Produced by Charles Wells.  Home    Website TOC    Website Index 

Back to top of Understanding Math chapter

Posted 14 November 2011

DEFINITIONS

The definition of a concept in math has properties that are different from definitions in other subjects:

¨ The definition gives a list of properties of the concept. 

¨ Any example of the concept must fit all the requirements of the definition (not just most of them).  

¨ Every math object that fits all the requirements of the definition is an example of the concept.

¨ Every correct statement about the concept follows logically from its definition.

¨ Definitions are crisp, not fuzzy.

¨ The definition gives a small amount of structural information and properties that are enough to determine the concept. 

¨ Usually, much else is known about the concept besides what is in the definition.

¨ The info in the definition may not be the most important things to know about the concept. 

¨ The same concept can have very different-looking definitions.  It may be difficult to prove they give the same concept.

¨ Math texts use special wording to give definitions.  Newcomers may not understand that the intent of an assertion is that it is a definition. 

Basics

A mathematical definition prescribes the meaning of a word or phrase in a very specific way.  The word or phrase is defined in terms of a list of required properties, although the list may be disguised by the wording (example). 

 In this website, what is being defined is called the definiendum.  (This is to avoid repeatedly saying “the word or phrase being defined”).   The phrase that gives the definition is called the defining phrase.   (A special case is the defining formula of a function.)

The definiendum can refer to either of these:

¨  a type of math object

¨  a property that a math object can have.

Examples

Nonsense example

Here is a nonsense example (I give some real examples below):

“A quilgo is a torca that is wabic and frumious”.

The definiendum is “quilgo” and the defining phrase is: “is a torca that is wabic and frumious”.  The list of required properties of a quilgo are:  (1) It must be a quilgo.  (2) It must be wabic.  (3) It must be frumious.

What the definition tells you

¨  If you have a quilgo then it is a torca and  it is wabious and it is frumious:

¨  If you have a torca which is wabic and frumious then it is a quilgo:

¨  If you have a torca which is wabic but not frumious then it is not a quilgo.  It is not “almost a quilgo” or anything of the sort.  It is NOT a quilgo.

¨  If you have an object which is wabic and frumious but it is not a torca then it is NOT a quilgo:

¨  In a proof, you can use any of the facts in the definition by just saying “by definition”.

·   If T is a quilgo, then you can say “T is wabious by definition.” 

·   If T is a wabious, frumious torca then you can say “T is a quilgo by definition”.

·   If T is a torca that is not frumious then you can say “By definition of quilgo, T is not a quilgo.”

Baby example

For any integer n:

¨  n is positive if n > 0.

¨  n is negative if  n < 0.

¨  n is nonnegative if .

We know  and , so by definition of “positive”,  is positive.  This argument depends on the fact that “3” and “  ” are two different names for the same object.

Example: square root

The facts about an object given in the definition may not be the ones most important to you.  Example:

 Definition: the symbol  denotes the unique positive real number whose square is 2.  

Everything that is true about  follows from this definition, but it takes a bit of work to determine that the decimal expansion of  begins 1.414… and that may have been what you really need to know.  See definition as constitution.

Example: domain

This is the definition of “domain” in topology:  "A domain is a connected open set."  (See also here.)

The definiendum is "domain".  The list of properties: “is a set”, “connected” and “open”. 

The definition assumes that you are working inside a topological space, so that the requirement “is a set” really means “is a subset of the space we are talking about”. It is like many definitions in that you have to include the context of the definition into the requirements. 

You may not be familiar with words such as “connected” and “open”, but in this chapter I am writing about the form that a definition takes and what that form tells you about the meaning.  Here this means a subset of a space is a domain if it is connected and open, whatever  “connected” and “open” mean!  

How definitions are worded  

There are many different ways to word a definition, and this long section describes a great many of them.  You may think that only a grammarian or a dictionary editor would appreciate such infinite attention to detail, but I recommend that you glance through the possibilities listed.  You may discover

¨  Some wordings that you had not recognized as definitions (also discussed here), and

¨  Other wordings that misled you as to what was being required.

Direct definitions

You can define "domain" in point set topology directly by saying:

"A domain is a connected open subset of a topological space."

The definiendum is "domain" and the defining phrase is "is a connected open set". Similarly:

"An even integer is an integer that is divisible by 2."

