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
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

8747

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

12. Knowledge Sources / C. Rationalism / 1. Rationalism
8725

Rationalism tries to apply mathematical methodology to all of knowledge
