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All the ideas for 'Structures and Structuralism in Phil of Maths', 'Philosophy of Mathematics' and 'Liberalism: the basics'

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78 ideas

2. Reason / D. Definition / 2. Aims of Definition
Definitions should be replaceable by primitives, and should not be creative [Brown,JR]
3. Truth / F. Semantic Truth / 2. Semantic Truth
While true-in-a-model seems relative, true-in-all-models seems not to be [Reck/Price]
4. Formal Logic / F. Set Theory ST / 3. Types of Set / d. Infinite Sets
Set theory says that natural numbers are an actual infinity (to accommodate their powerset) [Brown,JR]
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
ZFC set theory has only 'pure' sets, without 'urelements' [Reck/Price]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / d. Naïve logical sets
Naïve set theory assumed that there is a set for every condition [Brown,JR]
Nowadays conditions are only defined on existing sets [Brown,JR]
4. Formal Logic / F. Set Theory ST / 5. Conceptions of Set / e. Iterative sets
The 'iterative' view says sets start with the empty set and build up [Brown,JR]
4. Formal Logic / F. Set Theory ST / 7. Natural Sets
A flock of birds is not a set, because a set cannot go anywhere [Brown,JR]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
If a proposition is false, then its negation is true [Brown,JR]
5. Theory of Logic / G. Quantification / 5. Second-Order Quantification
Three types of variable in second-order logic, for objects, functions, and predicates/sets [Reck/Price]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
Axioms are either self-evident, or stipulations, or fallible attempts [Brown,JR]
5. Theory of Logic / L. Paradox / 4. Paradoxes in Logic / c. Berry's paradox
Berry's Paradox finds a contradiction in the naming of huge numbers [Brown,JR]
6. Mathematics / A. Nature of Mathematics / 1. Mathematics
Mathematics is the only place where we are sure we are right [Brown,JR]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
'Analysis' is the theory of the real numbers [Reck/Price]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / n. Pi
π is a 'transcendental' number, because it is not the solution of an equation [Brown,JR]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
Mathematics represents the world through structurally similar models. [Brown,JR]
6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
There is no limit to how many ways something can be proved in mathematics [Brown,JR]
Computers played an essential role in proving the four-colour theorem of maps [Brown,JR]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
Mereological arithmetic needs infinite objects, and function definitions [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
Peano Arithmetic can have three second-order axioms, plus '1' and 'successor' [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Set-theory gives a unified and an explicit basis for mathematics [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
Set theory may represent all of mathematics, without actually being mathematics [Brown,JR]
When graphs are defined set-theoretically, that won't cover unlabelled graphs [Brown,JR]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
Structuralism emerged from abstract algebra, axioms, and set theory and its structures [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
Relativist Structuralism just stipulates one successful model as its arithmetic [Reck/Price]
There are 'particular' structures, and 'universal' structures (what the former have in common) [Reck/Price]
Pattern Structuralism studies what isomorphic arithmetic models have in common [Reck/Price]
There are Formalist, Relativist, Universalist and Pattern structuralism [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / c. Nominalist structuralism
Formalist Structuralism says the ontology is vacuous, or formal, or inference relations [Reck/Price]
Maybe we should talk of an infinity of 'possible' objects, to avoid arithmetic being vacuous [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism
To see a structure in something, we must already have the idea of the structure [Brown,JR]
Universalist Structuralism is based on generalised if-then claims, not one particular model [Reck/Price]
Universalist Structuralism eliminates the base element, as a variable, which is then quantified out [Reck/Price]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
Sets seem basic to mathematics, but they don't suit structuralism [Brown,JR]
The existence of an infinite set is assumed by Relativist Structuralism [Reck/Price]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
The irrationality of root-2 was achieved by intellect, not experience [Brown,JR]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
There is an infinity of mathematical objects, so they can't be physical [Brown,JR]
Numbers are not abstracted from particulars, because each number is a particular [Brown,JR]
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
