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Single Idea 8958

[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers ]

Full Idea

Field's nominalist version of science develops a version of Newtonian gravitational theory, where no quantifiers range over mathematical entities, and space-time points and regions play the role of surrogates for real numbers.

Gist of Idea

In Field's version of science, space-time points replace real numbers

Source

report of Hartry Field (Science without Numbers [1980]) by Zoltán Gendler Szabó - Nominalism 5.1

Book Ref

'The Oxford Handbook of Metaphysics', ed/tr. Loux,M /Zimmerman,D [OUP 2005], p.36

A Reaction

This seems to be a very artificial contrivance, but Field has launched a programme for rewriting science so that numbers can be omitted. All of this is Field's rebellion against the Indispensability Argument for mathematics. I sympathise.

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]
18220 Both philosophy and physics now make substantivalism more attractive [Field,H]
18219 Relational space is problematic if you take the idea of a field seriously [Field,H]
18222 Beneath every extrinsic explanation there is an intrinsic explanation [Field,H]
18221 'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space [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]