Combining Philosophers

All the ideas for Hermarchus, Brian Clegg and Hugo Grotius

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30 ideas

4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
10859 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
10857 Set theory made a closer study of infinity possible [Clegg]
10864 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
10872 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
10875 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
10876 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
10878 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
10877 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
10879 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
10871 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
10874 Specification: a condition applied to a set will always produce a new set [Clegg]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
10880 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
10861 Beyond infinity cardinals and ordinals can come apart [Clegg]
10860 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
10854 Transcendental numbers can't be fitted to finite equations [Clegg]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / k. Imaginary numbers
10858 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
10853 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
10866 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
10869 The Continuum Hypothesis is independent of the axioms of set theory [Clegg]
10862 The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg]
20. Action / C. Motives for Action / 5. Action Dilemmas / c. Omissions
7404 Nations are not obliged to help one-another, but are obliged not to harm one another [Grotius, by Tuck]
24. Political Theory / A. Basis of a State / 3. Natural Values / c. Natural rights
7402 Everyone has a right of self-preservation, and harming others is usually unjustifiable [Grotius, by Tuck]
24. Political Theory / D. Ideologies / 5. Democracy / a. Nature of democracy
21938 Democracy needs respect for individuality, but the 'community of friends' implies strict equality [Grotius]
25. Social Practice / A. Freedoms / 7. Freedom to leave
19845 A person is free to renounce their state, as long as it is not a moment of crisis [Grotius, by Rousseau]
25. Social Practice / D. Justice / 2. The Law / c. Natural law
22133 Grotius and Pufendorf based natural law on real (rather than idealised) humanity [Grotius, by Ford,JD]
7406 A natural right of self-preservation is balanced by a natural law to avoid unnecessary harm [Grotius, by Tuck]
25. Social Practice / D. Justice / 2. The Law / d. Legal positivism
7403 Grotius ignored elaborate natural law theories, preferring a basic right of self-preservation [Grotius, by Tuck]
25. Social Practice / E. Policies / 1. War / b. Justice in war
23585 It is permissible in a just cause to capture a place in neutral territory [Grotius]
25. Social Practice / F. Life Issues / 6. Animal Rights
6005 Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley]
28. God / A. Divine Nature / 6. Divine Morality / b. Euthyphro question
6892 Moral principles have some validity without a God commanding obedience [Grotius, by Mautner]