Combining Texts

All the ideas for 'The Case for Closure', 'Continuity and Irrational Numbers' and 'fragments/reports'

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9 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
17611 We want the essence of continuity, by showing its origin in arithmetic [Dedekind]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / i. Reals from cuts
10572 A cut between rational numbers creates and defines an irrational number [Dedekind]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
17612 Arithmetic is just the consequence of counting, which is the successor operation [Dedekind]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / l. Limits
18087 If x changes by less and less, it must approach a limit [Dedekind]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
13007 Archimedes defined a straight line as the shortest distance between two points [Archimedes, by Leibniz]
11. Knowledge Aims / B. Certain Knowledge / 2. Common Sense Certainty
19553 Commitment to 'I have a hand' only makes sense in a context where it has been doubted [Hawthorne]
13. Knowledge Criteria / A. Justification Problems / 2. Justification Challenges / c. Knowledge closure
19551 How can we know the heavyweight implications of normal knowledge? Must we distort 'knowledge'? [Hawthorne]
19552 We wouldn't know the logical implications of our knowledge if small risks added up to big risks [Hawthorne]
19554 Denying closure is denying we know P when we know P and Q, which is absurd in simple cases [Hawthorne]