Combining Philosophers

All the ideas for Brian Clegg, Dag Prawitz and Andrew Bowie

expand these ideas     |    start again     |     specify just one area for these philosophers

33 ideas

2. Reason / A. Nature of Reason / 1. On Reason
20955 Art can make reason more all-inclusive, by articulating what seemed inexpressible [Bowie]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
10859 A set is 'well-ordered' if every subset has a first element [Clegg]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
10857 Set theory made a closer study of infinity possible [Clegg]
10864 Any set can always generate a larger set - its powerset, of subsets [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
10872 Extensionality: Two sets are equal if and only if they have the same elements [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
10875 Pairing: For any two sets there exists a set to which they both belong [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
10876 Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
10878 Infinity: There exists a set of the empty set and the successor of each element [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
10877 Powers: All the subsets of a given set form their own new powerset [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
10879 Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / k. Axiom of Existence
10871 Axiom of Existence: there exists at least one set [Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / l. Axiom of Specification
10874 Specification: a condition applied to a set will always produce a new set [Clegg]
5. Theory of Logic / A. Overview of Logic / 1. Overview of Logic
13831 Logic is based on transitions between sentences [Prawitz]
5. Theory of Logic / B. Logical Consequence / 1. Logical Consequence
13827 Logical consequence isn't a black box (Tarski's approach); we should explain how arguments work [Prawitz]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
13825 Natural deduction introduction rules may represent 'definitions' of logical connectives [Prawitz]
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
13823 In natural deduction, inferences are atomic steps involving just one logical constant [Prawitz]
5. Theory of Logic / J. Model Theory in Logic / 1. Logical Models
13826 Model theory looks at valid sentences and consequence, but not how we know these things [Prawitz]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10880 Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
10861 Beyond infinity cardinals and ordinals can come apart [Clegg]
10860 An ordinal number is defined by the set that comes before it [Clegg]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
10854 Transcendental numbers can't be fitted to finite equations [Clegg]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / k. Imaginary numbers
10858 By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
10853 Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
10866 Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
10869 The Continuum Hypothesis is independent of the axioms of set theory [Clegg]
10862 The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg]
11. Knowledge Aims / C. Knowing Reality / 3. Idealism / b. Transcendental idealism
22049 Transcendental idealism aims to explain objectivity through subjectivity [Bowie]
11. Knowledge Aims / C. Knowing Reality / 3. Idealism / d. Absolute idealism
20950 German Idealism says our thinking and nature have the same rational structure [Bowie]
22055 The Idealists saw the same unexplained spontaneity in Kant's judgements and choices [Bowie]
22054 German Idealism tried to stop oppositions of appearances/things and receptivity/spontaneity [Bowie]
22056 Crucial to Idealism is the idea of continuity between receptivity and spontaneous judgement [Bowie]
16. Persons / E. Rejecting the Self / 2. Self as Social Construct
20942 Nazis think race predetermines the self [Bowie]
19. Language / F. Communication / 1. Rhetoric
20946 Rhetoric is built into language, so it cannot be stripped from philosophy [Bowie]