Combining Philosophers

All the ideas for Adam Smith, George Cantor and Robert Pasnau

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105 ideas

1. Philosophy / C. History of Philosophy / 1. History of Philosophy
Original philosophers invariably seek inspiration from past thinkers [Pasnau]
Philosophy consists of choosing between Plato, Aristotle and Democritus [Pasnau]
1. Philosophy / C. History of Philosophy / 3. Earlier European Philosophy / b. Early medieval philosophy
The commentaries of Averroes were the leading guide to Aristotle [Pasnau]
Modernity begins in the late 12th century, with Averroes's commentaries on Aristotle [Pasnau]
1. Philosophy / C. History of Philosophy / 3. Earlier European Philosophy / c. Later medieval philosophy
Once accidents were seen as real, 'Categories' became the major text for ontology [Pasnau]
In 1347, the Church effectively stopped philosophy for the next 300 years [Pasnau]
1. Philosophy / C. History of Philosophy / 3. Earlier European Philosophy / d. Renaissance philosophy
Renaissance Platonism is peripheral [Pasnau]
Plato only made an impact locally in 15th century Italy [Pasnau]
After c.1450 all of Plato was available. Before that, only the first half of 'Timaeus' was known [Pasnau]
1. Philosophy / C. History of Philosophy / 4. Later European Philosophy / b. Seventeenth century philosophy
Philosophy could easily have died in 17th century, if it weren't for Descartes [Pasnau]
The 17th century is a metaphysical train wreck [Pasnau]
2. Reason / B. Laws of Thought / 6. Ockham's Razor
Anti-Razor: if you can't account for a truth, keep positing things until you can [Pasnau]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
Trying to represent curves, we study arbitrary functions, leading to the ordinals, which produces set theory [Cantor, by Lavine]
Cantor developed sets from a progression into infinity by addition, multiplication and exponentiation [Cantor, by Lavine]
A set is a collection into a whole of distinct objects of our intuition or thought [Cantor]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / c. Basic theorems of ST
Cantor's Theorem: for any set x, its power set P(x) has more members than x [Cantor, by Hart,WD]
Cantor proved that all sets have more subsets than they have members [Cantor, by Bostock]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / c. Unit (Singleton) Sets
If a set is 'a many thought of as one', beginners should protest against singleton sets [Cantor, by Lewis]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
Cantor showed that supposed contradictions in infinity were just a lack of clarity [Cantor, by Potter]
The continuum is the powerset of the integers, which moves up a level [Cantor, by Clegg]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
Cantor gives informal versions of ZF axioms as ways of getting from one set to another [Cantor, by Lake]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / d. Axiom of Unions III
The Axiom of Union dates from 1899, and seems fairly obvious [Cantor, by Maddy]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / b. Combinatorial sets
Cantor's sets were just collections, but Dedekind's were containers [Cantor, by Oliver/Smiley]
5. Theory of Logic / K. Features of Logics / 8. Enumerability
There are infinite sets that are not enumerable [Cantor, by Smith,P]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / b. Cantor's paradox
Cantor's Paradox: the power set of the universe must be bigger than the universe, yet a subset of it [Cantor, by Hart,WD]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / e. Mirimanoff's paradox
The powerset of all the cardinal numbers is required to be greater than itself [Cantor, by Friend]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Cantor named the third realm between the finite and the Absolute the 'transfinite' [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Cantor proved the points on a plane are in one-to-one correspondence to the points on a line [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
Cantor took the ordinal numbers to be primary [Cantor, by Tait]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / d. Natural numbers
Cantor presented the totality of natural numbers as finite, not infinite [Cantor, by Mayberry]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
Cantor introduced the distinction between cardinals and ordinals [Cantor, by Tait]
Cantor showed that ordinals are more basic than cardinals [Cantor, by Dummett]
Ordinals are generated by endless succession, followed by a limit ordinal [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
A cardinal is an abstraction, from the nature of a set's elements, and from their order [Cantor]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
