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

[one_true_leroy]

13. A few points...

You're flirting around with the definition if a limit, but making incorrect arguments. Zeno's paradox is no longer a paradox. The idea of a limit (which is central to analysis, the theoretical basis for calculus and the study of functions) is that if you can get as close as you want to a value- and then still get closer- then the sequence you are building actually IS that number. The number you constructed IS pi, just as .99999... IS 1. I remember ripping my hair out about Taylor series and infinite series. Then came Fourier series, a beast conceived in Hell (or at least in thermodynamics). If you want a book to focus on infinity, look for The Pleasures of Pi,e by YEO Adrian. It is very germane to this discussion.

Infinity is a strange beast. The Greeks, because of Zeno, were very suspicious of infinity ( and irrational numbers) and it was not until the 19th century that the field of real numbers and infinity were well enough elucidated to build a rigorous foundation for calculus.

I hear you when you talk of simple and down home understanding, which works 99% of the time for 90% of the people. However, it's precisely in those 1% moments in those 10% of minds that math reveals its beauty and wonder.

Math is a house of cards, and has been since Euclid. At the foundation are very careful, VERY precise definitions. These truly eliminate ambiguity, but must be accepted exactly as stated. The next layer up are the axioms, the 'rules' obeyed by the objects defined. Then from these two all else in that field of math must be built. Every theorem, every 'truth', must reduce to these axioms and definitions. To 'tweak' these definitions for a down home interpretation is to kick over the foundation. While your down home definition may be more satisfying in the short term, we pointy heads (and remember that everyone begins math with a simple approach) will reject those because the pointy head approach has and continues to yield beautiful results, while the simple approach collapses under the rigor needed to approach and understand that beauty.

Now, that's not saying the simpler definitions prevent you from appreciating that beauty, but they are not going to be sufficient to build that beauty.

Get the Eves book. It's of medium rigor, and your writing suggests that you'd be able to follow and hopefully it can convince you better than I can about the need for pointy heads to make math something more than a bunch of operations. Also, do some research on the history of geometry from Euclid to Hilbert ( which Eves covers in his first three chapters). In that history, you should come to appreciate the liberation of math from ambiguity.