Combining Texts

All the ideas for 'Defending the Axioms', 'Reference and Essence (1st edn)' and 'Philosophy of Mathematics'

expand these ideas     |    start again     |     specify just one area for these texts

17 ideas

2. Reason / D. Definition / 8. Impredicative Definition
10882 Predicative definitions only refer to entities outside the defined collection [Horsten]
2. Reason / D. Definition / 11. Ostensive Definition
18889 Ostensive definitions needn't involve pointing, but must refer to something specific [Salmon,N]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / h. System S5
14627 S4, and therefore S5, are invalid for metaphysical modality [Salmon,N, by Williamson]
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 / J. Model Theory in Logic / 2. Isomorphisms
10884 A theory is 'categorical' if it has just one model up to isomorphism [Horsten]
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 / 2. Proof in Mathematics
10885 Computer proofs don't provide explanations [Horsten]
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]
10881 The concept of 'ordinal number' is set-theoretic, not arithmetical [Horsten]
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]
9. Objects / D. Essence of Objects / 7. Essence and Necessity / a. Essence as necessary properties
18888 Essentialism says some properties must be possessed, if a thing is to exist [Salmon,N]
19. Language / B. Reference / 1. Reference theories
18886 Frege's 'sense' solves four tricky puzzles [Salmon,N]
19. Language / B. Reference / 3. Direct Reference / a. Direct reference
18887 The perfect case of direct reference is a variable which has been assigned a value [Salmon,N]
26. Natural Theory / B. Natural Kinds / 5. Reference to Natural Kinds
18891 Nothing in the direct theory of reference blocks anti-essentialism; water structure might have been different [Salmon,N]