Combining Texts

All the ideas for 'Defending the Axioms', 'Intro to 'The Reason's Proper Study'' and 'Aristotle and Kant on the Source of Value'

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20 ideas

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]
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / c. Grelling's paradox
10631 If 'x is heterological' iff it does not apply to itself, then 'heterological' is heterological if it isn't heterological [Hale/Wright]
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 / 4. Axioms for Number / g. Incompleteness of Arithmetic
10624 The incompletability of formal arithmetic reveals that logic also cannot be completely characterized [Hale/Wright]
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 / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
10629 If structures are relative, this undermines truth-value and objectivity [Hale/Wright]
10628 The structural view of numbers doesn't fit their usage outside arithmetical contexts [Hale/Wright]
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 / c. Neo-logicism
10622 The neo-Fregean is more optimistic than Frege about contextual definitions of numbers [Hale/Wright]
9. Objects / A. Existence of Objects / 2. Abstract Objects / c. Modern abstracta
10626 Objects just are what singular terms refer to [Hale/Wright]
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
10630 Abstracted objects are not mental creations, but depend on equivalence between given entities [Hale/Wright]
19. Language / E. Analyticity / 2. Analytic Truths
10627 Many conceptual truths ('yellow is extended') are not analytic, as derived from logic and definitions [Hale/Wright]
22. Metaethics / B. Value / 1. Nature of Value / f. Ultimate value
18225 If we can't reason about value, we can reason about the unconditional source of value [Korsgaard]
18228 An end can't be an ultimate value just because it is useless! [Korsgaard]
22. Metaethics / C. The Good / 1. Goodness / b. Types of good
18224 Goodness is given either by a psychological state, or the attribution of a property [Korsgaard]
23. Ethics / C. Virtue Theory / 3. Virtues / g. Contemplation
18233 Contemplation is final because it is an activity which is not a process [Korsgaard]
18226 For Aristotle, contemplation consists purely of understanding [Korsgaard]