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All the ideas for 'Mathematical Methods in Philosophy', 'Introduction to the Philosophy of Mathematics' and 'works'

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33 ideas

4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
17925 Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan]
17926 Rejecting double negation elimination undermines reductio proofs [Colyvan]
5. Theory of Logic / A. Overview of Logic / 9. Philosophical Logic
18739 Three stages of philosophical logic: syntactic (1905-55), possible worlds (1963-85), widening (1990-) [Horsten/Pettigrew]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
17924 Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan]
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
18741 Logical formalization makes concepts precise, and also shows their interrelation [Horsten/Pettigrew]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
18744 Models are sets with functions and relations, and truth built up from the components [Horsten/Pettigrew]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
17929 Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17930 Axioms are 'categorical' if all of their models are isomorphic [Colyvan]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
17928 Ordinal numbers represent order relations [Colyvan]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
17923 Intuitionists only accept a few safe infinities [Colyvan]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
17941 Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
17922 Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
17936 Transfinite induction moves from all cases, up to the limit ordinal [Colyvan]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
17940 Most mathematical proofs are using set theory, but without saying so [Colyvan]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
17931 Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
17932 If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan]
7. Existence / A. Nature of Existence / 1. Nature of Existence
18740 If 'exist' doesn't express a property, we can hardly ask for its essence [Horsten/Pettigrew]
10. Modality / E. Possible worlds / 1. Possible Worlds / a. Possible worlds
18745 A Tarskian model can be seen as a possible state of affairs [Horsten/Pettigrew]
18747 The 'spheres model' was added to possible worlds, to cope with counterfactuals [Horsten/Pettigrew]
10. Modality / E. Possible worlds / 1. Possible Worlds / b. Impossible worlds
18748 Epistemic logic introduced impossible worlds [Horsten/Pettigrew]
10. Modality / E. Possible worlds / 1. Possible Worlds / e. Against possible worlds
18746 Possible worlds models contain sets of possible worlds; this is a large metaphysical commitment [Horsten/Pettigrew]
18750 Using possible worlds for knowledge and morality may be a step too far [Horsten/Pettigrew]
14. Science / C. Induction / 6. Bayes's Theorem
17943 Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan]
14. Science / D. Explanation / 2. Types of Explanation / e. Lawlike explanations
17939 Mathematics can reveal structural similarities in diverse systems [Colyvan]
14. Science / D. Explanation / 2. Types of Explanation / f. Necessity in explanations
17938 Mathematics can show why some surprising events have to occur [Colyvan]
14. Science / D. Explanation / 2. Types of Explanation / m. Explanation by proof
17934 Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan]
17933 Reductio proofs do not seem to be very explanatory [Colyvan]
17935 If inductive proofs hold because of the structure of natural numbers, they may explain theorems [Colyvan]
17942 Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan]
15. Nature of Minds / C. Capacities of Minds / 5. Generalisation by mind
17937 Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan]
19. Language / A. Nature of Meaning / 6. Meaning as Use
20949 Study the use of words, not their origins [Herder]
22. Metaethics / B. Value / 1. Nature of Value / f. Ultimate value
7669 We cannot attain all the ideals of every culture, so there cannot be a perfect life [Herder, by Berlin]
24. Political Theory / D. Ideologies / 7. Communitarianism / a. Communitarianism
7668 Herder invented the idea of being rooted in (or cut off from) a home or a group [Herder, by Berlin]