Not being a mathematician I can say this with great arrogance that mathematics is something quite above and beyond mere calculation. Mathematics is much more than computation.
Yes, of course – but then again perhaps not “of course” but rather, “why?”
Why did Carl Friedrich Gauss say:
“Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.”
Well, calculators simply follow rules. But mathematics determines those rules, yes? No? Yes?
Wait a minute.
But have we determined, or (merely) discovered?
Plato, we have not surpassed you and hardly do we even presume to begin to catch up to you.
Crap! And now we have unwittingly entered the quagmire know as philosophy! Yipes! We must be lost? How did we get here? I simply wanted to know whether the cow I traded to my neighbor was worth the triangle of corn crop in the field and now I have stepped into a predicament that though thousands of years of thinking have been applied to the task, no man has resolved.
Whether we invented number or discovered it, the question has split the mathematicians into opposing, murderous and warring camps. Bullets and blood tried but failed to resolve the cosmic argument: http://www.amazon.com/Great-Feuds-Mathematics-Liveliest-Disputes/dp/0471648779
And voila. From the most certain foundation that man’s feeble reason can produce we step immediately into the deepest mud which he cannot escape. War is upon the minds of men, even in mathematics.
But the calculators crunch their numbers in (ignorant?) peaceful bliss.
In the end, mathematics is ruled and determined by its usefulness. Therefore it is ruled by reality. And this is the reason why you cannot divide by zero and why a set cannot contain itself: it is just not productive. It is not a law of mathematics that determines this, but rather a bias that mathematics should correspond to reality. Mathematics is above all, pragmatic. The theory that grounds mathematics is not merely pie in the sky, but rather, the most practical thing man has ever set forth to do. It is this practicality that demands that mathematics be consistent and complete. And it is this demand that is threatened by Godel, which is why it is such a scandal. Which is why a resolution is not insignificant.
But calculators are of no help here. What is needed is something half mathematical and half philosophical. In the end, reality decides. When Paul Dirac told all the world’s most talented physicists that he, a mathematician, knew more about reality than they did, no one could believe him. But when his equations led to a reality no physicist had ever dreamed of, mathematics was confirmed queen indeed. The positron exits, just as the mathematician said it must.