Combining Philosophers

All the ideas for Michael V. Wedin, Ernst Zermelo and Jennifer Fisher

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34 ideas

2. Reason / A. Nature of Reason / 1. On Reason
We reach 'reflective equilibrium' when intuitions and theory completely align [Fisher]
2. Reason / D. Definition / 8. Impredicative Definition
Predicative definitions are acceptable in mathematics if they distinguish objects, rather than creating them? [Zermelo, by Lavine]
4. Formal Logic / E. Nonclassical Logics / 3. Many-Valued Logic
Three-valued logic says excluded middle and non-contradition are not tautologies [Fisher]
4. Formal Logic / E. Nonclassical Logics / 4. Fuzzy Logic
Fuzzy logic has many truth values, ranging in fractions from 0 to 1 [Fisher]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
Set theory investigates number, order and function, showing logical foundations for mathematics [Zermelo]
We take set theory as given, and retain everything valuable, while avoiding contradictions [Zermelo]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
ZFC: Existence, Extension, Specification, Pairing, Unions, Powers, Infinity, Choice [Zermelo, by Clegg]
Zermelo published his axioms in 1908, to secure a controversial proof [Zermelo, by Maddy]
Set theory can be reduced to a few definitions and seven independent axioms [Zermelo]
Zermelo made 'set' and 'member' undefined axioms [Zermelo, by Chihara]
For Zermelo's set theory the empty set is zero and the successor of each number is its unit set [Zermelo, by Blackburn]
Zermelo showed that the ZF axioms in 1930 were non-categorical [Zermelo, by Hallett,M]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / c. Axiom of Pairing II
Zermelo introduced Pairing in 1930, and it seems fairly obvious [Zermelo, by Maddy]
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 [Zermelo, by Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / i. Axiom of Foundation VIII
Zermelo used Foundation to block paradox, but then decided that only Separation was needed [Zermelo, by Maddy]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / m. Axiom of Separation
Not every predicate has an extension, but Separation picks the members that satisfy a predicate [Zermelo, by Hart,WD]
The Axiom of Separation requires set generation up to one step back from contradiction [Zermelo, by Maddy]
5. Theory of Logic / A. Overview of Logic / 6. Classical Logic
Classical logic is: excluded middle, non-contradiction, contradictions imply all, disjunctive syllogism [Fisher]
5. Theory of Logic / C. Ontology of Logic / 2. Platonism in Logic
Logic formalizes how we should reason, but it shouldn't determine whether we are realists [Fisher]
5. Theory of Logic / L. Paradox / 3. Antinomies
The antinomy of endless advance and of completion is resolved in well-ordered transfinite numbers [Zermelo]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
In ZF, the Burali-Forti Paradox proves that there is no set of all ordinals [Zermelo, by Hart,WD]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / e. Countable infinity
Zermelo realised that Choice would facilitate the sort of 'counting' Cantor needed [Zermelo, by Lavine]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Zermelo believed, and Von Neumann seemed to confirm, that numbers are sets [Zermelo, by Maddy]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
Different versions of set theory result in different underlying structures for numbers [Zermelo, by Brown,JR]
7. Existence / D. Theories of Reality / 9. Vagueness / g. Degrees of vagueness
We could make our intuitions about heaps precise with a million-valued logic [Fisher]
8. Modes of Existence / B. Properties / 3. Types of Properties
A 'categorial' property is had by virtue of being or having an item from a category [Wedin]
9. Objects / B. Unity of Objects / 2. Substance / d. Substance defined
Substance is a principle and a kind of cause [Wedin]
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
Vagueness can involve components (like baldness), or not (like boredom) [Fisher]
9. Objects / C. Structure of Objects / 2. Hylomorphism / a. Hylomorphism
Form explains why some matter is of a certain kind, and that is explanatory bedrock [Wedin]
10. Modality / B. Possibility / 1. Possibility
We can't explain 'possibility' in terms of 'possible' worlds [Fisher]
10. Modality / B. Possibility / 8. Conditionals / c. Truth-function conditionals
If all truths are implied by a falsehood, then not-p might imply both q and not-q [Fisher]
10. Modality / B. Possibility / 8. Conditionals / d. Non-truthfunction conditionals
In relevance logic, conditionals help information to flow from antecedent to consequent [Fisher]
18. Thought / A. Modes of Thought / 6. Judgement / a. Nature of Judgement
We should judge principles by the science, not science by some fixed principles [Zermelo]