What is the difference between classification and characterization in reference to mathematical objects ? Some examples will be appreciated.
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These notions are not comparable at all. A classification applies to a whole class of objects (finite simple groups, for instance) while a characterisation applies to a single object, value, or property (the determinant of a matrix for instance, or the invertibility of that matrix). – Marc van Leeuwen May 30 '14 at 05:25
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You would probably get a more helpful answer if you gave some examples: provide some quotes to show where you have seen these terms and why they confuse you. – David May 30 '14 at 05:44
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1I do not understand the close votes here - although I do agree with David's comment, I think this question is fine as it stands. As really Marc van Leeuwen's comment is an answer. So where is the issue here? – user1729 May 30 '14 at 08:23
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A type of mathematical object can be classified into different categories or types; each example of the object lies in exactly one category (or is of exactly one type). This is what we call a classification. Given a type of mathematical object, it can also be classified in different ways. Here are some examples:
- Classification of finite simple groups.
- Classification of semisimple Lie algebras over an algebraically closed field.
- Classification of one-dimensional manifolds.
- Classification of closed surfaces (connected, compact, two-dimensional manifolds without boundary).
- Classification of simply connected Riemann surfaces.
A mathematical object (or collection of mathematical objects) can be characterised by a collection of properties; that is, it is determined by those properties alone. This is what we call a characterisation. In particular, a type of mathematical object is characterised by its definition, but we are often interested in other characterisations. Here are some examples:
- The prime numbers are characterised by their definition: numbers greater than one which have only two factors. However, they are also characterised by the following property: numbers $p$ greater than one such that if $p \mid ab$, then $p \mid a$ or $p \mid b$.
- The invertible matrices are characterised by the properties of being square and having non-zero determinant.
- Infinite sets are characterised by the property of having a proper subset with which it is equinumerous (i.e. there is a bijection between the subset and the set itself).
- Harmonic functions are characterised by the mean value property (assuming local integrability).
There are many more examples for both lists. If I think of any more instructive ones I'll add them.
Added Later: You might find the Wikipedia pages on classification and characterisation useful as well.
Michael Albanese
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