by Eliezer Yudkowsky Jan 15 2016 updated Jun 22 2016

Mathematics is the study of numbers and other ideal objects that can be described by axioms.

Mathematics is the study of crisply specified formal objects — for example, numbers — and the ways of knowing their [mathematical_properties properties] — such as [-proofs]. We can see "Logic" as the study of "which conclusions follow with certainty from which premises". Using this definition of logic, we can also see mathematics as the study of logical objects in logical universes — entities whose properties follow from specifications about them, rather than from observation of the real world. The number 3 is a logical object because its behavior follows from axioms about addition and multiplication; Mount Everest is a physical object because we learn about it by physically measuring Mount Everest.