Formal calculation with Maple

The spectacular performance of electronic calculation has long since led scientists to entrust numerical calculation to computers, resulting in a division of tasks: while machines were allowed to handle numerical applications, i.e. approximations, man, believing himself to be the only person capable of reasoning and carrying out an algebraic calculation, reserved control of accuracy for himself. The emergence of formal calculus systems, capable of performing algebraic calculations far beyond human capabilities, has challenged this reassuring division.

The introduction in 1995 of symbolic calculation software in science preparatory classes hastened this development in France. After hesitating between Mathematica and Maple, it was the latter, less expensive and easier to learn, that was adopted by the majority.

Without any prior knowledge, this gradual discovery of Maple software is not a simple instruction manual: as we explore the software, we want to emphasize the general characteristics of formal calculus by raising a few questions about it:

• reliability: can you prove a theorem using Maple? How is an algebraic expression represented in formal calculus? How does the software simulate a mathematical activity?

• how to use it: should we prefer to execute instructions one by one, interactively, or write programs? Which programming style is best suited to formal calculus? What type of data should be used for geometry, analysis or linear algebra?

• its impact: how does formal calculation change the way we work? Do we still need to know mathematics? Can Maple handle everything?

Our ambition is to show that symbolic calculation can significantly modify the practice of scientific work, and so we've drawn on a few examples, which are few in number but thorough. Although they only call on the mathematical knowledge of first-year higher education students, they nevertheless require careful reading.