The text of a 1997 address “On teaching mathematics”, by the great Russian mathematician V. I. Arnold who died in 2010, opens with a provocative statement:
Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap.
The full text of the address is worth a read, whether or not you agree with all of Arnold’s claims. One other passage in particular will resonate with many of us:
Not even mentioning the relative character of initial axioms, one cannot forget about the inevitability of logical mistakes in long arguments… Every working mathematician knows that if one does not control oneself (best of all by examples), then after some ten pages half of all the signs in formulae will be wrong and twos will find their way from denominators into numerators.
The technology of combatting such errors is the same external control by experiments or observations as in any experimental science and it should be taught from the very beginning to all juniors in schools.