Why Mathematics Cannot Be Merely Invented Nor Purely Hypothetico-Deductive

This paper argues that mathematics cannot be regarded as either a mere invention or a purely hypothetico-deductive enterprise. By extending Goodman’s “grue” problem into the mathematical domain, it is shown that definitional interequivalence between predicates does not preserve epistemic properties such as decidability. Even when two predicates are formally interdefinable, only one may be epistemically admissible within a consistent formal system. This constraint reveals that mathematical language is not a free invention but is shaped by deeper semantic and computational necessities. Mathematics thus occupies a unique epistemological position: it is a discipline in which formal equivalence does not entail epistemic equivalence, and where the very possibility of truth and knowledge is conditioned by the structure of the system itself.