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6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics

[maths as a necessity for empirical investigation]

17 ideas
If it can't be expressed mathematically, it can't occur in nature? [Heisenberg]
Mathematics is part of science; transfinite mathematics I take as mostly uninterpreted [Quine]
Nearly all of mathematics has to quantify over abstract objects [Quine]
Science requires more than consistency of mathematics [Putnam]
Indispensability strongly supports predicative sets, and somewhat supports impredicative sets [Putnam]
We must quantify over numbers for science; but that commits us to their existence [Putnam]
It is spooky the way mathematics anticipates physics [Weinberg]
Actual measurement could never require the precision of the real numbers [Bostock]
Physics requires the existence of properties, and also the abstract objects of arithmetic [Rey]
The application of mathematics only needs its possibility, not its truth [Shapiro on Field,H]
Hilbert explains geometry, by non-numerical facts about space [Field,H]
Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR on Field,H]
We must treat numbers as existing in order to express ourselves about the arrangement of planets [Yablo]
Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy]
Maybe applications of continuum mathematics are all idealisations [Maddy]
If a notion is ontologically basic, it should be needed in our best attempt at science [Schaffer,J]
Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend]