Interactive programming and automated mathematics

Abstract

The theme of these symposium proceedings is Interactive Systems for Experimental Applied Mathematics. Certainly, there can be little confusion about the basic meaning or broad fundamentals of applied mathematics. However, questions as to the value of the experimental approach toward mathematics are an entirely different matter. The present state-of-the-art is so scant in empirical or theoretical guidelines that this approach must be acknowledged as an expression of faith that a computer, used to explore ill-defined mathematical constructs and problems, might yield powerful insights and a fruitful methodology. The term interactive is difficult to define and I cannot pretend that I fully understand the relevance of the term as applied to some specific systems. However, in some sense, we would suppose that what we mean is close to the definition used by the physicist. When two effects interact, they do so in a nonseparable and usually nonlinear way. Usually, the effect of the interaction is that the whole is not simply equal to the sum of its parts, and that the characteristics of the interacted system can be surprisingly different from the qualities of its constituent parts. Therefore, in the man-machine interaction, we would expect more than in the old process of inputting a well-formulated set of directions with the machine performing in its capacity as an idiot savant. Obviously we can expect that any serious attempt at man-machine interaction will involve on-line response. However, I would not entirely exclude the possibility that significant interaction can occur off-line.