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Single Idea 18199
[filed under theme 6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
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Full Idea
We may say that indispensability is a pretty strong argument for the existence of at least predicative sets, and a pretty strong, but not as strong, argument for the existence of impredicative sets.
Gist of Idea
Indispensability strongly supports predicative sets, and somewhat supports impredicative sets
Source
Hilary Putnam (The Philosophy of Logic [1971], p.346), quoted by Penelope Maddy - Naturalism in Mathematics II.2
Book Ref
Maddy,Penelope: 'Naturalism in Mathematics' [OUP 2000], p.105
Related Ideas
Idea 18135
If mathematics rests on science, predicativism may be the best approach [Bostock]
Idea 8747
Realists are happy with impredicative definitions, which describe entities in terms of other existing entities [Gödel, by Shapiro]
Idea 18134
Predicativism makes theories of huge cardinals impossible [Bostock]
The
17 ideas
with the same theme
[maths as a necessity for empirical investigation]:
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17536
|
If it can't be expressed mathematically, it can't occur in nature?
[Heisenberg]
|
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18198
|
Mathematics is part of science; transfinite mathematics I take as mostly uninterpreted
[Quine]
|
|
9556
|
Nearly all of mathematics has to quantify over abstract objects
[Quine]
|
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9941
|
Science requires more than consistency of mathematics
[Putnam]
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18199
|
Indispensability strongly supports predicative sets, and somewhat supports impredicative sets
[Putnam]
|
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8857
|
We must quantify over numbers for science; but that commits us to their existence
[Putnam]
|
|
8732
|
It is spooky the way mathematics anticipates physics
[Weinberg]
|
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18150
|
Actual measurement could never require the precision of the real numbers
[Bostock]
|
|
3143
|
Physics requires the existence of properties, and also the abstract objects of arithmetic
[Rey]
|
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10261
|
The application of mathematics only needs its possibility, not its truth
[Field,H, by Shapiro]
|
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18218
|
Hilbert explains geometry, by non-numerical facts about space
[Field,H]
|
|
9623
|
Field needs a semantical notion of second-order consequence, and that needs sets
[Brown,JR on Field,H]
|
|
8863
|
We must treat numbers as existing in order to express ourselves about the arrangement of planets
[Yablo]
|
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18204
|
Scientists posit as few entities as possible, but set theorist posit as many as possible
[Maddy]
|
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18207
|
Maybe applications of continuum mathematics are all idealisations
[Maddy]
|
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14604
|
If a notion is ontologically basic, it should be needed in our best attempt at science
[Schaffer,J]
|
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8712
|
Mathematics should be treated as true whenever it is indispensable to our best physical theory
[Friend]
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