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Ideas for 'talk', '17: Epistle to Titus' and 'Grundlagen der Arithmetik (Foundations)'

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88 ideas

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / e. Ordinal numbers
8640 We cannot define numbers from the idea of a series, because numbers must precede that [Frege]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / l. Zero
9838 Treating 0 as a number avoids antinomies involving treating 'nobody' as a person [Frege, by Dummett]
9564 For Frege 'concept' and 'extension' are primitive, but 'zero' and 'successor' are defined [Frege, by Chihara]
10551 If objects exist because they fall under a concept, 0 is the object under which no objects fall [Frege, by Dummett]
8653 Nought is the number belonging to the concept 'not identical with itself' [Frege]
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / m. One
8636 We can say 'a and b are F' if F is 'wise', but not if it is 'one' [Frege]
8654 One is the Number which belongs to the concept "identical with 0" [Frege]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / a. Units
8641 You can abstract concepts from the moon, but the number one is not among them [Frege]
9989 Units can be equal without being identical [Tait on Frege]
17429 Frege says only concepts which isolate and avoid arbitrary division can give units [Frege, by Koslicki]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / d. Counting via concepts
17427 Frege's 'isolation' could be absence of overlap, or drawing conceptual boundaries [Frege, by Koslicki]
17437 Non-arbitrary division means that what falls under the concept cannot be divided into more of the same [Frege, by Koslicki]
17438 Our concepts decide what is countable, as in seeing the leaves of the tree, or the foliage [Frege, by Koslicki]
17426 A concept creating a unit must isolate and unify what falls under it [Frege]
17428 Frege says counting is determining what number belongs to a given concept [Frege, by Koslicki]
6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / e. Counting by correlation
15916 Frege's one-to-one correspondence replaces well-ordering, because infinities can't be counted [Frege, by Lavine]
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / h. Ordinal infinity
10034 The number of natural numbers is not a natural number [Frege, by George/Velleman]
6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
16883 Arithmetical statements can't be axioms, because they are provable [Frege, by Burge]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers
17460 A statement of number contains a predication about a concept [Frege]
16890 Frege's problem is explaining the particularity of numbers by general laws [Frege, by Burge]
8630 Individual numbers are best derived from the number one, and increase by one [Frege]
11029 'Exactly ten gallons' may not mean ten things instantiate 'gallon' [Rumfitt on Frege]
10013 Numerical statements have first-order logical form, so must refer to objects [Frege, by Hodes]
18181 The Number for F is the extension of 'equal to F' (or maybe just F itself) [Frege]
18103 Numbers are objects because they partake in identity statements [Frege, by Bostock]
10625 Frege had a motive to treat numbers as objects, but not a justification [Hale/Wright on Frege]
13871 Frege claims that numbers are objects, as opposed to them being Fregean concepts [Frege, by Wright,C]
13872 Numbers are second-level, ascribing properties to concepts rather than to objects [Frege, by Wright,C]
9816 For Frege, successor was a relation, not a function [Frege, by Dummett]
9953 Numbers are more than just 'second-level concepts', since existence is also one [Frege, by George/Velleman]
9954 "Number of x's such that ..x.." is a functional expression, yielding a name when completed [Frege, by George/Velleman]
10139 Frege gives an incoherent account of extensions resulting from abstraction [Fine,K on Frege]
10028 For Frege the number of F's is a collection of first-level concepts [Frege, by George/Velleman]
17636 A cardinal number may be defined as a class of similar classes [Frege, by Russell]
10029 Numbers need to be objects, to define the extension of the concept of each successor to n [Frege, by George/Velleman]
9973 The number of F's is the extension of the second level concept 'is equipollent with F' [Frege, by Tait]
16500 Frege showed that numbers attach to concepts, not to objects [Frege, by Wiggins]
9990 Frege replaced Cantor's sets as the objects of equinumerosity attributions with concepts [Frege, by Tait]
7738 Zero is defined using 'is not self-identical', and one by using the concept of zero [Frege, by Weiner]
23456 Frege said logical predication implies classes, which are arithmetical objects [Frege, by Morris,M]
13887 Frege started with contextual definition, but then switched to explicit extensional definition [Frege, by Wright,C]
13897 Each number, except 0, is the number of the concept of all of its predecessors [Frege, by Wright,C]
9856 Frege's account of cardinals fails in modern set theory, so they are now defined differently [Dummett on Frege]
9902 Frege's incorrect view is that a number is an equivalence class [Benacerraf on Frege]
17814 The natural number n is the set of n-membered sets [Frege, by Yourgrau]
17819 A set doesn't have a fixed number, because the elements can be seen in different ways [Yourgrau on Frege]
