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All the ideas for 'works', 'The View from Nowhere' and 'Understanding the Infinite'

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58 ideas

1. Philosophy / A. Wisdom / 3. Wisdom Deflated
There is more insight in fundamental perplexity about problems than in their supposed solutions [Nagel]
1. Philosophy / D. Nature of Philosophy / 1. Philosophy
Philosophy is the childhood of the intellect, and a culture can't skip it [Nagel]
1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / b. Philosophy as transcendent
It seems mad, but the aim of philosophy is to climb outside of our own minds [Nagel]
2. Reason / A. Nature of Reason / 5. Objectivity
Realism invites scepticism because it claims to be objective [Nagel]
Views are objective if they don't rely on a person's character, social position or species [Nagel]
Things cause perceptions, properties have other effects, hence we reach a 'view from nowhere' [Nagel, by Reiss/Sprenger]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
Those who reject infinite collections also want to reject the Axiom of Choice [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
The Power Set is just the collection of functions from one collection to another [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
Replacement was immediately accepted, despite having very few implications [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
Pure collections of things obey Choice, but collections defined by a rule may not [Lavine]
The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / c. Logical sets
The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The iterative conception of set wasn't suggested until 1947 [Lavine]
The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine]
The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine]
4. Formal Logic / F. Set Theory ST / 6. Ordering in Sets
A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Mathematical proof by contradiction needs the law of excluded middle [Lavine]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Every rational number, unlike every natural number, is divisible by some other number [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
Cauchy gave a necessary condition for the convergence of a sequence [Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
Counting results in well-ordering, and well-ordering makes counting possible [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine]
The infinite is extrapolation from the experience of indefinitely large size [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / c. Potential infinite
The intuitionist endorses only the potential infinite [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
Ordinals are basic to Cantor's transfinite, to count the sets [Lavine]
Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Set theory will found all of mathematics - except for the notion of proof [Lavine]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Intuitionism rejects set-theory to found mathematics [Lavine]
12. Knowledge Sources / B. Perception / 2. Qualities in Perception / b. Primary/secondary
Modern science depends on the distinction between primary and secondary qualities [Nagel]
We achieve objectivity by dropping secondary qualities, to focus on structural primary qualities [Nagel]
13. Knowledge Criteria / B. Internal Justification / 2. Pragmatic justification
Epistemology is centrally about what we should believe, not the definition of knowledge [Nagel]
13. Knowledge Criteria / D. Scepticism / 6. Scepticism Critique
Scepticism is based on ideas which scepticism makes impossible [Nagel]
14. Science / C. Induction / 4. Reason in Induction
Observed regularities are only predictable if we assume hidden necessity [Nagel]
16. Persons / B. Nature of the Self / 4. Presupposition of Self
Personal identity cannot be fully known a priori [Nagel]
The question of whether a future experience will be mine presupposes personal identity [Nagel]
16. Persons / D. Continuity of the Self / 4. Split Consciousness
I can't even conceive of my brain being split in two [Nagel]
22. Metaethics / B. Value / 1. Nature of Value / c. Objective value
Total objectivity can't see value, but it sees many people with values [Nagel]
22. Metaethics / B. Value / 2. Values / e. Death
We don't worry about the time before we were born the way we worry about death [Nagel]
22. Metaethics / B. Value / 2. Values / f. Altruism
If our own life lacks meaning, devotion to others won't give it meaning [Nagel]
22. Metaethics / C. The Good / 1. Goodness / f. Good as pleasure
Pain doesn't have a further property of badness; it gives a reason for its avoidance [Nagel]
23. Ethics / D. Deontological Ethics / 1. Deontology
Something may be 'rational' either because it is required or because it is acceptable [Nagel]
23. Ethics / D. Deontological Ethics / 2. Duty
If cockroaches can't think about their actions, they have no duties [Nagel]
23. Ethics / D. Deontological Ethics / 3. Universalisability
If we can decide how to live after stepping outside of ourselves, we have the basis of a moral theory [Nagel]
We should see others' viewpoints, but not lose touch with our own values [Nagel]
23. Ethics / D. Deontological Ethics / 6. Motivation for Duty
We find new motives by discovering reasons for action different from our preexisting motives [Nagel]
23. Ethics / E. Utilitarianism / 3. Motivation for Altruism
Utilitarianism is too demanding [Nagel]
26. Natural Theory / D. Laws of Nature / 9. Counterfactual Claims
An event causes another just if the second event would not have happened without the first [Lewis, by Psillos]