Before I posted Extensional and Intensional, I had emailed a draft to F. Kafi. The following was his response. –cw
In your example, “Suppose you set out to prove that if $f(x)$ is a differentiable function and $f(a)=0$ and the graph going from left to right goes UP to $f(a)$ and then DOWN after that then $a$ has to be a maximum of the function”, could we have the graph of the function $f(x)$ without being aware of the internal structure of the function; i.e., the mathematical formulation of $f(x)$ such as $f(x):=-(x-a)^2$ or simply its intensional meaning? Certainly not.
Furthermore, what paves the way for the comparison with our real world experiences leading to the metaphoric thinking is nothing but the graph of the function. Therefore, it is the intensional meaning of the function which makes the metaphoric mode of thinking possible.
The intensional meaning is specially required if we are using a grounding metaphor. A grounding metaphor uses concepts from our physical and real world life. As a result we require a medium to connect such real life concepts like “going up” and “going down” to mathematical concepts like the function $f(x)$. The intensional meaning of function $f(x)$ through providing numbers opens the door of the mind to the outer world. This is possible because numbers themselves are the result of a kind of abstraction process which the famous educational psychologist Piaget calls empirical abstraction. In fact, through empirical abstraction we transform the real world experience to numbers.
Let’s consider an example. We see some racing cars in the picture above, a real world experience if you are the spectator of a car match. The empirical abstraction works something like this:
Now we may choose a symbol like "$5$" to denote our understanding of "|||||".
It is now clear that the metaphoric mode of thinking is the reverse process of “empirical abstraction”. For example, in comparing “|||||||||||” with “||||” we may say “A car race with more competing cars is much more exciting than a much less crowded one.” Therefore, “|||||||||||”>“||”, where “>” is the abstraction of “much more exciting than”.
In the rigorous mode of thinking, the idea is almost similar. However, there is an important difference. Here again we have a metaphor. But this time, the two concepts are mathematical. There is no outer world concept. For example, we want to prove a differentiable function is also a continuous one. Both concepts of “differentiability” and “continuity” have rigorous mathematical definitions. Actually, we want to show that differentiability is similar to continuity, a linking metaphor. As a result, we again require a medium to connect the two mathematical concepts. This time there is no need to open the door of the mind to the outer world because the two concepts are in the mind. Hence, the intensional meaning of function $f(x)$ through providing numbers is not helpful. However, we need the intensional meanings of differentiability and continuity of $f(x)$; i.e., the logical definitions of differentiability and continuity.
In the case of comparing the graph of $f(x$) with a real hill we associated dots on the graph with the path on the hill. Right? Here we need to do the the same. We need to associate the $f(x)$’s in the definition of differentailblity to the $f(x)$’s used in the definition of continuity. The $f(x)$’s play the role of dots on the graph. As the internal structure of dots on the graph are unimportant to the association process in the grounding metaphor, the internal structure of $f(x)$’s in the logical definition are unimportant to the association process in the linking metaphor. Therefore, we only need the extensional meaning of the function $f(x)$; i.e., syntactically valid roles it can play in expressions.
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