The Mind of the Mathematician

Chapter 3. The Dynamics of Mathematical Creation

42 Chapter 3 } The Dynamics of Mathematical Creation Mathematical Cognition Cognitive activities are the active processes through which knowledge is acquired, such as perception, attention, memory, and learning. Interest in these activities has a long history, stretching back to classical times; today there is a vast literature on the subject. We cannot possibly cover this here, but fortunately a handbook of mathematical cognition has recently been published, containing twenty-seven essays on di√erent aspects of the subject (Campbell 2005). The authors of the essays seek to throw light on such questions as: ⭈ How does the mind represent numbers and make mathematical calculations? ⭈ What underlies the cognitive development of numerical and mathematical abilities? ⭈ What factors a√ect the learning of numerical concepts and procedures? ⭈ What are the biological bases of number knowledge? ⭈ Do humans and other animals share similar representations and processes? ⭈ What underlies numerical and mathematical disorders, and what is the prognosis for rehabilitation? Only some of the essays are directly relevant to the present work, but the reader who wishes for an up-to-date overview of research on mathematical cognition will find that the handbook provides it. Incidentally, the term ‘‘mathematics,’’ in cognitive science, does not usually extend beyond simple arithmetic; much of the research is limited to young children. The Dynamics of Mathematical Creation 43 Most abstract concepts are metaphysical in nature, drawing on the inferential structure of everyday bodily experience to reason about abstractions . Time, for example, is primarily conceptualized in terms of motion— either the motion of future times towards the observer or the motion of an observer toward a ‘‘time landscape.’’ For centuries the mathematical concept of continuity was based on the idea of motion—the motion of a physical object with definite direction and speed. Such motion proceeds without gaps, interruptions, or discontinuities. For example, Euler described a continuous curve as ‘‘a curve described by freely leading the hand.’’ This simple idea proved to be extremely rich and powerful and helped generate one of the most beautiful and productive branches of all mathematics: seventeenthcentury calculus. Formally the mathematical function does not move, but cognitively the function does move, does approach limits. Further discussion of such matters may be found in Where Does Mathematics Come From? by Lako√ and Nunez (2000). Cognitive styles are self-consistent modes of functioning, shown by individuals in their perceptual and intellectual activities, that correspond to habitual ways in which individuals organize and process information, solve problems, and make decisions. Although there are many variants, the two main cognitive styles are the intuitive and the logical, or analytic, styles. An intuitive person seeks to obtain a broad perspective on a problem and get an overall feel for it, reaching a conclusion fairly rapidly. An analytic person tends to take more of a logical step-by-step approach before deciding on a solution after a period of reflection. Neither cognitive style is generally preferable to the other, although one may be better than the other for certain tasks. It has been suggested that the left side of the brain is the analytic side, the right side of the brain the intuitive side, but this is an oversimplification. Most people use a mixture of cognitive styles, varying according to the matter under consideration. For over a century psychologists have been interested in cognitive styles: they distinguish between visual thinkers and verbal thinkers, between intuitive thinkers and logical thinkers. It seems to be generally assumed that most creative mathematicians think mainly in pictures, although no survey to establish this conclusively has yet been carried out. We conjecture that the answer might be di√erent for mathematicians working in di√erent branches of the discipline, thus one might expect geometers to be visual thinkers, analysts to be verbal thinkers, and so on. Even today, despite all that has been written on the subject, there does 44 Tour of the Literature not seem to be a clear answer as to what proportion of mathematicians are mainly visual thinkers and what proportion are mainly verbal. Most mathematicians are intuitive thinkers, relying on the unconscious mind to a large extent, and there is some evidence to support this idea. In the case of mathematicians of genius, such as Poincaré, a good deal of research activity appears to take place at a relatively unconscious level. Although the di√erence between intuitive thought and logical Aristotelian thought was already recognized in the...