17820 If you can subdivide objects many ways for counting, you can do that to set-elements too [Yourgrau on Frege]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / d. Hume's Principle
9956 'The number of Fs' is the extension (a collection of first-level concepts) of the concept 'equinumerous with F' [Frege, by George/Velleman]
13527 Frege's cardinals (equivalences of one-one correspondences) is not permissible in ZFC [Frege, by Wolf,RS]
22292 Hume's Principle fails to implicitly define numbers, because of the Julius Caesar [Frege, by Potter]
17442 Frege thinks number is fundamentally bound up with one-one correspondence [Frege, by Heck]
6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / e. Caesar problem
11030 The words 'There are exactly Julius Caesar moons of Mars' are gibberish [Rumfitt on Frege]
10030 'Julius Caesar' isn't a number because numbers inherit properties of 0 and successor [Frege, by George/Velleman]
8690 From within logic, how can we tell whether an arbitrary object like Julius Caesar is a number? [Frege, by Friend]
10219 Frege said 2 is the extension of all pairs (so Julius Caesar isn't 2, because he's not an extension) [Frege, by Shapiro]
13889 Fregean numbers are numbers, and not 'Caesar', because they correlate 1-1 [Frege, by Wright,C]
18142 One-one correlations imply normal arithmetic, but don't explain our concept of a number [Frege, by Bostock]
9046 Our definition will not tell us whether or not Julius Caesar is a number [Frege]
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / b. Mathematics is not set theory
16896 If numbers can be derived from logic, then set theory is superfluous [Frege, by Burge]
6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / e. Structuralism critique
8639 If numbers are supposed to be patterns, each number can have many patterns [Frege]
6. Mathematics / C. Sources of Mathematics / 1. Mathematical Platonism / a. For mathematical platonism
13874 Numbers seem to be objects because they exactly fit the inference patterns for identities [Frege]
13875 Frege's platonism proposes that objects are what singular terms refer to [Frege, by Wright,C]
7731 How can numbers be external (one pair of boots is two boots), or subjective (and so relative)? [Frege, by Weiner]
7737 Identities refer to objects, so numbers must be objects [Frege, by Weiner]
8635 Numbers are not physical, and not ideas - they are objective and non-sensible [Frege]
8652 Numbers are objects, because they can take the definite article, and can't be plurals [Frege]
6. Mathematics / C. Sources of Mathematics / 2. Intuition of Mathematics
17816 Frege's logicism aimed at removing the reliance of arithmetic on intuition [Frege, by Yourgrau]
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
8633 There is no physical difference between two boots and one pair of boots [Frege]
6. Mathematics / C. Sources of Mathematics / 5. Numbers as Adjectival
9951 It appears that numbers are adjectives, but they don't apply to a single object [Frege, by George/Velleman]
9952 Numerical adjectives are of the same second-level type as the existential quantifier [Frege, by George/Velleman]
11031 'Jupiter has many moons' won't read as 'The number of Jupiter's moons equals the number many' [Rumfitt on Frege]
8637 The number 'one' can't be a property, if any object can be viewed as one or not one [Frege]
9999 For science, we can translate adjectival numbers into noun form [Frege]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / a. Early logicism
7739 Arithmetic is analytic [Frege, by Weiner]
9945 Logicism shows that no empirical truths are needed to justify arithmetic [Frege, by George/Velleman]
8782 Frege offered a Platonist version of logicism, committed to cardinal and real numbers [Frege, by Hale/Wright]
13608 Mathematics has no special axioms of its own, but follows from principles of logic (with definitions) [Frege, by Bostock]
16905 Arithmetic must be based on logic, because of its total generality [Frege, by Jeshion]
5658 Numbers are definable in terms of mapping items which fall under concepts [Frege, by Scruton]
8655 Arithmetic is analytic and a priori, and thus it is part of logic [Frege]
6. Mathematics / C. Sources of Mathematics / 6. Logicism / d. Logicism critique
10831 Frege only managed to prove that arithmetic was analytic with a logic that included set-theory [Quine on Frege]
13864 Frege's platonism and logicism are in conflict, if logic must dictates an infinity of objects [Wright,C on Frege]
10033 Why should the existence of pure logic entail the existence of objects? [George/Velleman on Frege]
10010 Frege's belief in logicism and in numerical objects seem uncomfortable together [Hodes on Frege]
6. Mathematics / C. Sources of Mathematics / 7. Formalism
9631 Formalism fails to recognise types of symbols, and also meta-games [Frege, by Brown,JR]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
9875 Frege was completing Bolzano's work, of expelling intuition from number theory and analysis [Frege, by Dummett]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / c. Conceptualism
8642 Abstraction from things produces concepts, and numbers are in the concepts [Frege]
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / e. Psychologism
8621 Mental states are irrelevant to mathematics, because they are vague and fluctuating [Frege]