In both these cases the definiendum is the subject of the sentence.

Definitions using conditionals

It is more common to word definitions using "if", in a conditional sentence.  (See more about “if”).  In this case the subject of the sentence is a noun phrase giving the type of object or property being defined and the definiendum is given in the conclusion of the conditional sentence. The conditional sentence, like any such, may be worded with hypothesis first or with conclusion first (more here). Part of the hypothesis may be stated first in a separate sentence, called the precondition of the definition.  (See more about preconditions here.)  All this is illustrated in the list of examples following, which is not exhaustive.

¨  A set is a domain if it is open and connected.

¨  If a set is open and connected, it is a domain.

¨   

¨  A set D is a domain if D is open and connected.  (Or “…if it is open and connected”.)

¨  The set D is a domain if D is open and connected.

¨  Let D be a set [this is a precondition]. Then D is a domain if it is open and connected.

¨  Let D be a set. Define D to be a domain if it is open and connected.  

The definition of “even” can be done in most of these ways as well:

¨  An integer is even if it is divisible by 2.

¨  An integer n is even if n is divisible by  2.

¨  The integer n is even if n is divisible by  2. 

¨  Let n be an integer. Then n is even if it is divisible by  2.

¨  Let n be an integer. Define n to be even if it is divisible by  2.

¨  If n is an integer, then it is even if it is divisible by  2.

Sometimes a constraint is put on the variable in the definition after the definition is stated (this is called a postcondition). (See also where.)  For example,

¨  n is even if it is divisible by 2 (  ). 

Marking the definiendum

¨  All the definitions above are given with the definiendum marked by being in boldface.  That is standard practice on this website.

¨  Italics is often used instead to mark the definiendum.  This is more common in books than using boldface.

¨  You may be able to tell that a statement is a definition only because a word or phrase is in italics or boldface.

¨  Some authors do not mark the definiendum at all, but include it in a paragraph marked “definition”, for example:  “Definition: An integer is even if it is divisible by 2.” 

¨  Sometimes the author commands you to define something, as in   “Define an integer to be even if it is divisible by 2” or “Call an integer even if…” or “Say an integer is even if…”  This is not  telling you to do something, it is just telling you what it means for an integer is even.  Call and say are discussed at more length under their entries.

¨  Another way of marking the definiendum is to use a phrase such as “said to be”:  “An integer is said to be even if it is divisible by 2.”  

¨  Sometimes no indication at all is given that the statement is a definition.   This is an evil thing to do, but it does happen. 

Defining symbolic expressions

Symbolic expressions may be defined using the same terminology and styles as in definitions of words and phrases. 

When defining a word or phrase the scope of the definition is usually the entire document (the definition will stay in effect to the end).  Occasionally the author will say something like, “Just for the rest of this proof, say that a number is frumious if…” 

However, symbolic expressions are commonly defined for quite narrow scopes, a paragraph or a section.  Besides the ways I have already mentioned there are many other ways to say it the case of narrow scope:

Examples

¨  Let .  What is the derivative of f ?

¨  Put .  What is the derivative of f ?

¨  Say [suppose, assume] .  What is the derivative of f ?

¨  Define  to be . What is the derivative of f ?  (As I said above, this is not a command.)

¨   .   This symbol “:=” is called colon equals and originated in computer science.  Some mathematicians now routinely use it at the blackboard.

More about “if”

The standard definition of even says:

You can then prove a theorem:

Because of the definition, it is correct to say both of these things:

¨  If an integer is divisible by 2 then it is even.

¨  If an integer is even then it is divisible by 2.

But the theorem only justifies this one statement:

¨  If an integer is divisible by 4 then it is even.

It does not justify saying

¨  If an integer is even then it is divisible by 4.

In fact that statement is false.  (Consider 6.) 

The word “if”

 goes both ways inside a definition

  goes only one way inside a theorem

Because of this, some authors have begun using "if and only if" in definitions instead of "if", as in:

 More about this in the entry for if .  See also context-sensitive interpretation.

Properties of mathematical definitions

Mathematical definitions and concepts

Here are some seemingly contradictory points about this bit of purple prose:

¨  The special logical status of a definition (everything follows from it) is the reason that rewriting according to the definition is the reasonable first step in coming up with a proof.

¨  The definition of a concept is nevertheless not the only source of understanding the concept.   The info that is in the definition may not  include the most important aspects of the concept.  This point is amplified below.

