Combining Texts

All the ideas for 'How Things Might Have Been', 'What Required for Foundation for Maths?' and 'Computing Machinery and Intelligence'

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51 ideas

2. Reason / D. Definition / 2. Aims of Definition
17774 Definitions make our intuitions mathematically useful [Mayberry]
2. Reason / E. Argument / 6. Conclusive Proof
17773 Proof shows that it is true, but also why it must be true [Mayberry]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
17795 Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry]
17796 There is a semi-categorical axiomatisation of set-theory [Mayberry]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
17800 The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
17801 The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
17803 Limitation of size is part of the very conception of a set [Mayberry]
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
17786 The mainstream of modern logic sees it as a branch of mathematics [Mayberry]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
17788 First-order logic only has its main theorems because it is so weak [Mayberry]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
17791 Only second-order logic can capture mathematical structure up to isomorphism [Mayberry]
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
17787 Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
17790 No Löwenheim-Skolem logic can axiomatise real analysis [Mayberry]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
17779 'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry]
17778 Axiomatiation relies on isomorphic structures being essentially the same [Mayberry]
17780 'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry]
5. Theory of Logic / K. Features of Logics / 6. Compactness
17789 No logic which can axiomatise arithmetic can be compact or complete [Mayberry]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
17784 Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
17782 Greek quantities were concrete, and ratio and proportion were their science [Mayberry]
17781 Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
17799 Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry]
17797 Cantor extended the finite (rather than 'taming the infinite') [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
17775 If proof and definition are central, then mathematics needs and possesses foundations [Mayberry]
17776 The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry]
17777 Foundations need concepts, definition rules, premises, and proof rules [Mayberry]
17804 Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
17792 1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
17793 It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
17794 Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry]
17802 We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry]
17805 Set theory is not just another axiomatised part of mathematics [Mayberry]
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
17785 Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry]
9. Objects / A. Existence of Objects / 5. Individuation / a. Individuation
11897 A principle of individuation may pinpoint identity and distinctness, now and over time [Mackie,P]
11898 Individuation may include counterfactual possibilities, as well as identity and persistence [Mackie,P]
9. Objects / A. Existence of Objects / 5. Individuation / d. Individuation by haecceity
11883 A haecceity is the essential, simple, unanalysable property of being-this-thing [Mackie,P]
9. Objects / D. Essence of Objects / 1. Essences of Objects
11889 Essentialism must avoid both reduplication of essences, and multiple occupancy by essences [Mackie,P]
9. Objects / D. Essence of Objects / 3. Individual Essences
11877 An individual essence is the properties the object could not exist without [Mackie,P]
11882 No other object can possibly have the same individual essence as some object [Mackie,P]
11886 There are problems both with individual essences and without them [Mackie,P]
9. Objects / D. Essence of Objects / 5. Essence as Kind
11909 Unlike Hesperus=Phosophorus, water=H2O needs further premisses before it is necessary [Mackie,P]
11899 Why are any sortals essential, and why are only some of them essential? [Mackie,P]
9. Objects / D. Essence of Objects / 8. Essence as Explanatory
11906 The Kripke and Putnam view of kinds makes them explanatorily basic, but has modal implications [Mackie,P]
9. Objects / E. Objects over Time / 12. Origin as Essential
11894 Origin is not a necessity, it is just 'tenacious'; we keep it fixed in counterfactual discussions [Mackie,P]
10. Modality / E. Possible worlds / 3. Transworld Objects / a. Transworld identity
11887 Transworld identity without individual essences leads to 'bare identities' [Mackie,P]
10. Modality / E. Possible worlds / 3. Transworld Objects / c. Counterparts
11890 De re modality without bare identities or individual essence needs counterparts [Mackie,P]
11892 Things may only be counterparts under some particular relation [Mackie,P]
11893 Possibilities for Caesar must be based on some phase of the real Caesar [Mackie,P]
10. Modality / E. Possible worlds / 3. Transworld Objects / d. Haecceitism
11884 The theory of 'haecceitism' does not need commitment to individual haecceities [Mackie,P]
14. Science / D. Explanation / 2. Types of Explanation / k. Explanations by essence
11905 Locke's kind essences are explanatory, without being necessary to the kind [Mackie,P]
18. Thought / B. Mechanics of Thought / 6. Artificial Thought / b. Turing Machines
3979 The Turing Machine is the best idea yet about how the mind works [Fodor on Turing]
18. Thought / B. Mechanics of Thought / 6. Artificial Thought / c. Turing Test
5321 In 50 years computers will successfully imitate humans with a 70% success rate [Turing]
26. Natural Theory / B. Natural Kinds / 6. Necessity of Kinds
11907 Maybe the identity of kinds is necessary, but instances being of that kind is not [Mackie,P]