Cantor tried to prove points on a line matched naturals or reals - but nothing in between [Cantor, by Lavine]
Cantor's diagonal argument proved you can't list all decimal numbers between 0 and 1 [Cantor, by Read]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
A real is associated with an infinite set of infinite Cauchy sequences of rationals [Cantor, by Lavine]
Irrational numbers are the limits of Cauchy sequences of rational numbers [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
It was Cantor's diagonal argument which revealed infinities greater than that of the real numbers [Cantor, by Lavine]
Irrationals and the Dedekind Cut implied infinite classes, but they seemed to have logical difficulties [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
Cantor proposes that there won't be a potential infinity if there is no actual infinity [Cantor, by Hart,WD]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / f. Uncountable infinities
The naturals won't map onto the reals, so there are different sizes of infinity [Cantor, by George/Velleman]
Cantor needed Power Set for the reals, but then couldn't count the new collections [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
The Continuum Hypothesis says there are no sets between the natural numbers and reals [Cantor, by Shapiro]
CH: An infinite set of reals corresponds 1-1 either to the naturals or to the reals [Cantor, by Koellner]
Cantor: there is no size between naturals and reals, or between a set and its power set [Cantor, by Hart,WD]
Cantor's Continuum Hypothesis says there is a gap between the natural and the real numbers [Cantor, by Horsten]
Continuum Hypothesis: there are no sets between N and P(N) [Cantor, by Wolf,RS]
Continuum Hypothesis: no cardinal greater than aleph-null but less than cardinality of the continuum [Cantor, by Chihara]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities [Cantor, by Maddy]
Cantor's theory concerns collections which can be counted, using the ordinals [Cantor, by Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
Cardinality strictly concerns one-one correspondence, to test infinite sameness of size [Cantor, by Maddy]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
Property extensions outstrip objects, so shortage of objects caused the Caesar problem [Cantor, by Shapiro]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Pure mathematics is pure set theory [Cantor]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
Cantor says that maths originates only by abstraction from objects [Cantor, by Frege]
7. Existence / C. Structure of Existence / 1. Grounding / a. Nature of grounding
Priority was a major topic of dispute for scholastics [Pasnau]
7. Existence / C. Structure of Existence / 8. Stuff / b. Mixtures
In mixtures, the four elements ceased to exist, replaced by a mixed body with a form [Pasnau]
8. Modes of Existence / B. Properties / 3. Types of Properties
17th C qualities are either microphysical, or phenomenal, or powers [Pasnau]
8. Modes of Existence / B. Properties / 6. Categorical Properties
17th century authors only recognised categorical properties, never dispositions [Pasnau]
8. Modes of Existence / B. Properties / 8. Properties as Modes
The biggest question for scholastics is whether properties are real, or modes of substances [Pasnau]
8. Modes of Existence / C. Powers and Dispositions / 4. Powers as Essence
There is no centralised power, but we still need essence for a metaphysical understanding [Pasnau]
8. Modes of Existence / C. Powers and Dispositions / 6. Dispositions / a. Dispositions
Instead of adding Aristotelian forms to physical stuff, one could add dispositions [Pasnau]
8. Modes of Existence / C. Powers and Dispositions / 6. Dispositions / b. Dispositions and powers
Scholastics reject dispositions, because they are not actual, as forms require [Pasnau]
9. Objects / A. Existence of Objects / 5. Individuation / a. Individuation
Scholastics say there is a genuine thing if it is 'separable' [Pasnau]
9. Objects / A. Existence of Objects / 5. Individuation / b. Individuation by properties
Scholastics thought Quantity could be the principle of individuation [Pasnau]
If you reject essences, questions of individuation become extremely difficult [Pasnau]
9. Objects / B. Unity of Objects / 2. Substance / a. Substance
Corpuscularian critics of scholasticism say only substances exist [Pasnau]
Scholastics wanted to treat Aristotelianism as physics, rather than as metaphysics [Pasnau]
If crowds are things at all, they seem to be Substances, since they bear properties [Pasnau]
Corpuscularianism promised a decent account of substance [Pasnau]
