Ideas of Michèle Friend, by Theme

[American, fl. 2007, Professor at George Washington University, Washington D.C.]

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2. Reason / D. Definition / 8. Impredicative Definition
8721 An 'impredicative' definition seems circular, because it uses the term being defined
2. Reason / D. Definition / 10. Stipulative Definition
8680 Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects
2. Reason / E. Argument / 5. Reductio ad Absurdum
3678 Reductio ad absurdum proves an idea by showing that its denial produces contradiction
3. Truth / A. Truth Problems / 8. Subjective Truth
8705 Anti-realists see truth as our servant, and epistemically contrained
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
8713 In classical/realist logic the connectives are defined by truth-tables
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
8708 Double negation elimination is not valid in intuitionist logic
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
8694 Free logic was developed for fictional or non-existent objects
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
8665 A 'proper subset' of A contains only members of A, but not all of them
8672 A 'powerset' is all the subsets of a set
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
8677 Set theory makes a minimum ontological claim, that the empty set exists
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
8666 Infinite sets correspond one-to-one with a subset
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
8682 Major set theories differ in their axioms, and also over the additional axioms of choice and infinity
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
8709 The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false
5. Theory of Logic / G. Quantification / 7. Unorthodox Quantification
8711 Intuitionists read the universal quantifier as "we have a procedure for checking every..."
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / a. Set theory paradoxes
8675 Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets'
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. Burali-Forti's paradox
8674 The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
8667 The 'integers' are the positive and negative natural numbers, plus zero
8668 The 'rational' numbers are those representable as fractions
8670 A number is 'irrational' if it cannot be represented as a fraction
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
8661 The natural numbers are primitive, and the ordinals are up one level of abstraction
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
8664 Cardinal numbers answer 'how many?', with the order being irrelevant
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
8671 The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
8663 Raising omega to successive powers of omega reveal an infinity of infinities
8662 The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
8669 Between any two rational numbers there is an infinite number of rational numbers
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
8676 Is mathematics based on sets, types, categories, models or topology?
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
8678 Most mathematical theories can be translated into the language of set theory
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
8701 The number 8 in isolation from the other numbers is of no interest
8702 In structuralism the number 8 is not quite the same in different structures, only equivalent
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
8699 Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)?
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
8696 Structuralist says maths concerns concepts about base objects, not base objects themselves
8695 Structuralism focuses on relations, predicates and functions, with objects being inessential
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
8700 'In re' structuralism says that the process of abstraction is pattern-spotting
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
8681 The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts?
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
8712 Mathematics should be treated as true whenever it is indispensable to our best physical theory
6. Mathematics / C. Sources of Mathematics / 7. Formalism
8716 Formalism is unconstrained, so cannot indicate importance, or directions for research
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
8706 Constructivism rejects too much mathematics
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
8707 Intuitionists typically retain bivalence but reject the law of excluded middle
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
8704 Structuralists call a mathematical 'object' simply a 'place in a structure'
17. Mind and Body / E. Mind as Physical / 2. Reduction of Mind
8685 Studying biology presumes the laws of chemistry, and it could never contradict them
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
8688 Concepts can be presented extensionally (as objects) or intensionally (as a characterization)