Ideas from 'Thinking About Mathematics' by Stewart Shapiro [2000], by Theme Structure
[found in 'Thinking About Mathematics' by Shapiro,Stewart [OUP 2000,0192893068]].
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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. 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. Numbers / h. Reals from Cauchy
18249

Cauchy gave a formal definition of a converging sequence.

6. Mathematics / A. Nature of Mathematics / 4. The Infinite / g. Continuum Hypothesis
8733

The Continuum Hypothesis says there are no sets between the natural numbers and reals

6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
8764

Categories are the best foundation for mathematics

6. Mathematics / B. Foundations for Mathematics / 4. 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 / 6. 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 nonstarter if (as is widely held) logic has no ontology of its own

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

8749

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

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
8747

Realists are happy with impredicative definitions, which describe entities in terms of other existing entities

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
