What is the true definition of a number?

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In summary, the concept of a number is not easily defined and can vary depending on the context in which it is used. Different reference texts and sources may have differing definitions and there is no one concise, comprehensive, and exclusive definition of the term. Mathematicians are more concerned with how numbers behave rather than what they are, and the concept of a number can be extended to include quantities that may not be traditionally considered numbers. Ultimately, the definition of a number is not a fixed concept and can be subjective.
  • #1
cmb
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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
cmb said:
I
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
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
Chronos said:
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
cmb said:
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
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
AlephZero said:
. 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
logics said:
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
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
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
How can two numbers be different if they are equal?
 
  • #12
pwsnafu said:
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
disregardthat said:
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
cmb said:
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
disregardthat said:
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
Chronos said:
A number is an independent representation of quantity..
disregardthat said:
How can two numbers be different if they are equal?
disregardthat said:
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
logics, you are confusing 'number' with 'numeral', or 'symbol'.
 
  • #18
cmb said:
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
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
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
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
AlephZero said:
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
Thetes said:
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 separate 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
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
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.
 
  • #26
logics said:
It is mandatory for any scientific system/theory to "clearly define its terms". If we do not do that problems arise
First off, mathematics is not science. Science is constrained in that it must agree with reality. Mathematics isn't constrained in that way. The hypothesis that space and time are continuous is, at least conceptually, a falsifiable hypothesis. Suppose that some future scientific experiment shows this conjecture to be false at some very tiny scale. Will this invalidate the mathematics of the real numbers? No. It won't. It will merely mean that our use of the reals to describe the universe is not quite correct.

Secondly, mathematicians do describe their terms, very precisely -- in the form of undefined terms and axioms. The natural numbers are described by a certain collection of axioms, the integers by another collection of axioms, and the reals by yet another collection of axioms, and so on.

Lay people and scientists typically use the term "number" to mean the a member of the reals. Unless they mean something else. Mathematicians are precise and use terms such as integer, rational number, real number, complex number, p-adic number, quaternion, etc. There is no one concept in mathematics that definitively constitutes "number".
 
  • #27
Lay people and scientists typically use the term "number" to mean the a member of the reals. Unless they mean something else.

Wonderful!
 
  • #28
HallsofIvy said:
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.

I was told in another thread that I was confused over the meaning of 'number'. This reply means that I still am!

So my question was 'is there a PF-accepted meaning for the term "numbers" ?'.

So I remain confused: Is it;
A) OK for me to use the term 'number' in a way that *I* go on to qualify (namely, that numbers are representations of value, and there may be different representations of the same value)

or,

B) it is not OK for me to qualify 'number' with the qualification in (A), which therefore implies there are some non-negotiable hard-and-fast attributes of numbers - which I'd like to read some reference material on if it is so

?

This is 'not a contest' I'm pushing with anyone, of whether my proposition is correct or not, I'm just genuinely interested to know if the field of mathematics has already gone over this, and whether material already exists to discuss this, that I can build up my comprehension of.


I think the link Thetes provides, in post #21 is very useful to this end. That was ultimately the type of thing I was expecting to be directed to, and for the thread to go on to discuss.

It is particularly interesting because it completely dissociates the concept of 'number' *away* from both 'representations' AND 'quantities'. I find that interesting and curious, because if the only way to define numbers is in some other way divorced from the original purpose, then I think we are beginning to dig away at some important philosophical aspects of the notion of '[mathematical] concepts' itself.

I'm sure that link is not the only way of defining numbers, and it would be good to see the extensions of that type of mathematical construction to reals and irrationals, so I'd be very grateful if folks have other or further such axiomatic constructions they can post or link to. Maybe we can get on to discuss some of the 'surprising results' Thetes is hinting at?
 
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  • #29
Yes, the link that thetis gives is very good but if you look closely you will see that there us no definition of "number" there. It begins by saying "There exists a set N, whose elements are called natural numbers" so "natural numbers", at least, are whatever is in that set. But there are many different sets that would qualify as satisfying the "axioms" given there.
 
