AFTER THOUGHT

Computer-aided math is now at a point where unaided human intelligence cannot follow the proofs, a fact that has profound implications for future science, according to James (a former executive at Thinking Machines Corp.). He illustrates this thesis by summarizing the role of different forms of math in the history of science and philosophy. Ptolemy constructed his astronomical theory on a geometrical basis of perfect circles. But when astronomers (notably Tycho Brahe) began to collect data that failed to fit the theory, new mathematical tools became necessary to construct a more accurate model of the cosmos: first algebra, then calculus. Descartes's step-by-step sequential method was matched to the strengths of the human mind and gained its most impressive results from a miserly amount of data. But physical scientists came to scorn ``mere'' data collection. A true scientist worked to discover abstract theoretical principles; collecting data and doing arithmetic were the jobs of assistants. The earliest computers mimicked the methods of human calculators; their main advantages were increased speed and almost perfect accuracy. Advanced computers change all that, handling incredible floods of data with ridiculous ease—and in many cases, in parallel streams. It is no longer unthinkable to simply pile up huge quantities of fact and analyze the resulting patterns. The implications of this are most profound in disciplines to which the sequential maths were least adaptable: meteorology, biology, and economics, all of which generate enormous masses of seemingly chaotic data. The computers can analyze these data and discover patterns, even though the programmers can no longer follow their ``reasoning.'' What this finally means is that we humans will increasingly have to accept computers as equal partners in the enterprise of science—and to accept as valid computer-generated results we cannot begin to understand. A fascinating tour of scientific history, concluding with a vision of a future that is at once exhilarating and profoundly unsettling.