Combining Texts

All the ideas for 'Defending the Axioms', 'talk' and 'Empiricism, Semantics and Ontology'

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19 ideas

1. Philosophy / E. Nature of Metaphysics / 2. Possibility of Metaphysics
13939 No possible evidence could decide the reality of numbers, so it is a pseudo-question [Carnap]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
17610 The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy]
5. Theory of Logic / C. Ontology of Logic / 3. If-Thenism
17620 Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17605 Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy]
17625 If two mathematical themes coincide, that suggest a single deep truth [Maddy]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
17615 Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
17618 Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
17614 The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
13936 Questions about numbers are answered by analysis, and are analytic, and hence logically true [Carnap]
8748 Logical positivists incorporated geometry into logicism, saying axioms are just definitions [Carnap, by Shapiro]
7. Existence / A. Nature of Existence / 4. Abstract Existence
8960 Internal questions about abstractions are trivial, and external ones deeply problematic [Carnap, by Szabó]
7. Existence / D. Theories of Reality / 1. Ontologies
13933 Existence questions are 'internal' (within a framework) or 'external' (concerning the whole framework) [Carnap]
7. Existence / D. Theories of Reality / 3. Reality
13934 To be 'real' is to be an element of a system, so we cannot ask reality questions about the system itself [Carnap]
7. Existence / D. Theories of Reality / 11. Ontological Commitment / a. Ontological commitment
13938 A linguistic framework involves commitment to entities, so only commitment to the framework is in question [Carnap]
9. Objects / A. Existence of Objects / 6. Nihilism about Objects
13935 We only accept 'things' within a language with formation, testing and acceptance rules [Carnap]
12. Knowledge Sources / D. Empiricism / 1. Empiricism
13932 Empiricists tend to reject abstract entities, and to feel sympathy with nominalism [Carnap]
12. Knowledge Sources / D. Empiricism / 3. Pragmatism
13937 New linguistic claims about entities are not true or false, but just expedient, fruitful or successful [Carnap]
14. Science / B. Scientific Theories / 3. Instrumentalism
13940 All linguistic forms in science are merely judged by their efficiency as instruments [Carnap]
24. Political Theory / B. Nature of a State / 5. Culture
21432 Culture is the struggle to agree what is normal [Gibson,A]