Introducing Three Worlds of Mathematics

For several years I have been working with Eddie Gray and others on the ways in whichwe conceptualize different kinds of mathematical concept (Tall, 1995; Gray, Pitta, Pinto& Tall, 1999; Tall et al, 2000; Tall, Thomas, Davis, Gray, Simpson, 2000). Eddie and Iwere particularly interested in the distinction between objects formed in geometry (suchas points, lines, circles, polyhedra) and concepts studied in arithmetic, algebra andsymbolic calculus (numbers, algebraic expressions, limits). We concluded that thedevelopment of geometric concepts followed a natural growth of sophistication ablydescribed by van Hiele (1986) (though subject to over-elaboration by others) in whichobjects were first perceived as whole gestalts, then roughly described, with languagegrowing more sophisticated so that descriptions became definitions suitable fordeduction and proof. However, numbers and algebra began through compressing theprocess of counting to the concept of number and grew in sophistication through thedevelopment of successive concepts where processes were symbolised and used duallyas concepts (sum, product, exponent, algebraic expression as evaluation and manipulableconcept, limit as potentially infinite process of approximation and finite concept oflimit). We were also intrigued by the way in which experiences in elementarymathematics were reconceptualised from concepts that necessarily