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Re: When the going gets tough
<Don't give up on calculus; it really is easier to learn than it looks.
Think of it as a natural extension of algebra (which in turn is a natural
extension of arithmetic).>
I disagree, but forgive me a moment for a tangent to a tangent.
I like to say that calculus was invented at least 3 times: By Newton
(working from geometry), Leibniz, and my grandfather. He was a skilled
machinist who grew up on a farm and never went to high school. After his
heart forced retirement, he pursued roses and math. With the roses, he
rediscovered Mendel well enough to have much of the population of Bridgeport
show up to admire his yard.
With the math, he read high school algebra textbooks from the library and
extended the logic, inventing his own terms. When a friend of my father's
was having trouble with calculus in college, he asked what the problem was.
Recognizing the ideas, though not the terminology, he was able to do a good
job of tutoring.
I'm not my grandfather. For something like 30 years now I've argued with a
friend (a math Ph.D. but for the thesis; he decided to take a break to learn
ballroom dancing and labor relations, now teaches stats in college) that the
area of a circle cannot be determined. The formula assumes squares small
enough to fit the outline of a curved surface, and that can't happen until
the squares become points, in which case they can't measure... And so many
functions don't work until you reach infinity, and by definition infinity
can never be reached.
I really enjoy algebra, though.
What's the difference? The logic. Calculus and other higher forms have
their own, and if you're flexible enough, you can use two logic systems
depending on what you're thinking about, the same way that some people can
think in two languages. Languages also incorporate varying logics. I'm
rigorous, ok rigid, enough that I'm hardwired not to accept that what's
contradictory in one logic is perfectly acceptable in another. Quantum
mechanics was invented to vex me.
Many people can learn alternate logics without agonizing over every
contradiction from verbal/everyday logic, but I break arguments into single
steps to worry each one. For you and my grandfather calculus may be
enjoyably self-consistent and self-evident, but for me and a few other
people the inherent assumptions are downright painful.