Combining Texts

All the ideas for 'The Varieties of Necessity', 'Thinking About Mathematics' and 'Interview with Baggini and Stangroom'

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24 ideas

1. Philosophy / H. Continental Philosophy / 1. Continental Philosophy
Analytic philosophy has much higher standards of thinking than continental philosophy [Williamson]
4. Formal Logic / E. Nonclassical Logics / 4. Fuzzy Logic
Fuzzy logic uses a continuum of truth, but it implies contradictions [Williamson]
5. Theory of Logic / A. Overview of Logic / 3. Value of Logic
Formal logic struck me as exactly the language I wanted to think in [Williamson]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
Cauchy gave a formal definition of a converging sequence. [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
Categories are the best foundation for mathematics [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro]
6. Mathematics / C. Sources of Mathematics / 7. Formalism
Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
'Impredicative' definitions refer to the thing being described [Shapiro]
7. Existence / D. Theories of Reality / 10. Vagueness / c. Vagueness as ignorance
Close to conceptual boundaries judgement is too unreliable to give knowledge [Williamson]
9. Objects / B. Unity of Objects / 3. Unity Problems / e. Vague objects
What sort of logic is needed for vague concepts, and what sort of concept of truth? [Williamson]
10. Modality / C. Sources of Modality / 1. Sources of Necessity
Each area of enquiry, and its source, has its own distinctive type of necessity [Fine,K]
12. Knowledge Sources / B. Perception / 1. Perception
How can one discriminate yellow from red, but not the colours in between? [Williamson]
12. Knowledge Sources / C. Rationalism / 1. Rationalism
Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro]
13. Knowledge Criteria / C. External Justification / 7. Testimony
Unsupported testimony may still be believable [Fine,K]
26. Natural Theory / D. Laws of Nature / 8. Scientific Essentialism / a. Scientific essentialism
Causation is easier to disrupt than logic, so metaphysics is part of nature, not vice versa [Fine,K]