TAILIEUCHUNG - The Oxford Companion to Philosophy Part 60

The Oxford Companion to Philosophy Part 60. The book is alphabetized by the whole headings of entries, as distinct from the first word of a heading. Hence, for example, abandonment comes before a priori and a posteriori. It is wise to look elsewhere if something seems to be missing. At the end of the book there is also a useful appendix on Logical Symbols as well as the appendices A Chronological Table of Philosophy and Maps of Philosophy. | 570 mathematics philosophy of problems of singular term and prime number is a common noun. If the surface grammar of this sentence reflects its logical form and if there is means there exists then the sentence entails that both the number 1 000 000 and a greater prime number exist. For the realist-in-truth-value this existence is objective and so we are led to realism-in-ontology. In sum if one is a realist-in-truth-value then realism-in-ontology is the result of taking mathematical assertions at face value. Conversely realist-in-ontology at least suggests a straightforward Tarskian semantics. If mathematical objects exist independently of the mathematician then there is no impediment to a straightforward model-theoretic semantics which would presumably render assertions true or false objectively. Despite these natural alliances the literature reveals no consensus on logical relationships between the two varieties of realism. Each of the four possible positions is articulated and defended by established and influential philosophers of mathematics. Kurt Godel and Penelope Maddy adopt both forms of realism Hartry Field and the traditional intuitionists L. E. J. Brouwer and Arend Heyting reject both forms Geoffrey Heilman and Charles Chihara defend realism-in-truth-value anti-realism-in-ontology and Neil Tennant adopts the reverse realismin-ontology anti-realism-in-truth-value see the selected bibliography especially the Benacerraf and Putnam anthology . Typical forms of realism-in-ontology nicely account for the necessity of mathematics and give impetus to the traditional view that mathematical knowledge is a priori. If mathematics is about a realm of eternal abstract objects then mathematical truth is surely independent of any contingencies of the material world around us and mathematics is not known through sensory experience. Of course this alone does not account for the a priori nature of mathematics until the realist provides an epistemology. It is not clear .

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