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What is a number?

  1. Oct 21, 2011 #1

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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;

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


    McGraw-Hill says;

    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';

    Wikipedia possibly muddies the waters further with;

    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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  3. Oct 21, 2011 #2

    pwsnafu

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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.

    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 most important concept in mathematics would be toss up between "set" or "axiomatic system". Probably both actually.
     
  4. Oct 21, 2011 #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.
     
  5. Oct 21, 2011 #4

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    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.
     
  6. Oct 21, 2011 #5

    D H

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    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.
     
  7. Oct 21, 2011 #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.
     
  8. Oct 21, 2011 #7
    pardon me, Aleph: Science describes reality, how reality behaves, or tells the world how to bevave?
     
  9. Oct 21, 2011 #8

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    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.
     
  10. Oct 21, 2011 #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.
     
  11. Oct 21, 2011 #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?
     
  12. Oct 21, 2011 #11

    disregardthat

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    How can two numbers be different if they are equal?
     
  13. Oct 21, 2011 #12

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    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?
     
  14. Oct 21, 2011 #13

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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.
     
  15. Oct 21, 2011 #14

    disregardthat

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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.
     
  16. Oct 21, 2011 #15

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    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.
     
  17. Oct 21, 2011 #16
    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."
     
    Last edited: Oct 21, 2011
  18. Oct 21, 2011 #17

    disregardthat

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    logics, you are confusing 'number' with 'numeral', or 'symbol'.
     
  19. Oct 21, 2011 #18

    D H

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    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.
     
  20. Oct 21, 2011 #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.
     
  21. Oct 21, 2011 #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.
     
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