Combining Texts

All the ideas for 'A Résumé of Metaphysics', 'works' and 'Introduction to the Philosophy of Mathematics'

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32 ideas

4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
17926 Rejecting double negation elimination undermines reductio proofs [Colyvan]
17925 Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan]
4. Formal Logic / G. Formal Mereology / 1. Mereology
10397 Abelard's mereology involves privileged and natural divisions, and principal parts [Abelard, by King,P]
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 / 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]
8. Modes of Existence / E. Nominalism / 1. Nominalism / b. Nominalism about universals
10396 If 'animal' is wholly present in Socrates and an ass, then 'animal' is rational and irrational [Abelard, by King,P]
10395 Abelard was an irrealist about virtually everything apart from concrete individuals [Abelard, by King,P]
8. Modes of Existence / E. Nominalism / 3. Predicate Nominalism
15384 Only words can be 'predicated of many'; the universality is just in its mode of signifying [Abelard, by Panaccio]
10. Modality / A. Necessity / 4. De re / De dicto modality
8481 The de dicto-de re modality distinction dates back to Abelard [Abelard, by Orenstein]
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]
18. Thought / E. Abstraction / 8. Abstractionism Critique
15385 Abelard's problem is the purely singular aspects of things won't account for abstraction [Panaccio on Abelard]
19. Language / C. Assigning Meanings / 3. Predicates
15383 Nothing external can truly be predicated of an object [Abelard, by Panaccio]
22. Metaethics / C. The Good / 3. Pleasure / a. Nature of pleasure
5049 Intelligent pleasure is the perception of beauty, order and perfection [Leibniz]
26. Natural Theory / B. Natural Kinds / 7. Critique of Kinds
10398 Natural kinds are not special; they are just well-defined resemblance collections [Abelard, by King,P]
28. God / A. Divine Nature / 3. Divine Perfections
5048 Perfection is simply quantity of reality [Leibniz]
29. Religion / D. Religious Issues / 3. Problem of Evil / b. Human Evil
5050 Evil serves a greater good, and pain is necessary for higher pleasure [Leibniz]