A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form

by Paul Lockhart

The historical development of mathematics (especially in the past couple of centuries) exhibits a consistent, undeniable pattern: first come the problems, whose sources are many and varied, often inspired by the real world. Eventually, connections are made between diverse problems, usually due to common elements that appear in various proofs. Abstract structures are then devised that can “carry” the kind of information that forms the connection (the classic example being the “group” concept, which captures abstractly the idea of a closed system of activities, e.g., algebraic operations like addition, or systems of geometric or combinatorial transformations such as rotation or permutation). New questions then arise concerning the behavior of the new abstract structures—classification problems, construction of invariants, structure of sub-objects, et cetera. And the process continues with the discovery of new connections among the abstract structures themselves, generating even more powerful abstractions. Thus mathematics moves further and further away from its “naïve” origins. Some areas of mathematics, such as logic and category theory, concern themselves with spaces (so to speak) whose “points” are themselves mathematical theories!