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WILLIAM P. THURSTON ｗｗｗ（＾＾
（参考）
https://arxiv.org/pdf/math/9404236.pdf
APPEARED IN BULLETIN OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 30, Number 2, April 1994, Pages 161-177
ON PROOF AND PROGRESS IN MATHEMATICS
WILLIAM P. THURSTON
(抜粋)
2. How do people understand mathematics?

This is a very hard question. Understanding is an individual and internal matter
that is hard to be fully aware of, hard to understand and often hard to communicate.
We can only touch on it lightly here.
People have very different ways of understanding particular pieces of mathematics. To illustrate this, it is best to take an example that practicing mathematicians
understand in multiple ways, but that we see our students struggling with. The
derivative of a function fits well. The derivative can be thought of as:
(1) Infinitesimal: the ratio of the infinitesimal change in the value of a function
to the infinitesimal change in a function.
(2) Symbolic: the derivative of x^n is nx^(n-1), the derivative of sin(x) is cos(x),
the derivative of f ・ g is f′ ・ g * g′, etc.
(3) Logical: f′(x) = d if and only if for every ε there is a δ such that when
0 < |Δx| < δ,
|{(f(x + Δx) - f(x))/Δx}- d |< δ.
(4) Geometric: the derivative is the slope of a line tangent to the graph of the
function, if the graph has a tangent.
(5) Rate: the instantaneous speed of f(t), when t is time.
(6) Approximation: The derivative of a function is the best linear approximation to the function near a point.
(7) Microscopic: The derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power.

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