Combining Texts

All the ideas for 'Defending the Axioms', 'Causation and Laws of Nature' and 'Elements of Geometry'

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27 ideas

1. Philosophy / F. Analytic Philosophy / 3. Analysis of Preconditions
14600 Analysis aims at secure necessary and sufficient conditions [Schaffer,J]
2. Reason / E. Argument / 6. Conclusive Proof
8623 Proof reveals the interdependence of truths, as well as showing their certainty [Euclid, by Frege]
2. Reason / F. Fallacies / 1. Fallacy
14603 'Reification' occurs if we mistake a concept for a thing [Schaffer,J]
4. Formal Logic / C. Predicate Calculus PC / 2. Tools of Predicate Calculus / c. Derivations rules of PC
13907 If you pick an arbitrary triangle, things proved of it are true of all triangles [Euclid, by Lemmon]
4. Formal Logic / D. Modal Logic ML / 3. Modal Logic Systems / d. System T
14607 T adds □p→p for reflexivity, and is ideal for modeling lawhood [Schaffer,J]
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 / 2. Geometry
6297 Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Euclid, by Resnik]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
9603 An assumption that there is a largest prime leads to a contradiction [Euclid, by Brown,JR]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
9894 A unit is that according to which each existing thing is said to be one [Euclid]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
8738 Postulate 2 says a line can be extended continuously [Euclid, by Shapiro]
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 / 3. Axioms for Geometry
10302 Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Euclid, by Bernays]
22278 Euclid relied on obvious properties in diagrams, as well as on his axioms [Potter on Euclid]
8673 Euclid's parallel postulate defines unique non-intersecting parallel lines [Euclid, by Friend]
10250 Euclid needs a principle of continuity, saying some lines must intersect [Shapiro on Euclid]
14157 Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
1600 Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik]
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 / b. Indispensability of mathematics
14604 If a notion is ontologically basic, it should be needed in our best attempt at science [Schaffer,J]
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]
7. Existence / C. Structure of Existence / 2. Reduction
14599 Three types of reduction: Theoretical (of terms), Definitional (of concepts), Ontological (of reality) [Schaffer,J]
8. Modes of Existence / B. Properties / 13. Tropes / a. Nature of tropes
14605 Tropes are the same as events [Schaffer,J]
9. Objects / A. Existence of Objects / 5. Individuation / a. Individuation
14601 Individuation aims to count entities, by saying when there is one [Schaffer,J]
10. Modality / D. Knowledge of Modality / 4. Conceivable as Possible / a. Conceivable as possible
14606 Only ideal conceivability could indicate what is possible [Schaffer,J]