In his 1989 book "The Emperor's New Mind", Roger Penrose commented on the limitations on human knowledge with a striking example: He conjectured that we would most likely never know whether a string of 10 consecutive 7s appears in the digital expansion of the number pi. Just 8 years later, Yasumasa Kanada used a computer to find exactly that string, starting at the 22869046249th digit of pi. Penrose was certainly not alone in his inability to foresee the tremendous power that computers would soon possess. Many mathematical phenomena that not so long ago seemed shrouded and unknowable, can now be brought into the light, with tremendous precision.
In their article "Exploratory Experimentation and Computation," to appear in the November 2011 issue of the Notices of the American Mathematical Society, David H. Bailey and Jonathan M. Borwein describe how modern computer technology has vastly expanded our ability to discover new mathematical results. "By computing mathematical expressions to very high precision, the computer can discover completely unexpected relationships and formulas," says Bailey.
Mathematics, the Science of Patterns
A common misperception is that mathematicians' work consists entirely of calculations. If that were true, computers would have replaced mathematicians long ago. What mathematicians actually do is to discover and investigate patterns—patterns that arise in numbers, in abstract shapes, in transformations between different mathematical objects, and so on. Studying such patterns requires subtle and sophisticated tools, and, until now, a computer was either too blunt an instrument, or insufficiently powerful, to be of much use in mathematics. But at the same time, the field of mathematics grew and deepened so much that today some questions appear to require additional capabilities beyond the human brain.
"There is a growing consensus that human minds are fundamentally not very good at mathematics, and must be trained," says Bailey. "Given this fact, the computer can be seen as a perfect complement to humans—we can intuit but not reliably calculate or manipulate; computers are not yet very good at intuition, but are great at calculations and manipulations."