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Produced by Charles Wells     Revised 2013-06-05    Introduction to this website       website TOC      website index   blog	

DEFINITIONS

Summary

The definition of a concept in math has properties that are different from definitions in other subjects.These four rules are absolute requirements that all mathematical definitions obey:

Mnemonics:

  • LP: List of Properties.
  • EAP: Example has All Properties.
  • APE: All Properties means it's an Example.
  • COMP: Completeness

Math definitions have other properties as well. The list below describes aspects of definitions that people new to abstract math don't always understand:

A mathematical definition is fundamentally different from other sorts of defi­nitions, a fact that is not widely appre­ciated by mathe­maticians. The differences are discussed under definition, concept and dictionary definition in A Handbook of mathematical discourse.

Examples

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. 

 In this website, the word or phrase being defined is called the definiendum.  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:

Nonsense example

Here is a nonsense example. It uses words that (supposedly) have no meaning, to emphasize that when you see a mathematical definition the form of the definition gives information even though it may use words you don't know.

“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 torca.  (2) It must be wabic.  (3) It must be frumious.

What the definition tells you

Mathematical definitions are crisp:
something either is or is not an example, period

By definition: a magic formula

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

The definition is a totalitarian dictator.

Example: positive

This is a Baby example.

Definition

For any integer $n$:

We know $-(-3)=3$ and $3>0$, so by definition of “positive”, $-(-3)$ is positive.  This argument  depends on the fact that “$3$” and “$-(-3)$” are two different names for the same object.

Example: prime number

Definition

An integer $n$ is prime if $n>1$ and the only positive divisors of $n$ are $1$ and $n$.

Some who have just learned this may say, Then $1$ is a prime because then $n=1$ and the only positive divisors of $n$ are $1$ and $n$! But the definition is a dictator: it says $n$ must be greater than $1$, so $1$ is not a prime.

The paragraph above is a bit harsh, but it illustrates the point. Still, it is perfectly reasonable to ask, Why is $1$ excluded from being a prime? See the Wikipedia article on primes.

Example: square root

Definition

The symbol $\sqrt{2}$ denotes the unique positive real number whose square is 2.  

Everything that is true about $\sqrt{2}$ follows from this definition (COMP). That includes the fact that the decimal expansion of $\sqrt{2}$ begins $1.414\ldots$ and that may have been what you really need to know

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

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

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”. 

Very roughly, “connected” means it is not in two or more disjoint pieces and “open” means it does not include its boundary.  The area inside a circle, not including the boundary, is con­nected and open in the plane. You can find out more about these ideas in Wikipedia.

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!  

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.

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

Sometimes a constraint is put on the variable in the definition after the definition is stated, commonly in parentheses.  For example: n is even if it is divisible by 2 ($n\in \mathbb{Z}$).  This is called a postcondition. See also where.

Marking the definiendum

A statement in which one phrase is in italics or boldface
may be a definition of that phrase

Commandments

Sometimes the author commands you to define something, for example:  

 

This is not in fact telling you to do something, it is just telling you what it means for an integer to be even. 

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

More about “if”

The standard definition of even says:

Definition

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

 

You can then prove a theorem:

Theorem

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

 

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

But the theorem only justifies this one statement:

The theorem does not justify saying

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:

Definition

An integer is even if and only if it is divisible by 2.

 

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

Properties of mathematical definitions

Every proof originates solely in the definition

The definition of a math concept
is the fundamental fact about the concept
from which all other facts about it must be deduced

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

Here are some seemingly contradictory points about the purple prose above:

You generally have to use other theorems to prove a statement about the concept

The proof of any except the most elementary theorem about a concept will use other theorems about the concept. This does not contradict the idea that every true statement about the concept follows from the definition. In principle when the proof refers to a theorem you could replace the statement of the theorem by a proof of the theorem. If you do that over and over again the result will be a very long proof that really does assume only the definition. It is effectively impossible to do this by hand for the major deep theorems of math.

The definition is not the only source of understanding the concept

The definition must be taken literally.  

The notation and terminology used may suggest properties the definition does not actually require. For example, the standard definition of "subset of a set" allows the whole set to be a subset of itself, but the "sub" prefix in ordinary English may make you think a subset has to be a part of the set but not the whole thing. See semantic contamination.

The same concept can have very different-looking definitions

The definitions may have nothing at all in common with each other, and it may not be easy to prove they give the same concept. 

Two definitions are equivalent if anything that fits either definition fits the other one.

As I mentioned above, two equivalent definitions need not look at all alike.

Completeness

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

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

To fit a math definition,
a math object must have all the properties listed in the defining phrase.
Any object that has all those properties fits the definition.
In other words: DEFINITIONS WORK BOTH WAYS

No standardization

There is no Central Academy
of English Speaking Mathematicians
that determines the definition of a math word or phrase
and then enforces it.

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

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 this 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

 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.

Definition as just enough

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 just enough information to determine the concept, but not much more.

Definitions are not always absolutely as small as they can be. For example, the usual defi­nition of group given in under­graduate abstract algebra requires more than it needs to. See soojishin's expla­nation of one minimal defi­nition of group.

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

Definition is source of all truth

The definition determines everything about the concept
but it doesn't SAY everything about the concept.

Note that I am ignoring embarassing fine points such as degenerate triangles.

I need to clarify what determines everything means. One definition of triangle is that a triangle consists of three points connected by line segments. This definition more precisely determines every statement that is true about every triangle. For example, the angles at the corners of a triangle always add up to $2\pi$. It doesn't tell you that every triangle is isosceles.

The square root of two

Suppose you want to know the length $d$ of the diagonal of a square whose sides have length 1.  You apply the Pythagorean Theorem and conclude that $d=\sqrt{2}$. 

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: $\sqrt{r}$ is the unique positive real number s such that ${{s}^{2}}=r.$

 

So ${{d}^{2}}=2.$  Well big whoop.  You want to know how long the diagonal is.  That definition says nothing about length.  This is an example of the "just enough" nature of definitions.  The thing you are most interested in is approximately how long the diagonal is, and the definition of $\sqrt{r}$ says nothing about that. 

However, you can get an estimate of how big $\sqrt{2}$ is by using simple algebra facts, including the one that says: for positive $x$ and $y$, if ${{x}^{2}}<{{y}^{2}}$ then $x<y$. Now you start calculating:

By doing this over and over you can get many decimal places of $\sqrt{2}$. This shows that information about the magnitude of $\sqrt{2}$ is implied by the definition.

Definitions may be devious... 

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

Crisp and fuzzy

In many situations outside math, definitions are fuzzyFor 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 crispEither something fits the definition of a mathematical concept or it does not. 

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). 

 There is a sense in which a robin is a typical bird and a penguin is not a typical bird.   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                          \[\text{196},\text{883}\times \text{196},\text{883}\] The monster group deserves its name!

I can assert two things about this situation:

If you want to learn math,
listen carefully to what a mathematician says is rigorously true,
and listen carefully when they talk intuitively about some mathematical structure.
Both are vitally important to understanding math.

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:

The name “specification” is my own but many texts use what amounts to a specification for certain concepts without using the word “specification”. The meaning I use for specification is at least similar in spirit to the way computing scientists use the word.

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.


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