Talk:Natural number/Archive 2

This is an archive of past discussions about Natural number. Do not edit the contents of this page. If you wish to start a new discussion or revive an old one, please do so on the current talk page.

Archive 1

Archive 2

Archive 3

Archive 4

Maybe the Peano axiom "Every natural number a has a natural number successor, denoted by S(a)." should rather read "Every natural number a has exactly one natural number successor, denoted by S(a)." I know, that the last (mathematical induction) axiom implies this, but the proposed form (in contrast to the current form) is in accord with the use of the term successor here ("the successor").

Or change the use of the term successor so that it is not silently assumed that the successor is only one, could be even better. (something like a+1 instead of S(a) and "if a property is possessed by 'some' successor" instead of "... possessed by the successor" in the last axiom) --trosos 213.220.249.112 15:55, 13 January 2007 (UTC)

By the definition above, zero seems to be a natural number as it has a successor. Yet stretching this a bit further makes -1 a natural number as it too has a successor. I recall Peano got round the problem by first having an axiom defining zero as a natural number. Then the successor function is used to build the set and -1 never gets a look in. John H, Morgan (talk) 17:26, 2 December 2008 (UTC)

I've seen the set |N discussed in the sci.math newsgroup occasionally; is this another notation for N, or does it have some other special meaning? Or is this an attempt to represent a special character within the limitations of ASCII? Should it be mentioned in the article? I found this informal definition in a newsgroup post:

The informal definition |N = {1,2,3,...} is usually taken to mean that |N is a set S such that

(1) 1 is a member of S

(2) for each member n of S, n+1 is also a member of S, and

(3) |N is a subset of every set, S, with properties (1) and (2).

I'm not clear on (3), except that I think it means that S can be any set containing consecutive naturals (and possibly other members as well), and therefore |N is a subset of any such set. — Loadmaster 23:13, 9 April 2007 (UTC)

It's an attempt to mimic blackboard bold in ASCII. It's almost always more confusion than it's worth. No, I don't think it deserves mention in the article. --Trovatore 23:16, 9 April 2007 (UTC)

Couldn't the first sentence just be "A natural number is a number greater than zero with no decimal separator"? For people who just want to know the basic definition without reading the entire article? —Preceding unsigned comment added by 86.76.137.45 (talk)

No. That is a definition by non-essentials. You should not confuse a number with a particular representation of that number (in this case the decimal representation). JRSpriggs 10:38, 22 April 2007 (UTC)

The section on the Peano axioms claims that "All systems that satisfy these axioms are isomorphic". This would seem to contradict both the incompleteness theorem and the Löwenheim–Skolem theorem. 72.75.107.59 (talk) 01:29, 19 January 2008 (UTC)

Those refer to first-order logic. What the claim means is that all structures that satisfy the full Peano axioms, in the sense of second-order logic, are isomorphic. --Trovatore (talk) 01:33, 19 January 2008 (UTC)

Aren't the axioms listed there ("these axioms") all first-order? 72.75.107.59 (talk) 01:40, 19 January 2008 (UTC)

No, the full axiom of induction is not first-order. It becomes a first-order axiom schema if you limit the properties being considered to ones that can be defined by a first-order formula. --Trovatore (talk) 01:56, 19 January 2008 (UTC)

this is topical: Mathematics to Retire Number 4--Billymac00 (talk) 14:58, 4 April 2008 (UTC)

This is funny, but I don't think it should be in the article. Oleg Alexandrov (talk) 15:24, 4 April 2008 (UTC)

I am new to wiki and not sure if this is the proper way of posting a question, but I thought THE natural number was e (2.71828 18284 59045 23536...) since it appears in nature so often. (i.g. birthrates) Why is it not even noted on this page?

Just thought It should be noted since a single letter is hard to search for. see [1]

Fozforic (talk) 14:00, 23 September 2008 (UTC)

The term natural number, in mathematics, universally refers to the concept treated here (either the nonnegative integers or the positive integers, depending on your taste), and certainly does not include e, which however is the base of the natural logarithm function. The word natural appears in the names of both concepts, but that shouldn't be taken to indicate any close connection between them. This is in general the way mathematical nomenclature goes — multi-word terms mean what they're defined to mean, and their names should usually be taken as historical artifacts rather than as something you expect to be able to figure out the definition from. --Trovatore (talk) 17:02, 23 September 2008 (UTC)

This raises the question whether for the benefit of really ignorant people we should add a hat-note to disambiguate this, saying for example "For the base of the natural logarithms, see e (mathematical constant).". JRSpriggs (talk) 05:05, 24 September 2008 (UTC)

No, a search for e will get the reader to the desired page. --Salix alba 06:04, 24 September 2008 (UTC)

i removed the citation needed on "Some authors who exclude zero from the naturals use the term whole numbers, denoted \mathbb{W}, for the set of nonnegative integers. Others use the notation \mathbb{P} for the positive integers." if a citation is needed for that than there is quite a bit else that needs citation in this article. these symbols are very often found in math textbooks.

I thought that ${\displaystyle \mathbb {P} }$ referred to the set of prime numbers….
― Kinkydarkbird (Talk Page) 09:23, 9 January 2009 (UTC)

i've seen it for both, and i have at least one text with me that uses ${\displaystyle \mathbb {P} }$ for the positive integers. —Preceding unsigned comment added by 71.192.103.225 (talk) 06:05, 26 January 2009 (UTC)

The article mentions the alternative encoding of natural numbers as sets by 0 = {} and n+1 = {n}. I have heard that this encoding was used by Peano. Does someone know the reference? --Jan 91.180.52.246 (talk) 22:25, 5 January 2009 (UTC)

That would have been an odd thing for Peano to do, since his axiomata begin with ${\displaystyle \mathbb {N} _{1}}$. I believe that the construction that you here mention was proposed by Russell. —SlamDiego_←T 06:04, 20 January 2009 (UTC)

Wow, somebody really needs to learn more English morphology. Regardless of whether “definitionally” was superfluous (I was trying to capture that for which some other editor had been reaching with the mistaken claim that the inclusion of zero were “explicit”), it's a perfectly proper English word. —SlamDiego_←T 06:00, 20 January 2009 (UTC)

In the chapter "History of natural numbers and the status of zero" it says that 0 is the empty set. Sorry, but that is wrong. —Preceding unsigned comment added by 80.165.82.22 (talk) 19:49, 18 February 2009 (UTC)

What do you mean by "wrong"? There are multiple ways to define natural numbers in terms of set theory. Not all of them define 0 as the empty set, but the one that's generally considered "standard" for use in mathematics (whether you're doing mathematical logic or teaching undergraduates) very definitely does. And all of this is explained pretty clearly in the article. --75.36.134.30 (talk) 15:05, 26 February 2009 (UTC)

A question for you all: should we delete the text "This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is aleph-null" in the Notation section? I tried editing it out for the following reasons:

• The content is repeated in the Generalizations section, so it is redundant.
• It goes against a suggestion in the Wikipedia Manual of Style for mathematics articles: "A general approach is to start simple, then move toward more abstract and technical statements as the article proceeds."

My deletion was reverted by another Wikipedian (see the history), who later gave the following explanation for the revert:

• Since "countably infinite" and "aleph-null" are defined in terms of the set of natural numbers, it seems appropriate to me to mention that fact in the section dealing with notations for the natural numbers and related things.
• A little redundancy helps communication.
• The section on generalization seems too far down in the article to introduce these ideas.

Any opinions on this? FactSpewer (talk) 03:27, 15 April 2009 (UTC)

You deleted

Which one we should use depends on which is better suited for our purpose. E.g. in number theory, only the positive integers are denoted by natural numbers whereas in algebra it is convenient to denote the non-negative integers as the natural numbers.

and below

Of cource we can discuss or philosophize endless whether 0 is natural in the intuitive sense.

and commented in the history

18:54, 24 June 2009 Quaeler (talk | contribs) m (21,121 bytes) (Reverted good faith edits by Achim1999; Personalization of prose unencyclopedic; heavy mathematical drill down in opening paragraph makes the article less accessible to the average reader.. using [[WP:TW) (undo)

Well, I "personalization" with the words "we" and "our" because I thought this is good style! If you prefer you can use the passive form like

Which one should be used depends on which is better ....

And to your point "heavy mathematical drill down": You should immediateley delete the term "Ramsey theory" Argh! This is much heavier, why not using "combinatorics" ?

Moreover you (ther reader) could ask why this set is called "Natural" numbers? This never is done in this article! I open the eyes of the reader hopefully by stating ".... whether 0 is natural in the intuitive sense."

I hope in good faith you will take up these ideas and improve the article concerning these points and not only delete important(?) information. Regards, Achim1999 (talk) 19:33, 24 June 2009 (UTC)

I see by your talk page that you're no stranger to introducing verbiage that people have issue with; similarly, i found that your edits did nothing to improve the readability of the page and introduced no 'improvement'. Additionally, articles are not a place to wax philosophic as your text was doing. To be more to the point, i don't see that the article needs improvement in the areas which you attempted to introduce it and, as such, see no place to "take up these ideas and improve the article". Quaeler (talk) 20:52, 24 June 2009 (UTC)

The sentence "Of cource we can discuss or philosophize endless whether 0 is natural in the intuitive sense." has no meaningful content and is stylistically inappropriate for an encyclopedia. JRSpriggs (talk) 20:56, 24 June 2009 (UTC)

Are you pulling my leg? 2) The sentence "Of cource we can discuss or philosophize endless whether 0 is natural in the intuitive sense." should be taken, to tell the reader, somewhere in the main article if you like, WHY is this set of numbers called NATURAL numbers! 1) The sentence "Which one we should use depends on which is better suited for our purpose. E.g. in number theory, only the positive integers are denoted by natural numbers whereas in algebra it is convenient to denote the non-negative integers as the natural numbers." (you may reformulate it in passive form, if you dislike the we/our style) gives the reader an explanation why there are two different sets used. And therefore exactly this is the position to place it, IMHO. And this should be undoubtfully an improvement. Please judge because on the matters concerning the facts and not because your personal feelings are influenced by reading e.g. my talk-page or because you know I favour to behind-ask critically given states, if you feel to act like as a reviewer. :-( Regards, Achim1999 (talk) 21:17, 24 June 2009 (UTC)

To be pedantic, the order of my analysis was:

1. Notice new edits on this article because it's on my watchlist.
2. Reviewed the edits.
3. Thought to myself, 'wow - this shouldn't be in the article'
4. Rolled back the edits; due to my tools, this also opens your talk page.
5. Read your talk page.
6. Thought to myself, 'oh - that makes a little more sense'

and not:

1. Came across your talk page.
2. Read your talk page.
3. Thought to myself, 'i should go scrutinize his edits'.

So i have originally judged the matter based on solely on your work, and not the impression your talk page later propped up. Quaeler (talk) 00:22, 25 June 2009 (UTC)

It seems to me you favor meta-discussion to avoid discussion on contens like just happens. Sorry, therefore I have read your talk page and get the impression that you already are in the set of guys you are hunting for in the eyes of some other faithful WP-users. Therefore my advice: think twice about your article-editing-habit here on WP! BTW: I still hope that "Quaeler" is not the name/word in german for your intention here on WP. :-/ Regards, Achim1999 (talk) 10:04, 25 June 2009 (UTC)

In the article I read "To be unambiguous about whether zero is included or not, sometimes an index "0" is added ..." The funny thing is this isn't unambiguous as in my country (Belgium), zero is a natural number, and this symbol ${\displaystyle \mathbb {N} _{0}}$ means without zero, because ${\displaystyle \mathbb {N} }$ represents the natural numbers, which is with zero. Also for other number sets a sub-index 0 means "without zero" (in Belgium). Something extra I forgot first to mention: some people may wonder why things like ${\displaystyle \mathbb {Z} ^{+}}$ includes zero, but zero is in Belgium a positive AND a negative number so it is logic that in that case things like ${\displaystyle \mathbb {Z} ^{+}}$ means the positive numbers, which is with zero. In other countries in Europe is this not always the case, it depends.193.190.253.144 (talk) 21:47, 7 June 2008 (UTC)

Interesting -- it strikes me as really counterintuitive to write ${\displaystyle \mathbb {N} _{0}}$ for the specific purpose of excluding zero. However I also have not come across this notation as a way of allowing zero, which is what the article currently claims -- can anyone find this attested somewhere? --Trovatore (talk) 22:49, 7 June 2008 (UTC)

193's statement is at least partially confirmed by the Dutch Wikipedia. In the article Natuurlijk getal (Natural number) the notations ${\displaystyle \mathbb {Z} ^{+}}$ and ${\displaystyle \mathbb {Z} ^{-}}$ are given as including 0, while ${\displaystyle \mathbb {Z} _{0}^{+}}$ and ${\displaystyle \mathbb {Z} _{0}^{-}}$ are given as excluding 0. Also Positief getal (Positive number), Negatief getal (Negative number), and 0 (getal) (0 (number)) state that in Belgium the number 0 is considered both positive and negative. Dutch is one of the official languages of Belgium. The French Wikipedia is equivocal; for example, it defines Nombre positif (Positive number) as: un nombre qui est supérieur (supérieur ou égal) à zéro ("a number that is greater than (greater than or equal to) zero"), without explaining when the parenthesis is supposed to be in effect. Next thing, the French article states: Zéro est un nombre réel positif, et est un entier naturel. Lorsqu'un nombre est positif et non nul, il est dit strictement positif. ("Zero is a real positive number, and is a natural number. When a number is positive and non-zero, it is called strictly positive.") The Spanish Wikipedia, at Número positivo, has a similar text but is slightly clearer in expressing that the meaning is ambiguous. In Germany and Italy the number 0 is unequivocally neither positive nor negative.  --Lambiam 07:56, 8 June 2008 (UTC)

In French-speaking European countries, the word "positive" used to be used much as it is in English. "x is positive" meant x > 0. Now there are competing conventions. What used to be called positif ou nul (nonnegative) is now often positif (positive), and what used to be positif is now often strictement positif (strictly positive). (The word nul is an adjective that means "being zero".) I suspect that the change came about mostly because of the choices made by Bourbaki. The newer terminology is not universal, but it does predominate in France. Claims that the older usage has disappeared are probably exaggerations. Also, Canada has not followed French-speaking European countries with regard to the meaning of "positive". 128.32.238.145 (talk) 22:49, 16 November 2008 (UTC)

Since there is a lot of confusion in this area, I suggest we follow ISO so long as there is no reason to deviate from it. In particular, ISO 31-11 says that ${\displaystyle \mathbb {N} }$ includes 0, and also defines ${\displaystyle \mathbb {N} ^{*}=\mathbb {N} \setminus \{0\}}$. Normally I would have suggested to use ${\displaystyle \mathbb {N} _{0}}$ instead of ${\displaystyle \mathbb {N} }$, for clarity. I believe this is relatively standard practice in Germany. But as it seems this would only increase the confusion for Belgian and Dutch readers, it's probably best not to do that. --Hans Adler (talk) 11:10, 28 April 2009 (UTC)

There's an ISO for math??? That's terrible; that should not exist.

Of course I personally include zero in the naturals, but it's not because of any silly standards org that no one's ever heard of (in mathematics). I think the correct solution for the article is just to report that some include 0 and some don't, which is the truth. ISO should just be ignored, as is the practice of mathematicians generally. --Trovatore (talk) 18:07, 28 April 2009 (UTC)

I think such standards actually have the potential to be quite influential. What I learned at school seemed to be directly inspired by DIN 5473. I agree that such standards don't have much influence on university maths, and note my disclaimer about reasons to deviate from them. But keep in mind that most people who deal with concepts such as the natural numbers after school are engineers, for whom industry standards generally are relevant. Another point is that the committees behind such standards often do research that is similar to what we as Wikipedia editors are doing when we try to standardise our notation across articles: They try to find out which usage is more common, and they try to identify historical trends. Taken all this together, I think all else being equal we should prefer the usage prescribed by an ISO standard.

The DIN norms for maths, which no doubt influenced the ISO norms, are due to the Ausschuss für Einheiten und Formelgrößen, which was founded in 1907 by 10 scientific organisations including the Deutsche Physikalische Gesellschaft and the Verein Deutscher Ingenieure. [2] The mathematical norms function like a manual of style: If they manage to convince people or organisations, they will follow them. After reading our article ISO 31-11 I must say it seems to be perfectly sane. It would seem strange for me to not follow them simply to prove one's independence. Of course one problem with these standards is that the practice of only making them available for a lot of money is incompatible with scientific culture. --Hans Adler (talk) 20:17, 28 April 2009 (UTC)

Our standards should be the same as they always are — usage in the literature and the mathematical community, period. What some standards body claims is irrelevant. I have no specific gripe about any of the choices I saw in a brief scan; that isn't the point. Our job is to reflect the prevailing usage, and at the moment I believe it is still correct to say that both usages of natural number are current. Therefore we should say so. --Trovatore (talk) 20:44, 28 April 2009 (UTC)

I didn't want to say otherwise, but I see now how what I wrote can be misunderstood that way. I came here because anonymous editors at countable set have been removing the 0 from the natural numbers. (An ISO standard is probably a much more convincing argument for such users than a literature survey, but that's of course not relevant here.) However, I do think that we shouldn't have to define what ${\displaystyle \mathbb {N} }$ means in every single article that uses this symbol; we should have a general convention for this, and ${\displaystyle \mathbb {N} /\mathbb {N} ^{*}}$ is probably the best choice.

This discussion seems to have established (to my surprise) that ${\displaystyle \mathbb {N} _{0}}$ is ambiguous. Therefore it seems that the current text introducing the symbols needs changing. I am proposing to say that the natural numbers with/without 0 are denoted by ${\displaystyle \mathbb {N} /\mathbb {N} ^{*}}$, and that it is also common to denote them by ${\displaystyle \mathbb {N} _{0}/\mathbb {N} }$. If it can be sourced, I would also say that ${\displaystyle \mathbb {N} _{0}}$ is ambiguous because it can denote ${\displaystyle \mathbb {N} ^{*}}$. --Hans Adler (talk) 21:17, 28 April 2009 (UTC)

Since we seem to agree that in current use the term "natural number" has become ambiguous, so is necessarily the notation ${\displaystyle \mathbb {N} .}$ We should not try to hide that unfortunate fact. −Woodstone (talk) 22:39, 28 April 2009 (UTC)

I don't want to hide the ambiguity. But like every good textbook we can say, the symbol is ambiguous and this is how I use it. Only that we need to be a bit more subtle and can't say explicitly how we use the symbol, both for stylistic reasons and because we will never get all articles consistent with whatever convention we choose. When we use the symbol ${\displaystyle \mathbb {N} }$ in Wikipedia, then in many cases (probably the majority) it makes a difference whether 0 is included or not. See WP:WikiProject Mathematics/Conventions for how we have dealt with some similar problems. I believe that use of the symbol (but not of the term "natural number") needs to be standardised across Wikipedia. This talk page seems to be the best place to talk about this, although depending on the outcome of the discussion here it should of course be proposed on the Conventions page as well. Perhaps something like the following works:

A recent trend is to denote the natural numbers including 0 by ${\displaystyle \mathbb {N} }$ and the natural numbers without 0 by ${\displaystyle \mathbb {N} ^{*}}$. [3] This trend is reflected in recommendations for scientific writings such as ISO 31-11. However, the traditional practice of writing ${\displaystyle \mathbb {N} }$ for the natural numbers without 0 is still widespread. --Hans Adler (talk) 23:44, 28 April 2009 (UTC)

Yesterday evening, I stumpled about a paragraph in "Numbers" from Ebbinghaus et a., Springer, 1991, which states that 1stly R. Dedekind liked to start the Natural Numbers with one and 2ndly that G. Peano acknowledged that he was influenced by Dedekind's postulates when he defined his axioms for the Natural Numbers. Thus it appears to me that already in the beginning of the foundation of formal logic / axiomatization (around 1880) this "0 is / is not in N"-war started. ;-) Regards,Achim1999 (talk) 15:12, 8 July 2009 (UTC)

I know this isn't the Simple wiki, but even so, is it really necessary to say "Either a member of the set of the positive integers, or a member of the set of the non-negative integers", instead of leaving "sets" out of it and just saying "Either a positive integer or a non-negative integer"? 174.46.172.13 (talk) 10:35, 30 June 2009 (UTC)

I understand your point, and I tried to address it. Hans Adler 10:55, 30 June 2009 (UTC)

Although some number theory texts do start with 1, there are also many distinguished number theory texts that start with 0, and some of them are not so recent. For instance, see

• Serre, A course in arithmetic, Springer-Verlag, 1973, p. 115.
• Weil, Basic number theory, second edition, Springer, 1973, p. XIII.
• Ramakrishnan and Valenza, Fourier analysis on number fields, Springer-Verlag, 1999, p. 138.
• Rotman, Advanced modern algebra, Pearson, 2002, p.1. (This isn't really exclusively a number theory text, but the definition occurs in a section called "Some number theory".)

