Combining Texts

All the ideas for 'fragments/reports', 'Infinity: Quest to Think the Unthinkable' and 'The Origin of Forms and Qualities'

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31 ideas

2. Reason / D. Definition / 4. Real Definition
Essential definitions show the differences that discriminate things, and make them what they are [Boyle]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
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
Set theory made a closer study of infinity possible [Clegg]
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
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
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
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
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
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
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
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
Specification: a condition applied to a set will always produce a new set [Clegg]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
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
Beyond infinity cardinals and ordinals can come apart [Clegg]
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
Transcendental numbers can't be fitted to finite equations [Clegg]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / k. Imaginary numbers
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
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
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
The Continuum Hypothesis is independent of the axioms of set theory [Clegg]
The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg]
8. Modes of Existence / C. Powers and Dispositions / 1. Powers
Boyle attacked a contemporary belief that powers were occult things [Boyle, by Alexander,P]
8. Modes of Existence / C. Powers and Dispositions / 6. Dispositions / a. Dispositions
In the 17th century, 'disposition' usually just means the spatial arrangement of parts [Boyle, by Pasnau]
9. Objects / C. Structure of Objects / 2. Hylomorphism / a. Hylomorphism
Form is not a separate substance, but just the manner, modification or 'stamp' of matter [Boyle]
To cite a substantial form tells us what produced the effect, but not how it did it [Boyle]
12. Knowledge Sources / B. Perception / 2. Qualities in Perception / d. Secondary qualities
Boyle's term 'texture' is not something you feel, but is unobservable structures of particles [Boyle, by Alexander,P]
Boyle's secondary qualities are not illusory, or 'in the mind' [Boyle, by Alexander,P]
14. Science / D. Explanation / 2. Types of Explanation / i. Explanations by mechanism
Explanation is deducing a phenomenon from some nature better known to us [Boyle]
21. Aesthetics / C. Artistic Issues / 7. Art and Morality
Musical performance can reveal a range of virtues [Damon of Ath.]
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / g. Atomism
The corpuscles just have shape, size and motion, which explains things without 'sympathies' or 'forces' [Boyle, by Alexander,P]
26. Natural Theory / A. Speculations on Nature / 7. Later Matter Theories / b. Corpuscles
The corpuscular theory allows motion, but does not include forces between the particles [Boyle, by Alexander,P]