Combining Texts

All the ideas for 'Three Varieties of Knowledge', 'Introduction to the Philosophy of Mathematics' and 'Diffrance'

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29 ideas

2. Reason / A. Nature of Reason / 5. Objectivity
Objective truth arises from interpersonal communication [Davidson]
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]
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]
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 / 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 / f. Mathematical induction
Transfinite induction moves from all cases, up to the limit ordinal [Colyvan]
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]
11. Knowledge Aims / A. Knowledge / 4. Belief / e. Belief holism
A belief requires understanding the distinctions of true-and-false, and appearance-and-reality [Davidson]
13. Knowledge Criteria / E. Relativism / 2. Knowledge as Convention
Objectivity is intersubjectivity [Davidson]
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 / A. Nature of Mind / 4. Other Minds / b. Scepticism of other minds
If we know other minds through behaviour, but not our own, we should assume they aren't like me [Davidson]
15. Nature of Minds / A. Nature of Mind / 4. Other Minds / c. Knowing other minds
Knowing other minds rests on knowing both one's own mind and the external world [Davidson, by Dummett]
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]
19. Language / A. Nature of Meaning / 1. Meaning
'Différance' is the interwoven history of each sign [Derrida, by Glendinning]
19. Language / F. Communication / 4. Private Language
Content of thought is established through communication, so knowledge needs other minds [Davidson]
19. Language / F. Communication / 6. Interpreting Language / c. Principle of charity
The principle of charity attributes largely consistent logic and largely true beliefs to speakers [Davidson]