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Single Idea 18249
[filed under theme 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
]
Full Idea
A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε.
Gist of Idea
Cauchy gave a formal definition of a converging sequence.
Source
Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4)
Book Ref
Shapiro,Stewart: 'Thinking About Mathematics' [OUP 2000], p.181
A Reaction
The sequence is 'Cauchy' if N exists.
The
15 ideas
from 'Thinking About Mathematics'
8725
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Rationalism tries to apply mathematical methodology to all of knowledge
[Shapiro]
|
8730
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'Impredicative' definitions refer to the thing being described
[Shapiro]
|
8729
|
Intuitionists deny excluded middle, because it is committed to transcendent truth or objects
[Shapiro]
|
8731
|
Conceptualist are just realists or idealist or nominalists, depending on their view of concepts
[Shapiro]
|
8744
|
Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own
[Shapiro]
|
8749
|
Term Formalism says mathematics is just about symbols - but real numbers have no names
[Shapiro]
|
8750
|
Game Formalism is just a matter of rules, like chess - but then why is it useful in science?
[Shapiro]
|
8752
|
Deductivism says mathematics is logical consequences of uninterpreted axioms
[Shapiro]
|
8753
|
Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions
[Shapiro]
|
8760
|
Numbers do not exist independently; the essence of a number is its relations to other numbers
[Shapiro]
|
8761
|
A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them
[Shapiro]
|
8763
|
The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex
[Shapiro]
|
8762
|
Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3
[Shapiro]
|
8764
|
Categories are the best foundation for mathematics
[Shapiro]
|
18249
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Cauchy gave a formal definition of a converging sequence.
[Shapiro]
|