Combining Texts

All the ideas for 'On the Question of Absolute Undecidability', 'Concluding Unscientific Postscript' and 'Introduction to the Philosophy of Mathematics'

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42 ideas

1. Philosophy / D. Nature of Philosophy / 5. Aims of Philosophy / d. Philosophy as puzzles
I conceived it my task to create difficulties everywhere [Kierkegaard]
1. Philosophy / D. Nature of Philosophy / 8. Humour
Wherever there is painless contradiction there is also comedy [Kierkegaard]
3. Truth / A. Truth Problems / 2. Defining Truth
Kierkegaard's truth draws on authenticity, fidelity and honesty [Kierkegaard, by Carlisle]
3. Truth / A. Truth Problems / 3. Value of Truth
Pure truth is for infinite beings only; I prefer endless striving for truth [Kierkegaard]
3. Truth / A. Truth Problems / 8. Subjective Truth
The highest truth we can get is uncertainty held fast by an inward passion [Kierkegaard]
4. Formal Logic / E. Nonclassical Logics / 2. Intuitionist Logic
Rejecting double negation elimination undermines reductio proofs [Colyvan]
Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan]
4. Formal Logic / F. Set Theory ST / 1. Set Theory
Mathematical set theory has many plausible stopping points, such as finitism, and predicativism [Koellner]
'Reflection principles' say the whole truth about sets can't be captured [Koellner]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan]
5. Theory of Logic / J. Model Theory in Logic / 3. Löwenheim-Skolem Theorems
Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan]
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
Axioms are 'categorical' if all of their models are isomorphic [Colyvan]
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
We have no argument to show a statement is absolutely undecidable [Koellner]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
Ordinal numbers represent order relations [Colyvan]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
Intuitionists only accept a few safe infinities [Colyvan]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / i. Cardinal infinity
There are at least eleven types of large cardinal, of increasing logical strength [Koellner]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
PA is consistent as far as we can accept, and we expand axioms to overcome limitations [Koellner]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
Transfinite induction moves from all cases, up to the limit ordinal [Colyvan]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
Arithmetical undecidability is always settled at the next stage up [Koellner]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Most mathematical proofs are using set theory, but without saying so [Colyvan]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan]
14. Science / C. Induction / 6. Bayes's Theorem
Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan]
14. Science / D. Explanation / 2. Types of Explanation / e. Lawlike explanations
Mathematics can reveal structural similarities in diverse systems [Colyvan]
14. Science / D. Explanation / 2. Types of Explanation / f. Necessity in explanations
Mathematics can show why some surprising events have to occur [Colyvan]
14. Science / D. Explanation / 2. Types of Explanation / m. Explanation by proof
Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan]
Reductio proofs do not seem to be very explanatory [Colyvan]
If inductive proofs hold because of the structure of natural numbers, they may explain theorems [Colyvan]
Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan]
15. Nature of Minds / C. Capacities of Minds / 5. Generalisation by mind
Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan]
16. Persons / B. Nature of the Self / 2. Ethical Self
The real subject is ethical, not cognitive [Kierkegaard]
23. Ethics / F. Existentialism / 1. Existentialism
While big metaphysics is complete without ethics, personal philosophy emphasises ethics [Kierkegaard]
Speculative philosophy loses the individual in a vast vision of humanity [Kierkegaard]
23. Ethics / F. Existentialism / 6. Authentic Self
People want to lose themselves in movements and history, instead of being individuals [Kierkegaard]
Becoming what one is is a huge difficulty, because we strongly aspire to be something else [Kierkegaard]
28. God / A. Divine Nature / 2. Divine Nature
God does not think or exist; God creates, and is eternal [Kierkegaard]
28. God / B. Proving God / 3. Proofs of Evidence / d. Religious Experience
God cannot be demonstrated objectively, because God is a subject, only existing inwardly [Kierkegaard]
28. God / C. Attitudes to God / 2. Pantheism
Pantheism destroys the distinction between good and evil [Kierkegaard]
29. Religion / D. Religious Issues / 1. Religious Commitment / e. Fideism
Faith is the highest passion in the sphere of human subjectivity [Kierkegaard]
Without risk there is no faith [Kierkegaard]