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Full Idea
The problem for logicism was to find definitions of the primitive notions of Peano's theory, number, successor and 0, in terms of logical notions, so that the postulates could then be derived by logic alone.
Gist of Idea
The task of logicism was to define by logic the concepts 'number', 'successor' and '0'
Source
Bernard Linsky (Russell's Metaphysical Logic [1999], 7)
Book Ref
Linsky,Bernard: 'Russell's Metaphysical Logic' [CSLI 1999], p.115
A Reaction
Both Frege and Russell defined numbers as equivalence classes. Successor is easily defined (in various ways) in set theory. An impossible set can exemplify zero. The trouble for logicism is this all relies on sets.
Related Ideas
Idea 5897 0 is a non-successor number, all successors are numbers, successors can't duplicate, if P(n) and P(n+1) then P(all-n) [Peano, by Flew]
Idea 18179 For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy]
Idea 18178 For Zermelo the successor of n is {n} (rather than n U {n}) [Zermelo, by Maddy]
| 18774 | Definite descriptions, unlike proper names, have a logical structure [Linsky,B] |
| 18776 | Contextual definitions eliminate descriptions from contexts [Linsky,B] |
| 21704 | 'Impredictative' definitions fix a class in terms of the greater class to which it belongs [Linsky,B] |
| 21705 | Reducibility says any impredicative function has an appropriate predicative replacement [Linsky,B] |
| 21703 | Types are 'ramified' when there are further differences between the type of quantifier and its range [Linsky,B] |
| 21714 | The ramified theory subdivides each type, according to the range of the variables [Linsky,B] |
| 21713 | Did logicism fail, when Russell added three nonlogical axioms, to save mathematics? [Linsky,B] |
| 21715 | For those who abandon logicism, standard set theory is a rival option [Linsky,B] |
| 21719 | Extensionalism means what is true of a function is true of coextensive functions [Linsky,B] |
| 21721 | Higher types are needed to distinguished intensional phenomena which are coextensive [Linsky,B] |
| 21723 | The task of logicism was to define by logic the concepts 'number', 'successor' and '0' [Linsky,B] |
| 21727 | Definite descriptions theory eliminates the King of France, but not the Queen of England [Linsky,B] |
| 21729 | Construct properties as sets of objects, or say an object must be in the set to have the property [Linsky,B] |