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Single Idea 8760
[filed under theme 6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism
]
Full Idea
The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers.
Gist of Idea
Numbers do not exist independently; the essence of a number is its relations to other numbers
Source
Stewart Shapiro (Thinking About Mathematics [2000], 10.1)
Book Ref
Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.258
A Reaction
This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate.
The
15 ideas
from 'Thinking About Mathematics'
8725
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Rationalism tries to apply mathematical methodology to all of knowledge
[Shapiro]
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8730
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'Impredicative' definitions refer to the thing being described
[Shapiro]
|
8729
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Intuitionists deny excluded middle, because it is committed to transcendent truth or objects
[Shapiro]
|
8731
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Conceptualist are just realists or idealist or nominalists, depending on their view of concepts
[Shapiro]
|
8744
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Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own
[Shapiro]
|
8749
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Term Formalism says mathematics is just about symbols - but real numbers have no names
[Shapiro]
|
8750
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Game Formalism is just a matter of rules, like chess - but then why is it useful in science?
[Shapiro]
|
8752
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Deductivism says mathematics is logical consequences of uninterpreted axioms
[Shapiro]
|
8753
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Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions
[Shapiro]
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8760
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Numbers do not exist independently; the essence of a number is its relations to other numbers
[Shapiro]
|
8761
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A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them
[Shapiro]
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8763
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The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex
[Shapiro]
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8762
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Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3
[Shapiro]
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8764
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Categories are the best foundation for mathematics
[Shapiro]
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18249
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Cauchy gave a formal definition of a converging sequence.
[Shapiro]
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