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All the ideas for 'Improvement of Understanding', 'Vagueness' and 'Introducing the Philosophy of Mathematics'

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84 ideas

2. Reason / D. Definition / 2. Aims of Definition
All the intrinsic properties of a thing should be deducible from its definition [Spinoza]
2. Reason / D. Definition / 8. Impredicative Definition
An 'impredicative' definition seems circular, because it uses the term being defined [Friend]
2. Reason / D. Definition / 10. Stipulative Definition
Classical definitions attempt to refer, but intuitionist/constructivist definitions actually create objects [Friend]
2. Reason / E. Argument / 5. Reductio ad Absurdum
Reductio ad absurdum proves an idea by showing that its denial produces contradiction [Friend]
3. Truth / A. Truth Problems / 5. Truth Bearers
Truth and falsity apply to suppositions as well as to assertions [Williamson]
3. Truth / A. Truth Problems / 7. Falsehood
True and false are not symmetrical; false is more complex, involving negation [Williamson]
3. Truth / A. Truth Problems / 8. Subjective Truth
Anti-realists see truth as our servant, and epistemically contrained [Friend]
4. Formal Logic / B. Propositional Logic PL / 3. Truth Tables
In classical/realist logic the connectives are defined by truth-tables [Friend]
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
Double negation elimination is not valid in intuitionist logic [Friend]
4. Formal Logic / E. Nonclassical Logics / 3. Many-Valued Logic
Many-valued logics don't solve vagueness; its presence at the meta-level is ignored [Williamson]
4. Formal Logic / E. Nonclassical Logics / 6. Free Logic
Free logic was developed for fictional or non-existent objects [Friend]
4. Formal Logic / F. Set Theory ST / 2. Mechanics of Set Theory / b. Terminology of ST
A 'proper subset' of A contains only members of A, but not all of them [Friend]
A 'powerset' is all the subsets of a set [Friend]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / b. Empty (Null) Set
Set theory makes a minimum ontological claim, that the empty set exists [Friend]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
Infinite sets correspond one-to-one with a subset [Friend]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
Major set theories differ in their axioms, and also over the additional axioms of choice and infinity [Friend]
5. Theory of Logic / B. Logical Consequence / 4. Semantic Consequence |=
Formal semantics defines validity as truth preserved in every model [Williamson]
5. Theory of Logic / D. Assumptions for Logic / 1. Bivalence
'Bivalence' is the meta-linguistic principle that 'A' in the object language is true or false [Williamson]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Excluded Middle is 'A or not A' in the object language [Williamson]
The law of excluded middle is syntactic; it just says A or not-A, not whether they are true or false [Friend]
5. Theory of Logic / G. Quantification / 7. Unorthodox Quantification
Intuitionists read the universal quantifier as "we have a procedure for checking every..." [Friend]
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
Or-elimination is 'Argument by Cases'; it shows how to derive C from 'A or B' [Williamson]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / a. Set theory paradoxes
Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend]
5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. Burali-Forti's paradox
The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend]
5. Theory of Logic / L. Paradox / 6. Paradoxes in Language / b. The Heap paradox ('Sorites')
A sorites stops when it collides with an opposite sorites [Williamson]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
The 'integers' are the positive and negative natural numbers, plus zero [Friend]
The 'rational' numbers are those representable as fractions [Friend]
A number is 'irrational' if it cannot be represented as a fraction [Friend]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
The natural numbers are primitive, and the ordinals are up one level of abstraction [Friend]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / f. Cardinal numbers
Cardinal numbers answer 'how many?', with the order being irrelevant [Friend]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
The 'real' numbers (rationals and irrationals combined) is the Continuum, which has no gaps [Friend]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
Raising omega to successive powers of omega reveal an infinity of infinities [Friend]
The first limit ordinal is omega (greater, but without predecessor), and the second is twice-omega [Friend]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
Between any two rational numbers there is an infinite number of rational numbers [Friend]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
Is mathematics based on sets, types, categories, models or topology? [Friend]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Most mathematical theories can be translated into the language of set theory [Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
The number 8 in isolation from the other numbers is of no interest [Friend]
In structuralism the number 8 is not quite the same in different structures, only equivalent [Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
Are structures 'ante rem' (before reality), or are they 'in re' (grounded in physics)? [Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
Structuralist says maths concerns concepts about base objects, not base objects themselves [Friend]
Structuralism focuses on relations, predicates and functions, with objects being inessential [Friend]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
