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
f′◦ g ∗ g, etc.
(3) Logical: f (x) = d if and only if for every ǫ there is a δ such that when 0 < |∆x| < δ, |f(x + ∆x) − f(x)|/∆x− 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.
This is a list of different ways of <i>thinking about</i> or <i>conceiving of</i> the derivative, rather than a list of different <i>logical definitions</i>. Unless great efforts are made to maintain the tone and flavor of the original human insights, the differences start to evaporate as soon as the mental concepts are translated into precise, formal and
explicit definitions.
I can remember absorbing each of these concepts as something new and interesting, and spending a good deal of mental time and effort digesting and practicing
with each, reconciling it with the others. I also remember coming back to revisit these different concepts later with added meaning and understanding.
The list continues; there is no reason for it ever to stop.
http://mishap.sdf.org/by:thurston/thurston_On_proof_and_progress_in_mathematics.pdf