Combining Texts

All the ideas for 'Thinking About Mathematics', 'What is an Idea?' and 'The Folly of Trying to Define Truth'

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21 ideas

3. Truth / A. Truth Problems / 2. Defining Truth
23295 Truth cannot be reduced to anything simpler [Davidson]
3. Truth / C. Correspondence Truth / 3. Correspondence Truth critique
23298 Neither Aristotle nor Tarski introduce the facts needed for a correspondence theory [Davidson]
3. Truth / F. Semantic Truth / 1. Tarski's Truth / c. Meta-language for truth
23297 The language to define truth needs a finite vocabulary, to make the definition finite [Davidson]
3. Truth / G. Axiomatic Truth / 1. Axiomatic Truth
23296 We can elucidate indefinable truth, but showing its relation to other concepts [Davidson]
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]
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]
12. Knowledge Sources / C. Rationalism / 1. Rationalism
8725 Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro]
13. Knowledge Criteria / A. Justification Problems / 1. Justification / a. Justification issues
23294 It is common to doubt truth when discussing it, but totally accept it when discussing knowledge [Davidson]
18. Thought / C. Content / 2. Ideas
19423 By an 'idea' I mean not an actual thought, but the resources we can draw on to think [Leibniz]