Therefore I suggest removing the bit in the history section about number theorists keeping the older tradition of starting with 1. --FactSpewer (talk) 20:44, 8 November 2008 (UTC)

I don't understand in what way Woodstone's rewrite of the opening paragraph is "more balanced". I feel that the original wording matches actual usage better, because it is true that currently "natural number" can mean either an element of {1,2,...} or an element of {0,1,2,...}. The first sentence of the revision suggests that the former is a thing of the past, which is not the case. The second sentence of the revision suggests that all authors in mathematical logic, set theory, and computer science start with 0 (I would hesitate to make such a claim), and is ambiguous on whether starting at 1 is done by everyone else, or by no one else, or ... . --FactSpewer (talk) 04:34, 28 April 2009 (UTC)

After a neutral "0 is in or not", the preceding version had a phrase: "the latter is especially preferred in (some fields)". So only a positive statement for including 0. I wanted to add something positive for not including 0. It cannot be doubted that 0 is a relatively new concept compared to the positive natural numbers. So "originally start from 1" is true. Some specific fields now often include 0, is true as well. I agree this is not universal in those fields, so we could add a remark about that, and make explicit that other fields stick to the original definition. −Woodstone (talk) 07:44, 28 April 2009 (UTC)

OK, thank you; I understand your thinking now. Probably whoever wrote especially preferred meant not that it was better, but that it was more common in those fields; but I agree with you that that wording could be construed as making a value judgment, so let's avoid it. I'll just word the opening sentence to make it clear that both conventions are used now; and I'll leave the history to the history section, which agrees with what you say, and expands upon it in detail. --FactSpewer (talk) 04:15, 12 May 2009 (UTC)

Divisibility and 0 bites each other very much in theory. :) Thus 0 was never paid attention in classical number theory and it was not missed! Achim1999 (talk) 18:55, 24 June 2009 (UTC)

Division works somewhat differently in number theory and analysis. In analysis one is simply not permitted to divide by zero. In number theory division is replaced by the notions of divisibility and congruence, where divisibility and zero get along just fine. The positive integers under the divides relation m|n form a distributive lattice that is not complete, whereas the nonnegative integers form a complete distributive lattice with 1 at the bottom and 0 at the top (the opposite of the usual convention for naming top and bottom in a lattice). In particular, without zero one can take the GCD of any infinite set of positive integers but not the LCM, whereas with zero one can take both the GCD and LCM of any infinite set of nonnegative integers. Hence number theory is better off when it includes zero because doing so expands the available operations.

In computer science the natural numbers always start from 0, as a consequence of the convention of breaking the cycle ... < 110 < 111 < 000 < 001 < 010 < ... between 111 and 000 when interpreting bit strings as unsigned integers, more precisely the integers mod 2ⁿ for Ironically computer keyboards break it between 000 and 001 (so to speak) as a result of typewriters having always done so, putting 0 at the right. Rotary telephones also did so, but that was because dialling n produced n pulses and if 0 had produced no pulses instead of ten pulses the switchboard would never hear 0. --Vaughan Pratt (talk) 05:40, 17 August 2009 (UTC)

Sorry, do be pressed to open this section: But we dislike to see subjective, social unnecessary statments in the beginning (perhaps also in the wohle article?) of this scientific-supposed-to-be article! What N is, is stated precisely. No need for your personal oppinion to be added!

Regards, Achim1999 (talk) 11:49, 5 July 2009 (UTC)

I support the clarification there that the distinction is a matter of each author's convention. Without that clarification the "or" is more difficult to understand. — Carl (CBM · talk) 12:33, 5 July 2009 (UTC)

Achim, your argument makes no sense, and your attempt to enforce your removal of a stylistically necessary clarification by means of wiki-lawyering [4] was completely out of order. Given your obvious problems with the English language you should definitely not be edit-warring over style. You are probably not aware of it, but that's exactly what you are doing. Hans Adler 13:36, 5 July 2009 (UTC)

People who can not or want not to argue by contens but prefer to argue by pointing out wording, style and typos should be better ignored with their talk. :-(

Regards, Achim1999 (talk) 21:30, 5 July 2009 (UTC)

"..depending on context." is an empty phrase which is always correct. Thus adding this gives no further information but only blows up the writing. Escpecially we are here in a mathematical definition, hence one expects short, simple and clear wording, and no story-writers using fill-words. Argh!

Regards, Achim1999 (talk) 21:30, 5 July 2009 (UTC)

In dingo culture, a human being is either a man from asia or a woman from africa. What is unclear? And what becomes clearer by appending "depending on the context." to this sentence? *shaking my head* Regards, Achim1999 (talk) 21:35, 5 July 2009 (UTC)

Better wording-suggestion:

In mathematics, a natural number is either from the set {1, 2, 3, ...}, hence a positive integer, or from the set {0, 1, 2, ...}, then called a non-negative integer.

Regards, Achim1999 (talk) 21:58, 5 July 2009 (UTC)

That's not a good solution to the problem, and since your made-up example seems to confirm my suspicion that you are completely missing the issue, here it is: The purpose of "depending on context" or "depending on the author's convention" is to make it immediately clear that "either ... or" separates two different conventions rather than being part of a single, universally accepted definition of the natural numbers. This is an unusual situation, especially for such an elementary and well-known notion. We can expect that most readers 1) are not mathematicians, and 2) are familiar with only one convention and will be surprised to learn there are two. Without very clear hints they may well not understand this, and instead think that for some reason which they don't understand we are describing the convention they know in impenetrable mathematical jargon.

Note that your dingo example has exactly the same problem. Without "depending on context" it would be possible that a dingo can say "two human beings" to refer to a man from Asia and a woman from Africa. Adding "depending on context" or "depending on the dingo's convention" would clarify that there are two dingo dialects: one in which "human being" can only refer to a man from Asia, and one in which "human being" can only refer to woman from Africa.

Oh, and "depending on context" is of course not always correct. E.g. "an irreducible natural number is either the number 1 or a prime number" is correct, while "an irreducible natural number is either the number 1 or a prime number, depending on context" is plain wrong.

Your proposed reformulation is only marginally better in this respect than the version you have twice reverted to. It is also written in a very clumsy style, and any attempts to fix it would probably lead to what we have now (with or without the clarification).

By the way, I have two questions for you regarding "we dislike to see subjective, social unnecessary statments in the beginning":

1. Are you a single person or is "Achim1999" a group account? If you are a single person, who else do you think you are speaking for?
2. After my explanation, do you still believe that the explanation is unnecessary? If so, could you please elaborate what it is you don't like about it; I don't think there are many editors here who share your opinion.

Hans Adler 23:32, 5 July 2009 (UTC)

How about;

There are two conventions on what a natural number is:...? Septentrionalis PMAnderson 00:40, 6 July 2009 (UTC)

Thanks for finally give information of the contens and context which you think is ambigious. This appending "depending on the context." makes nothing clearer to me. Because this interpretation has nothing to do with english language, it occurs in german too and probably in many other language this is possible. But now I know what the misinterpretation could be. I only wonder if you think "Without very clear hints" is necessary why you are reluctant to give these very clear hints and make a better formulation which avoids exatly this supposed-to-be misunderstanding. E.g.:

In mathematics, there are two convention for a natural number: either it is a positive integer from the set {1, 2, 3, ...} or a non-negative integer from the set {0, 1, 2, ...}.

or perhaps

In mathematics, there are two convention for a natural number: either it is from the set {1, 2, 3, ...}, hence a positive integer, or from the set {0, 1, 2, ...}, then called a non-negative integer.

It should be easy, even for you ;-), to quickly generate contexts, such that "an irreducible natural number is either the number 1 or a prime number, depending on context." is a correct and useful statement. :)

To answer your many off-topic, non-mathematical social questions:

1. Achim1999 is no group-account. I even did not know that this is possible here.
2. I don't know. I have not made up my mind. But I was already pointed to this guidline(?) by other people / editors(?) here.
3. I never thought of your misinterpretation-possibility. I still have problems to realize how to interpretate this sentence as a single unique definition. Perhaps this is due to the fact, that one choice is a subset of the other.
4. I still don't know whether there is a clarification necessary, sorry. (see my answer 3).
5. I think appending "depending on context" makes nothing clearer -- I think you believe it should cause a grouping of some words in this sentence in your understanding. But this I can't realize, honestly.

Regards, Achim1999 (talk) 11:17, 6 July 2009 (UTC)

Group accounts are physically possible, in the sense that we cannot prevent someone else from sitting down with your password; they are forbidden, and grounds for blocking. Septentrionalis PMAnderson 16:32, 6 July 2009 (UTC)

BTW: I still wonder why you have pressed others (at least me) to make this discussion, and refuse to suggest a clear formulation which hits your point (two conventions are here in use) and therefore avoid such a waste of time. Regards, Achim1999 (talk) 11:41, 6 July 2009 (UTC)

I have pressed you? You reverted me twice for a reason that I suspect can be understood by nobody other than yourself ("we [sic!] dislike to see subjective, social unnecessary statments"), then you started the discussion here, and you unnecessarily notified me of it on my talk page. So far you have been reverted once by me, once (essentially) by JRSpriggs, and you have been told you are wrong by CBM. That's a score of 3 professional mathematicians and experienced Wikipedia editors disagreeing with you against a total of 1 editors (including you) agreeing with you. I think it's pretty clear at this stage that if you still don't understand things after I have taken the time to explain them to you, then it's entirely your problem. Wikipedia isn't school, and other editors here are not your teachers. We are not payed for explaining things to you that you don't want to understand. Hans Adler 15:05, 6 July 2009 (UTC)

Sadly, you seems really unwilling to stay factually. On such a base it makes no sense for me to discuss further! Sadly also, you like to judge wrong versus right by 3:1 opinnions getting in a few days. I can only hope that you act as a professional mathematician much more factually. Noone wanted you to act as a teacher -- at least I did not, but this seems also an off-topic attitude by you. Well, meanwhile other people took hand on this sentence and this very special interpretation-issue is gone. Unbelievable that you refused to make constructive changes which hit your point of (necessary?) clarification of those sentence. :-( Regards, Achim1999 (talk) 11:40, 7 July 2009 (UTC)

You were unable to express your concerns comprehensibly, which made it impossible for me to take them into account. Now the article has been changed to give slightly more detail. I think we had something very similar recently and somebody objected, but I don't have the time to look up the history now. In any case a first sentence with two parenthetical phrases is rubbish. And I have no idea why your strange objection doesn't apply to the present formulation.

If you have trouble communicating in English don't blame others for the fact. Thanks for the promise to end this silly discussion, by the way. Hans Adler 15:20, 7 July 2009 (UTC)

Yesterday a user added "... according to a more recent definition now also in common use." (I removed the parentheses). I'm really unhappy with the elastic wording "more recent". This should be made preciser. I believe it was first invented/defined by a formal approach in the 19th century to lay the foundation of numbers-notion (G. Frege / R. Dedekind). On the other hand the sentence should not become too long. Regards, Achim1999 (talk) 13:54, 8 July 2009 (UTC)

Well, just again this non-factual (I heard, there is a WP-guidline to generally not name people by calling their account names, especially if their action can be considered bad) liking author, disqualifing himself with this kind of attitude in the long run, sprang in action and triggers me to write this general comment regarding all WP-articles -- but I like to give two examples first to make my point hopefully clear:

Facts to this special example:

1. Since many months this section "algebraic properties" were here located without significant changes.
2. Some days ago, I noticed, that a,b, and c are in the mathematical interpretation so-called free variables. This I dislike, because in the langugual interpretation given by the context they should be considered to hold only natural numbers.
3. Therefore I added a very precise, short, mathematical statement as lead-in to all the other mathematical expressions in the following list which is in table-form and which is everything in this section.
4. Suddenly people spring in action, to avoid this mathematical clarification, which is also easily understandable by typcial to-be-expected non-mathematical readers, only to rephrase it with superfluous words like story-writes, using the interpretable word "properties" without any need! I did not expect others to act like very good mathematical writes, like say, D. E. Knuth, but thoughtless (in the most positive assumption) changes should be avoided which only blows up articles and even decrease readability.
5. Argh! :-( Sorry.

And a further example in this article which happend recently: Another guy/account disliked the word "computist". Even me, who probably had never heard this word, knew immeditely what the original text/author wants to express with this wording. Then this guy sprang into action who dislike this wording and must replace it by computer. But in common use this has today almost always a different meaning, thus this guy had to add an addtion to clarify this. Ironically he added in his comment to his change, that in pre-computer-age this word "computer" was used with a unique meaning of computist. But it now seems that he prefers fighting for the ambigusity of "computer" and supports it here by reusing it where totally unnecessary. My well reasoned revert, in fact I used his(!) reasons of usage in past to revert it, he reverted aagin but without any comments. Sorry, this I can not call faithful editing by heart. And to be frank, more the opposite creaps in my mind from learning other changes in other articles here in WP.

These kinds of attitude/behaviour of editing will surely not encourage more experts from special fields to sacrify their time to write WP-articles, a wish WP wants strongly.

BTW: Sorry, to sound in this writing to act like a well-sounded teacher. :-/

Regards, Achim1999 (talk) 10:48, 15 July 2009 (UTC)

No such guideline exists. Having to hunt through page histories to find out who you're talking about is very annoying, so giving names is almost always appropriate. Algebraist 10:52, 15 July 2009 (UTC)

I was told by a deserved editor on a user-talk page, that WP prefer to avoid explicit naming if criticizing editors at least. And it seems in retrospect that I criticized another deserved editor in that case. Maybe he meant only critizing of deserved editors and his wording looked to me more than a proposed policy. Sadly I have deleted all user-talk-pages (except my own) from my watch-list.

Regards, Achim1999 (talk) —Preceding undated comment added 11:19, 15 July 2009 (UTC).

You were told wrongly. Algebraist 11:22, 15 July 2009 (UTC)

Why do you think inline symbols are more readable than clear English text? Algebraist 11:49, 15 July 2009 (UTC)

Surely not generally, but you should think why they exist (resp. were invented). And in this case, section "Algebraic properties", they cause much better readability than prose. Anyway, I think it is not worth to discuss with people like CBM · talk who acts before thinking.

Have fun to change this article in the future without my support. Regards, Achim1999 (talk) 17:26, 18 July 2009 (UTC)

Sometimes making a small edit draws attention of other people to lingering problems in a page. I'm not thrilled at all with the table in the "algebraic properties" section. I don't see any reason why we cannot use prose, perhaps a bulleted list. I'll see what I can do. — Carl (CBM · talk) 11:46, 15 July 2009 (UTC)

Yes, prose would be an improvement. Algebraist

I made some changes. I'm still not thrilled with the "algebraic properties" section being followed by the "properties" section; they should probably be merged. Also, there should probably be a section on the key property of natural numbers, which is that they support proofs by induction. There is currently just a link to mathematical induction buried in the article. — Carl (CBM · talk) 12:04, 15 July 2009 (UTC)

Saying that starting at 1 is the traditional convention, and that starting at 0 is the convention following a formal definition in the 19th century, is not entirely accurate. The issue is more complicated than this, and is best left to the discussions later in the article, where it is amply explained. For example, Peano's original definition started at 1, not 0 (it is only that some logicians modified his definition later on to start at 0, out of convenience). And both definitions occur in the modern literature. --WardenWalk (talk) 19:20, 1 August 2009 (UTC)

Could someone with more experience of these things than I (or access to a good academic library) please add a reference to a work which treats 0 as a natural number? The article makes it sound as though this is the more common modern day convention, but my 2nd edition copy of the Penguin Dictionary of Mathematics (for example) only lists the ℤ>0 version (which it also identifies as ℤ⁺, to further muddy the waters). Thanks. Aoeuidhtns (talk) —Preceding undated comment added 12:46, 5 February 2010 (UTC).

Literally any book on set theory or mathematical logic. I checked Levy, Jech, and Kunen's set theory books just now, and also Shoenfield; Hinman; and Boolos, Boolos and Jeffrey on the logic side. All of these begin the natural numbers at 0. I don't think I have seen any text in logic that begins the natural numbers with 1, although I can't say I've read every single book in the area. — Carl (CBM · talk) 13:26, 5 February 2010 (UTC)

I think some mention of convention by country should be mentioned. For instance here in the UK it is traditional to start the natural numbers at 1 whereas in France it is traditional to include 0. (I do not know of any references for this but it seems to be common knowledge amongst my professors and doctors (I'm doing a post grad in maths)). Porkbroth (talk) 20:08, 2 March 2010 (UTC)

I move to change the article to be in agreement with MathWorld, the standard online reference for mathematics. This would involve using "+" to indicate positive (instead of non-negative) and "*" to indicate non-negative. We shouldn't have the top 2 references that come up when you Google "natural numbers" disagree! Also, see: The MathWorld page on Positive Integers. watson (talk) 02:32, 1 October 2010 (UTC)

MathWorld is very very bad at terminology. We should not rely on it whatsoever. --Trovatore (talk) 02:41, 1 October 2010 (UTC)

Agreed. I don't see how mathworld is a standard reference for mathematics any more than wiki. If anything the opposite is true, mathworld being by and large the work of a single individual, whereas wiki is a cooperative effort involving a number of individuals. Tkuvho (talk) 04:36, 1 October 2010 (UTC)

I agree with Hans Adler that nominal numbers should be dealt with separately. Note to Fullmetalactor: I would say that the 23 in "I live on 23rd street" is functioning as an ordinal number (by the way, be careful when correcting spelling!) Ebony Jackson (talk) 07:29, 17 August 2009 (UTC)

Would someone please mentions the symbol "ℕ" Unicode number next to it ? --DynV (talk) 07:16, 25 October 2009 (UTC)

You've made this request in at least two articles, maybe more, but I'm afraid I don't understand it. Could you please explain more clearly what you mean? --Trovatore (talk) 10:39, 25 October 2009 (UTC)

Certain symbols appear for some readers as a "ℕ" and for other readers as the intended symbol. When <math> is used, this problem does not occur. So in case of doubt, let us please use <math>. Bob.v.R (talk) 14:05, 25 October 2009 (UTC)

To Trovatore: Perhaps DynV means that he cannot read that character with his browser. It might look like an empty box or a question mark to him. (I presume he just cut and pasted the source to get it here.) He wants to know what numerical value is the code (the Unicode) for the symbol. JRSpriggs (talk) 19:43, 25 October 2009 (UTC)

That would be 2115 (hex), or entity "&#x2115;", showing as ℕ. −Woodstone (talk) 20:31, 25 October 2009 (UTC)

I think that Addition of natural numbers could be merged here; it has very little content and I don't forsee it growing much. — Carl (CBM · talk) 16:46, 21 January 2010 (UTC)

The "definition" of addition in Peano arithmetic (although called "in the natural numbers") should be moved there or to a related article, rather than here. I think perhaps a disambiguation might be left between addition in the natural numbers, pointing here, and addition in Peano arithmetic, pointing there. — Arthur Rubin (talk) 17:33, 21 January 2010 (UTC)

The recursive definitions of addition and multiplication certainly deserve to be in Peano arithmetic, but they are already there, in the section "Arithmetic". — Carl (CBM · talk) 17:37, 21 January 2010 (UTC)

Noted, sorry. I still think that addition of natural numbers should be left as a disambiguation page after the merger. — Arthur Rubin (talk) 17:52, 21 January 2010 (UTC)

Apparently I was grumpy this morning, I apologize for that. I do think a disambiguation page would be OK, and it will be good to save the edit history in any case. — Carl (CBM · talk) 19:11, 21 January 2010 (UTC)

When someone says "smallest group containing" or "smallest ring containing" they always mean in the sense of group or ring inclusion. For example, ${\displaystyle \mathbb {Q} [{\sqrt {2}}]}$ is the smallest ring containing ${\displaystyle \mathbb {Q} }$ and ${\displaystyle {\sqrt {2}}}$, although ${\displaystyle \mathbb {Q} [{\sqrt {2}}]}$ is still a countable ring. The integers are that smallest group containing the natural numbers in the sense that for any injective homorphism φ from the natural numbers to a group, there is an extension of φ which is an injective homorphism from the integers to that group. Indeed, that integers are precisely the group obtained from the natural numbers via the standard construction of extending a commutative cancellative semigroup into a group. — Carl (CBM · talk) 13:31, 5 February 2010 (UTC)

I'm slightly amazed by the mention of Cantor and set theory in the first paragraph. What does this have to do particularly with natural numbers? The article is called "natural number", not "the set of natural numbers". And even articles that are expressly about sets don't need to mention set theory explicitly. I am tempted to throw out the phase entirely (for now I just repaired it; it was calling the numbers themselves a set, which cannot be meant here), and to reformulate the following sentence so that it avoids the term set and the curly braces. Marc van Leeuwen (talk) 12:28, 22 April 2011 (UTC)

The reason for this is mainly historical: the page used to start by talking about the set of natural numbers, without any further explanation of "set". I am not sure if it would go over well to delete this altogether, though. Tkuvho (talk) 12:36, 22 April 2011 (UTC)

(edit conflict) I agree that the name-dropping of Cantor might be distracting. Since the next sentence in the lede already linked to set, I removed the sentence about Cantor entirely. Is that OK with everyone? — Carl (CBM · talk) 12:38, 22 April 2011 (UTC)

Well, perhaps not. Do we really need a further explanation of set there in the lede? — Carl (CBM · talk) 12:38, 22 April 2011 (UTC)

The lead says "natural numbers are the ordinary counting numbers 1, 2, 3, ... (sometimes zero is also included)," but the bulk of the article uses the usual mathematical definition including zero (e.g. Natural number#Algebraic properties, Natural number#Properties, Natural numbers#Formal definitions, etc.). I suggest rewording to indicate zero is almost always included. -- 202.124.74.200 (talk) 07:29, 28 August 2011 (UTC)

The inclusion of 0 in the term natural numbers is a relatively recent development. Traditionally it was not included. I propose to leave the lead neutral about this issue. −Woodstone (talk) 15:56, 28 August 2011 (UTC)

Relatively recent, but quite common for about two centuries now, surely? And it would be nice if the lead was rewritten to be neutral, rather than taking an exclude-zero stand which contradicted the include-zero body of the article. -- 202.124.72.202 (talk) 23:32, 28 August 2011 (UTC)

How about,

"The natural numbers are inconsistently defined. This inconsistency requires those using the term to be specific about which set of numbers they mean. Some define the natural numbers as the set of counting numbers, excluding zero {1, 2, 3, 4, ...}. Others define the set as including zero {0, 1, 2, 3, 4, ...}."

