It is a sad and threatening concept that mathematics has no connection to reality. This idea has discouraged many young folks from being interested in a living and mandatory field of study.
We study mathematics for very utilitarian reasons - to build things, to pass a class and get through school, to balance a check book, to calculate the prices of goods and serivces, and on and on and on.
We study mathematics for more passionate reasons - guys might want to impress girls with how smart they are (very sadly, I have met some otherwise bright young women who have deliberately done poorly in mathematics so as not to embarrass their boyfriends), someone might just be interested in a particular problem, collaboration with others, defend ourselves against liars (especially car salesmen) and on and on and on.
http://news.byu.edu/archive12-mar-mathletes.aspx
We MANDATE the study of mathematics for very important social and personal reasons - Learn how to Learn, learn how to organize information, learn how to think logically, learn how to solve problems that have not previously been solved, and on and on and on.
As far as your proof goes, line up five apples and write the number '1' next to each apple. Count the apples. Count the '1's. Did you get five both times? Can you assign apples to '1's in a one-to-one correspondence? I think we're done.
Anyway, that's the way I see it.
