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Ideas for 'Thinking About Mathematics', 'Review of 'Aenesidemus'' and 'Critique of Pure Reason'

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24 ideas

6. Mathematics / A. Nature of Mathematics / 2. Geometry
8739 Geometry studies the Euclidean space that dictates how we perceive things [Kant, by Shapiro]
8740 Geometry would just be an idle game without its connection to our intuition [Kant]
16899 Geometrical truth comes from a general schema abstracted from a particular object [Kant, by Burge]
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 / 5. The Infinite / c. Potential infinite
9632 Kant only accepts potential infinity, not actual infinity [Kant, by 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 / 3. Axioms for Geometry
3343 Euclid's could be the only viable geometry, if rejection of the parallel line postulate doesn't lead to a contradiction [Benardete,JA on Kant]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
8737 Kant suggested that arithmetic has no axioms [Kant, by Shapiro]
5557 Axioms ought to be synthetic a priori propositions [Kant]
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 / 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 / C. Sources of Mathematics / 2. Intuition of Mathematics
12421 Kant's intuitions struggle to judge relevance, impossibility and exactness [Kitcher on Kant]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
17617 Maths is a priori, but without its relation to empirical objects it is meaningless [Kant]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
12458 Kant taught that mathematics is independent of logic, and cannot be grounded in it [Kant, by Hilbert]
2795 If 7+5=12 is analytic, then an infinity of other ways to reach 12 have to be analytic [Kant, by Dancy,J]
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
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]
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]