By Ronald Larsen
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Additional info for Banach algebras: An introduction (Pure and applied mathematics)
Writing systems are the prime examples of cognitive technologies, but more specific to mathematics are mathematical notation systems and theories and, more recently, computational devices. Each cognitive technology for mathematics changes the way the subject is done and potentially transforms the practice of mathematical education as well. In some overt and covert ways, it may affect subject matter, curriculum, instruction, learning styles, and the types of cognitive demands made on students. Computer Algebra Systems (CAS)l are the most recent cognitive technologies to create new opportunities as well as dilemmas for mathematics education, particularly at the post-secondary level.
Neither one is satisfactory since these systems do more than manipulate symbols and more than just algebra. The term Computer Mathematics System (CMS) has been suggested (by D. Shumway) but has yet to be adopted. In this chapter we will use the abbreviation CAS to denote either a single system such as MAPLE or the whole family of such systems. 19 In this paper, we begin showing some very simple examples of the power of CAS and then we trace their historical genesis from a limited tool for specialized scientific problems into a cognitive technology for mathematics.
The authors emphasized that the main contribution of computer algebra to scientific work is that it expands the "frontiers of intractability" of a scientific theory. Since exploring the mathematical implications of scientific theories often involves a huge number of algebraic operations, subsequent results, even once they are published in journals, are nearly impossible to follow and verify. However, by using a computer algebra system, verification of such computations is feasible, thus rendering a particular theory more open to scrutiny.
Banach algebras: An introduction (Pure and applied mathematics) by Ronald Larsen