If you're having math problems, I feel bad for you, son...

[napoleon_in_rags]

15. Yes, I've always had something of a flirtation with limits...

Something of a dance, a desire to cross... Whoops, lost context. Its the Bukowski in me.

I was delighted to learn about limits in calculus in college, they answered some deep questions I had since childhood. But I disagree fundamentally that any argument I've made is incorrect. The limit a thing approaches is not thing thing approached. The classical examples of limits show a number where, at a certain value it is divided by zero, and is thus undefined at that point, but through limits, a value can be approximated through smaller and smaller intervals. But ultimately, we can't throw out the fundamental fact, the number is STILL undefined when divided by zero. The limit is not equal to the value at that point. Everything that comes after the axioms must obey the axioms.

then the sequence you are building actually IS that number

That's where I disagree, because I believe math has a logical basis. If we defined an interval which includes all numbers less than but not equal to C, than C better damn well not be in it, or our system is logically inconsistent. In fact I remember from my college days something called "proof by contradiction" where if we could show a hypothesis leads to a logical contradiction with that starting hypothesis, that hypothesis must be wrong.

But whatever. I see you're a person who cares about math, somebody who is trying to share ideas with me at least through books, and I care about that, I appreciate that. There's a lot of people who don't really have a concept of mathematical beauty. But what I resent is the mathematical purist who has confused rigor with an unnecessary ideological restriction of what the possibilities really are. I took you for a pointy hatter, but your vague reference to real analysis hasn't yet born that out, I must hear far more obscure pedantia to confirm my suspicion. Maybe you are, maybe you aren't. But the bottom line is, I don't care much for any of it.

Especially, infinity, especially the so called "real" numbers. Not too long ago, I read a really good write-up on elliptic curve cryptography. The questions, like many in cryptography, involved not questions about infinite numbers which have solutions in the realm of the Gods, but what real, mortal humans could solve on a real, finite computer. What happens when the system is represented in discrete terms. To me that's the real foundations of math: what us human mortals can know. So in that context, its valid to look at it as information, as a language, as a down home system that down home homo sapiens can represent.

If you're like me, you see mathematical beauty as having a spiritual quality, a language which brings you closer to God. I respect and value people with that insight. But within that context, please meditate on this term I heard some spiritual seeker in India make:

The crowning jewel of my wisdom is the knowledge of my own ignorance.

And ask this question: Could it be that the best math includes the mathematician in the equations?

PEace