I don't think we can truly comprehend
the infinite. I think our concepts of the infinite are ultimate finite. But then one might have to define "truly comprehend."
My current theory is that math is founded on an intuition of unity, and that this unity is systematically iterated in many many ways. We can't calculate with transcendental numbers except in closed form, or so it seems to me. Consider the use of radians. 2/pi is an answer, right? But if we want to apply this answer technologically, do we not have to approximate this answer? When bankers or physicist calculate with the number e, they have to use an approximation at some point. When one wants to apply a number, it has to be put in a closed form. But transcendental numbers aren't closed, really. We are never finished calculating them, even if we have all the precision we need. A transcendental number is something like an algorithm. An implied set of instructions for calculating digits.
Take e for example, defined as the limit n --> infinity (1+1/n)^n. Infinity is the instruction in this algorithm (metaphor, of course) to increase n
to the precision desire, with no upper bound.
We fly high into abstract realms, which are great, but must always return to a finite number of precise digits to interact with the real world.
---------- Post added 05-29-2010 at 04:23 PM ----------
How surprising! I enjoyed that paper. Yes, I'm in the intuitionist camp somewhere. I do
see the attractions of formalism, but I also see objections.
To look at a page of symbols and see particular symbols
is already, in my opinion, the work of that founding intuition I have argued for. I argue that human experience is automatically discrete. We break the world into pieces, into objects. We break a page of math into individual symbols visually, and then conceptually "decode" the relationships represented by/in these symbols.
In that paper, chess is talked of as if it is not considered a mathematical system. I personally consider it to be one. Because it's a precise abstract system.
Anyway, thanks for the links. I would like to hear your opinions in more depth sometime, if you ever feel like sharing them...