_Posted on August 27, 2024_ In my History of Mathematics course, the first assignment given was to write a short answer the title question. I post my response here: Mathematics is the study of the set of all implications (of the form $P$ implies $Q$, where $P$ and $Q$ are propositions) which may be formally reasoned about. In this formal reasoning there is a natural tendency toward a general study of abstraction. Formal reasoning _is_ in essence structured reasoning by way of relationship. Therefore, Mathematics is in this sense also the study of relationship between abstracted entities (via the formal reasoning about propositions involving such abstract entities.) ^c9ae31 The basic ideas of this "definition" come from a few places: Russel's _Principles of Mathematics_, Chapter 1 Personal experience (reading, philosophizing, _doing_ mathematics, etc...) Conversations