This describes the inconsistency and the requirement for clarity when dealing with this set, and takes no bias in definition. Cliff (talk) 18:43, 30 August 2011 (UTC)

I am not in favour of starting out with focus on a controversy in the lead. The current phrasing is clear and neutral enough. You may want to replace "sometimes", by something stronger, like "regularly". −Woodstone (talk) 10:10, 31 August 2011 (UTC)

I made a bold change to address the problem. The article now starts: "[...] are the ordinary counting numbers 0, 1, 2, 3, ... (traditionally zero is omitted)." Hans Adler 12:52, 31 August 2011 (UTC)

I tried another bold change. I rearranged the lede to put the stuff I find more important first, and put the long paragraph about zero last. Really the issue of zero is not the first thing we want people to worry about when they read the article. I also tried tweaking the wording some. I am afraid that a naive reader may not realize that "traditionally" means "in older texts, but not necessarily in newer ones" - they may think it means "since long ago, and continuing to today", i.e. "by tradition". — Carl (CBM · talk) 13:04, 31 August 2011 (UTC)

Brilliant! It's much better this way. Hans Adler 13:30, 31 August 2011 (UTC)

The article states " Many other mathematicians also include 0, although some have kept the older tradition and take 1 to be the first natural number.", with a footnote stating "This is common in texts about Real analysis. See, for example, Carothers (2000) p.3 [5] or Thomson, Bruckner and Bruckner (2000), p.2". Thanks to the miracle of Google books, I can see Carothers, page 3, which states "n is the set of natural numbers (positive integers)". There's nothing here about zero as a natural number. - Crosbie 07:43, 1 September 2012 (UTC)

Basically, "in numbers" is used for meaning "which come in more than one". Therefore the greek numbers started with two, but this does not mean that the Greeks did not know how to handle quantities one, or zero. If we are using the word "number" as meaning "a quantity", zero has to be included. If we are dealing with real numbers by contrast, zero is just another version of the infinite. Askedonty (talk) 20:49, 5 November 2012 (UTC)

In the article it is written that ${\displaystyle \,\mathbb {N} _{0}=\aleph _{0}=\omega }$, but should it not be ${\displaystyle |\,\mathbb {N} _{0}|=\aleph _{0}=\omega }$? If it is correct, then it should be more clarified that ${\displaystyle \aleph _{0}}$ can denote also a set of that size and not just the cardinality. — Preceding unsigned comment added by Tagib (talk • contribs) 11:29, 5 February 2012 (UTC)

Please read the sentence in which that formula occurs. — Carl (CBM · talk) 20:43, 21 July 2012 (UTC)

In the article:

others use whole number in a way that includes both 0 and the negative integers, i.e., as an equivalent of the integer term.^{[citation needed]}

The Hungarian term for numbers in {..., -2, -1, 0, 1, 2, ...} is egész, which means — see a Hungarian–English dictionary — whole. So at least Hungarians tend to interpret/use whole number as integer. Consider this a citation.— Preceding unsigned comment added by 46.107.101.192 (talk) 23:22, 26 March 2013‎ (UTC)

Well, the Czech word for integer (celé číslo) also literally translates as whole number, but it’s just that: a literal translation. This does not count as a use of the actual English expression, as literal translations of mathematical terms often give nonsensical results: for example, you cannot cite German Körper as evidence that body is a valid English synonym for field.—Emil J. 12:35, 25 April 2013 (UTC)

Speaking of German, ganze Zahl also literally means whole number, of course. Integers are in fact called “whole numbers” in quite a few (most?) languages. That’s how the English term came about in the first place, as the Latin adjective integer means whole.—Emil J. 12:47, 25 April 2013 (UTC)

This article misleadingly gave the impression that counting number is always defined to include zero, that whole number is always defined to include zero (sometimes with the addition of negative integers), and that integer is sometimes defined in a different way to the usual definition. MathWorld says that there are also authors that define counting number and whole number to exclude zero, the Wikipedia article for Whole number agrees that there are three possibilities for that term, and I think there is general agreement about integer. So I have edited this article.

Note also that:

• Counting number redirects to this article.
• Whole number has a link to Natural number#History of natural numbers and the status of zero, so it is good to put all relevant information in that section.
• There are additional references in a wikitext comment within the source of Whole number.
• If there are reliable sources, this article could explain what combinations of definitions are used (i.e. to explain how authors use the terms to distinguish the different sets of numbers). This is partly done in the Notation section, but without a citation.
• Although this article mentions the usage of the definitions in set theory, logic and computer science, it says nothing about the usage in number theory.

JonH (talk) 22:08, 6 August 2013 (UTC)

I like your changes, mostly because they're simpler than the previous text. I think, after a minimal mention to clear up confusions, the less time spent on the "status of zero" and on the locutions "whole number" and "counting number", the better. So that's basically to say I'm not enthusiastic about your last two bullets above — I think we should focus on the math, not the terminology. --Trovatore (talk) 22:12, 6 August 2013 (UTC)

I see what you mean. I think it is important that readers are told about the possibility of being confused; but perhaps the details of the confusion are not so important. JonH (talk) 22:26, 6 August 2013 (UTC)

Yes, that sounds about right. --Trovatore (talk) 07:32, 7 August 2013 (UTC)

FWIW - I don't know if my edit (based on a recent WP:Reliable Source) (recently reverted by User:Marc van Leeuwen) is entirely ok or not - but seems worth a discussion:

Copied from the Natural number lead:

Interestingly, the summation of all natural numbers to infinity is "minus one-twelfth".< ref name="NYT-20140203">Overbye, Dennis (February 3, 2014). "In the End, It All Adds Up to –1/12". New York Times. Retrieved February 3, 2014.</ref>

${\displaystyle \sum _{n=1}^{\infty }n=-{\frac {1}{12}}}$

ALSO - A relevant video (07:49) by Numberphile "proving" the notion is at the following => http://www.youtube.com/watch?v=w-I6XTVZXww - A related discussion is ongoing at Talk:Infinity#Minus one twelfth - Comments Welcome of course - in any case - Enjoy! :) Drbogdan (talk) 17:10, 4 February 2014 (UTC)

This is an obvious fallacy and hence does not belong in the article. — Anita5192 (talk) 17:16, 4 February 2014 (UTC)

Agree it does not belong here. But if you read 1 + 2 + 3 + 4 + ⋯ you see that the methods of zeta function regularization and Ramanujan summation assign the series a value of −1/12.--Salix alba (talk): 16:16, 6 February 2014 (UTC)

Not this again, please. Sławomir Biały (talk) 01:13, 10 February 2014 (UTC)

Early on: 'the Peano axioms (1889) begin the natural numbers with zero'. Later: 'Peano's original formulation, the first natural number was 1'. I think the second is correct. 31.52.252.138 (talk) 20:13, 10 May 2014 (UTC)

There is no contradiction. Note the difference in tense: the formulation changed after the original formulation. —Quondum 20:50, 10 May 2014 (UTC)

Reading a little more closely, the axioms appear to be agnostic to the interpretation of the first element; it is only when one defines the operation of addition that any difference emerges. The distinction arises from the choice of definition of the addition operation. As such, it seems that this article is making untrue statements about the Peano axioms (which do not define an addition operation, only a successor function). This seems to be a significant misrepresentation; I'd appreciate comment from those more familiar with working more formally in this area. —Quondum 21:06, 10 May 2014 (UTC)

The sentence in the lede "In 1763 W. Emerson's Method of Increments contains, on page 113, the phrase 'To find the product of all natural numbers from 1 to 100 ... .'" appears irrelevant to the point being discussed. Would anyone object if I deleted it? Maproom (talk) 23:12, 30 May 2014 (UTC)

I agree that it should be deleted, for a multitude of reasons, including that it does not allow us to deduce the first natural number. The subsequent sentence, "But the Peano axioms (1889) begin the natural numbers with zero." should also be deleted, as per my argument above. —Quondum 00:33, 31 May 2014 (UTC)

Right. I have deleted both sentences. Maproom (talk) 07:54, 31 May 2014 (UTC)

Is there agreement - at least in America - about whether the natural numbers include zero? (I don't care about university-level math in this context: I want to know what to tell my students so they'll "get the question right" on the high-stakes statewide tests.) --Uncle Ed (talk) 12:12, 25 April 2013 (UTC)

I don't know about America but in Australia the students get taught that natural numbers don't include zero, because whole numbers are natural numbers INCLUDING the number zero. — Preceding unsigned comment added by 121.45.228.1 (talk) 11:23, 5 February 2014 (UTC)

No test should ever rely on this; in most countries (including the US) different people use it differently. If the test in question has ever used the concept of "natural numbers" without defining them, you should raise an official complaint. 128.112.17.131 (talk) 19:09, 11 June 2014 (UTC)

Currently the image of apples that is used to illustrate counting has two problems: The apples look too identical, so one could say it is a picture of one apple; but ${\displaystyle 1\neq 6}$. There are six apples in the picture and they could be grouped by the eye in different ways, not only the intended one. The intended way of grouping the apples as (1 single apple, a pair of 2 apples, a row of 3 apples) could be highlighted by connecting them in a colored rectangular background or other helpful way.

I thought the same thing; their (apparently exact) similarity hides the issue of "differences among identical objects", which gets into some issues about the Peano axioms and the reflexive definition of equality: "For every natural number x, x = x. That is, equality is reflexive." http://en.wikipedia.org/wiki/Peano_axioms Does 1 apple = 1 apple? What if the apples are of different size? Or of different type? Then 1 apple might not equal 1 apple ... Bruce Schuman (talk) 14:28, 3 August 2013 (UTC)

I have replaced the image by a version, helpfully created by MjolnirPants, in which the apples within each row differ. I hope this deals with the identicality issue. Maproom (talk) 06:14, 13 June 2014 (UTC)

I don't know. To me, the caption says "one apple, two apples, three apples", but the picture just shows six apples, arranged in a triangle. The idea that the one, two, three are supposed to correspond to rows of the triangle is not really obvious.

Maybe if the rows were spaced farther apart? Or if we made them different things (one apple, two cats, three waterfalls)?

But I have to say I'm a little bit skeptical as to whether this notion is really well served by an image of this sort. Our readers pretty much know what it means to count; I don't know whether they really need a picture of counted things. --06:33, 13 June 2014 (UTC)

I agree that the image does nothing to help anyone understand the article – it just makes the article look prettier. But at least it is now technically correct. Maproom (talk) 08:36, 13 June 2014 (UTC)

I can make further changes to the image if anyone thinks it would be helpful. The image is being used to illustrates sets, right? I might be able to pull off some baskets to put the apples in. MjolnirPants Tell me all about it. 12:33, 13 June 2014 (UTC)

I think baskets is a good idea. Maproom (talk) 15:28, 13 June 2014 (UTC)

 Done File:Three Baskets.svg MjolnirPants Tell me all about it. 16:25, 13 June 2014 (UTC)

Someone added a {{dubious}} tag to the following sentence, without following up on the talk page:

Today some textbooks, especially tertiary textbooks, define the natural numbers to be the positive integers {1, 2, 3, ...}, while others, especially primary and secondary textbooks, define the term as the non-negative integers {0, 1, 2, 3, ...}.

Now, lots of times I just revert drive-by tags, but this sentence really has problems.

First of all, why textbooks, specifically? This is a mathematics article; we should be talking first and foremost about what mathematicians mean, not textbooks.

Also, I don't think it's true. At least in the United States, I believe primary and secondary textbooks usually start the natural numbers with 1, whereas by the time you get to college, you have a better chance of being exposed to the more modern (zero-including) convention. It's certainly possible that that has changed since I left high school, but I doubt it.

There used to be text about which fields of mathematics were more likely to use which convention; that would at least be more interesting than the "textbooks" angle, although the problem, again, was that I wasn't quite sure it was true.

Perhaps we should just say that some authors include zero and some do not, and leave it at that? It's not as interesting, but we can at least be sure it's true. --Trovatore (talk) 20:03, 16 September 2014 (UTC)

I agree. If you want to re-phrase it the way you described, go for it. I think it will be an improvement. MjolnirPants Tell me all about it. 20:35, 16 September 2014 (UTC)

If it is in fact true that school-level textbooks generally define the natural numbers as including zero while university-level textbooks generally define them as excluding zero, this is remarkable enough that it should be mentioned in the article. Even if it is only true of US textbooks. Maproom (talk) 17:27, 17 September 2014 (UTC)

Well, I don't really agree, but we don't need to agree on that point, because it isn't true in the first place. If anything it's the reverse. --Trovatore (talk) 17:32, 17 September 2014 (UTC)

Math Pedagogy has special challenges not found in the rest of mathematics. E.g. They go in a special order. So a teacher talking about whole numbers as not having a fractional part might define the set depending on whether the concepts of negative integers, zero have been introduced. If one were to describe whole as an adjective meaning the number has no fractional part, it skirts the issue as to which set is chosen and legitimizes the definition relative to the universe of discourse currently defined. I.e. an explanation of the use of whole numbers as a didactic tool would help readers understand why there are multiple conventions for the definition of the set of whole numbers. Thomas Walker Lynch (talk) 07:23, 12 October 2014 (UTC)

There is an article on Mathematics education. --50.53.38.70 (talk) 07:40, 12 October 2014 (UTC)

great, we can tie that in. I see three aspects/responsibilities for this article:

1. correct definition
2. description of convention
3. congruency with math education

It looks like we can fulfill all three.

Thomas Walker Lynch (talk) 08:55, 12 October 2014 (UTC)

WRT whole as an adjective resulting in different sets depending on the domain of discourse, there is a wiki on domain of discourse.Thomas Walker Lynch (talk) 12:03, 12 October 2014 (UTC)

This is a nice article with historical perspective. The mood nicely matches the tone in which number theorists talk about natural numbers. However, it goes a tiny bit too far in that direction by not providing the practical information of what is the current convention at the very top of the article. However, that information was buried further down - so I moved it up. I imagine that many readers will not be so interested in the romance of natural numbers, unfortunately, and will be glad to scan down a few lines to see what they came to find, the modern convention, and then to move on. — Preceding unsigned comment added by Thomas Walker Lynch (talk • contribs) 19:34, 2 October 2014 (UTC)

Excuse me, I see the text has been reverted without discussion or comment on this talk page. I pulled text up that was already in the article and documented, so it is hard to imagine that justification for this reversion. Furthermore the text pulled up for lower down in the article is well justified. The modern definition of natural number can be found in the basis of computers science, as found in the scheme descriptions used for teaching at MIT, CMU, and many other universities, the definition is provided by Wolfram Mathematica should be instructive to modern users, and that provided by the seminal works in modern set theory are all consistent.

Given the lack of discussion on this topic, and justification, I am going to replace the changes. However, I put in a paragraph after the description of the modern convention, "heartfelt", where a person who is familiar with a school of thought which has different conventions may expand and provide information about those different conventions.

I would ask that we resolve differences in discussion rather than in clobbering my edits. Please respect my time and expertise as I respect that of others.

Excuse me for not seeing history comments on the undo, I had expected to see discussion here. Likewise you all should have seen my talk section added in that same history transcript. Let me summarize:

---

>

Mainly I take objection to the statement in the head that there is 'no universal agreeement' because

1) it is too weak and is thus meaningless. No universal agreement only means that my Uncle Stan disagrees (and he recently changed his mind). If you think the head material should not be too wordy then why have a meaningless statement there?

2) later in the article it describes a convention for the definition, I doubt many have read that far, but if anything belongs in the head, it is a description of the convention of what the darn thing is. That is what people who come to this article want to know.

3) now there is an essential contradiction in the current article, it starts by saying there is no agreement of what it is, then it says there is a convention set theorists, logicians, and computer scientists agree on. I didn't write that, rather it is in the current article. Which is it? Disagreement or a modern convention? This is confusing to say the least.

4) the zero question is obviously of central importance for this article, this is what the discussion circles around. It belongs in the head. Furthermore, counting numbers and whole numbers now redirect to this page. I came to the page though such a link, read the header material and still had no idea why I was on the redirected page. That isn't right. If pages redirect here the topics need to be mentioned. With my edit they now are.

5) the current page fails the test of my bright now high school age kid being able to make sense of it. No wonder given the above. Problem is that non-sophisticated readers are not able to weigh through the mathematical verbiage to get to the sentence about the modern convention used by "set theorists, logicians, and computer scientists" - that needs to be known sooner.

6) if there is another convention besides the one used by, "set theorists, logicians, and computer scientists" then lets hear about, rather than deleting the information that is already there.

7) The head is 11 lines long, it is ridiculous to suggest it is too long. — Preceding unsigned comment added by Thomas Walker Lynch (talk • contribs) 14:32, 3 October 2014 (UTC)

— Preceding unsigned comment added by 218.187.100.54 (talk) 07:41, 3 October 2014 (UTC)

You should sign your posts with four tildes.

All professional mathematicians are familiar with the fact that some sources consider 0 a natural number while other sources don't.

"No universal agreement" clearly means "no universal agreement among professional mathematicians". Your Uncle Stan has nothing to do with it.

"No universal agreement" does not preclude agreements in certain areas, including set theorists, logicians, and computer scientists.

There are only two possibilities, a convention that 0 is a natural number and a convention that 0 is not a natural number. It would be difficult if not impossible to list which convention every single mathematical discipline accepts.

