On Nov 11, 2012, at 10:16 AM, Joe Niederberger <email@example.com> wrote:
> The common sense view of a continuous function is as I sate further below -- the graph of which I can draw without lifting the pencil. I think the conversion of that intuitive view into a definition involving limits was a great achievement for mathematics, and took some time to hit upon, nothing common sense about it. One must first struggle with Zeno's paradox to appreciate it.
I think you have that backwards. We don't convert common sense into theories. In fact our theories are generally in stark contrast to common sense. This is the dichotomy between concrete and formal thinking. We can't escape common sense and intuition but we must learn how to keep it in check. To prove my point, consider the following. Common sense tells us that a heavier object falls faster than a lighter one. Only after we develop a theory of gravity do we realize that this is not true. But even after knowing this, our common sense still tells us that a heavier object falls faster than a lighter one. It is not an easy task to control that urge. This is why we flounder in a technical subject when we run out of formal reasoning and experience, we revert back to common sense. I would say that we develop our theories in contrast with common sense, not in sync with it.