Is the mathematical avant-garde getting so abstruse that it stretches the limits of the human mind? Is it dangerous when a science becomes entirely dependent upon the calculations of computers? Here’s Sharon Begley in the WSJ:
Mathematicians have become increasingly vexed that some statements about numbers cannot be proved by humans. Worse, the proofs that computers do are so long and complicated that no one can say for sure that the statement being proved really is true, says Prof. Davies.
Two recent computer-aided proofs have this problem. One proved that to color any assembly of shapes, such as a map or a tiled floor, so that no adjoining shapes have the same color, you need only four colors. It’s called the four-color theorem. Another proved that to pack the most spheres in a big box, arrange them like oranges in a crate with oranges in each upper layer resting on the dimple formed by four oranges beneath them. A third proof, only partially complete, will likely run to 10,000 pages if it is ever written down in full, says Prof. Davies, “and would not be comprehensible to any single individual.”
No one has been able to check every line of these three proofs, as your 7th grade geometry teacher did. And the fact that a computer did them robs mathematicians of the joy of understanding how the necessary insight came about; the silicon ain’t talking. There will be more and more proofs that no human mind will be able to follow. As mathematicians try to understand the language of Galileo’s God, they may never be sure they have read it right.
I’m not too worried about the de-humanization of math. For one thing, it’s not as if you can google an answer to the “four-color theorem”. Mathematicians still have to design the computer program, which requires a subtle understanding of the problem they are trying to solve. Secondly, I see the increasing reliance on computers as an example of comparative advantage. We now know that the brain has very real cognitive limitations, especially when it comes to processing large amounts of information, like big numbers and long logic chains. (See George Miller, and “The Magical Number Seven“.) Given these neural constraints, it only makes sense to outsource to microchips the sort of calculations that we can’t do. Our comparative advantage is our imagination.