An irony in the nature of mathematical knowledge
During a recent discussion with Prof. Gangal, my PhD advisor, I noticed an irony of mathematical “counter-examples”: On the one hand, there are usually many more pathological cases than regular cases, but on the other hand they are difficult to construct.
Take rational numbers versus irrational ones for example. Rational ones are easy to imagine. Irrational ones need more explanation. But on the other hand, rational numbers are countable, whereas irrational numbers are uncountable. Even within the irrational numbers, there are simple ones like the square root of two, and then there are not so simple ones. Again the simpler ones are rarer than the rest.
Another example is that of smooth functions versus non-smooth ones. In some sense of cardinality, smooth functions are rare in the space of all possible functions, but they are the ones that are the easiest to imagine or construct.
My advisor looked at it from the other end: What we can easily imagine or construct is what we call regular, and we study them more extensively. We call others as pathological.
Illustrates that the human knowledge is finite, while the unknown is infinite!