'In re' structuralism says that the process of abstraction is pattern-spotting [Friend]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
The big problem for platonists is epistemic: how do we perceive, intuit, know or detect mathematical facts? [Friend]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / b. Indispensability of mathematics
Mathematics should be treated as true whenever it is indispensable to our best physical theory [Friend]
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Formalism is unconstrained, so cannot indicate importance, or directions for research [Friend]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
Constructivism rejects too much mathematics [Friend]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Intuitionists typically retain bivalence but reject the law of excluded middle [Friend]
7. Existence / D. Theories of Reality / 10. Vagueness / a. Problem of vagueness
When bivalence is rejected because of vagueness, we lose classical logic [Williamson]
Vagueness undermines the stable references needed by logic [Williamson]
A vague term can refer to very precise elements [Williamson]
7. Existence / D. Theories of Reality / 10. Vagueness / b. Vagueness of reality
Equally fuzzy objects can be identical, so fuzziness doesn't entail vagueness [Williamson]
7. Existence / D. Theories of Reality / 10. Vagueness / c. Vagueness as ignorance
Vagueness is epistemic. Statements are true or false, but we often don't know which [Williamson]
If a heap has a real boundary, omniscient speakers would agree where it is [Williamson]
The epistemic view says that the essence of vagueness is ignorance [Williamson]
If there is a true borderline of which we are ignorant, this drives a wedge between meaning and use [Williamson]
Vagueness in a concept is its indiscriminability from other possible concepts [Williamson]
7. Existence / D. Theories of Reality / 10. Vagueness / d. Vagueness as linguistic
The vagueness of 'heap' can remain even when the context is fixed [Williamson]
The 'nihilist' view of vagueness says that 'heap' is not a legitimate concept [Williamson]
We can say propositions are bivalent, but vague utterances don't express a proposition [Williamson]
If the vague 'TW is thin' says nothing, what does 'TW is thin if his perfect twin is thin' say? [Williamson]
7. Existence / D. Theories of Reality / 10. Vagueness / e. Higher-order vagueness
Asking when someone is 'clearly' old is higher-order vagueness [Williamson]
7. Existence / D. Theories of Reality / 10. Vagueness / f. Supervaluation for vagueness
Supervaluation keeps classical logic, but changes the truth in classical semantics [Williamson]
You can't give a precise description of a language which is intrinsically vague [Williamson]
Supervaluation assigns truth when all the facts are respected [Williamson]
Supervaluation has excluded middle but not bivalence; 'A or not-A' is true, even when A is undecided [Williamson]
Truth-functionality for compound statements fails in supervaluation [Williamson]
Supervaluationism defines 'supertruth', but neglects it when defining 'valid' [Williamson]
Supervaluation adds a 'definitely' operator to classical logic [Williamson]
Supervaluationism cannot eliminate higher-order vagueness [Williamson]
8. Modes of Existence / E. Nominalism / 1. Nominalism / a. Nominalism
Nominalists suspect that properties etc are our projections, and could have been different [Williamson]
9. Objects / A. Existence of Objects / 2. Abstract Objects / a. Nature of abstracta
Structuralists call a mathematical 'object' simply a 'place in a structure' [Friend]
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
If fuzzy edges are fine, then why not fuzzy temporal, modal or mereological boundaries? [Williamson]
9. Objects / E. Objects over Time / 8. Continuity of Rivers
A river is not just event; it needs actual and counterfactual boundaries [Williamson]
10. Modality / D. Knowledge of Modality / 1. A Priori Necessary
We can't infer metaphysical necessities to be a priori knowable - or indeed knowable in any way [Williamson]
11. Knowledge Aims / A. Knowledge / 1. Knowledge
We have inexact knowledge when we include margins of error [Williamson]
13. Knowledge Criteria / A. Justification Problems / 1. Justification / a. Justification issues
Knowing you know (KK) is usually denied if the knowledge concept is missing, or not considered [Williamson]
14. Science / D. Explanation / 2. Types of Explanation / k. Explanations by essence
To understand the properties we must know the essence, as with a circle [Spinoza]
17. Mind and Body / E. Mind as Physical / 2. Reduction of Mind
Studying biology presumes the laws of chemistry, and it could never contradict them [Friend]
18. Thought / A. Modes of Thought / 2. Propositional Attitudes
To know, believe, hope or fear, one must grasp the thought, but not when you fail to do them [Williamson]
18. Thought / D. Concepts / 1. Concepts / a. Nature of concepts
Concepts can be presented extensionally (as objects) or intensionally (as a characterization) [Friend]
18. Thought / D. Concepts / 4. Structure of Concepts / h. Family resemblance
'Blue' is not a family resemblance, because all the blues resemble in some respect [Williamson]
19. Language / B. Reference / 1. Reference theories
References to the 'greatest prime number' have no reference, but are meaningful [Williamson]
19. Language / C. Assigning Meanings / 2. Semantics
The 't' and 'f' of formal semantics has no philosophical interest, and may not refer to true and false [Williamson]
19. Language / D. Propositions / 2. Abstract Propositions / b. Propositions as possible worlds
It is known that there is a cognitive loss in identifying propositions with possible worlds [Williamson]