Visualizing in Arithmetic

Abstract

Geometry lends itself to visualization, but arithmetic, it seems, does not. Why not? One thought is this: geometry is peculiar among mathematical subjects because many geometrical items, especially shapes and transforma- tions, are visible and consequently visualizable; numbers, by contrast, are purely abstract and therefore not even potentially visualizable. This thought is definitely wrong. The objects of geometry are also purely abstract: no physical line is without breadth; no physical edge is perfectly straight; no physical surface is perfectly smooth and continuous; no motion of a physical body is perfectly rigid. Our initial concepts of geometrical ob- jects are constructed (by abstraction and idealization) from our experience of the physical world; and they are constructed so as to be applicable to observ- able features of it, applicable, that is, in the approximate way that suffices for practical purposes. The ability to use visual imagination in geometry derives at least in part from our visual experience of applications of geometrical con- cepts. Parallel remarks may be made about arithmetic: our experience of counting, of counting-adding-counting on, of subtracting and recounting, of arranging in rows and columns, etc., surely provides the possibility of visual- izing in arithmetic. Granting the possibility of visualizing in arithmetic, a more robust chal- lenge can be made to its utility. The thought again proceeds by contrast with the use of visual imagination in geometry. Here it is in outline. Visualizing may be an epistemically acceptable way, perhaps a reliable way, of arriving at belief in general theorems of geometry. For example, one may come to be- lieve a general theorem about circles by visualizing operations on a single circle, and this might be legitimate because all circles are the same in shape. But one cannot legitimately arrive at any general theorem about all numbers of a given kind if there are many of that kind, such as all primes, by visualiz- ing operations on one number of things; this is because no two numbers are numerically the same. Therefore, whatever rewards visualizing may bring in