¨  Facts about a concept that have been proved as theorems contribute greatly to understanding it, and can be used in proving things about it too.

¨   Images and metaphors associated with the concept, and the motivation behind the concept, contribute greatly to understanding the concept, but they cannot (directly) be used in proofs.

¨  The definition must be taken literally.   The notation and terminology used may suggest properties the definition does not actually require (semantic contamination).  Example.

¨  The same concept can have very different-looking definitions.  It may not be easy to prove they give the same concept.  Example:  You could define .  

A rilly rilly basic example of this is given in equivalence relations and partitions.  (See also the remark on two ways to define symmetric relation.)

Completeness

Mathematical definitions are complete, in the sense that a definition of quilgo, for example, lists some properties (the defining properties), and

¨  Every quilgo has those properties

¨  Every mathematical object that has those properties is a quilgo.

¨  These facts, particularly the second one, are frequently overlooked by people new to abstract math, and so is worth making purple:

No standardization

¨  There are certainly some words and phrases whose meaning is the same in almost any math text. 

¨  There are some very basic words with two common inequivalent definitions.  Examples: 

·    The natural numbers may or may not contain zero, and both these definitions occur commonly. 

·    A ring may or may not be required to have a multiplicative identity.

¨  There are many, many words and phrases that have equivalent definitions in most texts, but for which some texts give other definitions.  

·   Positive means greater than zero in almost all texts, except for certain European educational systems (perhaps only France), where it means “greater than or equal to zero”.

It is … quite hard to come up with good technical choices for formal definitions that will be valid in the variety of ways that mathematicians want to use them and that will anticipate future extensions of mathematics. If we were to continue to cooperate, much of our time would be spent with international standards commissions to establish uniform definitions and resolve huge controversies. William Thurston

¨  Certain words and phrases have a standard meaning in one branch of mathematics and a different meaning in another branch.

·   Field  means completely different things in abstract algebra and in mathematical physics.

·   People in fluid mechanics use continuum hypothesis with a completely different meaning from its use in set theory.

¨  Some math objects have two different standard symbols in different subjects (for example, i and j.)

¨  Some symbols such as  and log have just one meaning in high school but are used with many different meanings in post calculus math. 

Standardization is hopeless

When we gain a new understanding of a type of math object, we often realize that the names we have chosen don’t work well and need to change them.  Because of this common phenomenon, there are authors who deliberately set out to reform the terminology in a subject and redefine many of the terms in the subject or substitute others.  (Sometimes they do for other, mostly bad, reasons). Such attempts rarely work.  Bourbaki made the biggest effort of this sort and partly succeeded (but they failed with positive). 

Images and metaphors for definitions

Definition as presentation

In order to make it easy to show that some object is an example of the concept, the definition is minimal (or nearly so).  It includes (almost) as little information as possible that will still completely determine the concept.  (It is like a presentation of a group (MW, Wi), if you are familiar with that concept.)  Because of this, a mathematical definition hides the richness and complexity of the concept and as such may not be of much use if you want to understand it. Also,  if you are not used to the minimal nature of a mathematical definition you may gain an exaggerated idea of the importance of the items that the definition does include, particularly in the case of the many devious definitions in math.   

Examples

¨  A group is defined as a set with a binary operation satisfying certain properties.  But groups are important because their elements are symmetries.  The definition of group says nothing about the elements being symmetries, although it follows that the elements of any group are symmetries of in general several different structures.

¨  The real line have many different equivalent definitions, but most of the definitions start with the points on the line (equivalently, the real numbers).  This is confusing to the beginner because the points of the real line are hard to deal with both theoretically and conceptually.  It would be better to start by thinking about the real line as a line segment that has been  unboundedly in both directions.

¨  See also equivalence relations and partitions and literalist.

Definition as constitution

The definition of a concept is like the American Constitution.  It is the framework that justifies the operation of the federal government.  But reading it doesn’t contribute much to understanding the federal government.  (It contributes some, of course.)   A lot of the subtle interplay between the branches of the federal government, and between the federal government and the state governments, developed out of the constitution but is not visible in the constitution. 

Examples

Here is a baby example:

Suppose you want to know the length d of the diagonal of a square whose sides have length 1.  You apply Pythagorean Theorem and conclude that . 

