Ideas from 'Thinking About Mathematics' by Stewart Shapiro [2000], by Theme Structure

[found in 'Thinking About Mathematics' by Shapiro,Stewart [OUP 2000,0-19-289306-8]].

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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

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

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy

18249

Cauchy gave a formal definition of a converging sequence.

6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics

8764	Categories are the best foundation for mathematics

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

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

8761	A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them

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

6. Mathematics / C. Sources of Mathematics / 7. Formalism

8749	Term Formalism says mathematics is just about symbols - but real numbers have no names

8750	Game Formalism is just a matter of rules, like chess - but then why is it useful in science?

8752	Deductivism says mathematics is logical consequences of uninterpreted axioms

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism

8753	Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions

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

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism

8730	'Impredicative' definitions refer to the thing being described

12. Knowledge Sources / C. Rationalism / 1. Rationalism

8725	Rationalism tries to apply mathematical methodology to all of knowledge