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.||It's easy to become confused when you don't know the meanings of expressions used in a definition. So the idea is to avoid this kind of error.|||
|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.||Generally you want definitions to be "expansive" in the sense of applying to many different objects. But each assumption you introduce whittles down the number of objects the definition applies to. In other words, there is tension between (1) trying to have expansive definitions, and (2) adding in assumptions/restrictions in a definition. So you want to make sure each assumption pays its rent so that you don't make a definition narrower than it needs to be.||In the definition of convergence of a function at a point, Tao requires that must be adherent to . He then says that it is not worthwhile to define convergence when is not adherent to . (The idea is for the reader to make sure they understand why this assumption is good to have.)|
|Coming 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.|
|Coming up with counterexamples||As with coming up with examples, the idea is to train your intuition. But with counterexamples, you do it by making sure your conception of what the object "looks like" isn't too inclusive. |
|Writing out a wrong version of the definition||See this post by Tim Gowers (search "wrong versions" on the page).|
|Understanding 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).|
|Checking well-definedness||If the definition defines an operation||Checking that addition on the integers is well-defined.|
|Checking consistency with existing definition||If the definition supersedes an older definition or it clobbers up a previously defined notation|| Addition on the reals after addition on the rationals has 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.)
|Disambiguating 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.
In metric spaces, the difference between bounded and totally bounded. They are not the same concept in general, but one implies the other, so one should prove an implication and find a counterexample. However, in certain metric spaces (e.g. Euclidean spaces) the two concepts are identical, so one should prove the equivalence.
Sequantially compact vs covering compact: equivalent in metric spaces, but not true for more general topological spaces.
Cauchy sequence vs convergent sequence: equivalent in complete metric spaces, but not equivalent in general (although convergent implies Cauchy in general). However, even incomplete metric spaces can be completed, so the two ideas sort of end up blurring together.
|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.|
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.
|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.|
|Chunking/processing level by level||If a definition involves multiple layers of quantifiers.||See Tao's definitions for -close, eventually -close, -adherent, etc.|
|Asking some stock questions for a given field|| In computability theory, you should always be asking "Is this function total or partial?" or else you risk becoming confused.|
In linear algebra (when done in a coordinate-free way) one should always ask "is this vector space finite-dimensional?"
I think some other fields also have this kind of question that you should always be asking of objects.
Ways to speed things up
There are several ways to speed up/skip steps in the above table, so that one doesn't spend too much time on definitions.
One idea is to skip trying to really grok a definition at first, and see what bad things might happen. The idea is to then only come back to the definition when one needs details from it. This is similar to lazy evaluation in programming.
Building off similar definitions
If a similar definition has just been defined (and one has taken the time to understand it), a similar definition will not need as much time to understand (one only needs to focus on the differences between the two definitions). For instance, after one has understood set union, one can relatively quickly understand set intersection.
Relying on experience and intuition
Eventually, after one has studied a lot of mathematics, understanding definitions becomes more automatic. One can gain an intuition of which steps are important for a particular definition, or when to spend some time and when to move quickly. One naturally asks the important questions, and can let curiosity guide one's exploration.
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.
In definitions, it is traditional to use "if" to mean "if and only if". (Some authors use "iff" in definitions.)
- 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. See also Michael J. Schramm's Introduction to Real Analysis, which has the most complete "implication structure" between all the properties of the real line I've seen in a textbook. See also James Propp's paper "Real Analysis in Reverse".
- I think Tim Gowers's basic logic series of blog posts has some discussions about definitions