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Ideas for 'Thinking About Mathematics', 'On Signs (damaged)' and 'Philosophy of Mathematics'

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32 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics

9604	Mathematics is the only place where we are sure we are right [Brown,JR]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers

9622	'There are two apples' can be expressed logically, with no mention of numbers [Brown,JR]

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 / A. Nature of Mathematics / 3. Nature of Numbers / n. Pi

9648	π is a 'transcendental' number, because it is not the solution of an equation [Brown,JR]

6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics

9621	Mathematics represents the world through structurally similar models. [Brown,JR]

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 / 2. Proof in Mathematics

9646	There is no limit to how many ways something can be proved in mathematics [Brown,JR]

9647	Computers played an essential role in proving the four-colour theorem of maps [Brown,JR]

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 / 6. Mathematics as Set Theory / b. Mathematics is not set theory

9643	Set theory may represent all of mathematics, without actually being mathematics [Brown,JR]

9644	When graphs are defined set-theoretically, that won't cover unlabelled graphs [Brown,JR]

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 / B. Foundations for Mathematics / 7. Mathematical Structuralism / d. Platonist structuralism

9625	To see a structure in something, we must already have the idea of the structure [Brown,JR]

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique

9628	Sets seem basic to mathematics, but they don't suit structuralism [Brown,JR]

6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism

9606	The irrationality of root-2 was achieved by intellect, not experience [Brown,JR]

6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism

9612	There is an infinity of mathematical objects, so they can't be physical [Brown,JR]

9610	Numbers are not abstracted from particulars, because each number is a particular [Brown,JR]

6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival

9620	Empiricists base numbers on objects, Platonists base them on properties [Brown,JR]

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

9629	For nomalists there are no numbers, only numerals [Brown,JR]

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]

9639	Does some mathematics depend entirely on notation? [Brown,JR]

9630	The most brilliant formalist was Hilbert [Brown,JR]

6. Mathematics / C. Sources of Mathematics / 10. Constructivism / a. Constructivism

9608	There are no constructions for many highly desirable results in mathematics [Brown,JR]

9645	Constructivists say p has no value, if the value depends on Goldbach's Conjecture [Brown,JR]

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]