Combining Philosophers

All the ideas for William W. Tait, Brian Clegg and Anil Seth

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36 ideas

1. Philosophy / F. Analytic Philosophy / 7. Limitations of Analysis
9978 Analytic philosophy focuses too much on forms of expression, instead of what is actually said [Tait]
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 / b. Empty (Null) Set
9986 The null set was doubted, because numbering seemed to require 'units' [Tait]
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]
4. Formal Logic / F. Set Theory ST / 7. Natural Sets
9984 We can have a series with identical members [Tait]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
13416 Mathematics must be based on axioms, which are true because they are axioms, not vice versa [Tait, by Parsons,C]
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]
15. Nature of Minds / A. Nature of Mind / 8. Brain
23513 Single neurons can carry out complex functions [Seth]
23514 The cerbellum has a huge number of neurons, but little involvement in consciousness [Seth]
15. Nature of Minds / B. Features of Minds / 1. Consciousness / e. Cause of consciousness
23516 Maybe a system is conscious if the whole generates more information than its parts [Seth]
16. Persons / C. Self-Awareness / 2. Knowing the Self
23519 The self is embodied, perspectival, volitional, narrative and social [Seth, by PG]
18. Thought / B. Mechanics of Thought / 6. Artificial Thought / a. Artificial Intelligence
23518 Modern AI is mostly machine-based pattern recognition [Seth]
18. Thought / E. Abstraction / 2. Abstracta by Selection
9981 Abstraction is 'logical' if the sense and truth of the abstraction depend on the concrete [Tait]
9982 Cantor and Dedekind use abstraction to fix grammar and objects, not to carry out proofs [Tait]
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
9985 Abstraction may concern the individuation of the set itself, not its elements [Tait]
18. Thought / E. Abstraction / 8. Abstractionism Critique
9972 Why should abstraction from two equipollent sets lead to the same set of 'pure units'? [Tait]
9980 If abstraction produces power sets, their identity should imply identity of the originals [Tait]
20. Action / B. Preliminaries of Action / 2. Willed Action / a. Will to Act
23517 Volition is felt as doing what you want, with possible alternatives, and a source from within [Seth]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / d. Biological ethics
23515 Human exceptionalism plagues biology, and most other human thinking [Seth]