Posted by: Alexandre Borovik | September 14, 2009

## MK: A Childhood Story

I was studying at the FeMeSha 18 in Moscow around ’73. I recall being comfortable with the definition of derivative as a limit. On the other hand, the alternative definition that the instructor provided caused me no end of anxiety. Namely, he said the derivative is a number $D$ such that

$f(x)= f(a) + D (x-a) + o( x-a).$

As you correctly point out, it takes a considerable amount of mathematical training to formulate precisely what the problem was. The problem was that the definition says absolutely nothing about how one could find such a “$o()$“, or how to go about SIMULTANEOUSLY (in what sequence?) finding $D$ and “$o()$“. In retrospect, what I must have been bothered by is the non-constructive nature of this definition.

Actually I am currently writing a text on constructivism, and it could be that even after all these years I would still be unable to identify the source of the anxiety were it not for the fact of having understood constructivism better recently.

## Responses

1. Here we go again… “Comfortable with limits,” but not understanding what they mean…

2. As Bill Clinton had put it brilliantly, and when everything else about him is forgotten it will still shine, “it all depends on what ‘is’ is.” Just apply it to “for every epsilon there is delta bla-bla-bla…”

3. Human reading of mathematical formulas is directional: most people are aware of this when they use the equality sign. If one reads from left to right, the equality usually means the result of the computation, evaluation or reformulation of LHS is given by RHS. That’s why formulas like $0=\int_0^{+\infty}...$ usually look artificial.

Recently, teaching in Hebrew (with its right-to-left writing direction) I realized that the same is also true about the inclusion signs $x\in A$ is not the same as $A\owns x$. Poor students…