Empiricists base numbers on objects, Platonists base them on properties [Brown,JR]
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Does some mathematics depend entirely on notation? [Brown,JR]
For nomalists there are no numbers, only numerals [Brown,JR]
The most brilliant formalist was Hilbert [Brown,JR]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism
There are no constructions for many highly desirable results in mathematics [Brown,JR]
Constructivists say p has no value, if the value depends on Goldbach's Conjecture [Brown,JR]
7. Existence / C. Structure of Existence / 7. Abstract/Concrete / a. Abstract/concrete
David's 'Napoleon' is about something concrete and something abstract [Brown,JR]
8. Modes of Existence / E. Nominalism / 6. Mereological Nominalism
A nominalist might avoid abstract objects by just appealing to mereological sums [Reck/Price]
18. Thought / E. Abstraction / 1. Abstract Thought
'Abstract' nowadays means outside space and time, not concrete, not physical [Brown,JR]
The older sense of 'abstract' is where 'redness' or 'group' is abstracted from particulars [Brown,JR]
19. Language / A. Nature of Meaning / 7. Meaning Holism / c. Meaning by Role
A term can have not only a sense and a reference, but also a 'computational role' [Brown,JR]
24. Political Theory / A. Basis of a State / 4. Original Position / a. Original position
Rawls's theory cannot justify liberalism, since it presupposes free and equal participants [Charvet]
24. Political Theory / A. Basis of a State / 4. Original Position / b. Veil of ignorance
People with strong prior beliefs would have nothing to do with a veil of ignorance [Charvet]
24. Political Theory / D. Ideologies / 3. Conservatism
Societies need shared values, so conservatism is right if rational discussion of values is impossible [Charvet]
24. Political Theory / D. Ideologies / 4. Social Utilitarianism
The universalism of utilitarianism implies a world state [Charvet]
24. Political Theory / D. Ideologies / 6. Liberalism / a. Liberalism basics
Liberals value freedom and equality, but the society itself must decide on its values [Charvet]
24. Political Theory / D. Ideologies / 6. Liberalism / b. Liberal individualism
Modern libertarian societies still provide education and some housing [Charvet]
Liberalism needs people to either have equal autonomy, or everyone to have enough autonomy [Charvet]
Kant places a higher value on the universal rational will than on the people asserting it [Charvet]
24. Political Theory / D. Ideologies / 6. Liberalism / c. Liberal equality
Liberalism asserts maximum freedom, but that must be equal for all participants [Charvet]
Egalitarian liberals prefer equality (either of input or outcome) to liberty [Charvet]
24. Political Theory / D. Ideologies / 6. Liberalism / e. Liberal community
Liberals promote community and well-being - because all good societies need them [Charvet]
24. Political Theory / D. Ideologies / 6. Liberalism / f. Multiculturalism
Identity multiculturalism emerges from communitarianism, preferring community to humanity [Charvet]
24. Political Theory / D. Ideologies / 7. Communitarianism / b. Against communitarianism
For communitarians it seems that you must accept the culture you are born into [Charvet]
24. Political Theory / D. Ideologies / 9. Communism
Give by ability and receive by need, rather than a free labour market [Charvet]
25. Social Practice / A. Freedoms / 3. Free speech
Allowing defamatory speech is against society's interests, by blurring which people are trustworthy [Charvet]
25. Social Practice / A. Freedoms / 5. Freedom of lifestyle
'Freedom from' is an empty idea, if the freedom is not from impediments to my desires [Charvet]
Positive freedom can lead to coercion, if you are forced to do what you chose to do [Charvet]
First level autonomy is application of personal values; second level is criticising them [Charvet]
25. Social Practice / B. Equalities / 1. Grounds of equality
Mere equality, as in two trees being the same height, has no value at all [Charvet]
25. Social Practice / B. Equalities / 4. Economic equality
Inequalities are worse if they seem to be your fault, rather than social facts [Charvet]
Money allows unlimited inequalities, and we obviously all agree to money [Charvet]
25. Social Practice / D. Justice / 2. The Law / b. Rule of law
The rule of law is mainly to restrict governments [Charvet]
The 1689 Bill of Rights denied the monarch new courts, or the right to sit as judge [Charvet]
From 1701 only parliament could remove judges, whose decisions could not be discussed [Charvet]
Justice superior to the rule of law is claimed on behalf of the workers, or the will of the nation [Charvet]
The rule of law mainly benefits those with property and liberties [Charvet]
25. Social Practice / E. Policies / 3. Welfare provision
Welfare is needed if citizens are to accept the obligations of a liberal state [Charvet]
26. Natural Theory / A. Speculations on Nature / 5. Infinite in Nature
Given atomism at one end, and a finite universe at the other, there are no physical infinities [Brown,JR]