So, the lead tells the reader that the natural numbers are the positive whole numbers, but that some people include zero while others do not. That's all most people need to know. Rick Norwood (talk) 15:01, 3 October 2014 (UTC)

Rick, I agree with that general approach, but am not too happy about the zero-exclusive approach being presented as the default, with zero-inclusive being relegated to a passing line about "some mathematicians". --Trovatore (talk) 16:41, 3 October 2014 (UTC)

Rick, as Trovatore points out your position is inconsistent, as positive whole numbers do not include 0 and that is the lead in sentence. Thanks for the note about the four tildas. Uncle Stan is in fact a professional mathematician, and just having a quick look here his publication list is longer than yours ;-). I don't understand the adversity to bringing the "convention" sentence into the lead. And you say there are more conventions than you can enumerate? Help me understand that, perhaps give me three schools of thought that have a different convention than that used by the "set theorists, logicians, and computer scientists" mentioned in the article.

as the lead goes into the box on Google, it is important to provide the most common convention in the first paragraph instead of giving a decree that natural numbers are positive whole numbers 'period'. I just noticed that is what shows in that box. I moved the convention language there, though seems the wording could be improved. If there are other modern conventions they could be given next, or a 'it hasn't always been this way' could be added. IMHO Thomas Walker Lynch (talk) 17:21, 3 October 2014 (UTC)

The whole numbers page redirects here yet there is no definition for for whole numbers found here, even worse, the definition given for natural numbers builds from it. That seems a bit stressful giving to the readers who came to wikipedia to read about whole numbers. The most recent change still has the lead favoring a definition of natural numbers without zero, leaving open only a "possibility", when in fact modern convention as described lower down in this very article is the other way around. The inconsistency is confusing. Though the counting number page redirects here, there is no definition for counting numbers separate from natural numbers yet that is a common modern convention especially when zero is included in counting numbers. Editors emphasizing a definition of natural numbers different than the convention described in the very article have yet to identify a single modern school of study that uses this different convention only saying they are too numerous to enumerate. The prior text that was 'undone' had none of these shortcomings. I fail to understand why it was deleted. What was the reason? 218.187.84.185 (talk) 21:30, 3 October 2014 (UTC)

I would reject the idea that there is just one modern convention, based on this evidence:

• The pages at MathWorld for natural number, counting number and whole number.
• The page at the Encyclopedia of Mathematics for natural number.
• My original research in a library a few months ago when I checked each number theory book that I could find by looking in the index for "natural number". I forget the details, but the books were mainly published in the last 20 years, and some of them included zero, some of them excluded zero, and others did not define the term and instead used expressions such as "positive integers".
• A Google search quickly discovers that The Princeton Companion to Mathematics (published in 2008) says on page 17: "Some mathematicians prefer to include 0 as a natural number as well: for instance, this is the usual convention in logic and set theory. Both conventions are to be found in this book, but it should always be clear which one is being used."

To establish that the position has changed since 2008 would need some evidence from reliable sources, not just giving a definition but also saying that people have stopped using other conventions.

I agree that anyone looking for counting number, natural number or whole number should quickly get a clear statement of what the phrase means. Before 14 September, there was a "disambiguation page" [6] which explained that "whole number" has 3 different meanings, and it included a link to Natural number#History of natural numbers and the status of zero. For that reason, in August 2013 I concentrated all the information in this article about "whole number" into that history section, as explained at Talk:Natural_number/Archive_2#Counting_number_and_whole_number. But since 14 September, "whole number" redirects to "natural number" and the lead now needs to contain the information. I think that from just the lead it should be clear to the reader that they should not use any of these three phrases unless they state which definition they are using. JonH (talk) 04:09, 4 October 2014 (UTC)

I've done a rewrite based on JonH's comments, and removed some sentences that were vague or meaningless: "Natural numbers remain very important in modern times." I've also removed some unreferenced claims. It seems very unlikely to me that, when mathematicians coined the phrase "natural number", they were thinking about archeology. Rick Norwood (talk) 12:10, 4 October 2014 (UTC)

Rick Norwood, Independent of your agency relative to this subject your personal doubts should not be justification for deleting others edits. Now you do paint a comical picture of mathematicians practicing archeology - but that is your picture, not the one spoken of in the text that has been deleted. No mention of the mathematician who "coined" the term, etc. was made. Why would you use your craft to ridicule the work of another editor? What purpose does this serve? And note, you deleted more than just the point you make comment to here.

You should also note that the original article made the case that natural numbers are so named as these are some how organic to human mathematics - and the editor who wrote that is absolutely and unequivocally correct about this. As this thesis was already in the article, why take it to task now rather than before? I thought the prior editor made a good point and expanded upon it showing natural numbers that earned them their moniker, but didn't want to leave a reader with the impression that natural numbers are no longer relevant. It flowed nicely into the next section. Thomas Walker Lynch (talk) 15:49, 4 October 2014 (UTC)

The "conventional definition by set theorists .. " etc comment *comes from this very article* I simply moved it up. I moved it up for reasons given above, not withstanding that the prior article contradicted itself with a misleading statement in the first sentence stating that natural numbers unambiguously started with 1. That sentence then reflected in the summary box in google searches. Now editors involved with the article before take issue with something they did not take issue with before. Again, that is peculiar. (talk) 15:49, 4 October 2014 (UTC)

The original article stated, and I believe it still states further below, "the convention among set theorists, logicians, and computer scientists is to include zero in the set of natural numbers" I did not write this, but it was in the article at a prior date when the very editors taking issue with it now were active, in my understanding. However, I believe the statement can be defended, and present that defense here. (I did not know it was necessary to do so, as it was already in the article.):

Now another editor above points out some exceptions. Of course there are exceptions as it is a convention not a law. We need a more general approach to establish the convention rather than point references for or against. Here are a list of prime movers that have lead to the convention for zero being included in the denotation of the set of natural numbers in the aforementioned fields:

1. zero is the additive identity need for abstract algebra structures. You can't have a ring or group without it.

2. modulus arithmetic has a zero at the radix value. Hence, zero comes up in polynomial generators and in many other tools used in communications theory, cryptography, compression, and in other discrete systems.

3. computers implement modulus arithmetic, and thus all software is exposed to it

4. John Von Neumen included zero in his definition of natural number and it appears in w proofs etc.

5. zero is conventionally the axis origin ever since Descartes wrote of analytical geometry

6. The cardinality of the empty set is zero

7. The universally accept "count" when no items are present is zero.

It is hard to imagine mathematics without the above 7 things - does anyone disagree with this? You do math without these things? Please be careful to understand, I list these 7 compelling forces for including zero to explain to you what has lead to the convention of including zero in the set of natural numbers. I do not write it to convince you to do it yourselves. These are some of the things that have lead many of us to find the inclusion convenient, and in turn as many people do so, there is a convention. This convention was noted by a prior editor, and already included in the article.

Thomas Walker Lynch (talk) 15:49, 4 October 2014 (UTC)

I've pointed out the inconsistency of redirecting whole numbers here, and then instead of defining them, using them to define naturals. Another editor points out above that well this was not the way he would like the situation - and then put the circular definitions back in while deleting text that provided non-circular definitions. All I can say is, 'what they hey?'two circular paths or reasoning do not a linear reasoning make .. Isn't it the case there are only two ways to fix this issue: a) provide a page for whole numbers and turn off the redirect b) define them here? I did the latter, and the editor deleted it, but he did not do the former. Am I not justified in just putting the other text back? Thomas Walker Lynch (talk) 15:49, 4 October 2014 (UTC)

There are some other problems with the current article. For example the discussion of indices is naive. Fact is today in engineering, the hard sciences, and in computer science, the most common form of indexing is zero based. One can see this for example, in the linear algebra portrayed in any circuit theory book. The i, j, possibly k, indices go from zero to size minus 1. This is abstract work, circuit theory. In applied work there this is not just a happenstance of convention, rather there is a solid reason for it. It turns out that if one has a hierarchy of indexing, then the first element of the embedded object appears at the base of the containing object. Hence, using an equation such as base + size_of_object * index, then to not waste the area of the first object we must have an index of zero. In software languages this arithmetic is typically hidden and direct indexing is used. Now there are hedges on this. It may well be that practical issues have driven the change in convention for the abstract work, but so be it. This is an encyclopedia article, not a forum for changes. Thomas Walker Lynch (talk) 15:49, 4 October 2014 (UTC)

Is there any editor here who sees fault in the reasons provided above? Please be specific in any repliesThomas Walker Lynch (talk) 15:49, 4 October 2014 (UTC)

Much of the above is beside the point (following numbering):

1. the natural numbers are a group nor a ring
2. modulus n arithmetic is about equivalence classes: 0 and n are in the same class; it is just as good to take 1 to n as representatives.
3. 0 to n-1 as result of modulo division is an arbitrary software implementation choice (see line above)
4. many other mathematicians exclude 0
5. cartesian coordinates: this is rather more about real numbers, not naturals
6. empty set has no members
7. the acceptance of 0 is the first extension of the natural numbers

Conclusion: there is really no convincing argument to say that inclusion of 0 is conventional; many sources do not include it. The lead should make it clear from the beginning that there is no agreement whether to in- or exclude 0.

−Woodstone (talk) 16:20, 4 October 2014 (UTC)

I have restored the lead from before the recent thrashing about; I think it is better, or at least no worse, than any recent version. There is a preference for stability; changes, especially to the high-profile parts of the article, ought to be active improvements, or we should revert to the status quo ante.
That is not to say it can't change, but please, let's discuss changes incrementally and in detail. If there is a proposal for a non-incremental change, then please make the proposal on the talk page and wait for consensus. --Trovatore (talk) 16:56, 4 October 2014 (UTC)

OK, let's start with these problems with the current version:

• Counting number and Whole number, which are redirects to this article, do not appear in boldface in the lead per WP:R#PLA.
• The lead does not explicitly list the natural numbers in the first sentence. The quibble about whether zero is included is not mentioned until the third paragraph, but ${\displaystyle \{1,2,3,...\}}$ are always considered natural numbers, so the first sentence should say at least that much. And the quibble about zero should be supported by reliable sources.

--50.53.61.13 (talk) 17:22, 4 October 2014 (UTC)

Mr. Woodstone, 1) of course the natural numbers are not an algebraic structure to themselves, rather they are often the set elements over which such structures are built, and an additive identity is required. Hence anyone working in this area will include zero with their natural numbers. 2) Modulus arithmetic is a tool used in many areas of applied mathematics, I know I've been using it for decades. Yes, you can think about a modulus operation on a larger range number as creating equivalence classes, but that is does not change the fact that the most common convention by far used in such problems includes zero in the natural numbers. Also it is an ancillary observation rather than the answer to a given problem. 2) Yes, 0 to n-1 may be an arbitrary choice, but the point is, it is the arbitrary choice used. Please remember the point is about the common convention. "many other mathematicians exclude zero" I have twice in the talk pages above asked for a school of mathematics that does this, no example has been given. I'm not saying they don't exist, but given this many pages of talk about the subject and all these arbitrary deletes of my contributions - you would think someone would mention one or stopped deleting contributions. Anyway when one is pointed out we can add it to the article! *) many mathematicians exclude zero -- of course conventions are not universal, but whoever does this avoids the things in the 7 points I listed. Perhaps that is ok for the special problem area - if so that doesn't change what the common convention is. 5) I admit the point about analytical geometry is not a strong one, but people do by convention put axis at zero, and many mathematicians working in discrete math do make plot or create distance metrics. It is common to see scatter plots expecting all numbers to be above a horizontal line at zero or some such. Sometimes these represent error. 6) yes very good that is the point, the empty set has no members and its cardinality is *zero*. Hence any set theory type proof dealing with cardinality includes zero in the natural numbers. 7) yes natural numbers have been extended to include zero as the most common convention among set theorists, logicians, and computer scientists - as the prior editor wrote into the article as it was when I first came across it.

Should note, even if you chose a different set of digits instead of 0 to n-1, say n to n + r -1, you would still have a zero, it is just that your zero in the modulus. E.g. in modulus 10 if you chose to use 'a' - 'j' as your digits, 'a' would be your zero.Thomas Walker Lynch (talk) 12:04, 7 October 2014 (UTC)

Mr. Woodstone, abstract algebra, number theory, set theory and logic are the backbones of discrete math. You can't say we can take those out and it has no effect on the convention. Many many people work in these areas. Nor can you point out a few sources and say that a convention does not exist, as I can and have pointed out sources too, and there are those on the article. In order to establish or refute the existence of a convention will require a more general argument. I humbly submit, as described in detail in the prior paragraph, that your conclusion does not follow from your argument. It is not even close. Though please, if you see a flaw in the reasoning in my reply, please point it out. Please be very specific. Thomas Walker Lynch (talk) 17:33, 4 October 2014 (UTC)

Mr. Travoltore, you make an argument that stability is important - and then completely change the page. That is a bit confusing. You have offered direction for editing the page from it changed state. What was wrong with what was there? The last edit only changed whole to integer and swapped the order of the 'trivial' as you say inclusion of zero or not. Hey guys, this is beginning to look like you have a vested interest in the old text. Level with me, have you published something that you are trying to get support from the wiki pages for? Do you have a multipage revision plan I don't know about. As I am really confused by this last revision. The thing I would like to know first is how the whole number circular definition thing is to be fixed and why you reject the use of integer in the definition in its place. Thomas Walker Lynch (talk) 17:54, 4 October 2014 (UTC)

Please get consensus first for major changes. --Trovatore (talk) 04:22, 5 October 2014 (UTC)

OK, this behavior is totally unacceptable. Get consensus first. --Trovatore (talk) 04:41, 5 October 2014 (UTC)

You are absolutely correct. Crude reversions are no substitute for reading the edit history and looking at the diffs. Revert to this edit. --192.183.212.185 (talk) 04:58, 5 October 2014 (UTC)

Oh, I see now. I am not Thomas Walker Lynch. And I am only insisting that the citation that was added in this edit be preserved in your reversion. --192.183.212.185 (talk) 05:07, 5 October 2014 (UTC)

Fair enough. Would you go ahead and do it, please? I have reached 3RR. --Trovatore (talk) 05:15, 5 October 2014 (UTC)

Done. Is this diff what you expect? (NB: My IP address changed after I went offline.) --50.53.33.231 (talk) 06:17, 5 October 2014 (UTC)

I think for the discussion of the lead we should start further back, before all this started. At 2014-09-30T04:58:49 the lead looked like this:

"Whole number" redirects here. For other uses of the term, see Integer.

In mathematics, the natural numbers are those used for counting ("there are six coins on the table") and ordering ("this is the third largest city in the country"). These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively. A later notion is that of a nominal number, which is used only for naming.

Properties of the natural numbers related to divisibility, such as the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partition enumeration, are studied in combinatorics.

There is no universal agreement about whether to include zero in the set of natural numbers. Some authors begin the natural numbers with 0, corresponding to the non-negative integers {0, 1, 2, 3, ...}, whereas others start with 1, corresponding to the positive integers {1, 2, 3, ...}.

Rather concise and clear. Missing is the mention in bold of "whole number" and "counting number" which redirect here. What else exactly is wrong with this as a lead? −Woodstone (talk) 06:08, 5 October 2014 (UTC)

Could you please insert a link to the exact version you pasted? Comments:

• A list of the first few natural numbers should appear in the first sentence, since that is the most concise description possible. Compare the German and Italian versions.
• The lead should explain why they are called natural numbers. Instead it confusingly refers to counting. Is counting supposed to be natural?
• This sentence is fuzzy, pretentious, and too technical: "These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively." Grade school students should be able to read and understand the lead.

--50.53.33.231 (talk) 06:46, 5 October 2014 (UTC)

This lead is improved over what was here when I first saw the page. When I first saw the page it redirected from whole numbers, defined naturals in terms of them, with only the definition of naturals starting from one given. Also when I arrived on this page there was a very useful statement about modern convention. I copied it up, and it was deleted. Since then the others editors work using that sentence has also been deleted. In the article there were also some mistaken 'facts' given such that indices always start from one and a denial of the convention used in engineering and sciences. When I corrected those, the material as a whole was deleted rather than revised. I.e. this has been going on in the article as a whole, not just the lead. Thomas Walker Lynch (talk) 07:51, 5 October 2014 (UTC)

currently whole numbers redirects to this page, but they are not discussed. This could affect many readers, it is highly disrespectful to them. I changed the redirect to go to integers where whole numbers are discussed, and I see this morning that redirect has been deleted. I am putting the redirect back to integers. It can be changed later should a definition for whole numbers be added to this page.

218.187.84.185 (talk) 07:46, 5 October 2014

I tried to update the whole number redirect but it appears to be locked, so placed the blurb about counting and whole numbers for the sake of redirected readers. Thomas Walker Lynch (talk) 08:14, 5 October 2014 (UTC)

for the lead, it would seem appropriate to start the lead with the definition given by John Von Neumen, 0,... as used in set theory and number theory, as all major branches of mathematics today are founded upon set theory and definition of natural numbers is so important to number theory. A prior editors statement would be very useful in the lead for readers who come to this page, he wrote: "Including zero in the set of natural numbers is convention among set theorists, logicians, and computer scientists." I would suggest instead, "It is the convention among ..." and following with an explanation that the convention is not universal. We would further this explanation by explaining when zero is useful, as in the list of 7 fundamental reasons given above in this talk pages, and when it is not, for example when division by zero would unnecessarily become a burden. Such solid information would enrich the readers with a useful encyclopedia page.Thomas Walker Lynch (talk) 07:51, 5 October 2014 (UTC)

Hi Tom, I am not as prolific as you are, but I would like to comment on the set-theoretic definition by von Neumann (check your spelling). This would be inappropriate in the lead as it is too technical. Tkuvho (talk) 07:55, 5 October 2014 (UTC)

hello, well we don't need a detailed description, rather providing {0,1,2..} would suffice. The implication of the 7 items above (1. additive identity for algebraic structures, etc. as listed above) have lead to a more common definition for natural numbers. Also see my more general next remark. Oh, also note, please don't confuse 'prolific'for arguing with ghosts: my edits were consistently deleted within minutes of making them, typically with no explanation, or in some cases as you see above, with concise explanations being blown off.Thomas Walker Lynch (talk) 08:34, 5 October 2014 (UTC)

there is already another page on the set-theoretic definition of natural numbers (attempts to add it have been deleted). So a providing Von Neumann's definition could simply be linked to that, and other pages on number theory, computation theory, etc. Thomas Walker Lynch (talk) 08:45, 5 October 2014 (UTC)

The current language of 'universal agreement' carries no information, and the dearth of information creates the appearance of arbitrariness where one does not exist. A single exception negates universal agreement. What would give the reader information is a description of important cases of natural numbers defined one way or the other, and explaining conventions. Last night I added a paragraph with links to other wikipedia pages on fundamental subjects in mathematics that employee natural numbers. This took a while to create but it was deleted in less than five minutes with no explanation given - so count this paragraph among the ghost responses. Looking here, the prior editors sentence concerning conventions is still there, but it has been weakened and given a preface about the 19th century.Thomas Walker Lynch (talk) 08:34, 5 October 2014 (UTC)

"Last night I added a paragraph ..." Could you insert a link into your comment to the version you are referring to? --192.183.213.187 (talk) 03:48, 6 October 2014 (UTC)

Done already; see first line of section (after date-time). −Woodstone (talk) 04:14, 6 October 2014 (UTC)

Thanks, but I was asking Thomas for a link to his version, so that editors could comment on it. --192.183.213.187 (talk) 04:39, 6 October 2014 (UTC)

Hello ah .. I don't know how to make such a link .. The text in question had links to other wikipages set-theoretic natural number, number theory etc. and gave the conventional used on those pages (they were all {0,1,2...}). Let me ask again, is there anyone here who knows of a field of mathematics where the convention differs? (I can imagine problems involving division where I wouldn't want to have zero in my set, and it certainly is legal to take it out, but that does not negate convention. Also, it is possible to define the set of natural numbers different in a problem that one is working on, from the definition that was used for founding the set theory or number theory upon which the solution is being built. But these are not conventions.) In any case it looks like the conversation developing in the next section is going to bring is to a very nice lead sentence that has even stronger references than links to other wikipages. Thomas Walker Lynch (talk) 16:37, 6 October 2014 (UTC)

The properties paragraph has bubbled up above the definition again. Shouldn't the set be defined before its properties are discussed? Thomas Walker Lynch (talk) 16:40, 6 October 2014 (UTC)

One of the things that has come to light in the discussion below is that there are many redirects to this page. Does anyone know how to list them all instead of just running into them?

