Talk:Number

From Wikipedia, the free encyclopedia

This is the talk page for discussing improvements to the Number article.
Article policies
Archives: 1

[edit] But what _is_ a number?

"... an abstract idea used in counting and measuring" has no value as a definition, especially as the concept is not further visited. There are plenty of abstract ideas used in counting and measuring -- operators, equality, order -- the "definition" seems to hope that the reader already knows a number when she sees it.

I would propose the following definition: a numeric system is a monoid whose operator preserves a prescribed total order, and a number is an element of a numeric system, or more specifically one of the canonical numeric systems elevated by mathematicians over the centuries. This highlights the essential interface that we have had with numbers from pre-history: we can add them to each other and we can compare them against one another. Of course, in many numeric systems we can do more, but it seems the core that places numbers so fundamentally at the heart of our understanding of the world.

The "flaw" with this definition, of course, is that it excludes the complex field which has no inherent total order. I'm not willing to undertake WP:BOLD without feedback because it may be a heretical notion, but I don't believe this to actually be a flaw: I don't see that the elements of a complex field are much different than a matrix of numbers or a polynomial: an extension of a numeric system that uses the underlying numbers to form rich algebraic structures.

So shall I take a stab at editing the page, or is this a non-starter and people actually like numbers being an abstract idea used in counting and measuring? MatthewDaly 02:44, 6 November 2007 (UTC)

I'm afraid we're not permitted to make up our own definitions in WP articles. Doesn't matter whether they're good or not, so I won't address your proposal on the merits. Please review WP:NOR. --Trovatore 03:04, 6 November 2007 (UTC)
I am unclear on the scope of your rejection. The introduction is completely unsourced, so someone seems to have made up the "abstract idea" definition. It is hardly original research to observe that the concept of numbers historically have been about computability and order, as these are the whole of the core of numerical structures of Peano, Dedekind, and Conway, whose work I would intend to both leverage and reference were I to help on this page. I can appreciate that there is a lack of unanimity among mathematicians when it comes to the "numberness" of mathematical structures that are generated from the real numbers but have increasingly pathological behavior (complex numbers and polynomials are not ordered, matrices and quaternions don't have commutative multiplications, etc.), but the current alternative of devising a definition that is so vague as to not clearly indicate things that are universally regarded as not numbers strikes me as a poor one.MatthewDaly 05:14, 6 November 2007 (UTC)
Well, I was indeed rejecting your proposal (as it relates to the article), but I was not specifically defending the current text (which actually I hadn't read recently). There is no accepted general definition of "number", and the article certainly should not give the impression that there is.
However, now that I take a brief glance at the existing text, I don't see anything terribly wrong with it. I see the assertion "[a] number is an abstract idea used in counting and measuring" as being a descriptive assertion rather than a definition, and one that should be pretty uncontroversial -- that is, we all agree that the abstract idea of number is used in counting and measuring, whether or not we think that this usage precisely isolates what it means to be a number. Maybe you'd like to propose some text that makes more explicit that the sentence is not a definition?
Now that leaves open the question of whether the article should discuss definitions (necessarily plural, I think) that have been proposed for the notion of "number" in general. I don't think it's terribly necessary -- and I certainly wouldn't put it in the lead section -- but it might make an interesting sidelight somewhere in the body of the article. But any such definitions need to be sourced, and it should not be implied that any of them has general acceptance, because I think that none of them does. --Trovatore 06:35, 6 November 2007 (UTC)
Has MatthewDaly been reading Mathematics Made Difficult (ISBN 0-7234-0415-1)? That definition looks as if it was from there. — Arthur Rubin | (talk) 14:24, 6 November 2007 (UTC)
I have not, I'll have to look it up. It seems to me exactly the definition that any formalist would devise; I wonder why they do not seem to have made a point of doing so. Perhaps it doesn't have the same utility as we got from axiomatizing the previously abstract notions of "set" and "proof", but at least it would allow students to understand why some mathematical objects are universally understood to be numbers while others are universally excluded. Ah well, I am disappointed but sanguine.MatthewDaly 17:41, 7 November 2007 (UTC)

[edit] A number is

A number is a word(concept) that represents and contains a sequence of patterns or data. One word, equals one object.

For instance the numeral one, represents the mental object of one, as well as the geometric-symbol of one.

