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
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
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
Cauchy gave a formal definition of a converging sequence.
6. Mathematics / A. Nature of Mathematics / 4. The Infinite / g. Continuum Hypothesis
The Continuum Hypothesis says there are no sets between the natural numbers and reals
6. Mathematics / B. Foundations for Mathematics / 1. Foundations for Mathematics
Categories are the best foundation for mathematics
6. Mathematics / B. Foundations for Mathematics / 4. Definitions of Number / f. Zermelo numbers
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
Numbers do not exist independently; the essence of a number is its relations to other numbers
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
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
Term Formalism says mathematics is just about symbols - but real numbers have no names
Game Formalism is just a matter of rules, like chess - but then why is it useful in science?
Deductivism says mathematics is logical consequences of uninterpreted axioms
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
Conceptualist are just realists or idealist or nominalists, depending on their view of concepts
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
'Impredicative' definitions refer to the thing being described
Realists are happy with impredicative definitions, which describe entities in terms of other existing entities
12. Knowledge Sources / C. Rationalism / 1. Rationalism
Rationalism tries to apply mathematical methodology to all of knowledge