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All the ideas for 'Structures and Structuralism in Phil of Maths', 'Philosophy of Mathematics' and 'On the Genealogy of Morals'

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78 ideas

1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / a. Philosophy as worldly
The main aim of philosophy must be to determine the order of rank among values [Nietzsche]
1. Philosophy / G. Scientific Philosophy / 3. Scientism
Scientific knowledge is nothing without a prior philosophical 'faith' [Nietzsche]
2. Reason / A. Nature of Reason / 5. Objectivity
Objectivity is not disinterestedness (impossible), but the ability to switch perspectives [Nietzsche]
2. Reason / D. Definition / 2. Aims of Definition
Definitions should be replaceable by primitives, and should not be creative [Brown,JR]
2. Reason / D. Definition / 3. Types of Definition
Only that which has no history is definable [Nietzsche]
3. Truth / A. Truth Problems / 3. Value of Truth
Psychologists should be brave and proud, and prefer truth to desires, even when it is ugly [Nietzsche]
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]
11. Knowledge Aims / A. Knowledge / 5. Aiming at Truth
Philosophers have never asked why there is a will to truth in the first place [Nietzsche]
12. Knowledge Sources / E. Direct Knowledge / 4. Memory
Forgetfulness is a strong positive ability, not mental laziness [Nietzsche]
13. Knowledge Criteria / E. Relativism / 1. Relativism
There is only 'perspective' seeing and knowing, and so the best objectivity is multiple points of view [Nietzsche]
16. Persons / F. Free Will / 5. Against Free Will
Philosophers invented "free will" so that our virtues would be permanently interesting to the gods [Nietzsche]
18. Thought / A. Modes of Thought / 1. Thought
People who think in words are orators rather than thinkers, and think about facts instead of thinking facts [Nietzsche]
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]
20. Action / A. Definition of Action / 1. Action Theory
It is a delusion to separate the man from the deed, like the flash from the lightning [Nietzsche]
22. Metaethics / A. Ethics Foundations / 1. Nature of Ethics / h. Against ethics
We must question the very value of moral values [Nietzsche]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / f. Übermensch
The concept of 'good' was created by aristocrats to describe their own actions [Nietzsche]
A strong rounded person soon forgets enemies, misfortunes, and even misdeeds [Nietzsche]
22. Metaethics / A. Ethics Foundations / 2. Source of Ethics / g. Will to power
All animals strive for the ideal conditions to express their power, and hate any hindrances [Nietzsche]
23. Ethics / A. Egoism / 1. Ethical Egoism
Only the decline of aristocratic morality led to concerns about "egoism" [Nietzsche]
Nietzsche rejects impersonal morality, and asserts the idea of living well [Nietzsche, by Nagel]
23. Ethics / B. Contract Ethics / 1. Contractarianism
Basic justice is the negotiation of agreement among equals, and the imposition of agreement [Nietzsche]
A masterful and violent person need have nothing to do with contracts [Nietzsche]
23. Ethics / C. Virtue Theory / 3. Virtues / f. Compassion
Plato, Spinoza and Kant are very different, but united in their low estimation of pity [Nietzsche]
23. Ethics / D. Deontological Ethics / 2. Duty
Guilt and obligation originated in the relationship of buying and selling, credit and debt [Nietzsche]
23. Ethics / F. Existentialism / 1. Existentialism
If we say birds of prey could become lambs, that makes them responsible for being birds of prey [Nietzsche]
23. Ethics / F. Existentialism / 2. Nihilism
Modern nihilism is now feeling tired of mankind [Nietzsche]
24. Political Theory / A. Basis of a State / 1. A People / c. A unified people
Old tribes always felt an obligation to the earlier generations, and the founders [Nietzsche]
24. Political Theory / B. Nature of a State / 2. State Legitimacy / b. Natural authority
The state begins with brutal conquest of a disorganised people, not with a 'contract' [Nietzsche]
25. Social Practice / D. Justice / 3. Punishment / d. Reform of offenders
Punishment makes people harder, more alienated, and hostile [Nietzsche]
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]
29. Religion / D. Religious Issues / 1. Religious Commitment / a. Religious Belief
The truly great haters in world history have always been priests [Nietzsche]