# Changes

## Understanding mathematical definitions

, 00:22, 22 July 2019
List of steps
| Googling around/reading alternative texts || || Sometimes a definition is confusingly written (in one textbook) or the concept itself is confusing (e.g. because it is too abstract). It can help to look around for alternative expositions, especially ones that try to explain the intuitions/historical motivations of the definition. See also [[learning from multiple sources]]. || || In mathematical logic, the terminology for formal languages is a mess: some books define a structure as having a domain and an interpretation (so structure = (domain, interpretation)), while others define the same thing as interpretation = (domain, denotations), while still others define it as structure = (domain, signature, interpretation). The result is that in order to not be confused when e.g. reading an article online, one must become familiar with a range of definitions/terminology for the same concepts and be able to quickly adjust to the intended one in a given context.<br><br>To give another example from mathematical logic, there is the expresses vs captures distinction. But different books use terminology like arithmetically defines vs defines, represents vs expresses, etc. So again things are a mess.
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| Drawing a picture || Ideally ask this about every definition. But some subfields of math (e.g. analysis) are a lot more visual than others (e.g. mathematical logic). || || || Pugh's ''Real Mathematical Analysis'', Needham's ''Visual Complex Analysis''.
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| Chunking/processing level by level || If a definition involves multiple layers of quantifiers. || || || See Tao's definitions for $\varepsilon$-close, eventually $\varepsilon$-close, $\varepsilon$-adherent, etc.
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