Learn math by counting apples

Now if I have the two apples I counted yesterday, and you give me
three more apples, I can count them again and I know there will be
five more examples on free online math help. I don't need to know anything else
about the apples to know this; addition is a property of the numbers themselves,
independent of any other properties the apples may have.

This means I can forget
about the apples themselves, and even, if I wish, forget about the
process of counting and the perception involved in doing that. Here we
have entered the realm of mathematics, Here in math help where we deal with abstract
numbers rather than specific counted items. And within this realm,
since we are no longer dealing with perception, what we say can be
exactly true - although when we take our conclusions back to the
"real" world, we will have to deal with the possibility that our
perceptions are inaccurate: me math helper counted wrong, apples can disappear
spontaneously, or whatever.

I might try to apply addition to something
for which it doesn't work (adding, say, a liter of sugar to a liter of
water, and expecting 2 liters of sugar water); then the problem is not
in the math but in the application. It is the job of science (or
merely of experience) to determine what mathematical models apply to a
given situation.