Combining Texts

All the ideas for 'No Understanding without Explanation', 'Against Structural Universals' and 'Naturalism in Mathematics'

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46 ideas

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 [Maddy]
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 [Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
Axiom of Infinity: completed infinite collections can be treated mathematically [Maddy]
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 [Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / p. Axiom of Reducibility
Axiom of Reducibility: propositional functions are extensionally predicative [Maddy]
5. Theory of Logic / E. Structures of Logic / 1. Logical Form
'Propositional functions' are propositions with a variable as subject or predicate [Maddy]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
Completed infinities resulted from giving foundations to calculus [Maddy]
Cantor and Dedekind brought completed infinities into mathematics [Maddy]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
Infinity has degrees, and large cardinals are the heart of set theory [Maddy]
For any cardinal there is always a larger one (so there is no set of all sets) [Maddy]
An 'inaccessible' cardinal cannot be reached by union sets or power sets [Maddy]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
Theorems about limits could only be proved once the real numbers were understood [Maddy]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
The extension of concepts is not important to me [Maddy]
In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
Frege solves the Caesar problem by explicitly defining each number [Maddy]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Making set theory foundational to mathematics leads to very fruitful axioms [Maddy]
Unified set theory gives a final court of appeal for mathematics [Maddy]
Set theory brings mathematics into one arena, where interrelations become clearer [Maddy]
Identifying geometric points with real numbers revealed the power of set theory [Maddy]
The line of rationals has gaps, but set theory provided an ordered continuum [Maddy]
Mathematics rests on the logic of proofs, and on the set theoretic axioms [Maddy]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
Maybe applications of continuum mathematics are all idealisations [Maddy]
Scientists posit as few entities as possible, but set theorist posit as many as possible [Maddy]
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 [Maddy]
7. Existence / D. Theories of Reality / 11. Ontological Commitment / e. Ontological commitment problems
The theoretical indispensability of atoms did not at first convince scientists that they were real [Maddy]
8. Modes of Existence / B. Properties / 4. Intrinsic Properties
If you think universals are immanent, you must believe them to be sparse, and not every related predicate [Lewis]
8. Modes of Existence / B. Properties / 5. Natural Properties
I assume there could be natural properties that are not instantiated in our world [Lewis]
8. Modes of Existence / B. Properties / 13. Tropes / a. Nature of tropes
Tropes are particular properties, which cannot recur, but can be exact duplicates [Lewis]
8. Modes of Existence / D. Universals / 2. Need for Universals
Universals are meant to give an account of resemblance [Lewis]
8. Modes of Existence / E. Nominalism / 5. Class Nominalism
We can add a primitive natural/unnatural distinction to class nominalism [Lewis]
9. Objects / C. Structure of Objects / 1. Structure of an Object
The 'magical' view of structural universals says they are atoms, even though they have parts [Lewis]
If 'methane' is an atomic structural universal, it has nothing to connect it to its carbon universals [Lewis]
The 'pictorial' view of structural universals says they are wholes made of universals as parts [Lewis]
The structural universal 'methane' needs the universal 'hydrogen' four times over [Lewis]
Butane and Isobutane have the same atoms, but different structures [Lewis]
Structural universals have a necessary connection to the universals forming its parts [Lewis]
We can't get rid of structural universals if there are no simple universals [Lewis]
9. Objects / C. Structure of Objects / 5. Composition of an Object
Composition is not just making new things from old; there are too many counterexamples [Lewis]
9. Objects / C. Structure of Objects / 8. Parts of Objects / c. Wholes from parts
A whole is distinct from its parts, but is not a further addition in ontology [Lewis]
Different things (a toy house and toy car) can be made of the same parts at different times [Lewis]
11. Knowledge Aims / A. Knowledge / 2. Understanding
Scientific understanding is always the grasping of a correct explanation [Strevens]
We may 'understand that' the cat is on the mat, but not at all 'understand why' it is there [Strevens]
Understanding is a precondition, comes in degrees, is active, and holistic - unlike explanation [Strevens]
15. Nature of Minds / C. Capacities of Minds / 3. Abstraction by mind
Maybe abstraction is just mereological subtraction [Lewis]
15. Nature of Minds / C. Capacities of Minds / 6. Idealisation
Science idealises the earth's surface, the oceans, continuities, and liquids [Maddy]
18. Thought / E. Abstraction / 7. Abstracta by Equivalence
Mathematicians abstract by equivalence classes, but that doesn't turn a many into one [Lewis]