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