and the philosophical problem with the Big Bang hypothesis.
If you follow the (1) school, there was never a (0) or Nothing, before we get to Everything (1). So we have to start with something (1). However, if we start with Everyhting, how can there be any Beginning?
On the other hand if we start with (0) and Nothing, then we get to Everything (1), but we are stuck with beginning and an end.
So how can there be non-being? Nothing sounds like some meaningless statement. Doesn't meaninglessness have to exist, just to understand what a meaningful statement is?
See this is very ontic stuff. Hegel begins his Phenomenology of Mind, hashing out this dilemma of Being and Nothing. It was difficult to follow and so tedious that I just skipped after awhile and moved on. He basically says Nothing and Being need each and going around in circles, getting nowhere. So we are forced to procede. This duality and the need to procede further is the logical outcome that joins nothing and being to produce the synthesis of becoming, which is also the essence of Time
Personally, I am with the Nothing (0) crowd. There is a religious and exitential angle involved. I came from nothing. I am not nothing now. I will return to nothing. (Something from Adorno's probably) So then I am also a Heideggerian in some sense. I mean also to acknowledge that it was partly Heidegger who turned me on to the idea of doing philosophy as if we knew nothing about it...the whole idea of going back to before Socrates.
Cassirer starts his Substance and Function, with a discussion on set theory and its problems.
He begins with the identities Concept <-> Substance, and Substance <-> Space. He historizes these ideas as an Aristotlian view of the world that must be gotten around in order to understand the new discoveries in physics and math.
What he argues is that we have been acculturated into our most basic ideas that date back at least to Aristotle, such as the idea the world or space is made up of a substance and needs a container. And, that for each object of thought there corresponds an object in the world... These ideas need to be re-examined if we are to have an understanding of our emerging and different looking physical world... I forget how it goes from there...
CG