Ideas of Penelope Maddy, by Theme

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

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4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
'Forcing' can produce new models of ZFC from old models
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
A Large Cardinal Axiom would assert ever-increasing stages in the hierarchy
New axioms are being sought, to determine the size of the continuum
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / b. Axiom of Extensionality I
The Axiom of Extensionality seems to be analytic
Extensional sets are clearer, simpler, unique and expressive
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
Infinite sets are essential for giving an account of the real numbers
The Axiom of Infinity states Cantor's breakthrough that launched modern mathematics
Axiom of Infinity: completed infinite collections can be treated mathematically
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / g. Axiom of Powers VI
The Power Set Axiom is needed for, and supported by, accounts of the continuum
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / h. Axiom of Replacement VII
Replacement was added when some advanced theorems seemed to need it
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
The Axiom of Foundation says every set exists at a level in the set hierarchy
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
A large array of theorems depend on the Axiom of Choice
Modern views say the Choice set exists, even if it can't be constructed
The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres
Efforts to prove the Axiom of Choice have failed
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
Axiom of Reducibility: propositional functions are extensionally predicative
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The Iterative Conception says everything appears at a stage, derived from the preceding appearances
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Limitation of Size is a vague intuition that over-large sets may generate paradoxes
4. Formal Logic / F. Set Theory ST / 7. Natural Sets
Maddy replaces pure sets with just objects and perceived sets of objects
The master science is physical objects divided into sets
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Henkin semantics is more plausible for plural logic than for second-order logic
5. Theory of Logic / C. Ontology of Logic / 3. If-Thenism
Critics of if-thenism say that not all starting points, even consistent ones, are worth studying
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
'Propositional functions' are propositions with a variable as subject or predicate
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
If two mathematical themes coincide, that suggest a single deep truth
Hilbert's geometry and Dedekind's real numbers were role models for axiomatization
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / d. Actual infinite
Cantor and Dedekind brought completed infinities into mathematics
Completed infinities resulted from giving foundations to calculus
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / g. Continuum Hypothesis
Every infinite set of reals is either countable or of the same size as the full set of reals
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / h. Ordinal infinity
Cantor extended ordinals into the transfinite, and they can thus measure infinite cardinalities
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / i. Cardinal infinity
Cardinality strictly concerns one-one correspondence, to test infinite sameness of size
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / j. Large cardinals
For any cardinal there is always a larger one (so there is no set of all sets)
An 'inaccessible' cardinal cannot be reached by union sets or power sets
Infinity has degrees, and large cardinals are the heart of set theory
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / m. Limits
Theorems about limits could only be proved once the real numbers were understood
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / c. Fregean numbers
The extension of concepts is not important to me
In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / e. Caesar problem
Frege solves the Caesar problem by explicitly defining each number
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / f. Zermelo numbers
For Zermelo the successor of n is {n} (rather than n U {n})
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / g. Von Neumann numbers
Von Neumann numbers are preferred, because they continue into the transfinite
For Von Neumann the successor of n is n U {n} (rather than {n})
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / a. Mathematics is set theory
Set theory (unlike the Peano postulates) can explain why multiplication is commutative
Standardly, numbers are said to be sets, which is neat ontology and epistemology
Numbers are properties of sets, just as lengths are properties of physical objects
A natural number is a property of sets
Mathematics rests on the logic of proofs, and on the set theoretic axioms
Unified set theory gives a final court of appeal for mathematics
Identifying geometric points with real numbers revealed the power of set theory
Making set theory foundational to mathematics leads to very fruitful axioms
Set theory brings mathematics into one arena, where interrelations become clearer
The line of rationals has gaps, but set theory provided an ordered continuum
Set-theory tracks the contours of mathematical depth and fruitfulness
6. Mathematics / B. Foundations for Mathematics / 5. Mathematics as Set Theory / b. Mathematics is not set theory
Number theory doesn't 'reduce' to set theory, because sets have number properties
Sets exist where their elements are, but numbers are more like universals
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
If mathematical objects exist, how can we know them, and which objects are they?
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
Intuition doesn't support much mathematics, and we should question its reliability
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
We know mind-independent mathematical truths through sets, which rest on experience
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
Maybe applications of continuum mathematics are all idealisations
Scientists posit as few entities as possible, but set theorist posit as many as possible
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
The connection of arithmetic to perception has been idealised away in modern infinitary mathematics
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
Number words are unusual as adjectives; we don't say 'is five', and numbers always come first
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
We can get arithmetic directly from HP; Law V was used to get HP from the definition of number
7. Existence / D. Theories of Reality / 10. Ontological Commitment / e. Ontological commitment problems
The theoretical indispensability of atoms did not at first convince scientists that they were real
15. Nature of Minds / C. Capacities of Minds / 6. Idealisation
Science idealises the earth's surface, the oceans, continuities, and liquids