In light of the redirects for counting and whole numbers, I suggest that the title of the article be changed to "Counting, Whole, and Natural Numbers" Thomas Walker Lynch (talk) 21:37, 11 October 2014 (UTC)

Before proposing a title change, it would be a good idea to read Wikipedia:Article titles. --50.53.49.222 (talk) 14:17, 12 October 2014 (UTC)

I've gone over that, the title fits the guidelines of "short, natural, and recognizable" quite well, indeed, we have "countable, whole, and natural", which is about the same thing ;-)Thomas Walker Lynch (talk) 14:58, 12 October 2014 (UTC)

The title "Natural number" is more concise than "Counting, Whole, and Natural Numbers". These are concise article titles: China, Soviet Union, United Kingdom, United States. --50.53.41.238 (talk) 19:27, 13 October 2014 (UTC)

The current title is short, but it is not accurate. The title on the soviet union is not also the article on India. It is clear from DLazards recent Whole number edits that he would like to define whole numbers, counting numbers, as identical to the set of natural numbers, yet there has been no discussion of this. I would expect he would be strongly opposed to a title change for this reason.Thomas Walker Lynch (talk) 04:46, 14 October 2014 (UTC)

Please focus on content, not on editors. There are only two sentences in the lead concerning "whole numbers" and "counting numbers", and neither contains the word "identical". Nor do the cited sources say that. Both terms are ambiguous, and the two sentences reflect that. The words "... are also used ..." seem too strong, though. Can you suggest a way to fix that? --50.53.240.29 (talk) 05:24, 14 October 2014 (UTC)

"The terms whole number and counting number are also used to refer to the natural numbers" - no, the 'counting and whole numbers are examples of natural numbers' or perhaps 'counting numbers and whole numbers are natural'. We agreed to leave the lede alone until there was consensus. You don't have consensus. I will open a section on the equivalence of these sets so it can be discussed explicitly. DLazard has stated equivalence (I have no doubt he wouldn't hesitate to correct me, he isn't a man I perceive to need help in these matters), he also sourced the edit, how many times does to my name appear on this talk pages in similar circumstances?Thomas Walker Lynch (talk) 05:50, 14 October 2014 (UTC)

Ok, Dr. 50.53, I read that \{\{DISPLAYTITLE:Counting, Whole, and Natural Numbers \}\} (do not include backslashes .. and why do they show ???) would change the title heading on the article, but it does not. Is that formulated incorrectly? What is the interaction between the page name (appears in the link), the article title that displays, and the search box?Thomas Walker Lynch (talk) 15:21, 13 October 2014 (UTC)

The most beautiful introductory sentences I know of on the subject of cardinality:

A flock of four sheep and a grove of four trees are related to each other in a way in which neither is related to a pile of three stones or a grove of seven trees. Although the words for numbers have been used to

state this truism on the printed page, the relationship to which we refer underlies the concept of cardinal number. [S. C. Kleene, Introduction to MetaMathematics. New York: Elsevier Science Publishing Company, Inc., 1952-1991.]

Perhaps we should mention cardinal in our article, though this reaches beyond the countable sets.

I propose this as a new lead <-->

Counting Numbers, Natural Numbers, and Whole Numbers

The set of counting numbers are those used for counting objects, {1,2,3, ..}. Early on people realized that a flock of four sheep and a grove of four trees are related to each other in a way in which neither is related to a pile of three stones or a grove of seven trees.[] They began counting using marks on sticks and and piles of stones, and eventually with primitive numeral systems, and today with the Arabic numerals[][]. In modern times it is common to include zero in the set of counting numbers so as to have a count when no objects are present to be counted, for example, the count for a blank counting stick or an empty pile of rockszerohistory of numbers.

In 1899 the Mathematician Peano formalized and abstracted the concept of counting in mathematics by introducing the Peano Axioms.abstraction[] The Peano axioms give criteria for building a wide variety of countable sets that may be placed into correspondence with the set of counting numbers. We call such sets Natural numbers. Examples include the counting numbers, and other sets such as {{},{{}}, {{{}}}, ..}, (for an explanation of this latter set see Set-theoretic definition of natural numbers).

Whole numbers are numbers that can be counted to. Depending on the domain of discourse these may be the counting numbers, with or without zero, or when considering counting backwards is also allowed (using the Peano axioms with a start number of -1, and a successor function of n_{i+1} = n_{i} -1, the set of integers as a whole.

Cardinal numbers is a generalization of natural numbers used for counting the elements in a set.

Arithmetic is defined on top of counting[], and Algebra is defined on top of arithmetic.

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Each of the [] are references already mentioned on this talk page, which will be gathered when the text is placed into the article. As of the date of this writing, there are no open threads with citations counter to the statements of this proposed lead, (or provide relevant links here). Thomas Walker Lynch (talk) 05:12, 15 October 2014 (UTC)

Hi Thomas. I haven't read the whole proposed lede that you have written above, but you start the lede off with a story. This doesn't seem to get to the point the way the current lede does. The current lede delineates several definitions. That is helpful. I see your lede lacking in this respect.174.3.125.23 (talk) 09:44, 15 October 2014 (UTC)

Great, thanks for the constructive critism on that. The current lead speaks of counting apples and I like the idea of appealing to the mathematical philosophy of naturalism as a basis. The story is borrowed from a quote from Kleene who was influential in computation theory and wrote on meta-mathematics and the foundation of arithmetic. Here is is v2:

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Counting Numbers, Natural Numbers, and Whole Numbers

{quote|A flock of four sheep and a grove of four trees are related to each other in a way in which neither is related to a pile of three stones or a grove of seven trees. -Kleene [Kleene citing]}

The counting numbers are those used for counting objects, {1,2,3, ..} and have their origins in the earliest of mathematics. [link to history section, history of numbers wiki]. In modern times zero is often included in the set of counting numbers so to express a count when no objects are present to be counted.

In 1899 Peano formalized and abstracted the concept of counting.[Peano citations, abstraction citations] The Peano Axioms give criteria for building a wide variety of sets that may be placed into correspondence with the set of counting numbers. We call such sets Natural numbers. Examples include the counting numbers, e.g. N={0,1,2 ..}, and other sets such as N={{},{{}}, {{{}}}, ..}, [ link to the Peano section, to the Set-theoretic definition of natural numbers wiki, the transfinite numbers wiki, and Peano citations].

Whole numbers are numbers that can be counted to starting from zero, i.e. the same as counting numbers. Should zero be in the domain of discourse it is taken to be a whole number. If negatives are in the domain of discourse, see integer negative counting numbers are also taken to be whole. It is sometime observed that whole numbers are those that do not have a fractional part, but the concept of fraction part belongs to higher level constructions such as rationals or reals, so this observation cannot be the basis for a definition.

Arithmetic is constructed upon natural numbers, and Algebra is constructed on top of arithmetic.

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This work? Note also the title change.

Thomas Walker Lynch (talk) 04:16, 16 October 2014 (UTC)

We generally todo not start an article with a quote because we state facts, and quotes without an introduction does not provide the quote with context. As for the article title, the redirect system is in place so article titles are do not become infinitely long.174.3.125.23 (talk) 11:54, 16 October 2014 (UTC)

Ok, quote goes to the intro of the history section .. Quoting Kleene on counting is similar to quoting Einstein on physics.

I'll take up the title change discussion later.

addressing your comments, and some minor text changes:

<--> Natural Numbers

The counting numbers are those used for counting objects, {1,2,3, ..} and have their origins in the earliest of mathematics. [link to history section, history of numbers wiki]. In modern times zero is often included in the set of counting numbers so a count can be given when no objects are present to be counted.

In 1899 Peano formalized and abstracted the concept of counting.[Peano citations, abstraction citations] The Peano Axioms give criteria for building a wide variety of sets that may be placed into correspondence with the set of counting numbers. We call such sets Natural Numbers. Examples include the counting numbers, e.g. N={0,1,2 ..}, and many other sets, such as N={{},{{}}, {{{}}}, ..}, [ link to the Peano section, to the Set-theoretic definition of natural numbers wiki, the transfinite numbers wiki].

Whole numbers are those that can be counted to when starting from 1 and counting by 1, i.e. the same as counting numbers. Should zero be in the domain of discourse it is taken to be a whole number. If negatives are in the domain of discourse, see integer, the negative counting numbers are also taken to be whole. It is sometimes observed that whole numbers are those that do not have a fractional part, but the concept of fraction part belongs to higher level constructions such as rationals or reals, so this observation is not taken as a basis for definition.

Arithmetic is constructed upon the formalization of natural numbers, and Algebra is constructed on top of the formalization of arithmetic.

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this work?

Thomas Walker Lynch (talk) 03:54, 17 October 2014 (UTC)

Not quite. Our article states as one of the first few words "natural numbers". The reason is because the title of the article is about natural numbers. "natural number" is the more widely used term. We should indicate such prominence with it being some of the first words that appear in the article.174.3.125.23 (talk) 05:37, 17 October 2014 (UTC)

There has been a proposal to include a brief summary of von Neumann's definition of natural numbers in the lede. I would like to invite editor comments on this. I personally feel that natural numbers are prior to set theory as far as most readers of this page are concerned, and therefore including such material in the lede is not helpful. Including it later in the page may be appropriate. The set-theoretic definition of natural numbers serves the role of including them as part of the larger picture of modern mathematics, but this is not necessarily the role this page should play primarily, because it addresses a larger audience. Tkuvho (talk) 10:00, 5 October 2014 (UTC)

"The lead serves as an introduction to the article and a summary of its most important aspects." There is a whole section on "Formal definitions", so the lead should mention them. Here is a start: "The natural numbers can be formally defined in several ways." --192.183.213.187 (talk) 05:15, 6 October 2014 (UTC)

Actually there is something odd about the "formal definitions" section. It opens with the disclaimer "Main article: Set-theoretic definition of natural numbers" but then goes on to list the Peano axioms, which are certainly not a set-theoretic definition of natural numbers. There seems to be a confusion between a syntactic approach (Peano axioms) and semantic approach (set-theoretic construction e.g. von Neumann). Tkuvho (talk) 10:46, 6 October 2014 (UTC)

Thanks for pointing that out. The {{Main|Set-theoretic definition of natural numbers}} template should probably be moved into the "Constructions_based_on_set_theory" subsection. Also, the term "standard construction" is unsourced. --50.53.52.64 (talk) 12:09, 6 October 2014 (UTC)

Yes, I agree that the Peano Axioms which formally define arithmetic (and thus are important to computation theory and computer arithmetic) are *not* the same as the set-theoretic definition. Note there is a wikipage on the set-theoretic definition for natural numbers and John Von Neumann's definition can be found there. It would seem that one could then simply mention it inside square brackets without having to explain it. Thomas Walker Lynch (talk) 15:21, 6 October 2014 (UTC)

excuse me, I mean to say that the two derivations are different, they both arrive at the same resultThomas Walker Lynch (talk) 07:38, 7 October 2014 (UTC)

The Peano Axioms are not Peano's Axioms. He had nine, and did not specify a first number, just that there existed a number. The modern five Peano Axioms are named in honor of Peano and do begin with 0.Rick Norwood (talk) 13:34, 6 October 2014 (UTC)

That's interesting and should be mentioned at Peano axioms. Tkuvho (talk) 13:49, 6 October 2014 (UTC)

Peano's axiom 1 is "1 ∈ N". (Arithmetices principia: nova methodo exposita (1889), p. 1) --50.53.52.64 (talk) 14:40, 6 October 2014 (UTC)

Yes, at the library today I ran into quotes from the "La première version du system d'axioms de Peano" of 1798 in Jean Dieudonné's book. The first version of the axioms started with 1, but it was an evolving work, a second version soon after the first, the addition of zero and changes in the list of axioms themselves to arrive at the modern version.

There is another formal definition for Natural numbers given by George Pólya explained in "Ein Jahrhandert Mathematick 1890-1990", in the paper with the most appropriate title for this talk page! "Ideen Zur Abzahlung", the naturalichen zahlen: f:I -> N, f(i) = |Si| (that is S subscript i). That is to say he defines naturals as the absolute value of the integers. Thomas Walker Lynch (talk) 15:21, 6 October 2014 (UTC)

Hence all three modern formal definitions describe the same set. {0,1, 2, ...}Thomas Walker Lynch (talk) 15:27, 6 October 2014 (UTC)

Good work on your research. The year "1798" cannot be correct, since Peano lived from 1858 to 1932. Please clarify. --50.53.52.64 (talk) 15:41, 6 October 2014 (UTC)

Oh gosh thanks for pointing that out. Excuse me, I mixed up my notes, that is the date of the "Théorie de nombres", "publie pour le premier fois en 1798" by Legendre .. which I am looking for right now. — Preceding unsigned comment added by Thomas Walker Lynch (talk • contribs) 15:54, 6 October 2014 (UTC)

There is a link to Peano's book on the first point in this list. Yes the date shows 1889. There is a library check out stamp from 1903 on the second page - check it out! Thomas Walker Lynch (talk) 16:10, 6 October 2014 (UTC)

Hey look, there is already a wiki page for the Peano axioms, so that just should be referenced not reproduced on this page. https://en.wikipedia.org/wiki/Peano_axioms Thomas Walker Lynch (talk) 16:16, 6 October 2014 (UTC)

Ok building from "Special:Contributions/192.183.213.187" suggestion .. "The natural numbers can be formally defined in several ways." How about a lead sentence of: "The set of natural numbers can be formally defined using Peano Axioms, Set Theory, with the naturalichen zhalen to be {0,1,2..}" Where links are provided to https://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers, and https://en.wikipedia.org/wiki/Peano_axioms, and also to books and articles as mentioned above. This would follow with some statement that mathematicians are free to adopt their own conventions as is convenient for the problem they are working on and discuss the pros and cons of putting in zero (see the list of 7 above). The article would *not* have a formal definition section, as that is already covered by the other pages which are linked. We would have a section on the etymology of the term (see below). And as Poincaré and others argued the numbers come from the psyche, are somewhat intuitive, or God given, it would make sense to have a section on the intuitive meaning of natural numbers that pleases the elementary text writers (which was alluded to in an earlier version of the text) and even talk about how they came about 'naturally' so as to please Mr. Poincaré . Thomas Walker Lynch (talk) 17:18, 6 October 2014 (UTC)

Jean Dieudonné wrote a lot of books. Could you be more specific about where "La première version du system d'axioms de Peano" was published? --50.53.43.85 (talk) 00:58, 7 October 2014 (UTC)

This one is online: "Diskrete Mathematik: 1. Ideen Zur Abzählung" by Martin Aigner in Ein Jahrhundert Mathematik 1890-1990: Festschrift zum Jubiläum der DMV. --50.53.43.85 (talk) 01:57, 7 October 2014 (UTC)

Yes that is an image of the book I was referring to. ISBN 3-528-06326 matches up. See page 85. Thomas Walker Lynch (talk) 07:02, 7 October 2014 (UTC)

Thanks. In his book Discrete Mathematics, Aigner calls ${\displaystyle f}$ the counting function and denotes the codomain by ${\displaystyle \mathbb {N} _{0}}$. (p. 3) Can you figure out where he defines ${\displaystyle \mathbb {N} _{0}}$? --50.53.43.85 (talk) 07:45, 7 October 2014 (UTC)

In the German language text I was indexed to, and read there was a "naturlichen zahlen" function, and I understood integers going to naturals through the absolute value as part of the definitions, but seeing the English reference I believe my translation to be in error. ${\displaystyle \mathbb {N} _{0}}$ rather it must be understood from the notation.Thomas Walker Lynch (talk) 18:18, 7 October 2014 (UTC)

Unfortunately, I can't really help with a translation, but in Ideen Zur Abzählung, Aigner calls f "die Zählfunktion", so I am guessing that is translated in his book as "counting function". There isn't an article called counting function, but "the notation |A| means the number of elements in the set A." (Aigner, Discrete Mathematics, p. 3) --50.53.35.229 (talk) 19:25, 7 October 2014 (UTC)

If you were looking at "z. B. die natürlichen Zahlen", I believe that translates as "e.g. the natural numbers", so Aigner is giving an example of an index set ("eine Indexmenge"). --50.53.35.229 (talk) 20:19, 7 October 2014 (UTC)

Yes, I wish this citing had been a third construction example, but it is not.— Preceding unsigned comment added by Thomas Walker Lynch (talk • contribs) 00:13, 8 October 2014‎ (UTC)

Dieudonné work on the history of mathematics: Abrégé d'Histoire des Mathematiques, par Jean Dieudonné avec l'assistance de Pierre Dugac. See page 333. "La première version du system d'axioms de Peano .. traduite en langue modern"Thomas Walker Lynch (talk) 07:15, 7 October 2014 (UTC)

Thanks. The French WP has an article on the Abrégé d'histoire des mathématiques. Can you tell if this Google books preview is the same? --50.53.43.85 (talk) 08:20, 7 October 2014 (UTC)

The article on Pierre Dugac lists a German translation: Geschichte der Mathematik 1700-1900. Ein Abriß. (1985) --50.53.43.85 (talk) 09:37, 7 October 2014 (UTC)

https://fr.wikipedia.org/wiki/R%C3%A9f%C3%A9rence:Histoire_des_math%C3%A9matiques_(Dieudonn%C3%A9_(dir.)) My notes show the book the page number above appears from is ISBN 2 7056 5871 8 .. which is not identical to the one shown at the link, but that link also lists a 2 volume version apparently of the same book the ISBN I've noted being the ISBN for the second volume. I will go by the library tomorrow and try and clarify. I apologize for not being more attentive to the details of a two volumes. The Google Books excerpts you provided are apropos but google won't let read far enough down to know if they are the same. When I click or try to pull up the page it does nothing. Thomas Walker Lynch (talk) 10:50, 7 October 2014 (UTC)

Thanks for the link. The one-volume edition could be an abridgment of the two-volume edition. Google Books is displaying a snippet view of Abrégé d'histoire des mathématiques. That is all the publisher wants you to see. Sometimes, if you try a search term near the top or bottom, they will display different snippets. --50.53.35.229 (talk) 19:56, 7 October 2014 (UTC)

I have placed the pages from Diudonné on one of my webservers, see http://www.thomaswlynch.com/dieudonne.pdf There is no link, you have to type the full URL, the tab may say error 403, you can ignore that.

Note on page 333: "Dès le tome II de son Formulaire ([214], 1897-1899), Peano substitue l'ensemble N de tous les entiers naturels à celui N* des entiers positif non nuls, le 0 au 1 dans l'écriture des axiomes...", so I understand

"In the second edition of his Formulaire ([214], 1897-1899), Peano substituted all N of all the naturals and used N* for the entire positive non null, .." I.e. This is very important for the article, not only because Peano put 0 in N, and uses special notation for leaving out zero, but also because Dieudonné freely refers to N with zero in it as the Naturals. I sent this to a French colleague to verify the translation.

Indeed, that colleague pointed me at a wonderful reference that has a chapter on Natural numbers, I've cited it in the Etymology section.Thomas Walker Lynch (talk) 18:10, 8 October 2014 (UTC)

yes another French colleague confirms the translation from French showing Peano adn Dieudonné considered Naturals to include zero, he provides this: "In the second edition of his Formulaire ([214], 1897-1899), Peano substitutes the set N of all the Naturals Numbers for the set N* of the Whole Numbers positive non null, the 0 or 1 in the writing of the axioms, is t.." Thomas Walker Lynch (talk) 19:32, 8 October 2014 (UTC)

Thanks for all the excellent sources. The 1901 edition of Peano's Formulaire de Mathématiques uses ${\displaystyle \mathbb {N} _{0}}$ and ${\displaystyle \mathbb {N} _{1}}$. (p. 4, 39, 212) In his Preface, Peano says: "Selon l'order chronologique, les premiers symboles sont les chiffres 0, 1, 2,... dont l'origine est très ancienne." (p. iii) --50.53.47.9 (talk) 22:50, 8 October 2014 (UTC)

There is now an entry in the formal section on the Peano axioms that changes the standard modern definition of the first axiom and replace it with "starting from any number". I do not believe you can have an arithmetic without an additive or potentially under this definition a multiplicative identity. In any case it is a non-standard definition. Does anyone have a modern citation to a Peano axioms without 0? Furthermore we should not be competing with the editors of the other wikipage dedicated to that topic by putting a rivaling different definition here. I would like to delete the formal section and leave links to the appropriate other wikipages on the topics. Can anyone provide a reason not to do this? Thomas Walker Lynch (talk) 11:01, 7 October 2014 (UTC)

OK, I'm running to catch a plane right now so I don't have time to check what's happened recently; bear with me if I say something that's been made irrelevant by events.
But I want to remark on the notion of the natural numbers being "defined by the Peano axioms". No. The axioms do not define the natural numbers. It's true that in second-order logic, the original Peano axioms (allowing induction on arbitrary properties, not just ones defined by first-order formulas) do determine the natural numbers up to isomorphism. However, that's anachronistic; the natural numbers were understood before the Peano axioms, and second-order logic is a more advanced notion than the natural numbers.
As to whether the von Neumann definition should be included in the lead — it's not completely implausible that there could be a passing mention, as part of the summary of the formal notions. But I don't see the need for it, and I don't think we should strain to include it. In purpose, it is not so much a "definition" per se as it is a way of coding the naturals into the language of set theory, so that the machinery of set theory can be applied to the naturals. That's a very useful thing to do, and the definition is useful to that purpose. However, it is not foundational to the concept of the naturals. --Trovatore (talk) 10:59, 7 October 2014 (UTC)

This observation is similar in spirit to what I mentioned above about being able to define the set as one finds convenient for the problem one is working on. For example in the Real Analysis text book cited, Carothers desires to handle the Cardinality of the empty set separately, and thus zero cardinality is also separate. (However, on p18 there he does not define natural numbers, nor does he exclude the possibility of 0 from being in the natural numbers, rather he only uses them from 1. This is in fact a false citing and should be removed. .. p18 is what opens when the link is clicked on .. though I see it says p3 mentioned in the reference .. see if google shows that.) Though I suggest that an author could define Naturals to start from 1, or even start from 2, as the Greeks did, and I suggest saying that. Thomas Walker Lynch (talk) 11:51, 7 October 2014 (UTC)

Rather I am making a very harsh allegation against your wikipage -- that you [all or some] wrote zero out, provided a first sentenced that did this directly, used circular definitions, ignored other wikipages that had definitions that contained zero - going so far as to create rivaling material to other wikis, and deleted meaningful contributions of others so as to keep a false thesis.

