Combining Texts
Ideas for
'The Evolution of Logic', 'The Republic' and 'works'
expand these ideas
|
start again
|
choose
another area for these texts
display all the ideas for this combination of texts
13 ideas
6. Mathematics / A. Nature of Mathematics / 2. Geometry
|
8726
|
Geometry can lead the mind upwards to truth and philosophy [Plato]
|
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 / B. Foundations for Mathematics / 3. Axioms for Geometry
|
13474
|
Euclid has a unique parallel, spherical geometry has none, and saddle geometry has several [Hart,WD]
|
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
|
9863
|
We aim for elevated discussion of pure numbers, not attaching them to physical objects [Plato]
|
|
9864
|
In pure numbers, all ones are equal, with no internal parts [Plato]
|
|
8727
|
Geometry is not an activity, but the study of unchanging knowledge [Plato]
|
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
|
9861
|
The same thing is both one and an unlimited number at the same time [Plato]
|
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
|
13471
|
Mathematics makes existence claims, but philosophers usually say those are never analytic [Hart,WD]
|