Humans: we're too smart to really know what we're doing ...

The Science Essay in the Tues, May 25, 2004 New York Times is When Even Mathematicians Don't Understand the Math. (registration req'd)

It's pretty interesting and raises some issues regarding the nature of mathematics and programming discussed in a book by Donald MacKenzie, Mechanizing Proof.
From the essay:

In fact, it is difficult to explain what math is, let alone what it says. Math may be seen as the vigorous structure supporting the physical world or as a human idea in development. Some mathematicians say it is not in the same category as biology, astronomy or geology. While those fields have empirical systems of experimentation and discovery, some might say mathematicians rely on something more intuitive.

"It isn't science," said Dr. John L. Casti, the author of "Five Golden Rules: Great Theories of 20th-Century Mathematics and Why They Matter." "Mathematics is an intellectual activity - at a linguistic level, you might say - whose output is very useful in the natural sciences. I think the criteria that mathematicians use for what constitutes good versus bad mathematics is much more close to that of a poet or a sculptor or a musician than it is to a chemist."

And just as one cannot define what it is that makes a moving phrase played on a violin moving, the essence of the superb equation may also be ineffable.

This makes for a frustrating human dilemma. Our brains have the ability to compute the abstract mathematics they created to construct theories about reality, and yet they may never be smart enough to comprehend those theories, let alone explain them.

Despite his and his colleagues' tireless efforts, Dr. Greene concedes that this paradox ultimately makes sense.

"Our brains evolved so that we could survive out there in the jungle," he said. "Why in the world should a brain develop for the purpose of being at all good at grasping the true underlying nature of reality?"

There we have it ....