Combining Philosophers

Ideas for Archimedes, Ludwig Wittgenstein and Leon Horsten

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16 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
18281 In mathematics everything is algorithm and nothing is meaning [Wittgenstein]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
18738 We don't get 'nearer' to something by adding decimals to 1.1412... (root-2) [Wittgenstein]
15364 English expressions are denumerably infinite, but reals are nondenumerable, so many are unnameable [Horsten]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
18708 Infinity is not a number, so doesn't say how many; it is the property of a law [Wittgenstein]
6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
10885 Computer proofs don't provide explanations [Horsten]
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]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / a. Defining numbers
18153 A number is a repeated operation [Wittgenstein]
18160 The concept of number is just what all numbers have in common [Wittgenstein]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
10881 The concept of 'ordinal number' is set-theoretic, not arithmetical [Horsten]
15360 ZFC showed that the concept of set is mathematical, not logical, because of its existence claims [Horsten]
15369 Set theory is substantial over first-order arithmetic, because it enables new proofs [Horsten]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
18161 The theory of classes is superfluous in mathematics [Wittgenstein]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
11073 Two and one making three has the necessity of logical inference [Wittgenstein]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
6849 Wittgenstein hated logicism, and described it as a cancerous growth [Wittgenstein, by Monk]
23509 The logic of the world is shown by tautologies in logic, and by equations in mathematics [Wittgenstein]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / d. Predicativism
15370 Predicativism says mathematical definitions must not include the thing being defined [Horsten]