Combining Texts

Ideas for 'The Evolution of Logic', 'Critique of Pure Reason' and 'Abstract Objects: a Case Study'

expand these ideas     |    start again     |     choose another area for these texts

display all the ideas for this combination of texts

21 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 / e. Ordinal numbers
13459 The less-than relation < well-orders, and partially orders, and totally orders the ordinal numbers [Hart,WD]
13491 The axiom of infinity with separation gives a least limit ordinal ω [Hart,WD]
13463 There are at least as many infinite cardinals as transfinite ordinals (because they will map) [Hart,WD]
13492 Von Neumann's ordinals generalise into the transfinite better, because Zermelo's ω is a singleton [Hart,WD]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
13446 19th century arithmetization of analysis isolated the real numbers from geometry [Hart,WD]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
13509 We can establish truths about infinite numbers by means of induction [Hart,WD]
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 / 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]
13474 Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD]
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 / 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 / c. Neo-logicism
10580 Mathematics is both necessary and a priori because it really consists of logical truths [Yablo]
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]
13471 Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD]
6. Mathematics / C. Sources of Mathematics / 9. Fictional Mathematics
10579 Putting numbers in quantifiable position (rather than many quantifiers) makes expression easier [Yablo]