Ideas from 'Science without Numbers' by Hartry Field [1980], by Theme Structure
[found in 'Science without Number' by Field,Hartry [Blackwell 1980,0631130373]].
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4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
9570

In Field's Platonist view, set theory is false because it asserts existence for nonexistent things

5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
10260

Logical consequence is defined by the impossibility of P and ¬q

6. Mathematics / A. Nature of Mathematics / 3. Numbers / a. Numbers
8958

In Field's version of science, spacetime points replace real numbers

6. Mathematics / B. Foundations for Mathematics / 2. Axioms for Geometry
18217

Hilbert's geometry is interesting because it captures Euclid without using real numbers

18221

'Metric' axioms uses functions, points and numbers; 'synthetic' axioms give facts about space

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
8757

The Indispensability Argument is the only serious ground for the existence of mathematical entities

6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
18212

Nominalists try to only refer to physical objects, or language, or mental constructions

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
10261

The application of mathematics only needs its possibility, not its truth

18218

Hilbert explains geometry, by nonnumerical facts about space

9623

Field needs a semantical notion of secondorder consequence, and that needs sets

6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
18215

It seems impossible to explain the idea that the conclusion is contained in the premises

6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
18216

Abstractions can form useful counterparts to concrete statements

18214

Mathematics is only empirical as regards which theory is useful

18210

Why regard standard mathematics as truths, rather than as interesting fictions?

7. Existence / D. Theories of Reality / 10. Ontological Commitment / a. Ontological commitment
18211

You can reduce ontological commitment by expanding the logic

8. Modes of Existence / B. Properties / 12. Denial of Properties
8959

Field presumes properties can be eliminated from science

9. Objects / A. Existence of Objects / 2. Abstract Objects / d. Problems with abstracta
18213

Abstract objects are only applicable to the world if they are impure, and connect to the physical

14. Science / D. Explanation / 2. Types of Explanation / a. Types of explanation
18222

Beneath every extrinsic explanation there is an intrinsic explanation

18. Thought / E. Abstraction / 4. Abstracta by Example
9917

'Abstract' is unclear, but numbers, functions and sets are clearly abstract

26. Natural Theory / B. Concepts of Nature / 3. Space / c. Points in space
18223

In theories of fields, spacetime points or regions are causal agents

26. Natural Theory / B. Concepts of Nature / 3. Space / d. Substantival space
18220

Both philosophy and physics now make substantivalism more attractive

26. Natural Theory / B. Concepts of Nature / 3. Space / e. Relational space
18219

Relational space is problematic if you take the idea of a field seriously
