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Single Idea 8757

[filed under theme 6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism ]

Full Idea

There is one and only one serious argument for the existence of mathematical entities, and that is the Indispensability Argument of Putnam and Quine.

Clarification

The argument is that science requires mathematics

Gist of Idea

The Indispensability Argument is the only serious ground for the existence of mathematical entities

Source

Hartry Field (Science without Numbers [1980], p.5), quoted by Stewart Shapiro - Thinking About Mathematics 9.1

Book Ref

Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.227

A Reaction

Personally I don't believe (and nor does Field) that this gives a good enough reason to believe in such things. Quine (who likes 'desert landscapes' in ontology) ends up believing that sets are real because of his argument. Not for me.

The 21 ideas from 'Science without Numbers'

9570 In Field's Platonist view, set theory is false because it asserts existence for non-existent things [Field,H, by Chihara]
10260 Logical consequence is defined by the impossibility of P and ¬q [Field,H, by Shapiro]
8958 In Field's version of science, space-time points replace real numbers [Field,H, by Szabó]
10261 The application of mathematics only needs its possibility, not its truth [Field,H, by Shapiro]
8959 Field presumes properties can be eliminated from science [Field,H, by Szabó]
18212 Nominalists try to only refer to physical objects, or language, or mental constructions [Field,H]
18213 Abstract objects are only applicable to the world if they are impure, and connect to the physical [Field,H]
18215 It seems impossible to explain the idea that the conclusion is contained in the premises [Field,H]
18214 Mathematics is only empirical as regards which theory is useful [Field,H]
18216 Abstractions can form useful counterparts to concrete statements [Field,H]
18218 Hilbert explains geometry, by non-numerical facts about space [Field,H]
18219 Relational space is problematic if you take the idea of a field seriously [Field,H]
18220 Both philosophy and physics now make substantivalism more attractive [Field,H]
18221 'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space [Field,H]
18222 Beneath every extrinsic explanation there is an intrinsic explanation [Field,H]
9623 Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR on Field,H]
18223 In theories of fields, space-time points or regions are causal agents [Field,H]
9917 'Abstract' is unclear, but numbers, functions and sets are clearly abstract [Field,H]
8757 The Indispensability Argument is the only serious ground for the existence of mathematical entities [Field,H]
18211 You can reduce ontological commitment by expanding the logic [Field,H]
18210 Why regard standard mathematics as truths, rather than as interesting fictions? [Field,H]