Now we have an opportunity to provide a new lede and new article that stops the misplaced rivalry by providing links to those other pages, gives people real information about conventions used in important areas of mathematics, and has solid references. The question is the wording. What is on the page is already much improved, but we can do better. Thomas Walker Lynch (talk) 11:51, 7 October 2014 (UTC)

Rather I am making a very harsh allegation against your wikipage -- that you [all or some] wrote zero out, provided a first sentenced that did this directly, used circular definitions, ignored other wikipages that had definitions that contained zero - going so far as to create rivaling material to other wikis, and deleted meaningful contributions of others so as to keep a false thesis.

I have three points to make with regards to this:

1. You should pay attention when people attempt to correct you on your use of indenting. The conventions exist for a reason, and it can be difficult to parse to whom you are replying when you don't follow it. The convention is described at WP:INDENT, but in case you won't or can't follow the link, I will summarize: Keep replies indented one level from the comment they are replying to. You can indent replies by appending a number of colons equal to the number of colons in the comment you are replying to plus one to the beginning of each line of your reply. If the amount of indenting becomes too severe, you can use the template {{Outdent}} to reset the level of indentation. More information about this template can be found at its description page: Template:Outdent.
2. I don't understand where this sentiment comes from. The lede of this article had mention of the natural numbers possibly including zero since well before you began editing here.
3. You need to read the following: WP:NPA and WP:FOC. Your comments here are out of line and not conducive to a civil discussion. Accusing others of purposefully sabotaging wikipedia articles for any reason does nothing to further the discussion, and only serves to derail it. Furthermore, it will not accomplish anything; as there is no administrative action to be taken against those who have a bias, whereas those who engage in personal attacks against others can be blocked or banned from editing. This method of debate on your part can only lead to administrative action taken against you, not to a better article. MjolnirPants Tell me all about it. 13:41, 7 October 2014 (UTC)

Please excuse me, yes it is taking some getting the hang of the colons and all. Yes you are right I do need another level for the prior remark and have added that. The problem here I see is that a new section was needed for a new topic, so I have added one. I will certainly take note of the conventions. Thank you for pointing out how important it is to take note of the conventions. .. and I hope we will note the conventions used in defining Natural numbers in the article.

I wrote this not to incite, but to explain to others who may come later for the reason for the recent flurry of changes and the nature of the debate. As my edits were often deleted without explanation - it is a useful sign post. Thank you for the highlighting. Perhaps the comment should have been made earlier before some others joined in. I apologize and no insult intended, but the problem wasn't as apparent then.

And the point of my comment I think again avails itself "positive integer"s is redirected to this page. Is that your edit? As this page no longer defines Natural numbers in terms of whole numbers an the set {1,2,3 ..} is it reasonable to redirect positive integers here? Or the whole numbers? I would like to learn more about your talk comments that whole numbers are not even related to integers, and how you see them fitting in. Anyway, I hope you will speak to that in the section I opened on that topic. (see farther down) Thomas Walker Lynch (talk) 15:30, 7 October 2014 (UTC)

Okay, you're still not quite getting the formatting correct, but you do seem to be trying to, so credit where credit is due. I'll create a section on your talk page with an illustration of how it should look, and hopefully that will help. The section is here, and you can delete it once you've taken a look at the code if you want.

I'm not sure what edits you made which you think I deleted, but the only edit of your which I undid was the change to the redirect at Whole number. With regards to that, I stand willing to be convinced to do it another way, but so far, I have found your arguments lacking. My personal preference would be for Whole number to be a disambiguation that gives links to both Natural number and Integer. Another editor has espoused the same position on Talk:Whole number. All discussion of that subject should be undertaken there, as I just explained in the section below.

Your apology is commendable, but it should not be to me, but to the individual you were responding to. MjolnirPants Tell me all about it. 15:45, 7 October 2014 (UTC)

• Strictly speaking, what is the subtraction of natural numbers. Types of division. What is Euclidean division.
• Relations Order
• Cardinality aleph zero
• Comparison with continuous power
• Some topologies on the set of natural numbers. --190.117.197.235 (talk) 04:51, 27 July 2014 (UTC)

I've proposed that whole number be redirected here (Talk:Whole number#Redirect to natural number?). Further discussion (Talk:Whole number#A Whole Number Is...) was discussed as how to implement it. Just notifying all interested in some ideas being thrown around.174.3.125.23 (talk) 05:48, 1 September 2014 (UTC)

I've implemented the change.174.3.125.23 (talk) 23:37, 14 September 2014 (UTC)

Seems to me like a mistake, as many use "whole number" to include negative numbers. This was stated in the discussion. Maproom (talk) 06:12, 15 September 2014 (UTC)

:::Which you were a part of and agreed to redirect the article to this article.174.3.125.23 (talk) 09:35, 15 September 2014 (UTC) Sorry wrong person174.3.125.23 (talk) 09:38, 15 September 2014 (UTC)

There's no mistake here. "Natural number" and "whole number" are used similarly depending on the context and author. It doesn't makes less sense to fork material that doesn't need to be. You didn't object then. Why object now?174.3.125.23 (talk) 09:42, 15 September 2014 (UTC)

I wrote In my view it ought to be a disambiguation page, referring the reader to natural number for positive-only and for non-negative uses of "whole number", and to integer for uses of "whole number" which may be negative. That is still my view. Maproom (talk) 15:55, 15 September 2014 (UTC)

Subsequent to your response were evidence that references use it synonymously and refer to "whole number" in like meaning to "natural number". I see no mistake anywhere.174.3.125.23 (talk) 07:04, 16 September 2014 (UTC)

Sure, some sources use "whole number" to mean "natural number", I am not denying that. But others 12 say it is also used to mean "integer". A redirect to just one possible meaning is wrong. A disambiguation page is what we need. Maproom (talk) 07:21, 16 September 2014 (UTC)

No we don't need it. Per Natural number's lede "there is no universal agreement about whether to include zero in the set of natural numbers". This is equivalent to the definition of whole number. Disambiguation would confuse the topic.174.3.125.23 (talk) 09:31, 16 September 2014 (UTC)

Sure there is no agreement about whether "natural numbers" include zero. That is irrelevant. The point is that some reputable sources, including the two I cited above, consider that a "whole number" may be negative. There is universal agreement that a natural number can never be negative. So redirecting from "whole number" (possibly negative) to natural number (never negative) is misleading. Maproom (talk) 09:45, 16 September 2014 (UTC)

The term counting number is also used to refer to the natural numbers (either including or excluding 0). Likewise, some authors use the term whole number to mean a natural number including 0; some use it to mean a natural number excluding 0; while others use it in a way that includes both 0 and the negative integers, as an equivalent of the term integer.

The natural numbers are usually used as counting numbers. The second sentence starts with "Likewise", meaning that the rest of the content of the sentence would have a similar meaning in like fashion. This results in the article indicating that a natural number is used in like fashion as "whole number", meaning that natural numbers do include negative number according to some authors.174.3.125.23 (talk) 10:15, 16 September 2014 (UTC)

The passage you quote above states, correctly, that the term "whole number" is sometimes used to include negative integers. The article nowhere suggests that the term "natural number" can be used to include negative numbers. Can you quote any source that regards natural numbers as including negative numbers? Maproom (talk) 10:43, 16 September 2014 (UTC)

The prose in the article must be rewritten if this is not the case. Of course the redirection can be reversed, but lacking the burden of proof that you claim, I cannot agree to such an action.174.3.125.23 (talk) 11:27, 16 September 2014 (UTC)

I am not aware of any error in the article. If you know of one, please say what it is. And there are sources that say "natural numbers" do not include negative numbers, and none that say they can include negative numbers; so I plan to go ahead and replace the redirect by a disambiguation page. Maproom (talk) 12:06, 16 September 2014 (UTC)

I agree with Maproom on this. I can live with Whole number redirecting here, provided there is a clear enough hat note indicating that the term may refer to Integer. However, I don't find it ideal, and there is no such hat note. If Maproom doesn't think a hat note is sufficient (for which there are good arguments), then I'll support him. The quotes provided by the IP editor only support our side of the argument. I have yet to see any source which claims that 'whole number' always means 'natural number'. MjolnirPants Tell me all about it. 13:50, 16 September 2014 (UTC)

I agree with Maproom too. When 174.3.125.23 says "This results in the article indicating that a natural number is used in like fashion as "whole number", meaning that natural numbers do include negative number according to some authors" that is just mis-reading the article. What is said is that "whole number" can sometimes mean things that "natural number" can also mean, and moreover some people use "whole number" to include negative numbers. But there is (justly) no indication of anyone using "natural number" to include negatives. (If you search for all occurrences of "negative" in the article, you'll find that there is a sentence saying that it is popular to have N designate (only!) negative numbers, which is quite ridiculous, but entirely unrelated to this issue.) Marc van Leeuwen (talk) 14:58, 16 September 2014 (UTC)

There should absolutely be a hatnote. I thought that was part of the idea of the redirect; it was in my head, anyway.

So should we figure out what sort of hatnote, exactly? The best place to point people is the last sentence of the second-to-last paragraph of the "History and status..." section, but you can't really have a hatnote that points to that. It is a slightly awkward problem

Maybe the hatnote could point to Wiktionary? That really is sort of the basic problem with the whole long debate over the whole number search term — it's not about anything; it's just a word-usage question, which is not the purpose of an encyclopedia. --Trovatore (talk) 16:07, 16 September 2014 (UTC)

I thought it was pretty clear that any such hat note should link to integer. After all, "whole number" could mean non-negative integer (natural number), positive integer (natural number) or integer. Since the first two are covered by this page, the last one is the one that should be linked. I still think it's better to leave Whole number as a disambiguation page, but if the only consensus we can reach is a hat note, then hat note it to integer. MjolnirPants Tell me all about it. 17:00, 16 September 2014 (UTC)

So we have three options: disambiguation page, redirect to natural number with hatnote, redirect to integer with hatnote. That order is my order of preference. Maproom (talk) 17:35, 16 September 2014 (UTC)

I concur completely, with the addendum that I think redirecting to integer with a hat note would be worse than doing nothing. MjolnirPants Tell me all about it. 18:13, 16 September 2014 (UTC)

Ah, redirect here with hatnote to integer — I hadn't actually thought of that but I suppose it makes sense. Can we go ahead and do that, then? In my opinion the disambig page is more trouble than it's worth; it has to be constantly monitored to keep people from adding more verbiage to it. --Trovatore (talk) 19:20, 16 September 2014 (UTC)

I added the hat note. MjolnirPants Tell me all about it. 19:46, 16 September 2014 (UTC)

This is for Mr. Norwood ;-)

Towards the end of the 19th century there was a raging debate in Europe between the mathematical Naturalists and the Logicians. The Naturalists believed that numbers stemmed from the human mind. The Logicians believed they came from logic. (With this in mind we can see the irony of the Peano Axioms for arithmetic).

This comes from the book "History and Philosophy of Modern Mathematics": [naturalism philosophy in mathematics]

Minnesota Studies in the Philosophy of Science;Volume XI;Copyright 1988 by the University of Minnesota; William Aspray and Philip Pitcher, editors

p32

Pincaré criticized the logicist definitions of the numerals on the grounds they were ultimately circular, and he contended that that the proper resolution of the set-theoretic pradoxes should proceed by honoring the vicious circle principle. Goldfarb argues that the former criticism is not an elementary logical blunder, but the product of Poincareé's insistence that legitimate definitions must trace the obscure to th eclear, where the notions of clarity and obscurity are understood psychologically.

.. Ultimately, then the difference between Poincaré and his opponents comes down to a deep divergence in agendas for the philosophy of mathematics. Where Frege and later logicists saw the task of finding foundations as one of the showing how mathematics results from the most general conditions on rational thought, Poincareé saw mathematics as the product of natural objects – human beings – so that the task of finding foundations is intimately linked to bringing clarity (judged by the standards appropriate for such beings) to areas that are currently obscure (again, jugded by the standards appropriate for such beings). As Goldfarb hints, this contrast between Poincaré and the definders of the logicist program is not only useful for throsing into relief the central tenets of logicism, but it also enables us to see interesting parallels between the early criticisms of logicism and contemporary naturalistic approaches to the philosophy of mathematics.

Now in "über den Zhlbergriff" Kronecker writes

The difference in principles between geometry and the mechanics on th eone hand and the remaining mathematical disciplines, here comprised under the designation “arithmetic,” consists according to the Gauss in this, that the object of the latter, Number, is solely the product of our mind, whereas Space as well as Tim have also a reality, outside our mind, whose laws are unable to prescribe completelly a priori. [

Kronecker, Werke, vol 3, 1st half-volume, ed K. Hensel (Leipzig: Teubner, 1899), p 253 (emphasis original; Kronecker quotes, in a footnote a letter fom Gauss to Bessel, 9 April 1830).]

And of course we have Kronecker's quote:

God made the integers, all the rest is the work of man.

Quoted in "Philosophies of Mathematics" - Page 13 - by Alexander George, Daniel J. Velleman - Philosophy - 2002

Kronecker gives credit to the natural numbers to God, but this is also a naturalists quote, as this is the point that man works from. We also have the other Kronecker writing above to confirm this.

So yes, Mr. Norwood, that crazy Mathematician who coined this term was certainly thinking about the mindset of men independent of mathematics when he coined the term "natural numbers". You can set aside your doubts. Thomas Walker Lynch (talk) 15:42, 6 October 2014 (UTC)

Thank you for the information. Rick Norwood (talk) 19:34, 6 October 2014 (UTC)

I used the 'raging debate' imagery to parallel the imagery in you original comment ;-) Wouldn't it be nice if the raging debates today, and the focus of what is important, were about such questions in mathematics rather than the study in misery on the major media day after day... — Preceding unsigned comment added by Thomas Walker Lynch (talk • contribs) 07:06, 7 October 2014 (UTC)

I have placed "Philosophy of Mathematics and Logic" Oxford Scholarship 2005, ed. Stewart Shapiro, chapter 1 on my server http://www.thomaswlynch.com/Oxford_Handbook_of_Philosophy_of_Math_and_Logic.pdf for a short time. Note in that chapter he calls arithmetic the theory of natural numbers. There is much discussion about what natural numbers are.Thomas Walker Lynch (talk) 18:17, 8 October 2014 (UTC)

This section was moved to Talk:Whole number to comply with WP:TPG.

Hello, yesterday the section I opened on positive integers was quote "moved" though in fact it disappeared. Ok, so I gather the editor who did that wants to reboot that discussion elsewhere, and that is fine by me, but he didn't get the whole thread, so I fixed that part too. As there were only two of us, perhaps it is best for the talk pages. I'll delete this along with the other sectionThomas Walker Lynch (talk) 16:53, 8 October 2014 (UTC)

now it seems half of it has been put back .. 18:19, 8 October 2014 (UTC)

The section that was moved had two subjects: Positive integer and Whole number. Positive integer has its own talk page, although it is empty. The moved section is here: Talk:Whole number#Redirect Target, although the move did not preserve the section name, "Why does positive integers redirect here? Whole numbers not related to integers??", and the content was added as a subsection. That is a very confusing way to do a move. There are guidelines for refactoring talk pages. --50.53.47.9 (talk) 17:33, 8 October 2014 (UTC)

MjolnirPants (talk · contribs) has been informed that his comment move was done over the objections of another editor, and that doing so was a violation of the guidelines for refactoring talk pages. --50.53.47.9 (talk) 18:16, 8 October 2014 (UTC)

First off, I did not move the discussion "over the objection of another editor." No-one ever made any objection until well after the fact. Stop lying.

Second; That discussion is entirely off topic for this page as I have explained multiple times.

Third; THIS discussion is also off topic for this page. Both of you need to read WP:TPG. When you're done, read it again. Then, read it one more time. If you then have any questions, ask someone (even me). Talk pages are for discussion of ways to improve the article. They are not for discussion of policy (unless the talk page is that of a policy's page), they are not for complaining about perceived violations of policy, they are not for discussion of what to do on another page. From the WP:TPG page's opening paragraph:

The purpose of a Wikipedia talk page (accessible via the talk or discussion tab) is to provide space for editors to discuss changes to its associated article or project page

MjolnirPants Tell me all about it. 19:54, 8 October 2014 (UTC)

Instead of ordering other people to read Wikipedia policy, it might help for you to read Wikipedia:No personal attacks before calling someone a liar. Rick Norwood (talk) 20:07, 8 October 2014 (UTC)

Maybe that comment was out of line, but it has no bearing on the other party's adherence to policy, and it doesn't address the root of the problem, which was the insistence upon discussion changes to another article on this talk page. It certainly doesn't help that the IP editor's claims about there being objections to my move are demonstrably false. MjolnirPants Tell me all about it. 20:14, 8 October 2014 (UTC)

In what edit did you ask whether it would be OK to move another editor's comments to a different talk page? --50.53.47.9 (talk) 20:36, 8 October 2014 (UTC)

This edit, in which I said I would move it once I knew he'd had enough time to read it. He had plenty of time to object between then and when I moved it, judging by the 8 edits he made in the meantime, including edits to this section which included a reply to that very comment. You also seem to keep forgetting that the move brought this page in line with WP:TPG. I didn't violate policy with the move, I kept policy with it. MjolnirPants Tell me all about it. 20:49, 8 October 2014 (UTC)

Thanks. AFAICT, you didn't ask any questions in that edit. Could you quote yourself asking whether it would be OK to move another editor's comments to a different talk page? --50.53.47.9 (talk) 21:13, 8 October 2014 (UTC)

No, I didn't ask. I explained wikipedia policy, and I explained what I intended to do in accordance with that policy. Why don't you go back to that link you threw up earlier (Right Here) and quote me the part where it says one must poll all participants in a discussion for permission before moving the discussion? Wait, don't bother. I've got the entirety of that page's (note that it's not a policy) treatment of the subject of moving right here:

Material can be userfied or moved to a different page where it is more appropriate. If the refactoring is later reverted, the moved material should be deleted on the pages it was moved to prevent proliferation of the text.

So let's break this down: You are making the highly contentious assertion that I didn't follow the how to guide for moving talk page content based on your own interpretation of how you think it should be done, whereas I took steps to enforce a wikipedia policy after explaining that I would be taking these steps and waiting several hours for any objections (of which I received none). MjolnirPants Tell me all about it. 21:31, 8 October 2014 (UTC)

Rick, as you said before the "Peano axioms" can begin with an arbitrary "first number", so a person can do this, as for example:

N = { 2, 4, 16, 256 ..}

Here the first number is 2, and the successor function is square.

or could even do this:

N = { 5/2, 7/2, 9/2 ...}

Where the first number is 5/2 and the successor function adds one.