Now at this point I will make the (unrealistic) assumption that you know the basics of algebra but nothing at all about square roots and you don’t have a calculator.  You look up the definition of the radical sign:

Definition:  is the unique positive real number s such that  

So   Well big whoop.  You want to know how long the diagonal is.  That definition says nothing about length.   This is an example of the minimal nature of definitions.  The thing you are most interested in is approximately how long the diagonal is, and the definition of  says nothing about that. 

However, you can get an estimate of how big  is by using simple algebra facts, including the one that says: for positive x and y, if  then .

  and , so   

Hmm, , so . 

Well,  so . 

Now you are getting closer to knowing how big   is.

Actually, as you probably know, Newton came up with a Method that allows you to calculate it even faster.   And some folks, mostly old, know a way of doing it by hand that looks like a bizarre form of long division.  You can prove using the definition of  that both these methods work properly, but these methods are not at all obvious looking at the definition of square root! 

This example is kind of clunky and artificial, but it does illustrate the point: 

Definitions may be

  

Some apparently simple math concepts have really off-the-wall definitions.

¨  The most widely accepted definition of sets uses the Zermelo-Fraenkel axioms (MW, Wi). There are nine of them!   (Ten if you include the Axiom of Choice (MW, Wi)).  This complexity came about because of Russell’s Paradox.

¨  The natural numbers are defined by the Peano Axioms (MW, Wi).   They are not as complicated as the Zermelo-Fraenkel axioms but they are based on proof by induction and using them even the simplest facts about the natural numbers must be proved by induction.

¨  The real numbers may be defined by Dedekind cuts, which are difficult to understand and cumbersome to deal with.  There are several other ways to define the reals, also complicated.   

¨  Both functions and relations are defined in terms of sets of ordered pairs, which is not difficult or complicated but is hardly what you have in mind when you are thinking about functions and relations!  This modest deviousness is in both cases a ploy for allowing the functions or relations to be completely arbitrary.

Crisp and fuzzy

In many situations outside math, definitions are fuzzy.  For example, “warm weather” is a fuzzy concept.  Perhaps everyone will agree that if the temperature is 90 degrees F. then we have warm weather, and if it is 55 degrees F. we do not have warm weather.  But 70 degrees is sort of borderline.  Some will say it is warm and some will not. 

Mathematical concepts are crisp.  Either something fits the definition of a mathematical concept or it does not. 

Typical?

 There is a sense in which a robin is a typical bird and a penguin is not a typical bird.  (More).  A mathematical definition is simply a list of properties.  If an object has all the properties, it is an example of the definition.  If it doesn’t, it is not an example.  So in some basic sense no example is any more typical than any other.  This is in the sense of rigorous thinking described in the chapter on  images and metaphors. 

In fact mathematicians are often strongly opinionated about examples:  Some are typical, some are trivial, some are monstrous, some are surprising.  A dihedral group is a typical finite group.  The real numbers on addition are not typical; it has very special properties.  The monster group is the largest finite simple group; hardly typical.    Its smallest faithful representation as complex matrices involves  matrices that are of size .  The monster group deserves its name.

Nevertheless, the monster group is no more or less a group than the trivial group.

Specifications

Because the definition of a math concept can be devious, it may be hard to see how you can use it in a proof.  A specification of a mathematical concept is a set of statements that are all true of the concept and that suffice for many common uses, but which do not characterize the concept.   These are the main points about specifications:

¨  Everything the specification says about the concept is true of the concept and may be used in proofs.  

¨  You can’t use the specification to prove that something is an example of the concept.

The name “specification” is my own but many texts use what amounts to a specification for certain concepts without using the word “specification”. 

Examples

I give a specification for sets in the chapter on sets and a specification for functions in the chapter on functions. The list of properties of real numbers given in the chapter on real numbers amounts to a specification.

See literalism.  

Acknowledgments

Thanks to Dr. Hugh Porteous for corrections and suggestions.

Notes

More about “typical”.

The concept of “typical” refers to a genuine cognitive phenomenon that many literal-minded types sneer at.  

Vyvyan Evans and Melanie Green, Cognitive Linguistics: An Introduction.  Routledge, 2006, pages 273ff.

G. Lakoff, Women, Fire and Dangerous Things. University of Chicago Press, 1990.  (Look up radial concepts).