Combining Texts

All the ideas for 'Defending the Axioms', 'Introduction to Zermelo's 1930 paper' and 'The Nature of Mathematics'

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19 ideas

1. Philosophy / G. Scientific Philosophy / 3. Scientism
14782 Philosophy is an experimental science, resting on common experience [Peirce]
2. Reason / B. Laws of Thought / 3. Non-Contradiction
14787 Self-contradiction doesn't reveal impossibility; it is inductive impossibility which reveals self-contradiction [Peirce]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
17833 The first-order ZF axiomatisation is highly non-categorical [Hallett,M]
17834 Non-categoricity reveals a sort of incompleteness, with sets existing that the axioms don't reveal [Hallett,M]
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]
4. Formal Logic / F. Set Theory ST / 7. Natural Sets
17837 Zermelo allows ur-elements, to enable the widespread application of set-theory [Hallett,M]
5. Theory of Logic / C. Ontology of Logic / 3. If-Thenism
14783 Logic, unlike mathematics, is not hypothetical; it asserts categorical ends from hypothetical means [Peirce]
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]
17836 The General Continuum Hypothesis and its negation are both consistent with ZF [Hallett,M]
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
14788 Mathematics is close to logic, but is even more abstract [Peirce]
10. Modality / B. Possibility / 1. Possibility
14786 Some logical possibility concerns single propositions, but there is also compatibility between propositions [Peirce]
12. Knowledge Sources / D. Empiricism / 1. Empiricism
14789 Experience is indeed our only source of knowledge, provided we include inner experience [Peirce]
12. Knowledge Sources / D. Empiricism / 5. Empiricism Critique
14785 The world is one of experience, but experiences are always located among our ideas [Peirce]
22. Metaethics / A. Ethics Foundations / 1. Nature of Ethics / b. Defining ethics
14784 Ethics is the science of aims [Peirce]