John Von Neuman's natural numbers use the empty set as a first number, and union and nesting as the successor function, so they are natural numbers also, though they are sets and not composed of digit characters. Hence, this definition is a Peano definition - as all definitions must be.

'the set of Natural numbers' is in fact incorrect grammatical usage, rather we should say 'a set of Natural numbers'.

So the counting numbers are 'a set of natural numbers'. The counting numbers are all whole. We start counting at one - the naturalists assure us that making such a statement is ok. However, if we want to count elements in a set, and include the empty set, if we want an additive identity, or have a distance metric that includes a single point, then we had better put zero into our set of counting numbers. Zero will the be the 'first number' and the counting numbers with zero will still be natural. Note all these reasons for using zero are relative latecomers in the history of mathematics.

Whole numbers have no fractional part. Natural numbers may or may not be whole. The counting numbers are whole numbers and natural numbers.

I think this is a good summary of the literature, is mathematically precise, and ironically it appears to take something from all the editors who have commented here, even the b******s who keep deleting my comments ;-)Thomas Walker Lynch (talk) 21:14, 8 October 2014 (UTC)

Can you share some diffs showing where someone else deleted your comments? I've seen where you were reverted for deleting some of mine, and if someone were doing it to you previously, that might excuse it somewhat. Either way, if someone is deleting your comments it needs to stop right away.

For information on how to link to diffs, see here: WP:CDLG.MjolnirPants Tell me all about it. 21:35, 8 October 2014 (UTC)

see your own comments on my talk page for a description of some of them. If anyone can tell me how this is relevant to the subject of this section please do so now, otherwise, please move this back to the talk page, the etiquette section, or redact it. Of course I will also redact the sarcastic smiley remark, the humor of which apparently isn't universally accepted. Thomas Walker Lynch (talk) 10:34, 9 October 2014 (UTC)

note the integers are also natural numbers.

set 0 to be the first number

for a successor function use: if n > 0 then m = -n else m = -n + 1 ; n is the number operated on, m is the result

The rationals are natural also, have a two dimensional array of points (p,q), and run a spiral out from the

origin for the successor function. Thomas Walker Lynch (talk) 21:55, 8 October 2014 (UTC)

You are correct that the sets you list obey Peano Axioms (though not Peano's Axioms) but no mathematician calls them "natural numbers". They are in one-to-one correspondence with the natural numbers, but not isomorphic to the natural numbers. The natural numbers starting with zero are a semi-group under addition, but not a group. In any case, whether we begin with 0 or 1 is arbitrary, and all major sources agree that there is, currently, no bull goose mathematician who can make all mathematicians accept that one or the other is right.

On a related but different subject, see Bourbaki's definition of a ring.Rick Norwood (talk) 22:38, 8 October 2014 (UTC)

All sets built against the Peano Axioms will have isomorphic arithmetic. If one brings in algebraic structures as an supplemental constraint, then an additive identity is needed. The additive identity is not important for finding a set as being natural or not, but rather to qualify it for an algebraic structure. Such a requirement would eliminate the set {1, 2, ..} as well as the sets above.

better not to leave this dangling as declaration for those who might come later. To see this is true, first note that we can count a-priori, second note that the Peano axioms place the numbers that go into our N_s in a sequence n_0, n_1, n_2 .. , so place these numbers into correspondence with their count in the sequence n_i <=> i, now note that S(n_i) = n_(i+1) make the correspondence S(n_i) <=> S(i). Now any arithmetic built with S on N_a will necessarily be isomorphic to every other built on N_b. Now yes, wish there was a reference for it, common knowledge perhaps?Thomas Walker Lynch (talk) 07:59, 12 October 2014 (UTC)

Von Neuman's natural number definition is an example natural number set that is not based on whole numbers or even +1 as the successor, yet the arithmetic is isomorphic. An additive identity can be defined, but this is not made use of in proofs such as omega containing itself.

I believe it is clear from the mathematics philosophy citings I have provided, two of which I have uploaded for this group to see, that what is natural or not depends upon context. That context is namely the choice of a first number and of a successor function. We can not say generally that '23 is natural' rather we can only say it is natural relative to, say, a first number of 12 and a successor function of 1. Note the description in Stewart Shapiro's Philosophy of Mathematics and Logic 2005, where he calls arithmetic the study of natural numbers. That shows that what is natural depends on context.

Anyway, that is the implications of the foundation work, but if I'm misreading that I very much look forward to learning more as it has implications in computation theory. I.e. where is this lack of isomorphism between the arithmetics over differing sets chosen from Peano axioms. Showing one example would suffice to show a failure in the generalization.Thomas Walker Lynch (talk) 10:04, 9 October 2014 (UTC)

A discussion of conventions used might well be relevant, but not to the exclusion of correct definition. Thomas Walker Lynch (talk) 10:27, 9 October 2014 (UTC)

Accordingly a lead consistent with definition, given all the redirects to this page, would be titled "Whole, Counting, and Natural Numbers". And continue to say, the set of counting numbers are those used for counting objects. Whole numbers are those with no fractional part, and natural numbers are defined by the context of a base number and a successor function.

The first sentence is justified by the school of naturalism. This sentence would correspond to a section on counting numbers, most of which is already there. The beginning of that section would mention naturalism and provide references.

The second sentence is a bit more problematic as it refers to a concept not yet defined, and the only way to do that is to again appeal to naturalism and that of wholeness. Another section would do that. This section would also include a discussion of whole numbers as a didactic tool in math pedagogy. Math pedagogy has some special burdens that bare mentioning.

The third sentence would correspond to a modern discussion starting with a discussion of the school of Logicism and the 'coup' described by Dieudonné. Thus provide the Peano axioms, describe arithmetic, set theory, and number theory relevance. There would also be a discussion of conventions used. This would be an expansion of the current formal definitions section. Thomas Walker Lynch (talk) 10:27, 9 October 2014 (UTC)

[Comments copied from User_talk:D.Lazard by 50.53.53.206 (talk) 11:09, 9 October 2014 (UTC)]

Could you give us your translation of this quote from Abrégé d'histoire des mathématiques by Jean Dieudonné?

Note on page 333: "Dès le tome II de son Formulaire ([214], 1897-1899), Peano substitue l'ensemble N de tous les entiers naturels à celui N* des entiers positif non nuls, le 0 au 1 dans l'écriture des axiomes..."

Thomas Walker Lynch posted the quote on Talk:Natural_number in this edit.

--50.53.49.112 (talk) 03:59, 9 October 2014 (UTC)

"From volume II of his Formulaire ([214], 1897-1899) on, Peano substitutes the set N of all natural integers to the set N* of the nonzero positive integers, [and] the zero to the 1, ..."

The normal translation of "celui" would be "that", but I am not sure that "that N* of the positive integers" would be correct English. Adding "and" is required because of the truncation of the citation. D.Lazard (talk) 08:33, 9 October 2014 (UTC)

Note also the following: the fact that in French the set of natural integers contains zero does not imply that the same convention applies in English. The sentence contains an example where the English and French conventions differs: "nonzero positive integers" is redundant in English, not in French, where "positif" means "nonnegative". D.Lazard (talk) 08:48, 9 October 2014 (UTC)

Thanks. That's very helpful. In the 1901 edition of Formulaire de Mathématiques, Peano discusses what should be the first number ("le premier nombre"). (p. 39, Notes) --50.53.53.206 (talk) 12:13, 9 October 2014 (UTC)

thank you very much. I have received email back from a colleague expert in arithmetic at Lip-ens, you surely know him, and he throws this into the mix <<:

It seems to me that there is a slight inversion: "Peano substitue l'ensemble N de tous les entiers naturels à celui N* des entiers positif non nuls » means « Peano replaces the set N* of the positive nonzero integers by set set N of all natural numbers » and he also replaces the one by the zero in the writing of axioms.

>>Thomas Walker Lynch (talk) 12:20, 9 October 2014 (UTC)

I have used "substitute ... to" and not "substitute ... by". Thus, there is no inversion. By the way, it is impossible to use French texts for differentiating between "integer", "whole number" and "natural number". In fact, the literal translation of "entier naturel" is "natural whole", as "entier" is an adjective meaning "whole", which becomes a noun only in mathematics, where it means "integer". I suspect that, initially, "integer" and "whole number" were synonyms, one being a direct translation from Latin, and the other being a translation from the French translation of the same Latin word. By the way again, the term "natural number" seems much older than 19th century. It dates from the time where the negative numbers were not fully accepted in mathematics and were considered as "unnatural". The opposition between "natural" and "negative" is very similar to that of "real" and "imaginary" numbers. D.Lazard (talk) 13:40, 9 October 2014 (UTC)

The inversion the French professor of arithmetic is referring to is in the translation to English of the original French sentence. I.e. he is giving me advice relative to the translation. That is his email response verbatim after the top '<<' he is not speaking to you in that quote, but rather to me. He has only the original French text and my proposed translation, nothing more, none of this wikipage. He has not seen your entries here, nor mine or anyone else's. There is not bias. I can share the email correspondence if pressed because one finds that important, but he might be unhappy if I pulled him into the fray here.Thomas Walker Lynch (talk)

D.Lazard: 'I have used "substitute ... to" and not "substitute ... by".'

"substitute … to" is not standard English usage. The options are "substitute … for", "substitute … with", and "substitute … by". See the detailed usage note under "substitute" in the New Oxford American Dictionary and the entry for "substituer" in the Oxford-Hachette French Dictionary.

--50.53.45.210 (talk) 07:45, 23 October 2014 (UTC)

D.Lazard: "From volume II of his Formulaire ([214], 1897-1899) on, Peano substitutes the set N of all natural integers to the set N* of the nonzero positive integers, [and] the zero to the 1, ..."

If "for" were substituted for "to" in the translation, the sentence would be closer to standard English:

• "From volume II of his Formulaire ([214], 1897-1899) on, Peano substitutes the set N of all natural integers to for the set N* of the nonzero positive integers, [and] the zero to the 1, 0 for 1 in the written axioms ..."

Volume II of Peano's Formulaire is online at archive.org. His five axioms are here. Interestingly, Peano calls them "propositions primitives".

--50.53.35.240 (talk) 11:26, 25 October 2014 (UTC)

p39 "N_0 == <<nombre, (entier, positif ou nul)>>" =english=> " N_0 = number (whole, positive, or zero). Can entier be translated as integer?

Age of the natural number term - great. Can you point me at a source for that information? In the discussion I've been able to find on natural numbers, [afore mentioned texts Abrégé d'histoire des mathematiques, History and Philosophy of Modern Mathematics, and Philosophy of Mathematics and Logic] all first discuss natural numbers by name at the end of the 19th century either in the context of Poincaré and naturalism, or as a formalization by Peano. These two are placed in juxtaposition. I have gone so far as to write Dr. Shapiro and point blank ask him if the term was used sooner that that, and what for. If you have a reference for an earlier use, that would be wonderful. Though even if there is an earlier reference, the 'coup' of Peano is dominate in all the references above, he has provided the modern definition. Shapiro's discussion appears particularly poignant when he calls arithmetic the study of natural numbers.

When people equate whole numbers, counting numbers, and natural numbers, as this article appears to have originally done, one has great difficulty in making sense of the history of the terms and tracing etymology and evolution. It turns out to be important to place these terms into their appropriate schools of mathematical philosophy in order to see a continuity in going back and to have an understanding of what they are. When doing this, the above texts appear to run into a wall for natural numbers with Peano. In the lead sentence I am proposing, I have divided the terms by whether they are a property of a set, or a set itself, and placed them into their appropriate schools of thought: counting numbers goes to naturalism and is a unique set, whole numbers goes to math pedagogy and naturalism and as a didactic tool can take multiple forms (as noted in the current lede), and natural numbers goes to Logicism and Peano. One must have context to say a number is or isn't natural. That context is a first number and successor function with special qualities. Again this is described in the above texts, references linkedin into the wikipage for Peano Axioms, and via common usage.Thomas Walker Lynch (talk) 15:41, 9 October 2014 (UTC)

This page is not a forum to discuss the history of mathematics. The history of terminology is out of scope here. A section on the history of terminology for the integers would certainly be interesting in Wikipedia, but not here. The best place would be in Integer. However, to write such an history, it is required, by Wikipedia policy WP:OR, that a published text written by a specialist discusses this point. It is useless to discuss these questions here until some reliable source is found. As it is, the article is convenient, as it reports the various modern uses of these terms. It may certainly be improved, but nothing in your lengthy posts allows to improve it. Thus, again stop to use this page as a forum for your questions and thoughts about history and philosophy of mathematics. D.Lazard (talk) 16:36, 9 October 2014 (UTC)

The "history of numbers" page redirects here since 2005. There is a section in this article you are 'talking' about. Are you proposing to change this now? What will your edits be?

The definition of the natural numbers is very much part of the philosophy of mathematics even today. The Shapiro reference, "Philosophy of Mathematics and Logic" is from 2005, and he still publishes (perhaps you didn't see this? I have left a copy of the first chapter on my server, see the link above). It is a pretty good reference. The fact that there is a continuous understanding of its meaning going back to ~1900 is highly relevant, not a reason to dismiss the literature or be dismissive of my bringing this to light.

The lede section is much improved since I arrived here and criticized it for being circular and only providing {1,2,3, .. } in the lead sentence. Thank you for the positive feedback on this.Thomas Walker Lynch (talk) 19:10, 9 October 2014 (UTC)

Thomas Walker Lynch, can you please just stop arguing with people and accept correction? Please read WP:FORUM and WP:TPG. From WP:TPG:

The purpose of a Wikipedia talk page (accessible via the talk or discussion tab) is to provide space for editors to discuss changes to its associated article or project page. Article talk pages should not be used by editors as platforms for their personal views on a subject.

Emphasis addedMjolnirPants Tell me all about it. 19:33, 9 October 2014 (UTC)

The "history of numbers" page redirects here since 2005.
— User:Thomas Walker Lynch

Thanks for pointing that out. A better target for History of numbers would probably be History of mathematics or Number. --50.53.41.167 (talk) 04:56, 10 October 2014 (UTC)

Yes, this page is trying to be too much perhaps. The history of numbers section in this natural numbers article is well intended, and echos the mathematical school of naturalism discussions of the late nineteenth century [see "History and Philosophy of Modern Mathematics" book (discussed in the 'Origin of Natural Numbers" section on this talk page, "Philosophy of Mathematics and Logic" book full reference and link to first chapter given further up on this talk page, and the "Abégé d'Histore des Mathematiques" book discussed here and "von Neumann's definition in lede?" section on this talk page. (I introduced each of these references further above in the discussion)] I suggest the redirect be changed as you propose, and that the current history of numbers discussion on this page be converted to a short introduction to Logicism and Naturalism views on natural numbers as a summary of the reference material. Some of the material already there could be kept to buttress the naturalists position. This would lead nicely to the Peano axioms. Thomas Walker Lynch (talk) 08:32, 10 October 2014 (UTC)

The Number#History section doesn't have a subsection on natural numbers. --50.53.57.116 (talk) 12:57, 10 October 2014 (UTC)

The redirect from "history of numbers" does not promise the reader any history of natural numbers so it would be safe to change that now. The redirect here is probably based on a technically incorrect presumption of counting numbers being identical to natural numbers ["History and Philosophy .., "Philosophy of Mathematics ..", "Abégé d'Histore..", and the newer Peano citings added]. Accordingly, the history of numbers section in our article here should start out "the counting numbers" instead of the "the natural numbers", and needs to discuss Peano's coup [This is a quote from Dieudonné, see "Abégé d'Histore.."] in defining arithmetic when introducing the term natural numbers to be consistent with the references cited just above. (If someone has a reference to the term used before that, we would love to see it.) Seems we should update the history section here before adding something to the history of numbers page. Does someone have a reference to the 'dot' as an origin to mathematics? I added the picture of the counting stick, though the corresponding text was deleted. Can we change 'dot' to 'mark or stone in a pile', this is more accurate based on the citings we have, and it causes the bone picture to make sense. This bone is billed as the earliest example of mechanized counting (as opposed to using one's fingers which we suppose is much older) in discussion of it, there is a wiki page for it. Of course piles of stones don't survive time as do bones.Thomas Walker Lynch (talk) 14:33, 10 October 2014 (UTC)

The redirect from "history of numbers" does not promise the reader any history of natural numbers so it would be safe to change that now.
— User:Thomas Walker Lynch

The ball is in play here: Talk:History of numbers. --50.53.57.116 (talk) 14:56, 10 October 2014 (UTC)

To editor Thomas Walker Lynch: The distinction that you make between various kinds of numbers is wrong from a mathematical point of view as well from an historical point of view. Historically, until 15th century, there were only (whole) numbers and fractions. The fact of considering zero, negative numbers and real numbers as true numbers is thus relatively recent. The use of "whole" or "integer" to distinguish "natural" numbers from fraction does not implies any philosophical distinctions among integers. Your insistence of making conceptual differences is your own idea, which is not based on any reliable source. All the citations that you provide support only the fact that the name given to the positive integers or to the non-negative integers vary with the authors. You cite also Frege and Peano as if they have a different philosophical conception of the integers. This is possible but of low interest. The fact is that both were concerned by the need of a formal definition of the integers that is logically solid. They gave different formalisms that have been proved equivalent. This implies that for modern mathematicians as well as for ancient ones there is only one notion of natural numbers (positive integers). Thus your sentence "a technically incorrect presumption of counting numbers being identical to natural numbers" is completely wrong. Both are and have always been the same thing. Therefore, there is no valid reason to modify the article as you suggest.

It seems that your concern is the philosophical problem of the relationship between the mathematical and abstract concept of number and the real world. This a wide question, which is a special instance of the relationship between mathematics and the real world. This is completely out of the scope of this article. Moreover, very few may be written in an encyclopedia about this because of the lack of any consensus, except that the answer to this question is of low importance for mathematicians. For the question itself, you may hardly find two mathematicians or philosophers that have the same opinion. D.Lazard (talk) 15:30, 10 October 2014 (UTC)

There is an article on the philosophy of mathematics. --50.53.57.116 (talk) 15:39, 10 October 2014 (UTC)

D.Lazard just a few corrections here, I have not made any statements about zero, negatives, or reals being older that the 15th century. Though I did ask you for a reference or source material using the term natural number before the Peano Axioms as you said the term was used earlier. Do you know of such a reference??

As far as the integer bit I gave one citation from Dieudoné saying that in Peano's volume II, he placed 0 in N. The rest of the conversation on that was driven by you and and other editor, I only replied to a request for further translation. You do know that the 'history of numbers' comes to this article right?

I have not used 'whole' or 'integer'to distinguish natural from fraction. Again, I don't know where you are getting that, perhaps it is because of my remarks on whole numbers. You might not realize, but 'whole numbers' is also comes to this article. My input on whole numbers is now in the article is cited.

Yes I did summarize p32 from "History and Philosophy of Modern Mathematics" I'm sorry you find that of low interest. Gosh I find it fascinating, as apparently did the authors who took the time to have researched it and write about it.

The conceptual difference I am grappling with is between 'the set of counting numbers' and 'a set of natural numbers'. The references show that a definition of natural number has context of a starting number and a successor function (and that set choices such as {1,2,...} are conventions). How do we get from there to "It is positive integers." Here are a couple of questions I hope you will answer to help us understand that:

Are we to dismiss the set-theoretic definition of natural numbers as natural numbers. It is based on sets not counting numbers. I see they can be placed in correspondence, but that isn't equivalence because the successor functions are defined differently.

If the counting numbers and the natural numbers are identical, how is it that the Peano Axioms are numbered?

Thomas Walker Lynch (talk) 04:25, 11 October 2014 (UTC)

I saw in this article a few problems, people gave me a hard time here, but the article is much improved. There are still some issues with the history, and unresolved differences between your description of natural numbers and that in some of the source material. I'm trying to reconcile these sources to understand if you are describing convention or a profound equivalence, i.e. that arithmetic is no richer than counting. Don't you think that an opening sentence for this article that says that 'counting numbers are used for counting' would be better than 'natural numbers are used for counting'?Thomas Walker Lynch (talk) 04:25, 11 October 2014 (UTC)

DLazard, your indented comments here do not speak to the text above on the subject of changing the history redirect. Please take note of WP:INDENT. There is a section on this topic above.

