What is a number?

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In another thread, I was shot down for saying that a number was a representation of a value. I was told that the number was the value.

This is a point of confusion for me and I cannot marry this up with definitions of the term 'number' that I can find in learned texts.

Is the term itself just abit vague anyway, and can be used willy-nilly, or is there a concise, comprehensive and exclusive definition of the term?

My copy of Chambers Dictionary of Science and Technology says;

Number (Maths.). An attribute of objects or labels obtained according to a law or rule of counting.

which has been regarded here on Physics Forums as very imprecise.


McGraw-Hill says;

number [mathematics] Any real or complex number.

which is amusingly circular, but rather useless in this context.

General dictionaries often refer to 'number' as integer values only too, which is consistent with the term 'number theory' ("The study of integers and relations between them").

Here is a typical dictionary entry, though this one also extends the definition to other 'mathematical objects';

num·ber (nmbr)
n.
1. Mathematics
a. A member of the set of positive integers; one of a series of symbols of unique meaning in a fixed order that can be derived by counting.
b. A member of any of the further sets of mathematical objects, such as negative integers and real numbers.
2. numbers Arithmetic.
3.
a. A symbol or word used to represent a number.
b. A numeral or a series of numerals used for reference or identification: his telephone number; the apartment number.

Wikipedia possibly muddies the waters further with;

A number is a mathematical object used to count and measure. A notational symbol that represents a number is called a numeral but in common use, the word number can mean the abstract object, the symbol, or the word for the number.

which sounds to me like a number is neither the measure nor the representation but is some notional connection between the two.

Can we arrive at a 'Physics Forum' definition of 'number' that is a concise, comprehensive and exclusive definition of the term (whether or not it actually agrees with external reference texts) or is it just a bit nebulous and cannot be defined, which would be odd for the single most important concept in a subject as precise as mathematics!?
 

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  • #2
pwsnafu
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My copy of Chambers Dictionary of Science and Technology says;

This is a mathematics forum. If you want to cite dictionary definitions of terms, I would have have thought you would use a mathematics dictionary.

Can we arrive at a 'Physics Forum' definition of 'number' that is a concise, comprehensive and exclusive definition of the term

No. Don't believe me? At the very least, any definition of "number" must be general enough to include the integers and the complex numbers, so...
  • the integers are a principle ideal domain. What is special about the integers that distinguishes it from other PIDs?
  • the complex numbers are an algebraic closed field. What special property distinguishes it from other algebraic closed fields?
  • the real numbers are a vector space over rational numbers. What is special about the reals that is not shared by other rational vectors?
And remember, we need one property that is shared between integers, reals and complex numbers, and still specific be enough that it is a defining trait (we do want a definition no?).

the single most important concept in a subject as precise as mathematics!?

The most important concept in mathematics would be toss up between "set" or "axiomatic system". Probably both actually.
 
  • #3
Chronos
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A number is an independent representation of quantity. Our ancient ancestors dreamed up this idea to deal with bartering issues. It is crude, but, effective.
 
  • #4
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A number is an independent representation of quantity.

That's what I said, but the implicaton is that you could have different representations of the same quantity, thus [by that definition] no exclusion to having different numbers for the same quantity. This definition doesn't imply a one-one correspondence between number and quantity. But this observation was shot down here.
 
  • #5
D H
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Can we arrive at a 'Physics Forum' definition of 'number' that is a concise, comprehensive and exclusive definition of the term (whether or not it actually agrees with external reference texts) or is it just a bit nebulous and cannot be defined, which would be odd for the single most important concept in a subject as precise as mathematics!?
Short answer: No, we can't. There is not and cannot be a "concise, comprehensive and exclusive" definition of "number". Is zero a number? In the counting numbers, no, it isn't. In the naturals, yes, it is. How about -1, 1/3,√2, π, √-1 ? These quantities make perfect sense in some number systems, but absolutely no sense in others.
 
  • #6
AlephZero
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Mathematicans aren't too bothered about what a number (or any other mathematical concept) "is". The interesting thing is what it "does", in other words what mathematical operations you can do with it.

The question "what, if anything, is the connection between the mathmatical number 2 and the common-sense idea of two apples" is not part of math. Once mathematicians have defined how THEY want "numbers" to behave, whether or not you can use them for counting apples, or doing quantum mechanics, is somebody else's problem.
 
