Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

1Visualizing in MathematicsMARCUS GIAQUINTOVisual thinking in mathematics is widespread; it also has diverse kinds anduses. Which of these uses is legitimate? What epistemic roles, if any, canvisualization play in mathematics? These are the central philosophical questionsin this area. In this introduction I aim to show that visual thinkingdoes have epistemically significant uses. The discussion focuses mainly onvisual thinking in proof and discovery and touches lightly on its role inunderstanding.1.1 The context‘Mathematics can achieve nothing by concepts alone but hastens at once tointuition’ wrote Kant (1781/9,A715/B743), before describing the geometricalconstruction in Euclid’s proof of the angle sum theorem (Euclid, Book 1,proposition 32). The Kantian view that visuo-spatial thinking is essential tomathematical knowledge and yet consistent with its a priori status probablyappealed to mathematicians of the late 18th century. By the late 19th centurya different view had emerged: Dedekind, for example, wrote of an overpoweringfeeling of dissatisfaction with appeal to geometric intuitions in basicinfinitesimal analysis (Dedekind, 1872, Introduction). The grounds were felt tobe uncertain, the concepts employed vague and unclear. When such conceptswere replaced by precisely defined alternatives that did not rely on our sense ofspace, time, and motion, our intuitive expectations turned out to be unreliable:an often cited example is the belief that a continuous function on an intervalof real numbers is everywhere differentiable except at isolated points. Even ingeometry the use of figures came to be regarded as unreliable: ‘the theorem isonly truly demonstrated if the proof is completely independent of the figure’