Combining Texts

All the ideas for 'Mahaprajnaparamitashastra', 'Philosophies of Mathematics' and 'Penguin Dictionary of Philosophy'

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72 ideas

1. Philosophy / F. Analytic Philosophy / 5. Linguistic Analysis
Linguistic philosophy approaches problems by attending to actual linguistic usage [Mautner]
1. Philosophy / F. Analytic Philosophy / 7. Limitations of Analysis
Analytic philosophy studies the unimportant, and sharpens tools instead of using them [Mautner]
1. Philosophy / H. Continental Philosophy / 3. Hermeneutics
The 'hermeneutic circle' says parts and wholes are interdependent, and so cannot be interpreted [Mautner]
2. Reason / D. Definition / 4. Real Definition
'Real' definitions give the essential properties of things under a concept [Mautner]
2. Reason / D. Definition / 7. Contextual Definition
'Contextual definitions' replace whole statements, not just expressions [Mautner]
Contextual definitions replace a complete sentence containing the expression [George/Velleman]
2. Reason / D. Definition / 8. Impredicative Definition
Impredicative definitions quantify over the thing being defined [George/Velleman]
2. Reason / D. Definition / 9. Recursive Definition
Recursive definition defines each instance from a previous instance [Mautner]
2. Reason / D. Definition / 10. Stipulative Definition
A stipulative definition lays down that an expression is to have a certain meaning [Mautner]
2. Reason / D. Definition / 11. Ostensive Definition
Ostensive definitions point to an object which an expression denotes [Mautner]
2. Reason / F. Fallacies / 5. Fallacy of Composition
The fallacy of composition is the assumption that what is true of the parts is true of the whole [Mautner]
4. Formal Logic / E. Nonclassical Logics / 4. Fuzzy Logic
Fuzzy logic is based on the notion that there can be membership of a set to some degree [Mautner]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
The 'power set' of A is all the subsets of A [George/Velleman]
The 'ordered pair' <a, b>, for two sets a and b, is the set {{a, b},{a}} [George/Velleman]
Cartesian Product A x B: the set of all ordered pairs in which a∈A and b∈B [George/Velleman]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / e. Equivalence classes
Grouping by property is common in mathematics, usually using equivalence [George/Velleman]
'Equivalence' is a reflexive, symmetric and transitive relation; 'same first letter' partitions English words [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
Even the elements of sets in ZFC are sets, resting on the pure empty set [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
Axiom of Extensionality: for all sets x and y, if x and y have the same elements then x = y [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
Axiom of Pairing: for all sets x and y, there is a set z containing just x and y [George/Velleman]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
The Axiom of Reducibility made impredicative definitions possible [George/Velleman]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / a. Sets as existing
ZFC can prove that there is no set corresponding to the concept 'set' [George/Velleman]
4. Formal Logic / F. Set Theory ST / 8. Critique of Set Theory
As a reduction of arithmetic, set theory is not fully general, and so not logical [George/Velleman]
5. Theory of Logic / B. Logical Consequence / 6. Entailment
Entailment is logical requirement; it may be not(p and not-q), but that has problems [Mautner]
5. Theory of Logic / B. Logical Consequence / 7. Strict Implication
Strict implication says false propositions imply everything, and everything implies true propositions [Mautner]
5. Theory of Logic / B. Logical Consequence / 8. Material Implication
'Material implication' is defined as 'not(p and not-q)', but seems to imply a connection between p and q [Mautner]
A person who 'infers' draws the conclusion, but a person who 'implies' leaves it to the audience [Mautner]
5. Theory of Logic / D. Assumptions for Logic / 1. Bivalence
Vagueness seems to be inconsistent with the view that every proposition is true or false [Mautner]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Asserting Excluded Middle is a hallmark of realism about the natural world [George/Velleman]
5. Theory of Logic / G. Quantification / 1. Quantification
Quantifiers turn an open sentence into one to which a truth-value can be assigned [Mautner]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
A 'model' is a meaning-assignment which makes all the axioms true [George/Velleman]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
Differences between isomorphic structures seem unimportant [George/Velleman]
5. Theory of Logic / K. Features of Logics / 2. Consistency
Consistency is a purely syntactic property, unlike the semantic property of soundness [George/Velleman]
