Ideas from 'Science without Numbers' by Hartry Field [1980], by Theme Structure

[found in 'Science without Number' by Field,Hartry [Blackwell 1980,0-631-13037-3]].

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4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
In Field's Platonist view, set theory is false because it asserts existence for non-existent things
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
Logical consequence is defined by the impossibility of P and q
6. Mathematics / A. Nature of Mathematics / 3. Numbers / a. Numbers
In Field's version of science, space-time points replace real numbers
6. Mathematics / B. Foundations for Mathematics / 2. Axioms for Geometry
Hilbert's geometry is interesting because it captures Euclid without using real numbers
'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
The Indispensability Argument is the only serious ground for the existence of mathematical entities
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
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
The application of mathematics only needs its possibility, not its truth
Hilbert explains geometry, by non-numerical facts about space
Field needs a semantical notion of second-order consequence, and that needs sets
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
It seems impossible to explain the idea that the conclusion is contained in the premises
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
Abstractions can form useful counterparts to concrete statements
Mathematics is only empirical as regards which theory is useful
Why regard standard mathematics as truths, rather than as interesting fictions?
7. Existence / D. Theories of Reality / 10. Ontological Commitment / a. Ontological commitment
You can reduce ontological commitment by expanding the logic
8. Modes of Existence / B. Properties / 12. Denial of Properties
Field presumes properties can be eliminated from science
9. Objects / A. Existence of Objects / 2. Abstract Objects / d. Problems with abstracta
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
Beneath every extrinsic explanation there is an intrinsic explanation
18. Thought / E. Abstraction / 4. Abstracta by Example
'Abstract' is unclear, but numbers, functions and sets are clearly abstract
26. Natural Theory / B. Concepts of Nature / 3. Space / b. Points in space
In theories of fields, space-time points or regions are causal agents
26. Natural Theory / B. Concepts of Nature / 3. Space / c. Substantival space
Both philosophy and physics now make substantivalism more attractive
26. Natural Theory / B. Concepts of Nature / 3. Space / d. Relational space
Relational space is problematic if you take the idea of a field seriously