Combining Texts

All the ideas for 'works', 'Understanding the Infinite' and 'Discourse on the Origin of Inequality'

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101 ideas

1. Philosophy / A. Wisdom / 1. Nature of Wisdom
There is practical wisdom (for action), and theoretical wisdom (for deep understanding) [Aristotle, by Whitcomb]
2. Reason / A. Nature of Reason / 2. Logos
For Aristotle logos is essentially the ability to talk rationally about questions of value [Roochnik on Aristotle]
2. Reason / A. Nature of Reason / 4. Aims of Reason
Aristotle is the supreme optimist about the ability of logos to explain nature [Roochnik on Aristotle]
2. Reason / A. Nature of Reason / 9. Limits of Reason
Reason leads to prudent selfishness, which overrules natural compassion [Rousseau]
2. Reason / D. Definition / 4. Real Definition
Aristotelian definitions aim to give the essential properties of the thing defined [Aristotle, by Quine]
2. Reason / D. Definition / 5. Genus and Differentia
Aristotelian definition involves first stating the genus, then the differentia of the thing [Aristotle, by Urmson]
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]
4. Formal Logic / G. Formal Mereology / 1. Mereology
Aristotle relativises the notion of wholeness to different measures [Aristotle, by Koslicki]
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]
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
For Aristotle, the subject-predicate structure of Greek reflected a substance-accident structure of reality [Aristotle, by O'Grady]
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]
9. Objects / C. Structure of Objects / 2. Hylomorphism / a. Hylomorphism
The unmoved mover and the soul show Aristotelian form as the ultimate mereological atom [Aristotle, by Koslicki]
9. Objects / C. Structure of Objects / 2. Hylomorphism / d. Form as unifier
The 'form' is the recipe for building wholes of a particular kind [Aristotle, by Koslicki]
11. Knowledge Aims / A. Knowledge / 1. Knowledge
For Aristotle, knowledge is of causes, and is theoretical, practical or productive [Aristotle, by Code]
No one would bother to reason, and try to know things, without a desire for enjoyment [Rousseau]
12. Knowledge Sources / A. A Priori Knowledge / 1. Nature of the A Priori
The notion of a priori truth is absent in Aristotle [Aristotle, by Politis]
12. Knowledge Sources / C. Rationalism / 1. Rationalism
Aristotle is a rationalist, but reason is slowly acquired through perception and experience [Aristotle, by Frede,M]
12. Knowledge Sources / E. Direct Knowledge / 2. Intuition
Aristotle wants to fit common intuitions, and therefore uses language as a guide [Aristotle, by Gill,ML]
14. Science / B. Scientific Theories / 1. Scientific Theory
Plato says sciences are unified around Forms; Aristotle says they're unified around substance [Aristotle, by Moravcsik]
14. Science / D. Explanation / 1. Explanation / a. Explanation
Aristotelian explanations are facts, while modern explanations depend on human conceptions [Aristotle, by Politis]
14. Science / D. Explanation / 2. Types of Explanation / a. Types of explanation
Aristotle's standard analysis of species and genus involves specifying things in terms of something more general [Aristotle, by Benardete,JA]
14. Science / D. Explanation / 2. Types of Explanation / k. Explanations by essence
Aristotle regularly says that essential properties explain other significant properties [Aristotle, by Kung]
15. Nature of Minds / C. Capacities of Minds / 5. Generalisation by mind
General ideas are purely intellectual; imagining them is immediately particular [Rousseau]
Only words can introduce general ideas into the mind [Rousseau]
18. Thought / A. Modes of Thought / 5. Rationality / c. Animal rationality
Aristotle and the Stoics denied rationality to animals, while Platonists affirmed it [Aristotle, by Sorabji]
18. Thought / D. Concepts / 5. Concepts and Language / a. Concepts and language
Language may aid thinking, but powerful thought was needed to produce language [Rousseau]
19. Language / E. Analyticity / 2. Analytic Truths
The notion of analytic truth is absent in Aristotle [Aristotle, by Politis]
21. Aesthetics / A. Aesthetic Experience / 4. Beauty
Without love, what use is beauty? [Rousseau]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / b. Rational ethics
Rational morality is OK for brainy people, but ordinary life can't rely on that [Rousseau]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / e. Human nature
Aristotle never actually says that man is a rational animal [Aristotle, by Fogelin]
22. Metaethics / C. The Good / 1. Goodness / h. Good as benefit
If we should not mistreat humans, it is mainly because of sentience, not rationality [Rousseau]
23. Ethics / B. Contract Ethics / 2. Golden Rule
The better Golden Rule is 'do good for yourself without harming others' [Rousseau]
23. Ethics / C. Virtue Theory / 3. Virtues / f. Compassion
The fact that we weep (e.g. in theatres) shows that we are naturally compassionate [Rousseau]
24. Political Theory / A. Basis of a State / 1. A People / a. Human distinctiveness