  • #7
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. Once mathematicians have defined how THEY want "numbers" to behave...,

pardon me, Aleph: Science describes reality, how reality behaves, or tells the world how to bevave?
 
  • #8
D H
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pardon me, Aleph: Science describes reality, how reality behaves, or tells the world how to bevave?
The topic of this thread is mathematics, and in particular, numbers. While scientists use mathematics to describe reality, mathematics is not constrained by reality. Mathematics is not science.
 
  • #9
disregardthat
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What DH says. A number is a term used differently in different situations, and whether something is called a number or not depends on whether it catches on. Complex numbers are called numbers, but one is not automatically inclined to call any kind of extension of the real numbers for numbers. Infinitesimals might be called numbers, infinite cardinals might be called numbers. It isn't a matter of falling under the definition of a number.
 
  • #10
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So if I were to state;

"1/2 and 2/4 are the same number because they are the same quantity"

or

"1/2 and 2/4 are different numbers because they are different representations of a quantity"

then either statement is OK (neither right nor wrong) in Physics Forums, providing I am consistent to a definition of number I clarify the statement with?
 
  • #11
disregardthat
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How can two numbers be different if they are equal?
 
  • #12
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This is a mathematics forum. If you want to cite dictionary definitions of terms, I would have have thought you would use a mathematics dictionary.

I'd love to, but have never heard of a dictionary being devoted to mathematics alone, let alone own one for myself.

Do you have one, and could you post what it says, please?
 
  • #13
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How can two numbers be different if they are equal?

This is the point of my thread question.

If you define a number as "a representation of..." then two different representations are two different numbers.
 
  • #14
disregardthat
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This is the point of my thread question.

If you define a number as "a representation of..." then two different representations are two different numbers.

Would anyone accept a definition of a number which leads you to the conclusion that "the rational numbers 1/2 and 2/4 are two different numbers"? The question answer itself.
 
  • #15
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Would anyone accept a definition of a number which leads you to the conclusion that "the rational numbers 1/2 and 2/4 are two different numbers"? The question answer itself.

Yes, I consider them different numbers, especially if given the caveat "numbers are representations of..", which makes your answer ambiguous because I think you are implying no-one would.
 
  • #16
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A number is an independent representation of quantity..
How can two numbers be different if they are equal?
Would anyone accept a definition of a number which leads you to the conclusion that "the rational numbers 1/2 and 2/4 are two different numbers"?

it is necessary to remember that a word ["number" ',°] is a linguistic sign: a signifier : a 'sign and a signified: a °meaning, which the authority of Chronos tells us is: a quantity.
The most authoritative English dictionaries [Oxford: SOED and OALD] confirm that. OALD says : "a word [five] or a symbol [5] that represents an amount or a quantity".
Misunderstanding occurs when we forget this distinction: many symbols represent same quantity. They are synonyms, equivalences : 5 , 10/2, √25, 8-3, etc are different 'numbers'= 'symbols' for same °number= °amount, °quantity
I hope we can agree that Chronos' definition is not negotiable. Maths may elaborate on that 'independently' but only formally.

P.S. but another authority of PF [micromass] says [in thread 537605#11]: "what is a number anyway??.I have rarely seen a definition of a number in mathematics, and I doubt that such definition exists.
....definiton of "number" must include complex numbers."
 
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  • #17
disregardthat
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logics, you are confusing 'number' with 'numeral', or 'symbol'.
 
  • #18
D H
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So if I were to state;

"1/2 and 2/4 are the same number because they are the same quantity"

or

"1/2 and 2/4 are different numbers because they are different representations of a quantity"

then either statement is OK (neither right nor wrong) in Physics Forums, providing I am consistent to a definition of number I clarify the statement with?
Nonsense. You cannot come up with any meaningful, acceptable definition of number that allows the second.

Mathematicians already have a concept of what constitutes the "same number": Two numbers are in fact the same number if they are equal to one another. Equality is a central concept in any number system.
 
  • #19
lurflurf
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Well some equal numbers are more equal than others. When unclear by context it is best to state the specific equality being used. In mathematics there is value in considering things that are basically the same to be different, whilst simultaneously considering things that are basically different to be the same.
 
  • #20
mathwonk
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Is a "sage" a word or a person of understanding? Is it a representation of a concept, or a concept, or an example of that concept? or an herb for seasoning?

this discussion is bogus. mathematicians seldom use the word "number" in any precise discussion. there are many very precise types of objects called numbers of various sorts in mathematics, such as natural numbers, rational numbers, real numbers, complex numbers.

