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Single Idea 18214

[filed under theme 6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics ]

Full Idea

Mathematics is in a sense empirical, but only in the rather Pickwickian sense that is an empirical question as to which mathematical theory is useful.

Gist of Idea

Mathematics is only empirical as regards which theory is useful

Source

Hartry Field (Science without Numbers [1980], 1)

Book Ref

Field,Hartry: 'Science without Number' [Blackwell 1980], p.15

A Reaction

Field wants mathematics to be fictions, and not to be truths. But can he give an account of 'useful' that does not imply truth? Only in a rather dubiously pragmatist way. A novel is not useful.

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]