Introducing Three Worlds of Mathematics

For several years I have been working with Eddie Gray and others on the ways in which we conceptualize different kinds of mathematical concept were particularly interested in the distinction between objects formed in geometry (such as points, lines, circles, polyhedra) and concepts studied in arithmetic, algebra and symbolic calculus (numbers, algebraic expressions, limits). We concluded that the development of geometric concepts followed a natural growth of sophistication ably described by van Hiele (1986) (though subject to over-elaboration by others) in which objects were first perceived as whole gestalts, then roughly described, with language growing more sophisticated so that descriptions became definitions suitable for deduction and proof. However, numbers and algebra began through compressing the process of counting to the concept of number and grew in sophistication through the development of successive concepts where processes were symbolised and used dually as concepts (sum, product, exponent, algebraic expression as evaluation and manipulable concept, limit as potentially infinite process of approximation and finite concept of limit). We were also intrigued by the way in which experiences in elementary mathematics were reconceptualised from concepts that necessarily had properties to the formalism of advanced mathematics where specified properties are stated first as axioms and definitions, then other properties are deduced by formal proof. In Gray and Tall (2002), we presented the idea that there were three (or possibly four) fundamentally different types of object, those that arise through empirical abstraction (in the sense of Piaget) by which is meant the study of objects to discover their properties, those that arise from what Piaget termed pseudo-empirical abstraction from focusing on actions (such as counting) that are symbolised and mentally compressed as concepts (such as number), and those that arise from the study of properties and the logical deductions that follow from these found in the modern formalist approach to mathematics. Piaget also formulated the notion of reflective abstraction (which is essentially a more sophisticated version of pseudo-empirical abstraction) in which the focus is on actions on mental objects which are routinized, then conceptualised as processes and considered as mental objects at a higher level. Our possible 'fourth type of concept' described in that paper arose from distinguishing between abstract versions of pseudo-empirical abstraction (focusing on actions on mental objects) and abstract versions of empirical abstraction (focusing on the properties of abstract mental concepts). This could be considered as distinguishing between formal generalizations of arithmetic, algebra …