## Philosophy Lexicon of Arguments | |||

Author | Item | Excerpt | Meta data |
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Field, Hartry Books on Amazon |
Numbers | I 153 Numbers/Frege/Wright: Frege suggests that the fact that our arithmetical language has these qualities is sufficient to establish natural numbers as a sortal concept whose instances, if they have some, are the objects. - WrightVsFrege: but the objects do not have to exist. - Problem: Frege thus demands that empirical concerns are irrelevant. - Then there is also no possibility of an error. --- II 214 Numbers/BenacerrafVsReduction/Benacerraf/Field: there may be several correlations so that one cannot speak of "the" referent of number words. - Solution/Field: we have to extend "partially denoted" also to sequences of terms. - Then "straight", "prim", etc. become base-dependent predicates whose basis is the sequence of the numbers. - Then one can get mathematical truth (> truth preservation, truth transfer) - E.g. "The number two is Caesar" is neither true nor false. (without truth value). --- II 326 Definition natural numbers/Zermelo/Benacerraf/Field: 0 is the empty set and every natural number > 0 is the set that is the only element which includes the set which is n-1 - Definition natural numbers/von Neumann/Benacerraf/Field: Every natural number n is the set that has the sets as elements which are the predecessors of n as elements. Fact/Nonfactualism/Field: it is clear that there is no fact about whether Zermelos or von Neumann's approach "presents" the things "correctly" - there is no fact which decides whether numbers are sets. - That is what I call the Definition Structural Insight: it makes no difference what the objects of a mathematical theory are, if they are only in a right relationship with each other. |
Fie I H. Field Realism, Mathematics and Modality Oxford New York 1989 Fie II H. Field Truth and the Absence of Fact Oxford New York 2001 Fie III H. Field Science without numbers Princeton New Jersey 1980 |

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Ed. Martin Schulz, access date 2017-03-24