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

9570	In Field's Platonist view, set theory is false because it asserts existence for non-existent things [Chihara]

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

10260

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

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

8958	In Field's version of science, space-time points replace real numbers [Szabó]

6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry

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 [Shapiro]

18218

Hilbert explains geometry, by non-numerical facts about space

9623	Field needs a semantical notion of second-order consequence, and that needs sets [Brown,JR]

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

18210

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

18214

Mathematics is only empirical as regards which theory is useful

18216

Abstractions can form useful counterparts to concrete statements

7. Existence / D. Theories of Reality / 11. 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 [Szabó]

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

27. Natural Reality / B. Modern Physics / 2. Electrodynamics / b. Fields

18223

In theories of fields, space-time points or regions are causal agents

27. Natural Reality / C. Space / 4. Substantival Space

18220

Both philosophy and physics now make substantivalism more attractive

27. Natural Reality / C. Space / 5. Relational Space

18219

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