Combining Philosophers

Ideas for Prodicus, Aristotle and David Hilbert

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55 ideas

6. Mathematics / A. Nature of Mathematics / 1. Mathematics
560 Mathematical precision is only possible in immaterial things [Aristotle]
12377 Mathematics is concerned with forms, not with superficial properties [Aristotle]
9076 Mathematics studies the domain of perceptible entities, but its subject-matter is not perceptible [Aristotle]
8717 Hilbert wanted to prove the consistency of all of mathematics (which realists take for granted) [Hilbert, by Friend]
12456 I aim to establish certainty for mathematical methods [Hilbert]
12461 We believe all mathematical problems are solvable [Hilbert]
6. Mathematics / A. Nature of Mathematics / 2. Geometry
13472 Hilbert aimed to eliminate number from geometry [Hilbert, by Hart,WD]
9790 Geometry studies naturally occurring lines, but not as they occur in nature [Aristotle]
12372 The essence of a triangle comes from the line, mentioned in any account of triangles [Aristotle]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
1729 We perceive number by the denial of continuity [Aristotle]
10958 Perhaps numbers are substances? [Aristotle]
13273 Pluralities divide into discontinous countables; magnitudes divide into continuous things [Aristotle]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
11044 One is prior to two, because its existence is implied by two [Aristotle]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
22962 Two is the least number, but there is no least magnitude, because it is always divisible [Aristotle]
11042 Parts of a line join at a point, so it is continuous [Aristotle]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
12074 The one in number just is the particular [Aristotle]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
12273 Unit is the starting point of number [Aristotle]
12369 A unit is what is quantitatively indivisible [Aristotle]
17844 The unit is stipulated to be indivisible [Aristotle]
17845 If only rectilinear figures existed, then unity would be the triangle [Aristotle]
17859 Units came about when the unequals were equalised [Aristotle]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
17861 Two men do not make one thing, as well as themselves [Aristotle]
646 When we count, are we adding, or naming numbers? [Aristotle]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / a. The Infinite
18090 Without infinity time has limits, magnitudes are indivisible, and numbers come to an end [Aristotle]
9633 No one shall drive us out of the paradise the Cantor has created for us [Hilbert]
12460 We extend finite statements with ideal ones, in order to preserve our logic [Hilbert]
12462 Only the finite can bring certainty to the infinite [Hilbert]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / c. Potential infinite
22929 Aristotle's infinity is a property of the counting process, that it has no natural limit [Aristotle, by Le Poidevin]
13212 Infinity is only potential, never actual [Aristotle]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
12455 The idea of an infinite totality is an illusion [Hilbert]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
22930 Lengths do not contain infinite parts; parts are created by acts of division [Aristotle, by Le Poidevin]
18833 A continuous line cannot be composed of indivisible points [Aristotle]
12457 There is no continuum in reality to realise the infinitely small [Hilbert]
6. Mathematics / B. Foundations for Mathematics / 2. Proof in Mathematics
17967 To decide some questions, we must study the essence of mathematical proof itself [Hilbert]
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Geometry
9546 Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Hilbert, by Chihara]
18742 Hilbert's formalisation revealed implicit congruence axioms in Euclid [Hilbert, by Horsten/Pettigrew]
18217 Hilbert's geometry is interesting because it captures Euclid without using real numbers [Hilbert, by Field,H]
17965 The whole of Euclidean geometry derives from a basic equation and transformations [Hilbert]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
17964 Number theory just needs calculation laws and rules for integers [Hilbert]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / b. Greek arithmetic
17851 Number is plurality measured by unity [Aristotle]
17843 The idea of 'one' is the foundation of number [Aristotle]
17850 Each many is just ones, and is measured by the one [Aristotle]
11041 Some quantities are discrete, like number, and others continuous, like lines, time and space [Aristotle]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
9793 Mathematics studies abstracted relations, commensurability and proportion [Aristotle]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
13738 It is a simple truth that the objects of mathematics have being, of some sort [Aristotle]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / b. Against mathematical platonism
12339 Aristotle removes ontology from mathematics, and replaces the true with the beautiful [Aristotle, by Badiou]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / a. Mathematical empiricism
9974 Ten sheep and ten dogs are the same numerically, but it is not the same ten [Aristotle]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
17697 The existence of an arbitrarily large number refutes the idea that numbers come from experience [Hilbert]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
17698 Logic already contains some arithmetic, so the two must be developed together [Hilbert]
6. Mathematics / C. Sources of Mathematics / 7. Formalism
10113 The grounding of mathematics is 'in the beginning was the sign' [Hilbert]
10115 Hilbert substituted a syntactic for a semantic account of consistency [Hilbert, by George/Velleman]
22293 Hilbert said (to block paradoxes) that mathematical existence is entailed by consistency [Hilbert, by Potter]
12459 The subject matter of mathematics is immediate and clear concrete symbols [Hilbert]
6. Mathematics / C. Sources of Mathematics / 8. Finitism
10116 Hilbert aimed to prove the consistency of mathematics finitely, to show infinities won't produce contradictions [Hilbert, by George/Velleman]
18112 Mathematics divides in two: meaningful finitary statements, and empty idealised statements [Hilbert]