Peirce expressed his pragmatic maxim in the 1870s. If, as Peirce maintained, this original definition is a maxim of logic, it is mainly a maxim of the logic of science, as the title “Illustrations of the Logic of Science” indicates. Pure mathematical conceptions, and the logic of mathematics, if not totally excluded, have at least not been emphasized. During his years at Johns Hopkins University, pure mathematics became his subject of most concern, while logic was also conceived as semiotics during this time. So around the turn of the century, when the popular movement of pragmatism began with James’ “Berkeley Address”, Peirce found that the main difficulty with his original definition of the pragmatic maxim was how to make pure mathematical conceptions clear. He mentioned this problem repeatedly but only gave a tentative solution admitting that, at least according to his original definition, some meanings of pure mathematical conceptions cannot be clarified. This, I believe, is the most important reason for Peirce’s renaming and redefining the pragmatic maxim in semiotic terms. If other pragmatists, and scholars of pragmatism, had noticed this, then most criticisms of pragmatism could have been avoided and the history of pragmatism may have taken a different direction.