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### Ideas of Michèle Friend, by Text

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

 2007 Introducing the Philosophy of Mathematics
 p.113 8710 The powerset of all the cardinal numbers is required to be greater than itself
 p.128 8713 In classical/realist logic the connectives are defined by truth-tables
 1.4 p.12 8661 The natural numbers are primitive, and the ordinals are up one level of abstraction
 1.4 p.13 8663 Raising omega to successive powers of omega reveal an infinity of infinities
 1.4 p.13 8662 The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega
 1.5 p.14 8664 Cardinal numbers answer 'how many?', with the order being irrelevant
 1.5 p.15 8666 Infinite sets correspond one-to-one with a subset
 1.5 p.15 8665 A 'proper subset' of A contains only members of A, but not all of them
 1.5 p.16 8667 The 'integers' are the positive and negative natural numbers, plus zero
 1.5 p.17 8668 The 'rational' numbers are those representable as fractions
 1.5 p.17 8669 Between any two rational numbers there is an infinite number of rational numbers
 1.5 p.19 8671 The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps
 1.5 p.19 8670 A number is 'irrational' if it cannot be represented as a fraction
 1.5 p.21 8672 A 'powerset' is all the subsets of a set
 2.3 p.26 8674 The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal
 2.3 p.27 8675 Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets'
 2.3 p.29 8676 Is mathematics based on sets, types, categories, models or topology?
 2.3 p.32 8677 Set theory makes a minimum ontological claim, that the empty set exists
 2.3 p.33 8678 Most mathematical theories can be translated into the language of set theory
 2.3 p.34 3678 Reductio ad absurdum proves an idea by showing that its denial produces contradiction
 2.4 p.36 8680 Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects
 2.5 p.36 8681 The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts?
 2.6 p.42 8682 Major set theories differ in their axioms, and also over the additional axioms of choice and infinity
 3.1 p.51 8685 Studying biology presumes the laws of chemistry, and it could never contradict them
 3.4 p.64 8688 Concepts can be presented extensionally (as objects) or intensionally (as a characterization)
 3.7 p.77 8694 Free logic was developed for fictional or non-existent objects
 4.1 p.82 8695 Structuralism focuses on relations, predicates and functions, with objects being inessential
 4.1 p.82 8696 Structuralist says maths concerns concepts about base objects, not base objects themselves
 4.4 p.90 8699 Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)?
 4.4 p.91 8700 'In re' structuralism says that the process of abstraction is pattern-spotting
 4.4 p.93 8702 In structuralism the number 8 is not quite the same in different structures, only equivalent
 4.4 p.93 8701 The number 8 in isolation from the other numbers is of no interest
 4.5 p.97 8704 Structuralists call a mathematical 'object' simply a 'place in a structure'
 5.1 p.104 8705 Anti-realist see truth as our servant, and epistemically contrained
 5.1 p.106 8706 Constructivism rejects too much mathematics
 5.2 p.106 8707 Intuitionists typically retain bivalence but reject the law of excluded middle
 5.2 p.107 8708 Double negation elimination is not valid in intuitionist logic
 5.2 p.108 8709 The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false
 5.5 p.122 8711 Intuitionists read the universal quantifier as "we have a procedure for checking every..."
 6.1 p.128 8712 Mathematics should be treated as true whenever it is indispensable to our best physical theory
 6.5 p.145 8715 Infinities expand the bounds of the conceivable; we explore concepts to explore conceivability
 6.6 p.149 8716 Formalism is unconstrained, so cannot indicate importance, or directions for research
 Glossary p.172 8721 An 'impredicative' definition seems circular, because it uses the term being defined