 Brouwer and other intuitionists have shown how on this basis arithmetic, real analysis, and topology can be constructed. Moreover, Brouwer considers any exact thought that is not itself mathematics an application of mathematics. For whenever we consciously think of two things in an exact manner, that is, think them together while keeping them separate, we do so, according to Brouwer, by projecting the discrete parts of an empty two-ity onto them (Brouwer 1907: 179n.1 [1975: 97n.1]).