R
Roedy Green
This has been a bugbear of mine since university days. I have often
accused profs of making the simple sound complicated in order to
impress rather than inform. Did you ever read the Algol 68 report? It
is a classic case.
We studied proof after proof in mathematics, but we never saw how one
was actually constructed. It was like studying building construction
without ever seeing anything but a completed building. Mathematicians
were embarrassed about the way intuition and refinement went into a
polished elegant, minimalist proof. They wanted to hide every trace
the scaffolding had ever existed.
Similarly when we were studying group and ring theory, the profs would
smack us down if we ever tried to think about concrete examples
worked. This would contaminate your thinking they asserted. You would
make unwarranted assumptions that applied only to special cases. They
wanted you to argue and think purely from the abstract. I could see
insisting your final proof was purely abstract, but I could not see
why it was so wicked to use concrete examples to give your mind
something concrete to think about.
As a kid I used to sell "cheat sheets" to fellow students for 5 cents
each to give them rules of thumb, and intuitive hooks to help them
solve math, physics, chemistry and genetics problems. Human problem
solving is a sloppier trial-and-error business than we pretend.
One beauty of the Internet is anyone is free to compose instructional
materials with any degree of informality they please.
My contention is humans are superb at generalising from a well-chosen
set of examples. This is much easier than absorbing abstract rules.
This is what you would expect since experience teaches by example.
This is how all animals learn. Very rarely do you get presented with
an abstract theory. Consider how much easier arithmetic is for a
calculator than for a human. We humans are relatively inept even at
applying the relatively simple rules of arithmetic.
Yet a human can solve a difficult problem like who would make a
suitable roommate, one that can be thought of as reasoning from many
examples.