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**1 - 3**of**3**### Zero-Testing, Witness Conjectures and Differential Diophantine Approximation

, 2001

"... Consider a class of constants built up from the rationals using the field operations and a certain number of transcendental functions like exp. A central problem in computer algebra is to test whether such a constant, which is represented by an expression, is zero. The simplest approach to the zero- ..."

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Consider a class of constants built up from the rationals using the field operations and a certain number of transcendental functions like exp. A central problem in computer algebra is to test whether such a constant, which is represented by an expression, is zero. The simplest approach to the zero-test problem ist to evaluate...

### What’s experimental about experimental mathematics? ∗

, 2008

"... From a philosophical viewpoint, mathematics has often and traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments — one of the corner stones of most modern natural science — have had no role to play in mathematics. However, dur ..."

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From a philosophical viewpoint, mathematics has often and traditionally been distinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments — one of the corner stones of most modern natural science — have had no role to play in mathematics. However, during the last three decades, high speed computers and sophisticated software packages such as Maple and Mathematica have entered into the domain of pure mathematics, bringing with them a new experimental flavor. They have opened up a new approach in which computer-based tools are used to experiment with the mathematical objects in a dialogue with more traditional methods of formal rigorous proof. At present, a subdiscipline of experimental mathematics is forming with its own research problems, methodology, conferences, and journals. In this paper, I first outline the role of the computer in the mathematical experiment and briefly describe the impact of high speed computing on mathematical research within the emerging sub-discipline of experimental mathematics. I then consider in more detail the epistemological claims put forward within experimental mathematics and comment on some of the discussions that experimental mathematics has provoked within the mathematical community in recent years. In the second part of the paper, I suggest the notion of exploratory experimentation as a possible framework for understanding experimental mathematics. This is illustrated by discussing the so-called PSLQ algorithm.

### Computer-Mediated Thinking

, 2004

"... This paper discusses computer-mediated thinking and some of its possible implications for curriculum design in mathematics education. We begin with a discussion of today’s context and of ideas related to computermediated thinking. We continue with examples of the use of computermediated thinking in ..."

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This paper discusses computer-mediated thinking and some of its possible implications for curriculum design in mathematics education. We begin with a discussion of today’s context and of ideas related to computermediated thinking. We continue with examples of the use of computermediated thinking in modern applied mathematics. We then extract some suggestions for a curriculum in mathematics centred at the calculus level. We include specific suggestions for removing material from the current syllabus. We end with a discussion of the unintentional power of the calculus. 1