 Hempel accordingly drew a distinction of fundamental importance between pure and applied mathematics, which he emphasized by using a quotation from Einstein, who had observed, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality” (1921). The existence of alternative and incompatible formal systems, moreover, appears to affect Hempel’s defense of logicism from another direction. If mathematics is supposed to be reducible to logic and logic is supposed to be consistent, then how can alternative geometries be consistently reducible to logic? No one would dispute that they exist as distinct formal systems with their own axioms and primitives, but if these geometries are jointly reducible to logic only if logic is inconsistent, their existence suggests that, perhaps, as formalism claims, it is not the case there is one system of logic that is fundamental to all inquiries.