Context: Radical Constructivism is an issue that deeply divides the cognitive science community: most researchers reject it, but an increasing number do not. Problem: Constructivists stress that our knowledge starts from experience. Some (“ontic” constructivists) deny the existence of a mind-independent world, while others (“radical” constructivists) claim merely that, if such a world exists, we can know nothing about it. Both positions conflict with scientific realism. It is not clear that the conflict can be resolved. Method: This paper uses philosophical argument to ask whether constructivism can be rationally preferred over realism in science. Results: Ontic constructivism cannot be disproved by any knock-down argument. Nevertheless, it is irrational to accept it, because it ignores the strategy of “inference to the best explanation”: realism is the best explanation of the successes of science. Radical constructivism, too, fails to explain these successes. Some radical constructivists have tried to offer theories more sympathetic to realism. For instance, Ernst von Glasersfeld sees science as a coherent ordering of experience, and appeals to Piagetian psychology as support. There are close similarities. But Piaget was also caught in a constructivist anti-realism, despite his attempt to evade it. Implications: The constructivist’s claim that scientific concepts and theories are generated by human minds is correct. But this important insight should not be used to deny realism, which is the best explanation of the many undeniable successes of science and engineering.