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