Produced by Charles Wells Revised 2015-12-06 Introduction to this website website TOC website index blog Back to Understanding Math head
The definition of a concept in math has properties that are different from definitions in other subjects. The four rules below are absolute requirements that all mathematical definitions obey:
A mathematical definition is fundamentally different from other sorts of definitions, a fact that is not widely appreciated by mathematicians. Essentially, a math object is defined by accumulation of properties. This is different from definitions in other fields, even in other sciences. See the listings under definition, concept and dictionary definition in A Handbook of mathematical discourse.
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 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:
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.
Mathematical definitions are crisp: |
In a proof, you can use any of the facts in the definition by just saying "by definition".
A definition is a totalitarian dictator. |
For any integer $n$:
We know $-(-3)=3$ and $3>0$, so by definition of "positive", $-(-3)$ is positive. The fact that $-(-3)$ has a minus sign in front of it is irrelevant. This argument depends on the fact that "$3$" and "$-(-3)$" are two different names for the same object.
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 a definition is a dictator: this definition says $n$ must be greater than $1$, so it follows from EAP that $1$ 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.
The definition of a math object is defined by mathematicians and they can define it in any way they want, so naturally they make a definition that is useful, and they can change it any time they want. (They did, too; for a while in the nineteenth century $1$ was a prime.)
The definition of a word
is determined by the way it is used,
not by a higher power.
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 wanted to know.
The facts about an object given in the definition |
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
Difficulties like these are also discussed in Intent of Assertions.
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". This is an example of how you can be misled as to what is being required. It could have been worded this way: "A domain is a connected open subset of a topological space". But definitions are very commonly not that explicit. This example 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!
It is 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.
A statement in
which one phrase |
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.
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:
The standard definition of even says:
Definition: If an integer is divisible by $2$, then it is even.
You can then use the definition to prove a theorem:
Theorem: If an integer is divisible by $4$, then it is even.
Proof: If $n$ is divisible by $4$. then by the definition of "divides", there is an integer $k$ for which $n=4k$. But $4k=2(2k)$, and $2k$ is an integer, so $n$ is $2$ times an integer. So by definition of "divides", $n$ 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" |
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.
The definition
of a math concept |
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:
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 reduce a proof in the way just described by hand for the major deep theorems of math.
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 definitions may have nothing at all in common with each other, and it may not be easy to prove they give the same concept.
You can define $\sqrt{2}$ as the unique positive real number $r$ for which $r^2=2$, or by saying $\sqrt{2}=\frac{1}{\sin (\pi/4)}$. The second definition is totally lame, but it is in fact a correct definition (once you have defined "sine") and could be used to estimate the decimal places of $\sqrt{2}$ by drawing the appropriate right triangle and measuring.
A much more important example of two different-looking definitions is given in equivalence relations and partitions in Wikipedia. The usual definition of equivalence relation (a reflexive, symmetric and transitive relation) and a partition (a set of subsets of a set such that every element of a set is contained in exactly one of them) determine exactly the same structure on the set, even though the definitions look utterly different.
The point of this equivalence of definitions is that an equivalence relation determined a unique partition, and a partition determines a unique equivalence relation, and (take a deep breath) a partition determines an equivalence that always determines the partition you started with, and an equivalence relation determines a partition that always determines the equivalence relation you started with. It is worth working out a couple of small finite examples to understand this!
There is no Central Academy |
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).
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
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.
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.
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.
Note that I am ignoring fine points such as degenerate triangles.
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 $\sqrt{2}$ 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 ${{x}^{2}}\lt {{y}^{2}}$ then $x\lt y$. 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.
Some apparently simple math concepts have really off-the-wall definitions.
This modest deviousness is in both cases a ploy for allowing functions or relations to be completely arbitrary. For example, "$=$" and "$\lt$" and "divides" (as in "$4$ divides $96$") are all relations, but so is the completely arbitrary relation that says "$2$ is related to $3$ and $3$ is related to both $2$ and $5$ and that's all". This is definitely a relation, although useless, and is represented as the set $\{(2,3),(3,2),(3,5)\}$.
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 30 degrees C. then we have warm weather, and if it is 10 degrees C. we do not have warm weather. But 20 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.
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.
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).
If you want to learn math, |
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.
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.
Thanks to Dr. Hugh Porteous for corrections and suggestions.
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