Group:
sequences
Topic:
number as the extension of a class of equinumerous classes
Topic:
what is a number
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Summary
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 
| 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
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Subtopic: sequence and cardinality
| 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]
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Subtopic: tally
| Quote: use tally sticks or parts of your body to keep track of numbers; no need to count as we do [»ifraG_2000]
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Subtopic: recursion
| 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]
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Subtopic: proof by induction
| Quote: proof by induction rests upon the notion of a chain, i.e., a sequence of numbers
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Subtopic: numbers as logic
| Quote: to see if arithmetic was analytic, Frege reduced ordering in a sequence (number) to logical consequences from the laws of thought [»fregG_1879]
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Subtopic: philosophical problems with number
| 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]
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Related Topics
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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