"Riemann himself modestly apologizes for the philosophical shortcomings of his essay on the ground of his inexperience in philosophical matters. But the crudeness of his speculations affords a very striking illustration... of the well-known fact that exclusive devotion to the labors of the mathematical analyst has a tendency to develop certain special powers of the intellect at the expense of its general grasp and strength. Although Sir William Hamilton, no doubt, overstated the case against the mathematicians, I believe that his suggestions are not wholly unworthy of attention, and that there is force in the words of D'Alembert (referred to by Sir William Hamilton)... We have here five distinct propositions, which... may be stated in distinct form as follows: 1. That the nature of space is to be deduced from its concept. 2. That the concept of space can be formed and determined only by its subsumption under a higher concept. 3. That our space is a "triply extended Multiple or Aggregate," the higher concept under which its concept is to be subsumed being that of an "n-fold extended Multiple" or a "multiply extended Aggregate" (eine n-fach ausgedehnte Mannigfaltigkeit), and that—translating Riemann's phraseology into its plain logical import—the (logical) extension of this higher concept determines the number of the possible kinds of space. 4. That the conceptual possibility of space is coextensive with its empirical possibility, though not with its empirical reality. 5. That continuous quantities are coördinate with discrete quantities, i.e., are species of the same genus, both being in their nature multiples or aggregates."