Combining Philosophers

All the ideas for Michael Burke, Yale Kamisar and John Mayberry

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41 ideas

2. Reason / D. Definition / 2. Aims of Definition
Definitions make our intuitions mathematically useful [Mayberry]
2. Reason / E. Argument / 6. Conclusive Proof
Proof shows that it is true, but also why it must be true [Mayberry]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
Set theory can't be axiomatic, because it is needed to express the very notion of axiomatisation [Mayberry]
There is a semi-categorical axiomatisation of set-theory [Mayberry]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / f. Axiom of Infinity V
The misnamed Axiom of Infinity says the natural numbers are finite in size [Mayberry]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The set hierarchy doesn't rely on the dubious notion of 'generating' them [Mayberry]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Limitation of size is part of the very conception of a set [Mayberry]
5. Theory of Logic / A. Overview of Logic / 2. History of Logic
The mainstream of modern logic sees it as a branch of mathematics [Mayberry]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
First-order logic only has its main theorems because it is so weak [Mayberry]
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Only second-order logic can capture mathematical structure up to isomorphism [Mayberry]
5. Theory of Logic / G. Quantification / 2. Domain of Quantification
Big logic has one fixed domain, but standard logic has a domain for each interpretation [Mayberry]
5. Theory of Logic / J. Model Theory in Logic / 3. L÷wenheim-Skolem Theorems
No L÷wenheim-Skolem logic can axiomatise real analysis [Mayberry]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
Axiomatiation relies on isomorphic structures being essentially the same [Mayberry]
'Eliminatory' axioms get rid of traditional ideal and abstract objects [Mayberry]
'Classificatory' axioms aim at revealing similarity in morphology of structures [Mayberry]
5. Theory of Logic / K. Features of Logics / 6. Compactness
No logic which can axiomatise arithmetic can be compact or complete [Mayberry]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
Real numbers can be eliminated, by axiom systems for complete ordered fields [Mayberry]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / b. Quantity
Real numbers were invented, as objects, to simplify and generalise 'quantity' [Mayberry]
Greek quantities were concrete, and ratio and proportion were their science [Mayberry]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
Cantor's infinite is an absolute, of all the sets or all the ordinal numbers [Mayberry]
Cantor extended the finite (rather than 'taming the infinite') [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
If proof and definition are central, then mathematics needs and possesses foundations [Mayberry]
The ultimate principles and concepts of mathematics are presumed, or grasped directly [Mayberry]
Foundations need concepts, definition rules, premises, and proof rules [Mayberry]
Axiom theories can't give foundations for mathematics - that's using axioms to explain axioms [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
1st-order PA is only interesting because of results which use 2nd-order PA [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
It is only 2nd-order isomorphism which suggested first-order PA completeness [Mayberry]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Set theory is not just first-order ZF, because that is inadequate for mathematics [Mayberry]
We don't translate mathematics into set theory, because it comes embodied in that way [Mayberry]
Set theory is not just another axiomatised part of mathematics [Mayberry]
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
Real numbers as abstracted objects are now treated as complete ordered fields [Mayberry]
9. Objects / A. Existence of Objects / 5. Individuation / e. Individuation by kind
Persistence conditions cannot contradict, so there must be a 'dominant sortal' [Burke,M, by Hawley]
The 'dominant' of two coinciding sortals is the one that entails the widest range of properties [Burke,M, by Sider]
9. Objects / B. Unity of Objects / 1. Unifying an Object / b. Unifying aggregates
'The rock' either refers to an object, or to a collection of parts, or to some stuff [Burke,M, by Wasserman]
9. Objects / B. Unity of Objects / 3. Unity Problems / b. Cat and its tail
Tib goes out of existence when the tail is lost, because Tib was never the 'cat' [Burke,M, by Sider]
9. Objects / B. Unity of Objects / 3. Unity Problems / c. Statue and clay
Maybe the clay becomes a different lump when it becomes a statue [Burke,M, by Koslicki]
Burke says when two object coincide, one of them is destroyed in the process [Burke,M, by Hawley]
Sculpting a lump of clay destroys one object, and replaces it with another one [Burke,M, by Wasserman]
9. Objects / B. Unity of Objects / 3. Unity Problems / d. Coincident objects
Two entities can coincide as one, but only one of them (the dominant sortal) fixes persistence conditions [Burke,M, by Sider]
24. Applied Ethics / C. Death Issues / 5. Euthanasia
We only allow voluntary euthanasia to someone who is both sane and crazed by pain [Kamisar]
People will volunteer for euthanasia because they think other people want them dead [Kamisar]