Combining Texts

All the ideas for 'reports', 'Causality and Properties' and 'Understanding the Infinite'

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83 ideas

1. Philosophy / D. Nature of Philosophy / 1. Philosophy
He studied philosophy by suspending his judgement on everything [Pyrrho, by Diog. Laertius]
1. Philosophy / E. Nature of Metaphysics / 3. Metaphysical Systems
One system has properties, powers, events, similarity and substance [Shoemaker]
1. Philosophy / F. Analytic Philosophy / 4. Conceptual Analysis
Analysis aims at internal relationships, not reduction [Shoemaker]
2. Reason / A. Nature of Reason / 9. Limits of Reason
Sceptics say reason is only an instrument, because reason can only be attacked with reason [Pyrrho, by Diog. Laertius]
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]
8. Modes of Existence / B. Properties / 1. Nature of Properties
Formerly I said properties are individuated by essential causal powers and causing instantiation [Shoemaker, by Shoemaker]
8. Modes of Existence / B. Properties / 5. Natural Properties
Genuine properties are closely related to genuine changes [Shoemaker]
Properties must be essentially causal if we can know and speak about them [Shoemaker]
To ascertain genuine properties, examine the object directly [Shoemaker]
8. Modes of Existence / B. Properties / 10. Properties as Predicates
We should abandon the idea that properties are the meanings of predicate expressions [Shoemaker]
Some truths are not because of a thing's properties, but because of the properties of related things [Shoemaker]
8. Modes of Existence / C. Powers and Dispositions / 3. Powers as Derived
Things have powers in virtue of (which are entailed by) their properties [Shoemaker]
One power can come from different properties; a thing's powers come from its properties [Shoemaker]
Properties are functions producing powers, and powers are functions producing effects [Shoemaker]
8. Modes of Existence / C. Powers and Dispositions / 5. Powers and Properties
Shoemaker says all genuine properties are dispositional [Shoemaker, by Ellis]
A causal theory of properties focuses on change, not (say) on abstract properties of numbers [Shoemaker]
'Square', 'round' and 'made of copper' show that not all properties are dispositional [Shoemaker]
The identity of a property concerns its causal powers [Shoemaker]
Properties are clusters of conditional powers [Shoemaker]
Could properties change without the powers changing, or powers change without the properties changing? [Shoemaker]
If properties are separated from causal powers, this invites total elimination [Shoemaker]
The notions of property and of causal power are parts of a single system of related concepts [Shoemaker]
Actually, properties are individuated by causes as well as effects [Shoemaker]
8. Modes of Existence / C. Powers and Dispositions / 6. Dispositions / b. Dispositions and powers
Dispositional predicates ascribe powers, and the rest ascribe properties [Shoemaker]
8. Modes of Existence / D. Universals / 2. Need for Universals
Universals concern how things are, and how they could be [Shoemaker, by Bird]
8. Modes of Existence / E. Nominalism / 5. Class Nominalism
Triangular and trilateral are coextensive, but different concepts; but powers and properties are the same [Shoemaker]
9. Objects / D. Essence of Objects / 15. Against Essentialism
There is no subset of properties which guarantee a thing's identity [Shoemaker]
10. Modality / B. Possibility / 1. Possibility
Possible difference across worlds depends on difference across time in the actual world [Shoemaker]
10. Modality / D. Knowledge of Modality / 4. Conceivable as Possible / a. Conceivable as possible
'Conceivable' is either not-provably-false, or compatible with what we know? [Shoemaker]
10. Modality / D. Knowledge of Modality / 4. Conceivable as Possible / b. Conceivable but impossible
It is possible to conceive what is not possible [Shoemaker]
13. Knowledge Criteria / A. Justification Problems / 1. Justification / a. Justification issues
If we need a criterion of truth, we need to know whether it is the correct criterion [Pyrrho, by Fogelin]
13. Knowledge Criteria / D. Scepticism / 1. Scepticism
The Pyrrhonians attacked the dogmas of professors, not ordinary people [Pyrrho, by Fogelin]
13. Knowledge Criteria / D. Scepticism / 6. Scepticism Critique
Academics said that Pyrrhonians were guilty of 'negative dogmatism' [Pyrrho, by Fogelin]
13. Knowledge Criteria / E. Relativism / 1. Relativism
Judgements vary according to local culture and law (Mode 5) [Pyrrho, by Diog. Laertius]
Objects vary according to which sense perceives them (Mode 3) [Pyrrho, by Diog. Laertius]
Perception varies with viewing distance and angle (Mode 7) [Pyrrho, by Diog. Laertius]
Perception and judgement depend on comparison (Mode 10) [Pyrrho, by Diog. Laertius]
Individuals vary in responses and feelings (Mode 2) [Pyrrho, by Diog. Laertius]
Animals vary in their feelings and judgements (Mode 1) [Pyrrho, by Diog. Laertius]
Perception varies with madness or disease (Mode 4) [Pyrrho, by Diog. Laertius]
Perception of things depends on their size or quantity (Mode 8) [Pyrrho, by Diog. Laertius]
Perception of objects depends on surrounding conditions (Mode 6) [Pyrrho, by Diog. Laertius]
Perception is affected by expectations (Mode 9) [Pyrrho, by Diog. Laertius]
14. Science / C. Induction / 5. Paradoxes of Induction / a. Grue problem
Grueness is not, unlike green and blue, associated with causal potential [Shoemaker]
26. Natural Theory / C. Causation / 7. Eliminating causation
There are no causes, because they are relative, and alike things can't cause one another [Pyrrho, by Diog. Laertius]
26. Natural Theory / C. Causation / 8. Particular Causation / b. Causal relata
If causality is between events, there must be reference to the properties involved [Shoemaker]
26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / b. Scientific necessity
If causal laws describe causal potentialities, the same laws govern properties in all possible worlds [Shoemaker]
If properties are causal, then causal necessity is a species of logical necessity [Shoemaker]
If a world has different causal laws, it must have different properties [Shoemaker]
26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / d. Knowing essences
It looks as if the immutability of the powers of a property imply essentiality [Shoemaker]
27. Natural Reality / A. Classical Physics / 1. Mechanics / a. Explaining movement
Motion can't move where it is, and can't move where it isn't, so it can't exist [Pyrrho, by Diog. Laertius]