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All the ideas for 'fragments/reports', 'Philosophies of Mathematics' and 'Authority and the Individual'

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61 ideas

2. Reason / D. Definition / 7. Contextual Definition
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]
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 / 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 / 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]
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
Corresponding to every concept there is a class (some of them sets) [George/Velleman]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / e. Human nature
We divide mankind into friend and foe, and cooperate with one and compete with the other [Russell]
24. Political Theory / A. Basis of a State / 1. A People / c. A unified people
Gradually loyalty to a creed increased, which could even outweigh nationality [Russell]
Increasingly war expands communities, and unifies them through fear [Russell]
In early societies the leaders needed cohesion, but the rest just had to obey [Russell]
24. Political Theory / A. Basis of a State / 2. Population / b. State population
The economic and political advantages of great size seem to have no upper limit [Russell]
24. Political Theory / B. Nature of a State / 1. Purpose of a State
Government has a negative purpose, to prevent trouble, and a positive aim of realising our desires [Russell]
24. Political Theory / C. Ruling a State / 2. Leaders / b. Monarchy
A monarch is known to everyone in the group, and can thus unite large groups [Russell]
24. Political Theory / C. Ruling a State / 4. Changing the State / b. Devolution
Power should be with smaller bodies, as long as it doesn't restrict central powers [Russell]
24. Political Theory / D. Ideologies / 2. Anarchism
In an anarchy universities, research, books, and even seaside holidays, would be impossible [Russell]
A state is essential, to control greedy or predatory impulses [Russell]
24. Political Theory / D. Ideologies / 5. Democracy / f. Against democracy
In democracy we are more aware of being governed than of our tiny share in government [Russell]
24. Political Theory / D. Ideologies / 8. Socialism
Managers are just as remote from workers under nationalisation as under capitalism [Russell]
Socialists say economic justice needs some state control of industries, and of foreign trade [Russell]
Being a slave of society is hardly better than being a slave of a despot [Russell]
25. Social Practice / A. Freedoms / 1. Slavery
Slavery began the divorce between the work and the purposes of the worker [Russell]
25. Social Practice / B. Equalities / 1. Grounds of equality
Slaves can be just as equal as free people [Russell]
25. Social Practice / B. Equalities / 4. Economic equality
Scarce goods may be denied entirely, to avoid their unequal distribution [Russell]
25. Social Practice / D. Justice / 1. Basis of justice
Modern justice is seen as equality, apart from modest extra rewards for exceptional desert [Russell]
26. Natural Theory / A. Speculations on Nature / 5. Infinite in Nature
Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius]
27. Natural Reality / G. Biology / 3. Evolution
Archelaus said life began in a primeval slime [Archelaus, by Schofield]