Understanding mathematical definitions
Understanding mathematical definitions refers to the process of understanding the meaning of definitions in mathematics.
List of steps
Understanding a definition in mathematics is a complicated and laborious process. The following table summarizes some of the things one might do when trying to understand a new definition.
|Type-checking and parsing||Parse each expression in the definition and understand its type.|||
|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 well-defined||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 similar-seeming 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.|
When reading textbooks
Most textbooks will assume the audience is a competent mathematician, so won't bother to explain what you should be doing at each definition.
- 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_well-ordering_principle John Stillwell (who also wrote Mathematics and Its History) has a book called Reverse Mathematics that might explain this at an accessible level.