The numeral one could be considered the first letter of the mathematical alphabet. I think it's best to think of mathematics as a language that describes shapes and patterns. A number would be considered both data and a shape at the same time, a data-shape. In fact all numerals are geometric shapes. -- Yours truly BeExcellent2every1 (talk) 12:24, 21 November 2007 (UTC)

This is original research and not suitable for Wikipedia. Rick Norwood (talk) 15:30, 19 November 2007 (UTC)

[edit] Catalan deffinition of number

I suggest the definition given by Pompeu Fabra in his Dictionary of Catalan Language.

It can be translated into English like:

A number is the concept that arises from counting things which form a collection, or a generalization of this concept.

This definition has several advantages:

  • It is clear for everybody
  • Directly relates numbers with intuitive groundings of set theory (collection).
  • Includes all kinds of numbers, because all of them can be considered generalizations of natural numbers (those which arise from counting).
  • Excludes all things that are not considered numbers.
  • It is not original research. It is the definition given by an expert both on mathematics and on language. There is a clear bibliographic reference.

I don’t know if it is the best solution for the English Wikipedia. In Catalan there are three completely different words to express: a) "nombre", the abstract concept of number (the definition I am suggesting), b) "número", the representation of the number in a numeration system, and c) "xifra" the symbols used to represent the numbers. I fear this is not the same in English, but from a mathematical point of view when we talk about numbers we think on the abstract concept. --62.57.139.143 (talk) 12:27, 1 January 2008 (UTC)

It seems very similar to the current opening sentence, except that we refer to measurement as well as counting. I think this is a good thing, as measurement is significant in its own right, not just a generalisation of counting. JPD (talk) 11:25, 2 January 2008 (UTC)
As it has been said before: “There are plenty of abstract ideas used in counting and measuring”.
A lot of confusion comes from involving the measuring process in defining number.
Measuring can be reduced to counting how many times the units of measure are contained in the magnitude to be measured.
But in the process of measuring, several problems have to be solved:
  • How the units of measurement are combined to generate a higher magnitude? i.e. the units of length have to be put contiguously to other units of length on a straight line, it is forbidden to overlap, left gaps and put them on a curved line.
  • How to compare two magnitudes? You have to describe the experiment used to compare the weight to be measured against the units of measure using a mechanical device.
  • How to divide a magnitude in equally valued parts? How to divide the unit of measure to get factionary units?
They are mainly related to the physic properties of the magnitude to be measured.
Measuring can be related to the invention of rational numbers. This is because of the need of dividing the measuring unit in equally valued parts. This could be avoided if the unit of measure is small enough. It also can be related to invention of real numbers if it is admitted that magnitude can be divided in infinitesimally small parts (which is not clear from practical and even theoretical point of view when considering real magnitudes). But they can be introduced simply as generalizations of natural numbers.
But, the only new concept that arises from counting is the concept of natural number. Counting is establishing a bijection between the set being counted and a set of new entities, the set of natural numbers, the only meaning of its elements is that that have in common all the sets that give the same outcome when being counted. That’s why I think it is better to use “The concept that arises from counting” instead of “an abstract idea used in counting”.
I think that making reference to measuring has the intention of opening doors to generalizations of natural numbers, but there are other kinds of numbers that cannot be used in measuring. So I think it is preferable to use directly the expression “or a generalization of this concept”.--147.83.48.87 (talk) 18:17, 2 January 2008 (UTC)

I really wish people would drop this useless effort to find a general definition of "number" in the context of this article. You're not going to find it, because there's no such thing -- "number" is a word applied by divers sources to differing collections of concepts, with some commonalities, but no clear demarcation between what belongs and what doesn't. What the article needs to do is simply present the various things that some reasonable fraction of the literature takes to be "numbers", without trying to make them tidier than they actually are. Even the current first sentence overreaches in that direction (how, for example, are octonions "used in counting and measuring"?).

I think the lead sentence should not start a number is... at all, because that formula almost promises that we're going to give a definition, and we can't. A better lead might begin something like

In mathematics, the notion of number is used in various ways, including abstractions used to count objects and measure quantities

Needs polishing, but you can see where I'm going -- we should point in the direction of the most used senses of the word, but without straining to extract a commonality that may not be there, and most especially without giving any warrant to claim that we're excluding anything in particular from numberhood. --Trovatore (talk) 05:39, 3 January 2008 (UTC) "I am not a number, I am a free man!" Rick Norwood (talk) 13:28, 17 January 2008 (UTC)