Equivalent definitions of mathematical structures | Wikipedia audio article

This is an audio version of the Wikipedia Article:


00:00:20 1 Isomorphic implementations
00:00:31 2 Deduced structures and cryptomorphisms
00:00:41 3 Ambient frameworks
00:00:51 4 Structures according to Bourbaki
00:01:02 4.1 Transport of structures; isomorphism
00:01:12 4.2 Functoriality
00:01:22 5 Mathematical practice
00:01:43 6 Canonical, not just natural
00:02:04 7 See also
00:02:24 8 Notes
00:02:35 9 Footnotes
00:02:45 10 References
00:03:06 11 Further reading
00:03:26 12 External links
00:03:47 Transport of structures; isomorphism
00:04:08 Functoriality
00:04:28 Mathematical practice
00:04:49 Canonical, not just natural
00:05:10 { a0, a1, a2, ... } where S(an)
00:05:31 a0; and on the other hand, X
00:05:51 bm + bn, bm·n
00:06:12 bn for all n one gets the canonical equivalence between the two structures. However, one may also require a0
00:06:33 b0, and an
00:06:53 See also
00:07:04 Notes



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SUMMARY
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In mathematics, equivalent definitions are used in two somewhat different ways. First, within a particular mathematical theory (for example, Euclidean geometry), a notion (for example, ellipse or minimal surface) may have more than one definition. These definitions are equivalent in the context of a given mathematical structure (Euclidean space, in this case). Second, a mathematical structure may have more than one definition (for example, topological space has at least seven definitions; ordered field has at least two definitions).
In the former case, equivalence of two definitions means that a mathematical object (for example, geometric body) satisfies one definition if and only if it satisfies the other definition.
In the latter case, the meaning of equivalence (between two definitions of a structure) is more complicated, since a structure is more abstract than an object. Many different objects may implement the same structure.