Combining Philosophers

Ideas for Gottfried Leibniz, Richard Dedekind and Friedrich Schelling

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33 ideas

6. Mathematics / A. Nature of Mathematics / 2. Geometry
13163 Circles must be bounded, so cannot be infinite [Leibniz]
13008 Geometry, unlike sensation, lets us glimpse eternal truths and their necessity [Leibniz]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
9823 Numbers are free creations of the human mind, to understand differences [Dedekind]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / c. Priority of numbers
10090 Dedekind defined the integers, rationals and reals in terms of just the natural numbers [Dedekind, by George/Velleman]
17452 Ordinals can define cardinals, as the smallest ordinal that maps the set [Dedekind, by Heck]
7524 Order, not quantity, is central to defining numbers [Dedekind, by Monk]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
14131 Dedekind's ordinals are just members of any progression whatever [Dedekind, by Russell]
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]
14437 Dedekind's axiom that his Cut must be filled has the advantages of theft over honest toil [Dedekind, by Russell]
18094 Dedekind says each cut matches a real; logicists say the cuts are the reals [Dedekind, by Bostock]
18244 I say the irrational is not the cut itself, but a new creation which corresponds to the cut [Dedekind]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
12920 There is no multiplicity without true units [Leibniz]
9147 Number cannot be defined as addition of ones, since that needs the number; it is a single act of abstraction [Fine,K on Leibniz]
12956 Only whole numbers are multitudes of units [Leibniz]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / c. Counting procedure
9824 In counting we see the human ability to relate, correspond and represent [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 / 4. Using Numbers / g. Applying mathematics
19390 Everything is subsumed under number, which is a metaphysical statics of the universe, revealing powers [Leibniz]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / b. Mark of the infinite
9826 A system S is said to be infinite when it is similar to a proper part of itself [Dedekind]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / d. Actual infinite
19406 I strongly believe in the actual infinite, which indicates the perfections of its author [Leibniz]
13190 I don't admit infinite numbers, and consider infinitesimals to be useful fictions [Leibniz]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / j. Infinite divisibility
19375 The continuum is not divided like sand, but folded like paper [Leibniz, by Arthur,R]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / k. Infinitesimals
18081 Nature uses the infinite everywhere [Leibniz]
18080 A tangent is a line connecting two points on a curve that are infinitely close together [Leibniz]
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
12937 We shouldn't just accept Euclid's axioms, but try to demonstrate them [Leibniz]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
23026 We know mathematical axioms, such as subtracting equals from equals leaves equals, by a natural light [Leibniz]
13508 Dedekind gives a base number which isn't a successor, then adds successors and induction [Dedekind, by Hart,WD]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
18096 Zero is a member, and all successors; numbers are the intersection of sets satisfying this [Dedekind, by Bostock]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / e. Peano arithmetic 2nd-order
18841 Categoricity implies that Dedekind has characterised the numbers, because it has one domain [Rumfitt on Dedekind]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
14130 Induction is proved in Dedekind, an axiom in Peano; the latter seems simpler and clearer [Dedekind, by Russell]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
8924 Dedekind originated the structuralist conception of mathematics [Dedekind, by MacBride]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / b. Varieties of structuralism
9153 Dedekindian abstraction talks of 'positions', where Cantorian abstraction talks of similar objects [Dedekind, by Fine,K]