ABSTRACT
This chapter aims to substantiate the view that there are thought experiments in mathematics which involve visualization of physical situations or transformations, often with an idealized aspect. It has three main parts, corresponding to the mathematical areas from which the examples are drawn: knot theory, graph theory and geometric group theory. The visual thought experiments take place in the context of background knowledge about the links between the mathematical definitions and idealized physical objects and transformations that can be visualized. The visual thought experiments neither are, nor serve in place of, mathematical proofs of the conclusions reached, even when those conclusions are true and the thought experiments are reliable ways of reaching them. If visual thought experiments of the kinds can provide reasons for mathematical beliefs, they would provide empirical reasons. But mathematics, as opposed to the application of mathematics to non-mathematical subject matter, is an a priori science.
