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Single Idea 10020

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 5. Definitions of Number / c. Fregean numbers ]

Full Idea

The inconsistency of Grundgesetze was only a minor flaw. Its fundamental flaw was its inability to account for the way in which the senses of number terms are determined. It leaves the reference-magnetic nature of the standard numberer a mystery.

Gist of Idea

Frege's biggest error is in not accounting for the senses of number terms

Source

comment on Gottlob Frege (Grundgesetze der Arithmetik 2 (Basic Laws) [1903]) by Harold Hodes - Logicism and Ontological Commits. of Arithmetic p.139

Book Ref

-: 'Journal of Philosophy' [-], p.139

A Reaction

A point also made by Hofweber. As a logician, Frege was only concerned with the inferential role of number terms, and he felt he had captured their logical form, but it is when you come to look at numbers in natural language that he seem in trouble.

The 46 ideas with the same theme [Frege's view of numbers as extensions of classes]:

9992 The 'extension of a concept' in general may be quantitatively completely indeterminate [Cantor]
9949 There is the concept, the object falling under it, and the extension (a set, which is also an object) [Frege, by George/Velleman]
10623 Frege defined number in terms of extensions of concepts, but needed Basic Law V to explain extensions [Frege, by Hale/Wright]
9975 Frege ignored Cantor's warning that a cardinal set is not just a concept-extension [Tait on Frege]
10020 Frege's biggest error is in not accounting for the senses of number terms [Hodes on Frege]
10553 A number is a class of classes of the same cardinality [Frege, by Dummett]
10625 Frege had a motive to treat numbers as objects, but not a justification [Hale/Wright on Frege]
17820 If you can subdivide objects many ways for counting, you can do that to set-elements too [Yourgrau 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]
17636 A cardinal number may be defined as a class of similar classes [Frege, by Russell]
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]
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]
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]
17460 A statement of number contains a predication about a concept [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]
9586 In a number-statement, something is predicated of a concept [Frege]
3331 If '5' is the set of all sets with five members, that may be circular, and you can know a priori if the set has content [Benardete,JA on Frege]
14117 Numbers are properties of classes [Russell]
17817 Defining 'three' as the principle of collection or property of threes explains set theory definitions [Yourgrau]
13894 Sameness of number is fundamental, not counting, despite children learning that first [Wright,C]
12215 The existence of numbers is not a matter of identities, but of constituents of the world [Fine,K]
18182 The extension of concepts is not important to me [Maddy]
18177 In the ZFC hierarchy it is impossible to form Frege's set of all three-element sets [Maddy]
8297 Numbers are universals, being sets whose instances are sets of appropriate cardinality [Lowe]
17902 A successor is the union of a set with its singleton [George/Velleman]
17461 Some 'how many?' answers are not predications of a concept, like 'how many gallons?' [Rumfitt]