Difference between revisions of "Understanding mathematical definitions"
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Revision as of 21:16, 24 December 2018
Understanding a definition in mathematics is a pretty complicated and laborious process. The following table summarizes some of the things one might do when trying to understand a new definition.
Step  Condition  Description  Purpose  Example 

Typechecking and parsing  
Checking assumptions of objects introduced  Remove or alter each assumption of the objects that have been introduced in the definition to see why they are necessary.  
Come up with examples  Come up with some examples of objects that fit the definition. Emphasize edge cases.  Examples help to train your intuition of what the object "looks like".  For monotone increasing functions, an edge case would be the constant function.  
Come up with counterexamples  
Writing out a wrong version of the definition  See this post by Tim Gowers (search "wrong versions" on the page).  
Understand the kind of definition  Generally a definition will do one of the following things: (1) it will construct a brand new type of object (e.g. definition of a function); (2) it will take an existing type of object and create a predicate to describe some subclass of that type of object (e.g. take the integers and create the predicate even); (3) it will define an operation on some class of objects (e.g. take integers and define the operation of addition).  
Check that it is welldefined  If the definition defines an operation  
Check consistency with existing definition  If the definition supersedes an older definition or it clobbers up a previously defined notation  Addition on reals after addition on rationals have been defined. For any function and , the inverse image is defined. On the other hand, if a function is a bijection, then is a function, so its forward image is defined given any . We must check that these two are the same set (or else have some way to disambiguate which one we mean). (This example is mentioned in both Tao's Analysis I and in Munkres's Topology.)  
Disambiguate similarseeming concepts  The idea is that sometimes, two different definitions "step on" the same intuitive concept that someone has.  (Example from Tao) "Disjoint" and "distinct" are both terms that apply to two sets. They even sound similar. Are they the same concept? Does one imply the other? It turns out, the answer is "no" to both: and are distinct but not disjoint, and and are disjoint but not distinct. Partition of a set vs partition of an interval.  
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:Learning from multiple sources.  
Drawing a picture  
Chunking/processing level by level  This is for definitions that involve multiple layers of quantifiers.  See Tao's definitions for close, eventually close, adherent, etc. 
See also
External links
 https://www.maa.org/node/121566 lists some other steps for both theorems and definitions
 https://en.wikipedia.org/wiki/Reverse_mathematics  this one is more important for understanding theorems. But the idea is to think, for each theorem, its place in the structure of the theory/relationship to other theorems. see for example https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers#Forms_of_completeness and https://en.wikipedia.org/wiki/Axiom_of_choice#Equivalents and https://en.wikipedia.org/wiki/Mathematical_induction#Equivalence_with_the_wellordering_principle John Stillwell (who also wrote Mathematics and Its History) has a book called Reverse Mathematics that might explain this at an accessible level.