The Semantics of Mathematical Language
Nicolas Goodman Principal Investigator
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The central question which motivates Professor Goodman's study is what is the nature of mathematical reality. Professor Goodman is seeking in his research to develop a philosophy of mathematics which embodies the central insights of both the classical and the constructive traditions. Such a philosophy would be "Platonistic" in its ontology but constructivist in its epistemology. He found a way to accomplish this goal by using Lewis' "S4" semantics as the underlying logic which allowed for a reconciliation of the classical and intuitionistic positions. Under this grant, Professor Goodman will fill in his argument for the classical mathematical realism from the "hardness" and objectivity of mathematical statements and the empirical verifiability of their consequences. He will argue that experience with these features of mathematics allows us to move from assertability conditions to truth conditions, but that assertability conditions still play an indispensable role in the semantics of irreducibly intentional parts of mathematics. Using epistemic theories which Professor Goodman interprets as combining truth and assertability conditions, he will attempt a systematic account of actual mathematical discourse. This research project combines Professor Goodman's deep understanding of mathematics with subtle philosophical analysis. His prior work clearly demonstrates his ability to present clear, well-organized and provocative arguments for his positions. His study promises to make a significant contribution to our understanding of the nature of mathematical realism and its implication in ordinary mathematical language as it is found in textbooks, monographs and journals.