  • #30
cmb said:
I was told in another thread that I was confused over the meaning of 'number'. This reply means that I still am!

So my question was 'is there a PF-accepted meaning for the term "numbers" ?'.
No. There. Is. Not.


So I remain confused: Is it;
A) OK for me to use the term 'number' in a way that *I* go on to qualify (namely, that numbers are representations of value, and there may be different representations of the same value)

or,

B) it is not OK for me to qualify 'number' with the qualification in (A), which therefore implies there are some non-negotiable hard-and-fast attributes of numbers - which I'd like to read some reference material on if it is so

?
I strongly suggest that you don't go reinventing a wheel that took multiple generations of incredibly smart people to develop. The only way to make progress in math and science is to stand on the shoulders of those who have preceded us. Instead, read up on modern algebra. That's the algebra you take after learning calculus, not before.

What you are doing is asking us at PF to write a book, many books in fact. There are books and books on this topic. Asking us to write a book is not a fair question for a discussion forum such as this. You might be able to find some of that information on the internet, but only piecemeal. To find the information as a whole you need to read a book, take a class, or both.
 
  • #31
come on guys, this is embarrassing
 
  • #32
Numbers are just ways to capture variation. That is one of the most important central ideas in mathematics: to explain and analyze variation in many different useful ways.

Each different type of number has different properties for variation. Your complex numbers introduce more variation that your real numbers, and your real numbers introduce more variation than your integers.

What that variation corresponds to is another matter. It might be physical, it might not be. We don't care about that, we only care about how the variation can be described, analyzed, and how we can extrapolate useful properties from these things.

The thing that makes mathematics powerful is that we have a lot of results that apply to situations with a great amount of variation.

It is not useful for mathematicians to prove every individual scenario individually. There are potentially infinite numbers of these, even when you constrain the classes of things you wish to prove. The point is to prove properties of something that has a large amount of variation, and the higher the amount of variation, the more powerful the result tends to be.

If we did not focus on variation, then we would be proving every situation individually. A computer can do this, but the practical effect of doing this is, in many situations (not all though, since the state space for some problems might actually be manageable with a computer) is not to be considered.
 
  • #33
cmb said:
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?
Dictionary of Mathematics, by C.T. Baker, published by Hart Publishing Co, Inc. The price on the cover is $2.95. I think I bought it sometime in the late 60s.

It has a definition for Numbers, Cardinal and Ordinal, but doesn't bother to define Number.
 
  • #34
The closer you look at the foundation the more wobbly it seems. Set theory is how number systems are defined. But, this is like a shell game. You ask what is a number, so I tell you well it's made of smaller objects. Then you ask what are the smaller objects. So, I quit hiding behind numbers and systems and tell it to you straight, we don't know what they are but they are useful. There has to be a starting point to the definitions. Those are our axioms and postulates which are just assumptions.

This might seem a sad truth that all mathematicians have to face at some point, but it's the best we can do. Unfortunately the problems only increase from there. In the early years of set theory there were a number of http://www.cs.amherst.edu/~djv/pd/help/Russell.html" [Broken] the argument.

So mathematics is a language taken by faith to be correct. Don't worry lies in math are harder to spot than English.
 
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1. What is the most common definition of a number?

The most common definition of a number is a mathematical object used to quantify and measure quantities, such as size, amount, or value.

2. What distinguishes a number from other mathematical objects?

A number is unique in that it represents a specific quantity or value, rather than a concept or operation like other mathematical objects such as variables or functions.

3. Is zero considered a number?

Yes, zero is considered a number because it represents a quantity or value, specifically the absence of quantity or value.

4. Are all numbers real numbers?

No, not all numbers are real numbers. Real numbers include all rational and irrational numbers, but there are also imaginary numbers and complex numbers that are not considered real numbers.

5. Can numbers have different meanings in different contexts?

Yes, the definition of a number can vary depending on the context in which it is used. For example, in mathematics, a number may refer to a specific quantity, while in linguistics, a number may refer to the grammatical category of singular or plural.

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