Topic: number as a progression for counting and 1-1 relations

topics > philosophy > Group: philosophy of mathematics


number as the extension of a class of equinumerous classes
what is a number


The first axiomatization of number was by Dedekind [1888]. The classical formalization of Peano's axioms [1899] is similar. Both define numbers as a progression starting from 1 or 0 and satisfying the induction principle.

The importance of a number is its position in the progression, i.e., where it appears as we say the numbers in sequence. We measure the size of a set by placing the numbers in one-to-one correspondence with its elements. Note that any recursive progression will do. We agree to use one particular progression because it is useful.

While this viewpoint has been successful for number theory, it is not satisfying at a gut level. We want to know what numbers are, not what they might be. Saying that we count via a one-to-one correspondence sounds like number itself. Number appears too important a concept to leave to convention. One could ask, what is it that make number so useful?

Frege uses the idea of a progression to define number. (cbb 3/94)

Subtopic: number sequence -- Peano's axioms up

Quote: a number sequence is a sequence of elements, each with a unique successor and reachable from '1' [»dedeR2_1890]
Quote: number proceeds from unity to ten to 30 to 100 and on to infinity
Quote: Peano's axioms for number concern 0, unique successors, and induction [»russB_1919, OK]
Quote: any progression matches Peano's five axioms for numbers; so the axioms do not define '0', 'number' and 'successor' [»russB_1919, OK]
Quote: a sequence of number words is just that, a sequence with certain properties; not distinct from numbers [»benaP_1965]
Quote: arithmetic concerns the abstract structure of progression, and not particular objects--the numbers
Quote: intransitive counting is repeating number words in right order, as in counting sheep

Subtopic: sequence and cardinality up

Quote: transitive counting is measuring the cardinality of a set by counting; one-to-one correspondence [»benaP_1965]
Quote: a numeral directly specifies the position of its referent in the progression of numbers; i.e., a system of generating and counting [»ackeD_1979]
Quote: the value of a particular expression is the position of a number within a sequence and the corresponding rules for counting and cardinality [»benaP_1965]
Quote: QuineWV defines the numbers as any progression; but also need transitive counting [»benaP_1965]
Quote: a whole is one concept made of subparts with something in common; can be enumerated [»leibGW_1666]

Subtopic: tally up

Quote: use tally sticks or parts of your body to keep track of numbers; no need to count as we do [»ifraG_2000]

Subtopic: recursion up

Quote: less-than over natural numbers should be recursive so we can compare numbers and count them [»benaP_1965]
Quote: numbers have a recursive 'less-than' relation because numeric notation is generated recursively [»benaP_1965]

Subtopic: proof by induction up

Quote: proof by induction rests upon the notion of a chain, i.e., a sequence of numbers

Subtopic: numbers as logic up

Quote: to see if arithmetic was analytic, Frege reduced ordering in a sequence (number) to logical consequences from the laws of thought [»fregG_1879]

Subtopic: philosophical problems with number up

Quote: if numbers are abstractions for counting then numbers in different bases are different numbers [»cbb_1990, OK]
Quote: the concept of number should apply in the right way to common objects
Quote: saying that number is a one-one correlation between classes is just substituting another expression for number [»wittL_1939]
Quote: we fix our technique that 13 follows 12; the only discovery is that this is a valuable thing to do [»wittL_1939]
Quote: can count in letter numerals, e.g., fhiza, but they are not numbers
[»cbb_1990, OK]

Related Topics up

Group: sequences   (7 topics, 97 quotes)

Topic: number as the extension of a class of equinumerous classes (23 items)
Topic: what is a number
(55 items)

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