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Single Idea 18218

[filed under theme 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics ]

Full Idea

Facts about geometric laws receive satisfying explanations, by the intrinsic facts about physical space, i.e. those laid down without reference to numbers in Hilbert's axioms.

Gist of Idea

Hilbert explains geometry, by non-numerical facts about space

Source

Hartry Field (Science without Numbers [1980], 3)

Book Ref

Field,Hartry: 'Science without Number' [Blackwell 1980], p.27

A Reaction

Hilbert's axioms mention points, betweenness, segment-congruence and angle-congruence (Field 25-26). Field cites arithmetic and geometry (as well as Newtonian mechanics) as not being dependent on number.

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]