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|Mathematics||Field I 20
Mathematics/Identification/Interpretation/Benacerraf: (1965) Thesis: There is an abundance of arbitrariness in the identification of mathematical objects with other mathematical objects:
E.g. numbers: numbers can be identified with quantities, but with which?
Real numbers: for them, however, there is no uniform set theoretical explanation. You can identify them with Dedekind's cuts, with Cauchy's episodes,...
...with ordered pairs, with the tensor product of two vector spaces, or with tangent vectors at one point of a manifold.
Fact: there does not seem to be a fact that decides which identification to choose! (> Nonfactualism).
Field: the problem goes even deeper: it is then arbitrary what one chooses as fundamental objects, e.g. amounts?
Field I 21
Basis/Mathematics/Benacerraf: one can assume functions as fundamental and define sets as specific functions, or relations as basic building blocks and sets as a relation of additivity 1. (adicity).
Mathematics/Indeterminateness/Arbitrariness/Crispin Wright: (1983): Benacerraf's Paper creates no special problem for mathematics:
Benacerraf: "Nothing in our use of numerical singular terms is sufficient to specify which, if any amounts are they.
WrightVsBenacerraf: this also applies to the singular terms, which stand for the quantities themselves! And according to Quine also for the singular terms, which stand for rabbits!
FieldVsWright: this misses Benacerraf's argument. It is more against an anti-platonic argument: that we should be skeptical about numbers, because if we assume that they do not exist, then it seems impossible to explain how we have to refer to them or how we have beliefs about them.
According to Benacerraf's argument, our practice is sufficient to ensure that the entities to which we apply the word "number" forms a sequence of distinct objects under the relation we call "<". (less-than relation). But that's all. Perhaps, however, our use does not even determine this.
Perhaps they only form a sequence that fulfills our best axiomatic theory of the first level of sequences. That is, everything determined by the use would then be a non-standard model of such a theory. And that would also apply to quantities.
Philosophy of Mathematics 2ed: Selected Readings Cambridge 1984
Realism, Mathematics and Modality Oxford New York 1989
Truth and the Absence of Fact Oxford New York 2001
Science without numbers Princeton New Jersey 1980