  This paper addresses the actual practice of justifying definitions in
mathematics. First, I introduce the main account of this issue, namely
Lakatos's proof-generated definitions. Based on a case study of definitions of
randomness in ergodic theory, I identify three other common ways of justifying
definitions: natural-world-justification, condition-justification and
redundancy-justification. Also, I clarify the interrelationships between the
different kinds of justification. Finally, I point out how Lakatos's ideas are
limited: they fail to show that various kinds of justification can be found and
can be reasonable, and they fail to acknowledge the interplay between the
different kinds of justification.