9. Objects / B. Unity of Objects / 2. Substance / c. Types of substance
Scholastics use 'substantia' for thick concrete entities, and for thin metaphysical ones [Pasnau]
9. Objects / B. Unity of Objects / 2. Substance / e. Substance critique
For corpuscularians, a substance is just its integral parts [Pasnau]
9. Objects / B. Unity of Objects / 3. Unity Problems / c. Statue and clay
If clay survives destruction of the statue, the statue wasn't a substance, but a mere accident [Pasnau]
9. Objects / C. Structure of Objects / 2. Hylomorphism / a. Hylomorphism
Corpuscularianism rejected not only form, but also the dependence of matter on form [Pasnau]
9. Objects / C. Structure of Objects / 2. Hylomorphism / b. Form as principle
Hylomorphism may not be a rival to science, but an abstract account of unity and endurance [Pasnau]
9. Objects / C. Structure of Objects / 2. Hylomorphism / c. Form as causal
Hylomorphism declined because scholastics made it into a testable physical theory [Pasnau]
Scholastics made forms substantial, in a way unintended by Aristotle [Pasnau]
Scholastics began to see substantial form more as Aristotle's 'efficient' cause [Pasnau]
9. Objects / C. Structure of Objects / 2. Hylomorphism / d. Form as unifier
Aquinas says a substance has one form; Scotists say it has many forms [Pasnau]
9. Objects / C. Structure of Objects / 4. Quantity of an Object
Scholastic Quantity either gives a body parts, or spreads them out in a unified way [Pasnau]
9. Objects / C. Structure of Objects / 7. Substratum
There may be different types of substrate, or temporary substrates [Pasnau]
A substrate may be 'prime matter', which endures through every change [Pasnau]
A substratum can't be 'bare', because it has a job to do [Pasnau]
If a substrate gives causal support for change, quite a lot of the ingredients must endure [Pasnau]
9. Objects / D. Essence of Objects / 7. Essence and Necessity / b. Essence not necessities
Aristotelians deny that all necessary properties are essential [Pasnau]
9. Objects / E. Objects over Time / 6. Successive Things
Typical successive things are time and motion [Pasnau]
9. Objects / E. Objects over Time / 10. Beginning of an Object
Weak ex nihilo says it all comes from something; strong version says the old must partly endure [Pasnau]
14. Science / D. Explanation / 2. Types of Explanation / k. Explanations by essence
Essences must explain, so we can infer them causally from the accidents [Pasnau]
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability [Cantor, by Friend]
18. Thought / E. Abstraction / 2. Abstracta by Selection
Cantor says (vaguely) that we abstract numbers from equal sized sets [Hart,WD on Cantor]
We form the image of a cardinal number by a double abstraction, from the elements and from their order [Cantor]
22. Metaethics / B. The Good / 1. Goodness / i. Moral luck
A carelessly thrown brick is condemned much more if it hits someone [Smith,A, by Harman]
23. Ethics / B. Contract Ethics / 1. Contractarianism
People prepare our dinner from their own self-interest, not from humanity [Smith,A]
24. Political Theory / D. Ideologies / 11. Capitalism
Selfish profit-seeking increases collective wealth, so greed is good, and egoism is altruism [Smith,A, by Harari]
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / g. Atomism
Atomists say causation is mechanical collisions, and all true qualities are microscopic [Pasnau]
26. Natural Theory / A. Speculations on Nature / 7. Later Matter Theories / a. Early Modern matter
In the 17th C matter became body, and was then studied by science [Pasnau]
26. Natural Theory / A. Speculations on Nature / 7. Later Matter Theories / b. Corpuscles
Atomism is the commonest version of corpuscularianism, but isn't required by it [Pasnau]
If there are just arrangements of corpuscles, where are the boundaries between substances? [Pasnau]
26. Natural Theory / C. Causation / 2. Types of cause
Scholastic causation is by changes in the primary qualities of hot, cold, wet, dry [Pasnau]
26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / a. Scientific essentialism
Substantial forms were a step towards scientific essentialism [Pasnau]
27. Natural Reality / C. Space / 3. Points in Space
Cantor proved that three dimensions have the same number of points as one dimension [Cantor, by Clegg]
27. Natural Reality / E. Cosmology / 3. The Beginning
Scholastic authors agree that matter was created by God, out of nothing [Pasnau]
28. God / A. Divine Nature / 2. Divine Nature
Only God is absolutely infinite [Cantor, by Hart,WD]
29. Religion / B. Monotheistic Religion / 4. Christianity / b. Transubstantiation
Transubstantion says accidents of bread and wine don't inhere in the substance [Pasnau]