Rigid designators designate whatever they do in all possible worlds. Mathematical definite descriptions are usually considered paradigmatic examples of such expressions. The main aim of the present paper is to challenge this view. It is argued that mathematical definite descriptions cannot be rigid in the same sense as ordinary empirical definite descriptions because—assuming that mathematical facts are not determined by goings on in possible worlds—mathematical descriptions designate whatever they do independently of possible worlds. Nevertheless, there is a widespread practice of (...) treating mathematical definite descriptions as rigid. Apart from this, there might be theoretical reasons for admitting that they are rigid in some sense. The second part of the paper suggests a way out. Borrowing from ideas proposed by Kit Fine, it develops and defends an extended notion of rigidity, which can be applied to mathematical definite descriptions. Importantly, this notion is fully compatible with the claim that mathematical facts are independent of possible worlds. (shrink)