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Ideas of Penelope Maddy, by Text

[American, b.1950, Professor of Logic and Philosophy of Science at the University of California, Irvine.]

1981 Sets and Numbers
I p.345 If mathematical objects exist, how can we know them, and which objects are they?
II p.347 Set theory (unlike the Peano postulates) can explain why multiplication is commutative
II p.347 The master science is physical objects divided into sets
III p.347 Standardly, numbers are said to be sets, which is neat ontology and epistemology
III p.349 Numbers are properties of sets, just as lengths are properties of physical objects
III p.349 Sets exist where their elements are, but numbers are more like universals
IV p.350 Number words are unusual as adjectives; we don't say 'is five', and numbers always come first
V p.353 Number theory doesn't 'reduce' to set theory, because sets have number properties
1988 Believing the Axioms I
0 p.482 New axioms are being sought, to determine the size of the continuum
1.1 p.484 Extensional sets are clearer, simpler, unique and expressive
1.1 p.484 The Axiom of Extensionality seems to be analytic
1.3 p.485 The Iterative Conception says everything appears at a stage, derived from the preceding appearances
1.3 p.485 Limitation of Size is a vague intuition that over-large sets may generate paradoxes
1.5 p.486 The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics
1.5 p.486 Infinite sets are essential for giving an account of the real numbers
1.6 p.486 The Power Set Axiom is needed for, and supported by, accounts of the continuum
1.7 p.487 Efforts to prove the Axiom of Choice have failed
1.7 p.488 Modern views say the Choice set exists, even if it can't be constructed
1.7 p.488 A large array of theorems depend on the Axiom of Choice
1.8 p.489 Replacement was added when some advanced theorems seemed to need it
1990 Realism in Mathematics
p.191 We know mind-independent mathematical truths through sets, which rest on experience
p.223 Maddy replaces pure sets with just objects and perceived sets of objects
p.224 Intuition doesn't support much mathematics, and we should question its reliability
3 2 p.19 A natural number is a property of sets
1997 Naturalism in Mathematics
107 p.117 The extension of concepts is not important to me
I Intro p.1 Mathematics rests on the logic of proofs, and on the set theoretic axioms
I.1 p.5 Frege solves the Caesar problem by explicitly defining each number
I.1 p.7 We can get arithmetic directly from HP; Law V was used to get HP from the definition of number
I.1 p.9 'Propositional functions' are propositions with a variable as subject or predicate
I.1 p.11 Axiom of Reducibility: propositional functions are extensionally predicative
I.1 p.16 Infinity has degrees, and large cardinals are the heart of set theory
I.1 p.17 For any cardinal there is always a larger one (so there is no set of all sets)
I.1 p.17 Cardinality strictly concerns one-one correspondence, to test infinite sameness of size
I.1 p.17 Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities
I.1 n39 p.15 Cantor and Dedekind brought completed infinities into mathematics
I.2 p.23 In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets
I.2 p.26 Set theory brings mathematics into one arena, where interrelations become clearer
I.2 p.26 Making set theory foundational to mathematics leads to very fruitful axioms
I.2 p.26 Unified set theory gives a final court of appeal for mathematics
I.2 p.26 Identifying geometric points with real numbers revealed the power of set theory
I.2 p.27 The line of rationals has gaps, but set theory provided an ordered continuum
I.2 p.27 Theorems about limits could only be proved once the real numbers were understood
I.2 n8 p.24 For Zermelo the successor of n is {n} (rather than n U {n})
I.2 n8 p.24 For Von Neumann the successor of n is n U {n} (rather than {n})
I.2 n9 p.24 Von Neumann numbers are preferred, because they continue into the transfinite
I.3 p.51 Completed infinities resulted from giving foundations to calculus
I.3 p.52 Axiom of Infinity: completed infinite collections can be treated mathematically
I.3 p.60 The Axiom of Foundation says every set exists at a level in the set hierarchy
I.4 p.66 'Forcing' can produce new models of ZFC from old models
I.5 p.73 A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy
I.5 p.74 An 'inaccessible' cardinal cannot be reached by union sets or power sets
II.5 p.131 Scientists posit as few entities as possible, but set theorist posit as many as possible
II.6 p.143 The theoretical indispensability of atoms did not at first convince scientists that they were real
II.6 p.143 Science idealises the earth's surface, the oceans, continuities, and liquids
II.6 p.152 Maybe applications of continuum mathematics are all idealisations
2007 Second Philosophy
III.8 n1 p.299 Henkin semantics is more plausible for plural logic than for second-order logic
2011 Defending the Axioms
1.3 p.31 Hilbert's geometry and Dedekind's real numbers were role models for axiomatization
1.3 p.35 The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres
2.3 p.53 The connection of arithmetic to perception has been idealised away in modern infinitary mathematics
2.4 n40 p.56 Every infinite set of reals is either countable or of the same size as the full set of reals
3.3 p.99 Critics of if-thenism say that not all starting points, even consistent ones, are worth studying
3.4 p.82 Set-theory tracks the contours of mathematical depth and fruitfulness
5.3ii p.129 If two mathematical themes coincide, that suggest a single deep truth