Topic: number as the extension of a class of equinumerous classes

topics > philosophy > Group: philosophy of mathematics

number as a progression for counting and 1-1 relations
object and value equivalence
what is a number


The first rigorous definition of number was by Frege [1884]. It was rediscovered by Russell in 1901. They define a number as the extension of a class of equinumerous classes. For instance, 0 is the extension of the concept "unequal to itself" and 1 belongs to the concept "identical with 0" (among others), and 2 is the class of all pairs of objects. Frege defined cardinality through the ordering of a sequence which was in turn defined by logic. In this way number is reduced to logic.

The key ideas are that numbers apply to concepts, that words take their meaning from the sentences in which they occur, and that the recognition statment for number is a 1-1 relationship between concepts. Since 1-1 relationships are logically defined, number is logically defined, i.e., analytic a priori.

The principle difficulties with this definition are that the class of all classes of n objects is ill-defined in standard set theories, and that the concept of a 1-1 relationship is tied to that of number. Another objection is that numbers act as operators instead of predicates or adjectives of classes.

Frege's goal was to define number as an objectively real object. But, as observed by Benacerraf, Frege's construction is one of many definitions of number. If other constructions appear to be equally valid, how can any of them be what number is? (cbb 3/94)

Subtopic: number as equinumerous class up

Quote: the central notion of the cardinal numbers is "just as many as"; i.e., a one-to-one relationship between two sets [»dummM_1967]
Quote: a number is the number of some class, i.e., the class of those classes that are similar to it; discovered by Frege [»russB_1919, OK]
Quote: FregeG and RussellB see numbers as logical attributes of a concept; i.e., its cardinality [»carnR_1931]
Quote: define a 1-1 correlation as a relation that assigns an object for concept G to each object in concept F and vice versa [»fregG_1884]
Quote: two concepts are equinumerous if there is a one-to-one correlation; 'n is a number' is a concept that is equinumerous to a concept F [»fregG_1884]
Quote: 0 is the number that applies to the concept 'unequal to itself' [»fregG_1884]
Quote: 0 is a number because any relation, including identity, is a 1-1 relation for concepts under which no object falls [»fregG_1884]
Quote: define 1 as the Number belonging to the concept "identical with 0"; this concept only contains 0 [»fregG_1884]
Quote: define successor as the Number belonging to a concept of one additional element [»fregG_1884]
Quote: Frege sketches the proof that every natural number has a successor and therefore forms an infinite sequence [»fregG_1884]
Quote: Frege defines sequence through hereditary properties that hold for all remaining elements of a sequence [»fregG_1879]

Subtopic: numbers 'de re' up

Quote: if believe that 5 is the smallest perfect number than believe it de re; different than an existence predicate [»ackeD_1979]
Quote: a number is such that J believes 'n is F' de re iff J believes at least one member of the set of propositions for 'n is F' [»ackeD_1979]
Quote: suppose that J knows that 1100 base 2 is 1100 base 2 (de re) but not that 1100 base 2 is 12; should be know de re [»ackeD_1979]

Subtopic: numeric identity up

Quote: define numeric identity by a one-one relationship; need to use identity to define numeric identity [»fregG_1884]
Quote: define numeric identity in the same way as defining direction from the extension of the concept of parallel lines [»fregG_1884]
Quote: define number by fixing the sense of numerical identity, i.e., the recognition statement for numbers [»fregG_1884]
Quote: to define Number need to fix the sense of numerical identity; of the number which belongs to the concept F and to the concept G [»fregG_1884]

Subtopic: problems with number as class up

Quote: saying that number is a one-one correlation between classes is just substituting another expression for number [»wittL_1939]
Quote: are all classes of the same size, the same 'number'? if so, when do names refer to the same thing (e.g., overlapping shadows) [»wittL_1939]
Quote: FregeG defined number as an equivalence class, but numbers are neither predicates nor adjectives
Quote: consistent set theories do not include the class of all classes with 17 members; this avoids the paradoxes [»benaP_1965]
Quote: while FregeG felt numbers needed a reference, it does not make sense to identify '3' with '[[[0]]]'; their contexts differ

Related Topics up

Topic: number as a progression for counting and 1-1 relations (22 items)
Topic: object and value equivalence (60 items)
Group: sets   (7 topics, 148 quotes)
Topic: what is a number
(55 items)

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