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Single Idea 18210
[filed under theme 6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
]
Full Idea
Why regard the axioms of standard mathematics as truths, rather than as fictions that for a variety of reasons mathematicians have become interested in?
Gist of Idea
Why regard standard mathematics as truths, rather than as interesting fictions?
Source
Hartry Field (Science without Numbers [1980], p.viii)
Book Ref
Field,Hartry: 'Science without Number' [Blackwell 1980], p.-7
The
21 ideas
from 'Science without Numbers'
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9570
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In Field's Platonist view, set theory is false because it asserts existence for non-existent things
[Field,H, by Chihara]
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10260
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Logical consequence is defined by the impossibility of P and ¬q
[Field,H, by Shapiro]
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8958
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In Field's version of science, space-time points replace real numbers
[Field,H, by Szabó]
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10261
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The application of mathematics only needs its possibility, not its truth
[Field,H, by Shapiro]
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8959
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Field presumes properties can be eliminated from science
[Field,H, by Szabó]
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18212
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Nominalists try to only refer to physical objects, or language, or mental constructions
[Field,H]
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18213
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Abstract objects are only applicable to the world if they are impure, and connect to the physical
[Field,H]
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18215
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It seems impossible to explain the idea that the conclusion is contained in the premises
[Field,H]
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18214
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Mathematics is only empirical as regards which theory is useful
[Field,H]
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18216
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Abstractions can form useful counterparts to concrete statements
[Field,H]
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18218
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Hilbert explains geometry, by non-numerical facts about space
[Field,H]
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18219
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Relational space is problematic if you take the idea of a field seriously
[Field,H]
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18220
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Both philosophy and physics now make substantivalism more attractive
[Field,H]
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18221
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'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space
[Field,H]
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18222
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Beneath every extrinsic explanation there is an intrinsic explanation
[Field,H]
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9623
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Field needs a semantical notion of second-order consequence, and that needs sets
[Brown,JR on Field,H]
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18223
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In theories of fields, space-time points or regions are causal agents
[Field,H]
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9917
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'Abstract' is unclear, but numbers, functions and sets are clearly abstract
[Field,H]
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8757
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The Indispensability Argument is the only serious ground for the existence of mathematical entities
[Field,H]
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18211
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You can reduce ontological commitment by expanding the logic
[Field,H]
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18210
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Why regard standard mathematics as truths, rather than as interesting fictions?
[Field,H]
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