I focus on specific practices in twentieth- and twenty-first-century mathematics of articulating, barring, taming, and operating with what mathematicians widely call mathematical monsters. I describe how over centuries the quotidian procedures of the epitome of rational practice – mathematics – have produced beings outside the extant purified categories understood by theorems and proofs, despite, and sometimes as a consequence of, ever greater precision and rigor. However, mathematical monsters stand in a different relation to their makers than socio-economic and moral monsters (...) who are barred from the world. Indeed, we will see how mathematicians have made plural and productive accommodation with their monsters. Although some mathematicians have been persuaded by logicians to bar monsters from the temple of mathematics by simply forbidding the operations that engender beings against nature, most mathematicians have accommodated or simply cohabited with their monsters by extending axioms, definitions, theorems, and methods of proving theorems as a part of quotidianized practice. (shrink)