Abstraction as a Natural Process of Mental Compression

This paper considers mathematical abstraction as arising through a natural mechanism of the biological brain in which complicated phenomena are compressed into thinkable concepts. The neurons in the brain continually fire in parallel and the brain copes with the saturation of information by the simple expedient of suppressing irrelevant data and focusing only on a few important aspects at any given time. Language enables important phenomena to be named as thinkable concepts that can then be refined in meaning and connected together into coherent frameworks. Gray and Tall (1994) noted how this happened with the symbols of arithmetic, yielding a spectrum of performance between the more successful who used the symbols as thinkable concepts operating dually as process and concept (procept) and those who focused more on the step-by-step procedures and could perform simple arithmetic but failed to cope with more sophisticated problems. In this paper, we broaden the discussion to the full range of mathematics from the young child to the mature mathematician, and we support our analysis by reviewing a range of recent research studies carried out internationally by research students at the University of Warwick. The term “abstract” has its origins in the Latin ab (from) trahere (to drag) as: • a verb: to abstract (a process), • an adjective: to be abstract (a property), • noun: an abstract, for instance, an image in painting (a concept). The corresponding word “abstraction” is dually a process of drawing from a situation and also the concept (the abstraction) output by that process. It has a multi-modal meaning as process, property, or concept. Gray and Tall (2001) envisaged at least three distinct types of mathematical concept: one based on the perception of objects, a second based on processes that are symbolised and conceived dually as process or object (procept) and a third based on a list of properties that acts as a concept definition for the construction of axiomatic systems in advanced mathematical thinking. Each of these is an