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All the ideas for 'Thinking About Mathematics', 'The Theory of Knowledge' and 'How to Define Theoretical Terms'

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25 ideas

2. Reason / D. Definition / 2. Aims of Definition
15527 Defining terms either enables elimination, or shows that they don't require elimination [Lewis]
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
8729 Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro]
5. Theory of Logic / E. Structures of Logic / 2. Logical Connectives / a. Logical connectives
23476 Logical constants seem to be entities in propositions, but are actually pure form [Russell]
23477 We use logical notions, so they must be objects - but I don't know what they really are [Russell]
5. Theory of Logic / I. Semantics of Logic / 3. Logical Truth
18273 Logical truths are known by their extreme generality [Russell]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / b. Types of number
8763 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
18249 Cauchy gave a formal definition of a converging sequence. [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
8764 Categories are the best foundation for mathematics [Shapiro]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / f. Zermelo numbers
8762 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
8760 Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro]
8761 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
8744 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
8749 Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro]
8750 Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro]
8752 Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
8753 Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
8731 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
8730 'Impredicative' definitions refer to the thing being described [Shapiro]
7. Existence / D. Theories of Reality / 8. Facts / d. Negative facts
22315 There can't be a negative of a complex, which is negated by its non-existence [Potter on Russell]
10. Modality / E. Possible worlds / 3. Transworld Objects / b. Rigid designation
15530 A logically determinate name names the same thing in every possible world [Lewis]
12. Knowledge Sources / C. Rationalism / 1. Rationalism
8725 Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro]
14. Science / B. Scientific Theories / 8. Ramsey Sentences
15528 A Ramsey sentence just asserts that a theory can be realised, without saying by what [Lewis]
15526 There is a method for defining new scientific terms just using the terms we already understand [Lewis]
15529 It is better to have one realisation of a theory than many - but it may not always be possible [Lewis]
15531 The Ramsey sentence of a theory says that it has at least one realisation [Lewis]