Mary have each mastered different parts of the English language and P is related in different ways to the sentences in each part, P means one thing to Fred and something else to Mary. Moreover, if P derives its meaning from its relations with all the sentences of a language, as soon as the vocabulary of an individual changes by the addition or elimination of a sentence, the totality of relations changes, and therefore also the meaning of P. As this is a common phenomenon, the result is that P has two different meanings in two different moments in the life of the same person. Consequently, if one accepts the truth of a sentence and then rejects it later on, the meaning of what one rejected and what one accepted are completely different and therefore one cannot change opinions with regard to the same sentences. Philosophy of mathematics In the philosophy of mathematics, Putnam has utilized indispensability arguments to argue for a realist interpretation of mathematics. In his 1971 book Philosophy of Logic, he presented what has since been called the locus classicus of the Quine–Putnam indispensability argument. The argument, which he attributed to Willard Van Orman Quine, is presented in the book as "quantification over mathematical entities is indispensable for science, both formal and physical; therefore we should accept such quantification; but this commits us to accepting the existence of the mathematical entities in question." According to Charles Parsons, Putnam "very likely" endorsed this version of the argument in his early work, but later came to deny some of the views present in it. In 1975, Putnam formulated his own indispensability argument based on the no miracles argument in the philosophy of science, saying, "I believe that the positive argument for realism [in science] has an analogue in the case of mathematical realism. Here too, I believe, realism is the only philosophy that doesn't make the success of the science a miracle". According to Putnam, Quine's version of the argument was an argument for the existence of abstract mathematical objects, while Putnam's own argument was simply for a realist interpretation of mathematics, which he believed could be provided by a "mathematics as modal logic" interpretation that need not imply the existence of abstract objects. Putnam also held the view that mathematics, like physics and other empirical sciences, uses both strict logical proofs and "quasi-empirical" methods. For example, Fermat's Last