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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
17 ideas
with the same theme
[maths as a necessity for empirical investigation]:
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17536
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If it can't be expressed mathematically, it can't occur in nature?
[Heisenberg]
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18198
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Mathematics is part of science; transfinite mathematics I take as mostly uninterpreted
[Quine]
|
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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
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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]
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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]
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3143
|
Physics requires the existence of properties, and also the abstract objects of arithmetic
[Rey]
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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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18218
|
Hilbert explains geometry, by non-numerical facts about space
[Field,H]
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9623
|
Field needs a semantical notion of second-order consequence, and that needs sets
[Brown,JR on Field,H]
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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
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Mathematics should be treated as true whenever it is indispensable to our best physical theory
[Friend]
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