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All the ideas for 'Philosophy of Mathematics', 'A Defense of Abortion' and 'Truthmakers'

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77 ideas

2. Reason / D. Definition / 8. Impredicative Definition
Impredicative definitions are wrong, because they change the set that is being defined? [Bostock]
3. Truth / A. Truth Problems / 2. Defining Truth
We might define truth as arising from the truth-maker relation [MacBride]
3. Truth / B. Truthmakers / 1. For Truthmakers
Phenomenalists, behaviourists and presentists can't supply credible truth-makers [MacBride]
3. Truth / B. Truthmakers / 2. Truthmaker Relation
If truthmaking is classical entailment, then anything whatsoever makes a necessary truth [MacBride]
3. Truth / B. Truthmakers / 3. Truthmaker Maximalism
'Maximalism' says every truth has an actual truthmaker [MacBride]
Maximalism follows Russell, and optimalism (no negative or universal truthmakers) follows Wittgenstein [MacBride]
3. Truth / B. Truthmakers / 5. What Makes Truths / a. What makes truths
The main idea of truth-making is that what a proposition is about is what matters [MacBride]
3. Truth / B. Truthmakers / 6. Making Negative Truths
There are different types of truthmakers for different types of negative truth [MacBride]
There aren't enough positive states out there to support all the negative truths [MacBride]
3. Truth / B. Truthmakers / 8. Making General Truths
Optimalists say that negative and universal are true 'by default' from the positive truths [MacBride]
3. Truth / B. Truthmakers / 12. Rejecting Truthmakers
Does 'this sentence has no truth-maker' have a truth-maker? Reductio suggests it can't have [MacBride]
Even idealists could accept truthmakers, as mind-dependent [MacBride]
Maybe 'makes true' is not an active verb, but just a formal connective like 'because'? [MacBride]
Truthmaker talk of 'something' making sentences true, which presupposes objectual quantification [MacBride]
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
Classical interdefinitions of logical constants and quantifiers is impossible in intuitionism [Bostock]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
There is no single agreed structure for set theory [Bostock]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / a. Types of set
A 'proper class' cannot be a member of anything [Bostock]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
We could add axioms to make sets either as small or as large as possible [Bostock]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The Axiom of Choice relies on reference to sets that we are unable to describe [Bostock]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / f. Limitation of Size
Replacement enforces a 'limitation of size' test for the existence of sets [Bostock]
5. Theory of Logic / A. Overview of Logic / 5. First-Order Logic
First-order logic is not decidable: there is no test of whether any formula is valid [Bostock]
The completeness of first-order logic implies its compactness [Bostock]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
Connectives link sentences without linking their meanings [MacBride]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / c. not
'A is F' may not be positive ('is dead'), and 'A is not-F' may not be negative ('is not blind') [MacBride]
5. Theory of Logic / G. Quantification / 4. Substitutional Quantification
Substitutional quantification is just standard if all objects in the domain have a name [Bostock]
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
The Deduction Theorem is what licenses a system of natural deduction [Bostock]
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
Berry's Paradox considers the meaning of 'The least number not named by this name' [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
Each addition changes the ordinality but not the cardinality, prior to aleph-1 [Bostock]
ω + 1 is a new ordinal, but its cardinality is unchanged [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
A cardinal is the earliest ordinal that has that number of predecessors [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
Aleph-1 is the first ordinal that exceeds aleph-0 [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
Instead of by cuts or series convergence, real numbers could be defined by axioms [Bostock]
The number of reals is the number of subsets of the natural numbers [Bostock]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
For Eudoxus cuts in rationals are unique, but not every cut makes a real number [Bostock]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
Infinitesimals are not actually contradictory, because they can be non-standard real numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
Modern axioms of geometry do not need the real numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
The Peano Axioms describe a unique structure [Bostock]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
Hume's Principle is a definition with existential claims, and won't explain numbers [Bostock]
Many things will satisfy Hume's Principle, so there are many interpretations of it [Bostock]
There are many criteria for the identity of numbers [Bostock]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
Frege makes numbers sets to solve the Caesar problem, but maybe Caesar is a set! [Bostock]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
Numbers can't be positions, if nothing decides what position a given number has [Bostock]
Structuralism falsely assumes relations to other numbers are numbers' only properties [Bostock]
6. Mathematics / C. Sources of Mathematics / 3. Mathematical Nominalism
Nominalism about mathematics is either reductionist, or fictionalist [Bostock]
Nominalism as based on application of numbers is no good, because there are too many applications [Bostock]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
Actual measurement could never require the precision of the real numbers [Bostock]
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
Ordinals are mainly used adjectively, as in 'the first', 'the second'... [Bostock]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / b. Type theory
Simple type theory has 'levels', but ramified type theory has 'orders' [Bostock]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / c. Neo-logicism
Neo-logicists agree that HP introduces number, but also claim that it suffices for the job [Bostock]
Neo-logicists meet the Caesar problem by saying Hume's Principle is unique to number [Bostock]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
If Hume's Principle is the whole story, that implies structuralism [Bostock]
Many crucial logicist definitions are in fact impredicative [Bostock]
Treating numbers as objects doesn't seem like logic, since arithmetic fixes their totality [Bostock]
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
Higher cardinalities in sets are just fairy stories [Bostock]
A fairy tale may give predictions, but only a true theory can give explanations [Bostock]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
The best version of conceptualism is predicativism [Bostock]
Conceptualism fails to grasp mathematical properties, infinity, and objective truth values [Bostock]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
If abstracta only exist if they are expressible, there can only be denumerably many of them [Bostock]
Predicativism makes theories of huge cardinals impossible [Bostock]
If mathematics rests on science, predicativism may be the best approach [Bostock]
If we can only think of what we can describe, predicativism may be implied [Bostock]
The usual definitions of identity and of natural numbers are impredicative [Bostock]
The predicativity restriction makes a difference with the real numbers [Bostock]
7. Existence / A. Nature of Existence / 6. Criterion for Existence
Maybe it only exists if it is a truthmaker (rather than the value of a variable)? [MacBride]
7. Existence / C. Structure of Existence / 1. Grounding / a. Nature of grounding
Different types of 'grounding' seem to have no more than a family resemblance relation [MacBride]
Which has priority - 'grounding' or 'truth-making'? [MacBride]
7. Existence / C. Structure of Existence / 6. Fundamentals / d. Logical atoms
Russell allows some complex facts, but Wittgenstein only allows atomic facts [MacBride]
10. Modality / A. Necessity / 6. Logical Necessity
Wittgenstein's plan to show there is only logical necessity failed, because of colours [MacBride]
19. Language / F. Communication / 2. Assertion
In logic a proposition means the same when it is and when it is not asserted [Bostock]
25. Social Practice / F. Life Issues / 3. Abortion
The right to life is not a right not to be killed, but not to be killed unjustly [Thomson]
A newly fertilized ovum is no more a person than an acorn is an oak tree [Thomson]
Maybe abortion can be justified despite the foetus having full human rights [Thomson, by Foot]
It can't be murder for a mother to perform an abortion on herself to save her own life [Thomson]
The foetus is safe in the womb, so abortion initiates its death, with the mother as the agent. [Foot on Thomson]
Is someone's right to life diminished if they were conceived by a rape? [Thomson]
The right to life does not bestow the right to use someone else's body to support that life [Thomson]
No one is morally required to make huge sacrifices to keep someone else alive for nine months [Thomson]