A 'consistent' theory cannot contain both a sentence and its negation [George/Velleman]
5. Theory of Logic / K. Features of Logics / 3. Soundness
Soundness is a semantic property, unlike the purely syntactic property of consistency [George/Velleman]
5. Theory of Logic / K. Features of Logics / 4. Completeness
A 'complete' theory contains either any sentence or its negation [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Rational numbers give answers to division problems with integers [George/Velleman]
The integers are answers to subtraction problems involving natural numbers [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
Real numbers provide answers to square root problems [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
Logicists say mathematics is applicable because it is totally general [George/Velleman]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
The classical mathematician believes the real numbers form an actual set [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
Second-order induction is stronger as it covers all concepts, not just first-order definable ones [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
The Incompleteness proofs use arithmetic to talk about formal arithmetic [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
A successor is the union of a set with its singleton [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
Frege's Theorem shows the Peano Postulates can be derived from Hume's Principle [George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Set theory can prove the Peano Postulates [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Talk of 'abstract entities' is more a label for the problem than a solution to it [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
If mathematics is not about particulars, observing particulars must be irrelevant [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
In the unramified theory of types, the types are objects, then sets of objects, sets of sets etc. [George/Velleman]
The theory of types seems to rule out harmless sets as well as paradoxical ones. [George/Velleman]
Type theory has only finitely many items at each level, which is a problem for mathematics [George/Velleman]
Type theory prohibits (oddly) a set containing an individual and a set of individuals [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 8. Finitism
Bounded quantification is originally finitary, as conjunctions and disjunctions [George/Velleman]
Much infinite mathematics can still be justified finitely [George/Velleman]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
The intuitionists are the idealists of mathematics [George/Velleman]
Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman]
10. Modality / B. Possibility / 9. Counterfactuals
Counterfactuals presuppose a belief (or a fact) that the condition is false [Mautner]
Counterfactuals are not true, they are merely valid [Mautner]
Counterfactuals are true if in every world close to actual where p is the case, q is also the case [Mautner]
Counterfactuals say 'If it had been, or were, p, then it would be q' [Mautner]
Maybe counterfactuals are only true if they contain valid inference from premisses [Mautner]
10. Modality / C. Sources of Modality / 6. Necessity from Essence
Essentialism is often identified with belief in 'de re' necessary truths [Mautner]
11. Knowledge Aims / B. Certain Knowledge / 3. Fallibilism
Fallibilism is the view that all knowledge-claims are provisional [Mautner]
12. Knowledge Sources / B. Perception / 4. Sense Data / a. Sense-data theory
'Sense-data' arrived in 1910, but it denotes ideas in Locke, Berkeley and Hume [Mautner]
14. Science / C. Induction / 5. Paradoxes of Induction / a. Grue problem
Observing lots of green x can confirm 'all x are green' or 'all x are grue', where 'grue' is arbitrary [Mautner, by PG]
14. Science / C. Induction / 5. Paradoxes of Induction / b. Raven paradox
'All x are y' is equivalent to 'all non-y are non-x', so observing paper is white confirms 'ravens are black' [Mautner, by PG]
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
Corresponding to every concept there is a class (some of them sets) [George/Velleman]
19. Language / C. Assigning Meanings / 9. Indexical Semantics
The references of indexicals ('there', 'now', 'I') depend on the circumstances of utterance [Mautner]
20. Action / C. Motives for Action / 5. Action Dilemmas / b. Double Effect
Double effect is the distinction between what is foreseen and what is intended [Mautner]
Double effect acts need goodness, unintended evil, good not caused by evil, and outweighing [Mautner]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / e. Human nature
'Essentialism' is opposed to existentialism, and claims there is a human nature [Mautner]
23. Ethics / C. Virtue Theory / 3. Virtues / a. Virtues
The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna]