Combining Texts

All the ideas for 'Defending the Axioms', 'The Epistemology of Essence (draft)' and 'Later Letters to Dedekind'

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12 ideas

4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Cantor, by Lake]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
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
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
Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy]
If two mathematical themes coincide, that suggest a single deep truth [Maddy]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
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
Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]
10. Modality / D. Knowledge of Modality / 4. Conceivable as Possible / a. Conceivable as possible
If conceivability is a priori coherence, that implies possibility [Tahko]
14. Science / D. Explanation / 2. Types of Explanation / k. Explanations by essence
Essences are used to explain natural kinds, modality, and causal powers [Tahko]
26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / a. Scientific essentialism
Scientific essentialists tend to characterise essence in terms of modality (not vice versa) [Tahko]
26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / b. Scientific necessity
If essence is modal and laws are necessary, essentialist knowledge is found by scientists [Tahko]