Humans are less distinguished from other animals by understanding, than by being free agents [Rousseau]
24. Political Theory / A. Basis of a State / 1. A People / b. The natural life
Most human ills are self-inflicted; the simple, solitary, regular natural life is good [Rousseau]
Is language a pre-requisite for society, or might it emerge afterwards? [Rousseau]
I doubt whether a savage person ever complains of life, or considers suicide [Rousseau]
Leisure led to envy, inequality, vice and revenge, which we now see in savages [Rousseau]
Primitive man was very gentle [Rousseau]
Our two starting principles are concern for self-interest, and compassion for others [Rousseau]
Savages avoid evil because they are calm, and never think of it (not because they know goodness) [Rousseau]
Savage men quietly pursue desires, without the havoc of modern frenzied imagination [Rousseau]
24. Political Theory / A. Basis of a State / 3. Natural Values / a. Natural freedom
A savage can steal fruit or a home, but there is no means of achieving obedience [Rousseau]
24. Political Theory / A. Basis of a State / 3. Natural Values / b. Natural equality
In a state of nature people are much more equal; it is society which increases inequalities [Rousseau]
It is against nature for children to rule old men, fools to rule the wise, and the rich to hog resources [Rousseau]
24. Political Theory / B. Nature of a State / 2. State Legitimacy / a. Sovereignty
People accept the right to be commanded, because they themselves wish to command [Rousseau]
24. Political Theory / B. Nature of a State / 5. Culture
We seem to have made individual progress since savagery, but actually the species has decayed [Rousseau]
24. Political Theory / C. Ruling a State / 4. Changing the State / c. Revolution
Revolutionaries usually confuse liberty with total freedom, and end up with heavier chains [Rousseau]
24. Political Theory / D. Ideologies / 5. Democracy / b. Consultation
Plebiscites are bad, because they exclude the leaders from crucial decisions [Rousseau]
24. Political Theory / D. Ideologies / 5. Democracy / c. Direct democracy
In a direct democracy, only the leaders should be able to propose new laws [Rousseau]
25. Social Practice / A. Freedoms / 1. Slavery
Enslaved peoples often boast of their condition, calling it a state of 'peace' [Rousseau]
If the child of a slave woman is born a slave, then a man is not born a man [Rousseau]
People must be made dependent before they can be enslaved [Rousseau]
25. Social Practice / A. Freedoms / 5. Freedom of lifestyle
Like rich food, liberty can ruin people who are too weak to cope with it [Rousseau]
25. Social Practice / B. Equalities / 1. Grounds of equality
Three stages of the state produce inequalities of wealth, power, and enslavement [Rousseau]
25. Social Practice / B. Equalities / 4. Economic equality
The pleasure of wealth and power is largely seeing others deprived of them [Rousseau]
25. Social Practice / C. Rights / 4. Property rights
Persuading other people that some land was 'owned' was the beginning of society [Rousseau]
What else could property arise from, but the labour people add to it? [Rousseau]
Land cultivation led to a general right of ownership, administered justly [Rousseau]
If we have a natural right to property, what exactly does 'belonging to' mean? [Rousseau]
25. Social Practice / D. Justice / 2. The Law / c. Natural law
Writers just propose natural law as the likely useful agreements among people [Rousseau]
25. Social Practice / D. Justice / 3. Punishment / b. Retribution for crime
Primitive people simply redressed the evil caused by violence, without thought of punishing [Rousseau]
25. Social Practice / E. Policies / 1. War / e. Peace
A state of war remains after a conquest, if the losers don't accept the winners [Rousseau]
25. Social Practice / E. Policies / 5. Education / a. Aims of education
It is the mark of an educated mind to be able to entertain an idea without accepting it [Aristotle]
25. Social Practice / E. Policies / 5. Education / b. Education principles
Aristotle said the educated were superior to the uneducated as the living are to the dead [Aristotle, by Diog. Laertius]
25. Social Practice / F. Life Issues / 6. Animal Rights
Both men and animals are sentient, which should give the latter the right not to be mistreated [Rousseau]
26. Natural Theory / A. Speculations on Nature / 5. Infinite in Nature
There are potential infinities (never running out), but actual infinity is incoherent [Aristotle, by Friend]
26. Natural Theory / A. Speculations on Nature / 6. Early Matter Theories / a. Greek matter
Aristotle's matter can become any other kind of matter [Aristotle, by Wiggins]
26. Natural Theory / B. Natural Kinds / 2. Defining Kinds
Men started with too few particular names, but later had too few natural kind names [Rousseau]
27. Natural Reality / G. Biology / 3. Evolution
Small uninterrupted causes can have big effects [Rousseau]
29. Religion / A. Polytheistic Religion / 2. Greek Polytheism
The concepts of gods arose from observing the soul, and the cosmos [Aristotle, by Sext.Empiricus]