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Computers, Reasoning and Mathematical Practice
"... ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of ..."
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ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of r of R then R is commutative. Special cases of this, for example f(x) is x 2 \Gamma x or x 3 \Gamma x, can be given a first order proof in a few lines of symbol manipulation. The usual proof of the general result [20] (which takes a semester's postgraduate course to develop from scratch) is a corollary of other results: we prove that rings satisfying the condition are semisimple artinian, apply a theorem which shows that all such rings are matrix rings over division rings, and eventually obtain the result by showing that all finite division rings are fields, and hence commutative. This displays von Neumann's architectural qualities: it is "deep" in a way in which the symbol manipulati...
A Sceptic's Approach to Combining HOL and Maple
 Journal of Automated Reasoning
, 1997
"... . We contrast theorem provers and computer algebra systems, pointing out the advantages and disadvantages of each, and suggest a simple way to achieve a synthesis of some of the best features of both. Our method is based on the systematic separation of search for a solution and checking the solution ..."
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. We contrast theorem provers and computer algebra systems, pointing out the advantages and disadvantages of each, and suggest a simple way to achieve a synthesis of some of the best features of both. Our method is based on the systematic separation of search for a solution and checking the solution, using a physical connection between systems. We describe the separation of proof search and checking in some detail, relating it to proof planning and to the complexity class NP, and discuss different ways of exploiting a physical link between systems. Finally, the method is illustrated by some concrete examples of computer algebra results proved formally in the HOL theorem prover with the aid of Maple: the evaluation of trigonometric integrals. Key words: Proof checking, automated theorem proving, computer algebra, complexity theory JEL codes: ? 1. Theorem provers vs. computer algebra systems Computer algebra systems (CASs) seem superficially similar to computer theorem provers: both a...
MathXpert: Learning Mathematics in the 21 st Century
"... ABSTRACT: MathXpert is a computer program designed to help students learn algebra, trigonometry, and onevariable calculus. Its scope extends from elementary algebraic manipulations to the most complicated limits and integrals usually considered in University classes. It is capable of generating com ..."
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ABSTRACT: MathXpert is a computer program designed to help students learn algebra, trigonometry, and onevariable calculus. Its scope extends from elementary algebraic manipulations to the most complicated limits and integrals usually considered in University classes. It is capable of generating complete, stepbystep solutions to mathematical problems, but normally this capability is used to assist the student in developing his or her own stepbystep solution. The design of MathXpert follows principles dictated by its intended use to support learning; these principles and how they influenced MathXpert are discussed. The paper concludes with a discussion of the current and future use of technology in general, and MathXpert in particular, in mathematics education. KEY WORDS: mathematics education, computer algebra, calculus 1.