Computer-encouraged serendipity in pure mathematics

Abstract

Computer applications to research in pure mathematics during the past several years have concentrated on using the data processing and logical capabilities of computers to carry out exhaustive and exhausting combinatorial or logical manipulations. Computers can also be used to suggest entirely new theorems and results, but this sort of application has not yet been particularly widely investigated. The addition of an interactive graphics terminal to even a small-to medium-sized computer enables one to construct an "n-dimensional blackboard." On this device a mathematician can display and manipulate two-dimensional projections of n-dimensional geometrical constructs with much greater speed and accuracy than he can bring to his usual blackboard sketches. This sort of facility can be particularly helpful to the type of mathematician who thinks geometrically, and who relies a great deal on geometrical visualizations to suggest new theorems and proofs. Using control dials, joysticks, and other analog input devices, a mathematician can get immediate portrayal of the geometric effect of continuously varying parameters. He also has finger-tip control over the current values of these parameters, encouraging the development of a visceral feeling for the effect of these parameter variations. Early experience with such an "n-dimensional blackboard," implemented in a prototype version on the microprogrammed minicomputer BUGS system at Brown University, is extremely promising. This BUGS blackboard, used in the area of singularities of algebraic curves in complex two-space, has suggested several new theorems and areas for further investigation. A description is given of the facilities of the BUGS blackboard. Also, the observed interaction between the physical facilities available on the BUGS system is mentioned, as well as the sorts of mathematical results that the use of these facilities in the BUGS blackboard seems to encourage.