**ERIC Number:**EJ1107510

**Record Type:**Journal

**Publication Date:**2004

**Pages:**4

**Abstractor:**ERIC

**ISBN:**N/A

**ISSN:**ISSN-0228-0671

**EISSN:**N/A

Building Theories: The Three Worlds of Mathematics

Tall, David

For the Learning of Mathematics, v24 n1 p29-32 2004

In this commentary on Matthew Inglis' "Three Worlds and the Imaginary Sphere" (see EJ1106688), David Tall develops the theme that the building of theories is not an easy process. A theory in progress is a particularly delicate creation. Theories do not appear fully formed. There is a period of exploration and incubation that precedes the eventual formulation. In the case of the theory presented here (three worlds of mathematics), although it is moving towards a stable form, it is still in the process of being filled out and refined. Theories are built by reflecting on one's experiences and, "because we have different experiences, we naturally produce different theories or different aspects of a theory that can be made stronger by refinement." The development of the theory of "three worlds of mathematics" has gone beyond random specialization and on to systematic specialization in three areas: calculus (Tall, 2003), proof (Tall, 2002b) and vectors (Watson, Spyrou, and Tall, 2002). Tall presents the definition of the "three worlds of mathematics" as follows: The first grows out of our perceptions of the world and consists of our thinking about things that we perceive and sense, not only in the physical world, but in our own mental world of meaning. The second world is the world of symbols that we use for calculation and manipulation in arithmetic, algebra, calculus, and so on. The third world is based on properties, expressed in terms of formal definitions that are used as axioms to specify mathematical structures (such as "group," "field," "vector space," "topological space," and so on). In building a theory of different worlds of mathematics, Tall concludes, one cannot begin by stating definitions and proving theorems. There must be ideas that are tested out by trying out formulations to see if they make sense to others and to test the ideas in several different contexts (so far, calculus, vectors, and proof) to see if they have a useful practical meaning.

Descriptors: Theories, Mathematics, Mathematical Concepts, Perception, Symbols (Mathematics), Mathematics Instruction

FLM Publishing Association. 382 Education South, University of Alberta, Edmonton, Alberta T6G 2G5, Canada. e-mail: flm2@ualberta.ca; Web site: http://flm.educ.ualberta.ca

**Publication Type:**Journal Articles; Opinion Papers; Reports - Evaluative

**Education Level:**N/A

**Audience:**N/A

**Language:**English

**Sponsor:**N/A

**Authoring Institution:**N/A

**Grant or Contract Numbers:**N/A