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What is the difference between formal and informal definitions in mathematics ?

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    Formal definition: a definition. Informal definition: not a definition. – 2017-02-07
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    Welcome to the Math Stack Exchange. Are you looking for examples? Can you think of any? – 2017-02-07
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    For an informal definition are you thinking of like a characterization? A definition would be what something is, where as a characterization is what something acts like. – 2017-02-07
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    @did If informal definition is not a definition then why is it called a definition ? – 2017-02-07
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    Because some people are shy... – 2017-02-07
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    Informal would be a "working" definition, a "not fully formed" definition, a "tentative" definition etc – 2017-02-07
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    The notion of formality varies across different branches of mathematics. One often hears logicians say things like, "Most mathematics is informal," by which they mean that mathematics isn't usually done in a formal language like first-order logic. What sets mathematics apart, though, they say, is that large portions of it are straightforwardly but tediously _formalizable_ in their very stringent sense. So, anyway, the answer to your question depends on what you mean by "formal". – 2017-02-07
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    Sorry? Do not be fooled by the light tone, these assertions are quite serious (and you should soon see them more or less confirmed in more verbose formulations below...). But maybe you prefer "serious" and wrong answers such as the one equating characterization with informal definition (while a minute of thought should suffice to be convinced that *any* characterization must include a *formal* definition)... – 2017-02-07

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An "informal definition" is an intuitive description of some mathematical notion new to the reader. An example: "The derivative of a funcion $f$ at some point $p$ is the slope of the tangent to the graph of $f$ at the point $\bigl(p,f(p)\bigr)$." Such an informal definition does not allow you to mathematically work with the new notion.

In contrast, a "formal definition", definition for short, is an encoding of some mathematically precise situation into a single word, or code-word, like $e$ or $\pi$.