There are occasions when mathematicians say "number" when they believe the listener knows which type of numbers are being referenced. Ordinary dictionaries, on the other hand, attempt to list all uses which anyone anywhere might make of a word, without regard to mathematical precision.

If you ask a mathematician what a number is, he will possibly try to state what all those more precise examples have in common. I for example would suggest they are objects designed for calculation. I.e. they are susceptible to some sort of useful operation combining them such as addition or multiplication.

A mathematician seldom if ever refers to the symbol or representation, when he uses the word number, rather he means the abstract concept the symbol represents. Thus to him 1/2 and 2/4 are the same rational number. If he wants to refer to the pair of integers appearing in this symbol, he may call the object a "quotient", or an
indicated quotient", referring to the two integers being divided rather than the result of that division.

On the other hand mathematicians are human and subject to inconsistency and some may sometimes say rational number when they mean pair of integers representing a rational number. Communication is difficult even for scientists.

But the word "number" is not ordinarily in use by itself as a precise term in mathematics as far as I know.

A calculus teacher may mention numbers, thinking that the class is only thinking of one kind of numbers, real numbers. This is actually hazardous, since some students only know positive integers, and they think rules like (cf)' = c.f' only apply to integers c.
 
  • #21
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There is nothing bogus about asking what a number is. It's a valid question and something that is hidden from most people that use math. Thehttp://www-math.mit.edu/~katrin/100/notes/natural.pdf" is from set theory. You have to have axioms and the inductive property and you can build the natural numbers. From the natural numbers you can build the integers. From the integers you can build the rationals. From the rationals you can build the irrationals. And from the irrationals you can build the "real" numbers. From the real numbers the complex numbers. From the complex numbers the hypercomplex.

Each set is built from the definitions (or axioms) and operations (induction, limits, addition, subtraction). Now, there are some surprising results even for the natural numbers. Mathematics is exactly like any other language like English. There are just more rules applied to try and make the words consistent. Strangely enough we can't even guarantee that for the natural numbers.
 
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  • #22
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Mathematicans aren't too bothered about what a number (or any other mathematical concept) "is". The interesting thing is what it "does", in other words what mathematical operations you can do with it.

The question "what, if anything, is the connection between the mathmatical number 2 and the common-sense idea of two apples" is not part of math. Once mathematicians have defined how THEY want "numbers" to behave, whether or not you can use them for counting apples, or doing quantum mechanics, is somebody else's problem.

I agree with Aleph. Mathematician's have no interest in what a number "is", what it "is", is a philosophical question, impossible to answer. What is a set? Its the same question, we say a set is a collection, but still philosophers keep asking the question.
 
  • #23
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There is nothing bogus about asking what a number is. It's a valid question and something that is hidden from most people that use math. Thehttp://www-math.mit.edu/~katrin/100/notes/natural.pdf" is from set theory. You have to have axioms and the inductive property and you can build the natural numbers. From the natural numbers you can build the integers. From the integers you can build the rationals. From the rationals you can build the irrationals. And from the irrationals you can build the "real" numbers. From the real numbers the complex numbers. From the complex numbers the hypercomplex.

Each set is built from the definitions (or axioms) and operations (induction, limits, addition, subtraction). Now, there are some surprising results even for the natural numbers. Mathematics is exactly like any other language like English. There are just more rules applied to try and make the words consistent. Strangely enough we can't even guarantee that for the natural numbers.

Thetes it may be important to note that our axioms are formed via our logical system. Most of our mathematics is constructeed via the law of the exluded middle. I would postulate that our number sense is impossible to seperate from our logical formulations so we must accept that numbers as quite elusive. It is like asking why can't something be true, false, and thirty seven other "degrees". I would assume this has to do with our conception of equality, as previously mentioned.
 
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  • #24
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It is mandatory for any scientific system/theory to "clearly define its terms". If we do not do that problems arise
 
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  • #25
HallsofIvy
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No, it is not. In fact, in mathematics is is essential that there exist "undefined terms"- that is, terms that we can treat as "containers" into which we can put whatever meaning we like, as long as the relationships between the terms, given by "axioms" or "postulates" are still true. That is precisely why mathematics is so general.
 

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