How is error possible in mathematics?

Mathematicians do not study objects, but the relations between objects.

In one word, to draw the rule from experience, one must generalize; this is a necessity that imposes itself on the most circumspect observer.

In the old days when people invented a new function they had something useful in mind.

Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means. The geometer might be replaced by the "logic piano" imagined by Stanley Jevons; or, if you choose, a machine might be imagined where the assumptions were put in at one end, while the theorems came out at the other, like the legendary Chicago machine where the pigs go in alive and come out transformed into hams and sausages. No more than these machines need the mathematician know what he does.

All of mathematics is a tale about groups.

Thus, be it understood, to demonstrate a theorem, it is neither necessary nor even advantageous to know what it means.