Excellent article. I noticed recursively growing series all seemed to have the same ratio, but I didn't realize this was related to e. However, I did see e show up all over the place, which I didn't understand. Anyways, it's obvious I'll need to read more of your articles. I'd much prefer to understand the intuition behind mathematics than just learn the formulae so I can "get things done."
Regarding calc, I had a strange experience where I kept insisting calculus only logically works if it is based on infinitesimals, which I thought was obvious, but the people I was talking with insisted that it was based on limits. They couldn't understand that an infinitely small value has to be more than zero if an infinity of them is to add up to anything more than zero.
In general, my math education seems to have been particularly bad at dealing with infinity.
If d is an infinitesimal value, I'm not sure what d * infinity would be. If d is defined as 1 / infinity, then it would equal 1, but that doesn't seem right.
This textbook teaches calculus using the infinitesimal approach. To do rigorous infinitesimal calculus, you have to define the hyperreal number system.
The tricky thing is that infinity isn't a number as we normally think of numbers. There are different kinds of infinity, so infinity / infinity does not necessarily equal 1. It could be any number between 1 and infinity, inclusive.
To get this intuition, think of the integers. There are obviously an infinite number of them. Now think of the rationals. There are now an infinite number of numbers between 1 and 2, so there are more rational numbers than there are integers; in fact infinitely more.
I believe set theory is the only area of mathematics where different orders of infinite actually mean anything.
So, in set theory, cardinality is a measure of the elements of a set. The cardinality of the infinite set of natural numbers is aleph-0. An infinite set has cardinality aleph-0 if it can be put in one to one correspondence with the naturals. http://en.wikipedia.org/wiki/Bijection This counterintuitively includes the rational numbers also. http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
The real numbers can't be put into one-to-one correspondence with the rational numbers. So the infinite of the reals truly is bigger than the infinite of the rational numbers.
You don't think mathematics is unified, i.e. something true in one area may not be true in another? I don't think a lot of mathematicians believe this, and it isn't clear to me that "infinity" refers to two different things in the two different realms (the other possibility).
Regarding calc, I had a strange experience where I kept insisting calculus only logically works if it is based on infinitesimals, which I thought was obvious, but the people I was talking with insisted that it was based on limits. They couldn't understand that an infinitely small value has to be more than zero if an infinity of them is to add up to anything more than zero.
In general, my math education seems to have been particularly bad at dealing with infinity.