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All the ideas for 'The Roots of Reference', 'Thinking about Consciousness' and 'Philosophies of Mathematics'

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81 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]
8. Modes of Existence / C. Powers and Dispositions / 3. Powers as Derived
Dispositions are physical states of mechanism; when known, these replace the old disposition term [Quine]
12. Knowledge Sources / B. Perception / 7. Causal Perception
Perceptual concepts can't just refer to what causes classification [Papineau]
15. Nature of Minds / A. Nature of Mind / 1. Mind / e. Questions about mind
The only serious mind-brain theories now are identity, token identity, realization and supervenience [Papineau]
15. Nature of Minds / A. Nature of Mind / 3. Mental Causation
Maybe mind and body do overdetermine acts, but are linked (for some reason) [Papineau]
15. Nature of Minds / A. Nature of Mind / 4. Other Minds / c. Knowing other minds
Young children can see that other individuals sometimes have false beliefs [Papineau]
Do we understand other minds by simulation-theory, or by theory-theory? [Papineau]
15. Nature of Minds / A. Nature of Mind / 8. Brain
Researching phenomenal consciousness is peculiar, because the concepts involved are peculiar [Papineau]
15. Nature of Minds / B. Features of Minds / 1. Consciousness / a. Consciousness
Whether octopuses feel pain is unclear, because our phenomenal concepts are too vague [Papineau]
Our concept of consciousness is crude, and lacks theoretical articulation [Papineau]
We can’t decide what 'conscious' means, so it is undecidable whether cats are conscious [Papineau]
15. Nature of Minds / B. Features of Minds / 1. Consciousness / e. Cause of consciousness
Maybe a creature is conscious if its mental states represent things in a distinct way [Papineau]
15. Nature of Minds / B. Features of Minds / 1. Consciousness / f. Higher-order thought
The 'actualist' HOT theory says consciousness comes from actual higher judgements of mental states [Papineau]
Actualist HOT theories imply that a non-conscious mental event could become conscious when remembered [Papineau]
States are conscious if they could be the subject of higher-order mental judgements [Papineau]
Higher-order judgements may be possible where the subject denies having been conscious [Papineau]
17. Mind and Body / A. Mind-Body Dualism / 6. Epiphenomenalism
The epiphenomenal relation of mind and brain is a 'causal dangler', unlike anything else [Papineau]
Maybe minds do not cause actions, but do cause us to report our decisions [Papineau]
17. Mind and Body / C. Functionalism / 8. Functionalism critique
Role concepts either name the realising property, or the higher property constituting the role [Papineau]
17. Mind and Body / D. Property Dualism / 2. Anomalous Monism
If causes are basic particulars, this doesn't make conscious and physical properties identical [Papineau]
17. Mind and Body / D. Property Dualism / 5. Supervenience of mind
Supervenience can be replaced by identifying mind with higher-order or disjunctional properties [Papineau]
17. Mind and Body / E. Mind as Physical / 1. Physical Mind
The completeness of physics is needed for mind-brain identity [Papineau]
17. Mind and Body / E. Mind as Physical / 2. Reduction of Mind
Mind-brain reduction is less explanatory, because phenomenal concepts lack causal roles [Papineau]
Weak reduction of mind is to physical causes; strong reduction is also to physical laws [Papineau]
17. Mind and Body / E. Mind as Physical / 5. Causal Argument
It is absurd to think that physical effects are caused twice, so conscious causes must be physical [Papineau]
17. Mind and Body / E. Mind as Physical / 6. Conceptual Dualism
Accept ontological monism, but conceptual dualism; we think in a different way about phenomenal thought [Papineau]
17. Mind and Body / E. Mind as Physical / 7. Anti-Physicalism / c. Knowledge argument
Mary acquires new concepts; she previously thought about the same property using material concepts [Papineau]
18. Thought / A. Modes of Thought / 1. Thought
Thinking about a thing doesn't require activating it [Papineau]
Consciousness affects bodily movement, so thoughts must be material states [Papineau]
18. Thought / C. Content / 6. Broad Content
Most reductive accounts of representation imply broad content [Papineau]
If content hinges on matters outside of you, how can it causally influence your actions? [Papineau]
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 / A. Nature of Meaning / 5. Meaning as Verification
Verificationists tend to infer indefinite answers from undecidable questions [Papineau]
19. Language / C. Assigning Meanings / 2. Semantics
Teleosemantics equates meaning with the item the concept is intended to track [Papineau]
19. Language / C. Assigning Meanings / 8. Possible Worlds Semantics
Truth conditions in possible worlds can't handle statements about impossibilities [Papineau]
Thought content is possible worlds that make the thought true; if that includes the actual world, it's true [Papineau]
26. Natural Theory / C. Causation / 8. Particular Causation / b. Causal relata
Causation is based on either events, or facts, or states of affairs [Papineau]
Causes are instantiations of properties by particulars, or they are themselves basic particulars [Papineau]
26. Natural Theory / D. Laws of Nature / 10. Closure of Physics
The completeness of physics cannot be proved [Papineau]
Determinism is possible without a complete physics, if mental forces play a role [Papineau]
Modern biological research, especially into the cell, has revealed no special new natural forces [Papineau]
27. Natural Reality / A. Classical Physics / 2. Thermodynamics / c. Conservation of energy
Quantum 'wave collapses' seem to violate conservation of energy [Papineau]