DLazard, I went to the trouble of creating a talk section on history so that we could discuss your comments that history should not be discussed here, and adjust the "history of numbers" redirection accordingly. You have deleted that section thus leaving the proposal buried here were some may not have seen it. Can you please put it back?Thomas Walker Lynch (talk) 21:19, 11 October 2014 (UTC)

1. Counting number
2. Counting numbers
3. History of numbers
4. ℕ
5. Natural integer
6. Natural number (transclusion)
7. Natural Numbers
8. Natural numbers
9. Non-negative integer
10. Nonnegative integer
11. OTTFFSSENT
12. Positive integer
13. Positive integers
14. Unnatural number
15. Unnatural numbers
16. Von Neumann natural number
17. Von Neumann natural numbers
18. Whole number
19. Whole number (disambiguation)
20. Whole Numbers
21. Whole numbers

What is a transclusion?

Is it proper to have a Whole number dsimabiguation and a direction? Is that redundant or does it help in some way?

IMHO:

Will delete the capitalization variations as they are unnecessary.

The history of numbers and Von Neumann natural number to be moved to respective articles.

From math is fun: "Question : What are Unnatural Numbers ? Answer :There is no such term called unnatural numbers." Nor is this 'term' explained on the page, will delete this redirect.

OTTFFSSENT? One Two Three... should be put on the acronyms list and redirect there. Probably predates the acronyms list. I will remove redirect.

The integer terms to be forwarded to the integer article.

Natural Integer removed so it goes to disambiguation. How can we assure that the natural number page and the integer page both appear on that disambiguation page? Thomas Walker Lynch (talk) 15:35, 12 October 2014 (UTC)

Thomas Walker Lynch asked: "Is it proper to have a Whole number dsimabiguation and a direction? Is that redundant or does it help in some way?"

Good question. According to the edit history, Whole number (disambiguation) was created by SmackBot, and the two subsequent edits were by bots. In the last version of Whole number before the 16:52, 8 October 2008, bot-creation, "Whole number" was indeed a dab page. There are only two pages linking to it, and neither should be. --50.53.49.222 (talk) 18:20, 12 October 2014 (UTC)

The links to Whole number (disambiguation) have been changed to Whole number. (1, 2) --50.53.49.222 (talk) 18:35, 12 October 2014 (UTC)

Rick Norwood named three textbooks that he has used "to teach the Peano Axioms." I was planning to add all three as sources for the subsection on the Peano axioms. One of those, Naive Set Theory by Halmos, has a chapter called "The Peano Axioms", but the axioms appear to be set-theoretic. For example, Halmos says: "${\displaystyle 0=\varnothing }$" and "${\displaystyle n^{+}=n\cup \{n\}}$". Is Halmos using the term "Peano Axioms" in a significantly different sense than the article? --50.53.49.222 (talk) 15:52, 12 October 2014 (UTC)

There have been three aspects people have spoken from: a) mathematical definition/understanding b) convention c) math pedagogy. When I first saw this page I was concerned upon finding the 0 based convention was written out.

Rick's comments have been insightful, and I appreciate his patience. Thank you Rick. Particularly two comments A. he doubted a mathematician was thinking of archaeology when he coined natural numbers. B. That the first axiom should be defined in this article in terms of a 'first number'.

At the time when Rick said 'A' I understood as DLazard, that natural numbers we considered 'natural' exactly because they were what people originally used (no negatives etc.). But the math sources told a more sophisticated story, yes there were the naturalists (dignity saved..), but now 'arithmetic is the theory of natural numbers' [Shapiro p8], and Dieudonné called it Peano Axioms a 'coup', (among Peanos many coups) [citing goes here ;-)].

Rick's 'B' allows for a unified presentation of Von Neumann's definition with a first number of'{}'. Please forgive me Rick, but in trying to address the convention issue, I asked you to update the Peano Axioms with a zero as the first number, (a common convention, but not a full definition). Can you please put back the 'a first number' version back? We can then state Von Neumann's definition in terms of the Peano Axioms.

Rick you also made the argument about algebraic structure, the same one I had made earlier and discussed with another editor at length. Zero provides for algebraic structure (which goes beyond arithmetic structure) this is stated on the Peano Axioms page and we should mention it too.

Thomas Walker Lynch (talk) 19:03, 12 October 2014 (UTC)

Could you please answer the question or not comment? --50.53.49.222 (talk) 19:16, 12 October 2014 (UTC)

Excuse me, shall I redact that ;-) .. btw, can't see pages 41-43 in the reference you sent. Naturals are defined on page 44.Thomas Walker Lynch (talk) 19:43, 12 October 2014 (UTC)`

Blame Google Books and the publisher. They want you to buy a copy, not read it online. I find that counterproductive. Anyway, a large public library or an academic library would probably have it. You might be able see more if you can guess a search term: "peano axioms" or "natural numbers", say. --50.53.37.49 (talk) 00:06, 13 October 2014 (UTC)

In the conventional phrasing of the Peano Axioms, the context consists of a first number and a successor function. One can build many arithmetic systems by changing the parameters. In Halmos's definition both first number and the successor function are embedded in the definition of natural number. There are no free parameters, so we get exactly one arithmetic. One could call this a 'base' arithmetic, or some such, and then say any system that is isomorphic to it is 'representative'. The symbols in a the representative system 'represent' numbers, etc. But here is the key difference, Halmos's system does not automatically give us the isomorphism, rather we have to go find it. With the conventional statement of the Peano Axioms, if we use two different first numbers, those are in correspondence. If we use two different successor functions, those are in correspondence. It seems that Halmos's system is less rich. Said richness is what is added to counting numbers with the Peano Axioms, so daresay, Halmos's definition appears at first blush to be an identity function on the counting numbers and not a set of Peano Axioms.Thomas Walker Lynch (talk) 20:57, 12 October 2014 (UTC)

I don't recall saying that this article should state the first axiom in terms of "first number". I do recall saying that some books state the axiom that way. For this article, which should be readable by non-mathematicians, think Axiom One should just state "Zero is a natural number." After the axioms, we should mention that some authors start with 1. Whether we need to get into "first number" at all at this level of exposition I doubt. As for the Halmos quote. Halmos defined "0" as "{}", so his version starts with 0. Rick Norwood (talk) 22:44, 12 October 2014 (UTC)

I'm not going to misrepresent the source in the Notes. Could you please answer the question:

• Is Halmos using the term "Peano Axioms" in a significantly different sense than the article?

--50.53.37.49 (talk) 00:11, 13 October 2014 (UTC)

Answer: No. I thought I made that clear. {} is the symbol Halmos uses for 0. The main difference in the Halmos version of the Peano Axioms is that he uses the language of set theory instead of the language of arithmetic. Rick Norwood (talk) 00:17, 13 October 2014 (UTC)

Thanks. Your wording is exactly what I was looking for: "[Halmos] uses the language of set theory instead of the language of arithmetic." --50.53.37.49 (talk) 00:42, 13 October 2014 (UTC)

What of the fixed successor function? Halmos has shown counting numbers have recursive structure i.e. are natural. Other sets which the Peano Axioms could be used to show have recursive structure this set of axioms does not speak to. It would be analogous to defining a group on a specific set and saying there is only one group in the universe, rather than specifying the properties of a group - when talking about algebraic structure. A piece of abstraction is missing.Thomas Walker Lynch (talk) 07:59, 13 October 2014 (UTC)

Is it significant that he 'proves' IV and V rather than leaving them as axioms?Thomas Walker Lynch (talk) 08:12, 13 October 2014 (UTC)

Rick, I understand from WP:IDENT that a new subject in a section starts at the bottom outdented, so let me continue the above other subject on the formal section and first number rather than mix is in above. Just a question. Wouldn't it be better to give the definition you gave with the 'there is a first number' and then to start the Von Neumann section with '{} is taken as the first number' rather than starting with the demonstrably false, and very confusing to a younger reader, "0 = {}"? I do know people often write it this way, but that doesn't make it a requirement to do so here. Also I do understand the intended meaning is that the two entities are to be placed into correspondence, but to a person seeing this the first time with no prior context such would be found in a book .. just asking.Thomas Walker Lynch (talk) 08:35, 13 October 2014 (UTC)

As an issue that has been raised, integer uses "whole number" as an synonym to integer. Since it is relevant to this page as this "whole number" redirects to natural number, I am posting this notice that I will remove it as a synonym on integer. Additionally, the use of "whole number" as a synonym to "integer" is not cited.174.3.125.23 (talk) 08:58, 13 October 2014 (UTC)

Here are a few things to consider, whatever you do:

The definition of whole number is consistently given as "number without fraction". However, the domain of discourse varies, so sometimes 'the set of whole numbers' ends up being the integers, the non-negatives, or the positives. The citing to that definition of 'set of whole numbers' is in the lede here, Weisstein, Eric W., "Counting Number", and "Whole Number", MathWorld. This talk page has a section where the original redirect was discussed. It looks like you were involved in that.

The definition "number without fraction" is a bit problematic, as it requires knowing what a fraction is, a higher level construct. You could instead define a whole number as a number reachable with a successor function of ++1 or --1 from zero via the Peano Axioms. (or variations leaving out --1 or starting at 1). Such a definition would fit better on this page, though currently it is not given this way. I noticed that the integer construction discussion defines integers on top of naturals via the spiral over a plane. Hence, if you define wholes on integers, you get essentially the same definition as this one.

As a final note, as was discussed earlier though deleted, it seems the set of whole numbers is not commonly relied upon in serious works, but it is common in math textbooks. There is a section on this talk page requesting a math textbook sentence about wholes in the article here.

Thomas Walker Lynch (talk) 10:49, 13 October 2014 (UTC)

Hi Thomas, I love speaking with you. But which section requests this sentence about math textbooks?174.3.125.23 (talk) 11:22, 13 October 2014 (UTC)

likewise, thanks for the talk notes. Here is the link to section with request for a text book sentence here Thomas Walker Lynch (talk) 13:32, 13 October 2014 (UTC)

174.3.125.23 said: 'Additionally, the use of "whole number" as a synonym to "integer" is not cited.'

Please use {{cn}} or {{cn span}} to tag unsourced text, so that we can understand what you are referring to.

--50.53.41.238 (talk) 13:54, 13 October 2014 (UTC)

It is not necessary to always specifically cite everything. I'd have thought that knowing people sometimes refer to them as whole numbers was a sort of the sky is blue sort of thing. Dmcq (talk) 14:58, 13 October 2014 (UTC)

"Knowing people" are not verifiable sources, and the term "whole number" is very ambiguous, so it needs to be thoroughly sourced.(1, 2) --50.53.41.238 (talk) 18:46, 13 October 2014 (UTC)

Neither mathematicians nor college textbooks use the phrase "whole number" very often. "no fractional part" is a bad definition, since it assumes the reader already knows the difference between a whole number and a fraction. I would prefer something along the lines of "A whole number is a counting number, such as "1, 2, 3, ...". Some include 0 and, if the students have been introduced to negative number, include -1, -2, -3, ... . Professionals usually use the more technical terms natural number or integer." But this is off the top of my head. I'm not out at school right now, and reference books are not easily to hand, so I leave this edit to others who are interested. Rick Norwood (talk) 14:40, 13 October 2014 (UTC)

If the phrase "Professionals usually use ..." appears in the article, I will immediately tag it with {{pov-inline}}, because anyone can seek to use precise terminology. Sources that explain why a particular definition was chosen, would be very interesting, though. --50.53.41.238 (talk) 19:04, 13 October 2014 (UTC)

Some recent posts in #Discussion of lead discuss the title of the article. I strongly oppose to such a change. In fact, any user searching for "Natural number", "Whole number" or "Counting number" is automatically redirected to this article, and the lead of the article mention these three names. WP:POFRED says "Reasons for creating and maintaining redirects include: Alternative names ...". We are exactly in this case, and there is no reason to not follows the usual Wikipedia rules. D.Lazard (talk) 20:44, 13 October 2014 (UTC)

This is what I gather from the research regarding this article and topic, present and past:

Counting numbers is a concrete concept. Natural numbers is an abstraction of counting numbers provided by the Peano Axioms. Examples of natural numbers include the counting numbers and in addition other sets such as {{},{{}},{{},{{}}} ..}.

Shapiro, p8: arithmetic is the study of natural numbers.

The abstraction of natural numbers is important in computation theory where natural numbers may be represented as symbols on a tape, arithmetic constructed, and proofs performed as to the complexity of algorithms that calculate against these numbers. Implementation is the art of abstracting to physical observables, so this abstraction is important to the implementation of computers, particularly in the area of computer arithmetic.

Counting numbers are natural. Whole numbers are natural. Natural numbers are not counting numbers. Thomas Walker Lynch (talk) 06:11, 14 October 2014 (UTC)

The common convention is not incongruent with the formal definition. N={1,2,3 ...} for example is a perfectly fine specimen from the abstraction. Any set of natural numbers can be used for counting. Thomas Walker Lynch (talk) 07:47, 14 October 2014 (UTC)

Everything is wrong in the preceding posts: "number" is an abstract concept, "counting" is a mind operation and therefore an abstract operation. How "counting number" could be concrete? Moreover, "concrete concept" is a contradiction by itself, as a concept is, by definition, an abstraction.

You use "natural number" in a sense which is yours and only yours. For everybody else, "natural number" is that is described in the article, nothing else. In particular, there is only one set of natural numbers and the phrase "any set of natural numbers" is a nonsense. Peano axioms are not a definition of natural numbers, but a formalization (among several equivalent ones) of the much older (more than 2,000 years) concept of number.

The paragraph beginning by "the abstraction" consists in your own view on topics which are out of scope of this article. Thus it does deserve to be discussed here; such a discussion belong to a forum and this talk page is not a forum.

More generally, it is a waste of time of discussing if the terms denote or not the same concept, as there are no source stating that the concepts are different (except for the inclusion or not of zero), and all sources says that the three terms denote 1, 2, ... (and possibly zero), which means that the terms denote the same concept. The fact that some term may be preferred in some context (education, advanced mathematics, philosophy, ...) could be mentioned only if it would be attested by reliable sources. D.Lazard (talk) 08:32, 14 October 2014 (UTC)

Thomas said: "Shapiro, p8: arithmetic is the study of natural numbers."

The article on arithmetic says that, although it doesn't appear to be sourced. Perhaps you could tag it with {{Citation needed}}. There is no article on the Philosophy of number. Perhaps you could start one. As for this article, there is a "Properties" section that explains how the operations of arithmetic follow from the Peano axioms. The second and third sentences about monoids are misplaced, because they clutter up the list of properties. Perhaps you could tag that section with {{Copy edit-section}}.

BTW, Stewart Shapiro is a contributor to Meaning in Mathematics, edited by John Polkinghorne. That book is not mentioned in Philosophy of mathematics. Perhaps you could add it. --50.53.39.150 (talk) 14:37, 14 October 2014 (UTC)

Here are some more references:

1. concrete and abstract

In modern usage the best explanation I have seen is provided in this reference, found online, from the book "How to Design Programs": http://www.htdp.org/2003-09-26/Book/curriculum-Z-H-27.html#node_chap_21. The examples are in a formal language known as Scheme, and thus are formal. See also the wikipage on the subject. Abstraction (mathematics).

(accordingly (The counting numbers are used to count objects. The definition is is constant object {1,2,3 ..}. The Peano axioms provide a means of creating isomorphic counting systems (and thus arithmetic systems) using a much broader class of math entities, for example the set {{},{{}}, ..} set-theoretic natural numbers)

1. you use natural number in a manner that is yours alone, the abstraction of paragraph

There is no shortage of citations for the definition of Peano Axioms that show the abstraction of the successor function, Rick pointed out earlier of abstraction of zero , the first number. Does anyone deny this and require mores specifics? We do need to collect citations.

We have citations to examples of counting against other sets created against the Peano axioms but are not identical to the the counting numbers, e.g. {{},{{}}, ..} set-theoretic natural numbers. A very good reference showing the abstraction of counting numbers used in computation theory is: http://www.amazon.com/Elements-Theory-Computation-2nd-Edition/dp/0132624788 . In this book the construction of arithmetic is shown step by step using a first number of the ascii character 's'and the a successor function that appends an 's' to a string of 's'. From there arithmetic is created etc. A good collection of papers in the computer arithmetic and abstractions and implementation of counting is given by http://www.amazon.com/Computer-Arithmetic-Society-Press-Tutorial/dp/0818689315/.

1. "there are no source stating that the concepts are different" .. every single book we have found that discusses the Peano Axioms, whether it be historical, philosophical, or a serious math work calls the set constructed the natural numbers, not counting numbers, does any one need these enumerated? The counting numbers are natural numbers, so there is nothing wrong with that statement. The converse can probably be said as 'any set of natural numbers can be used for counting'. Can anyone provide a single citation that builds the "counting numbers" from five axioms, etc?

that provides citations in support of each point raised in contention.

As the lead is now open for edits, there are supporting citations, and there are no counter citations, suggest the above be worked into the lede. The word concrete can be dropped without loss of meaning.

Thomas Walker Lynch (talk) 15:49, 14 October 2014 (UTC)

As most of preceding discussions were about terminology, it seems useful to clarify the historical origin of the terminology. The assertions which follow are issued from my knowledge coming from more than 50 years of practice as a professional mathematician. I am presently unable to source them because I do not remember where I have learned this and that. However if or when sources will be found, what follows will probably deserve to be included in the article.

• Integer and whole number: before 16th century, the only numbers that were known were positive integers and fractions (ratio in Latin) of positive integers (which gave rational numbers). At that time (before America discovery), European mathematics were written only in Latin. In English, the Latin word "Integer" has been imported verbatim. It has also been translated either directly or through another language (possibly French) into "whole number". The French word for "integer" is "entier" which literally means "whole" (except for the meaning of integer "entier" is always an adjective). The fact that integers include negative integers and whole numbers do not is a much later convention. This explains why some authors still use "whole number" as an equivalent of "integer"
• Natural number: This term probably dates from 15th or 16th when negative integers did appear as strange and "unnatural" objects. Presently the word "natural" means here that these integers are primitive in the sense that all the other numbers are constructed from them. This term is mainly used when negative integers are not yet available, in an elementary classes or in the presentation of the foundations of arithmetic. When negative numbers have been defined, the terms "nonnegative integer" and "positive integer" are preferred a less ambiguous. Note that, in French, the literal equivalent of "natural number" ("nombre naturel") is not used, and "natural number" is commonly translated as "entier naturel", which literally means "natural integer".
• Counting number: This seems a recent term which may have been introduced for the need of pedagogy, for distinguish these numbers from "measuring numbers", which have commonly a decimal dot. In fact, kids know of these two kinds of numbers much before learning any mathematics.

D.Lazard (talk) 14:08, 14 October 2014 (UTC)

That is a good suggestion. The Oxford English Dictionary would be a good place to start. --50.53.39.150 (talk) 14:41, 14 October 2014 (UTC)

A discussion of usage would also be useful, assuming it could be sourced. Rick Norwood appears to have been suggesting the same thing here. --50.53.39.150 (talk) 15:22, 14 October 2014 (UTC)

There is a wikipedia page on counting. Counting numbers are used for counting. Counting is a 50,000 year old art. Do you think in the beginning that first people perhaps looked at their numbers and said "those numbers are for natural, so lets call them natural numbers", of that they said "those numbers are for counting, lets call them counting numbers .." Hmmm.Thomas Walker Lynch (talk) 16:08, 14 October 2014 (UTC)

Finding "counting sticks" "counting rods" "counting stones" "counting beads" .. but no mention of a "natural stick", "natural rods", "natural stones", or "natural beads" (relative to math). There is a wiki article on History_of_writing_ancient_numbers, oh here is one one on counting rods but don't see a wiki on "natural rod". Gee, perhaps a deepweb search will come up with something. I'll keep looking. ;-) Thomas Walker Lynch (talk) 16:26, 14 October 2014 (UTC)

I was using the term "usage" in the lexicographic sense. Dictionaries often have usage notes. --50.53.39.150 (talk) 16:36, 14 October 2014 (UTC)

Can we please start this article by saying "Counting numbers are used for counting." (or "Natural numbers are used for naturaling" ha guess not this one ;-) ) After all those Peano axioms are numbered .. Thomas Walker Lynch (talk) 18:05, 14 October 2014 (UTC)

This edit suggests that there are some problems with the hatnote, which currently says:

• "This article is about the elementary notion of number. For more advanced properties, see Integer."

1. This article discusses the Peano axioms, which are not "elementary".
2. The word "notion" is ambiguous. It also sounds pretentious.
3. Fundamentally, this article is about two sets, { 1, 2, 3, … } and { 0, 1, 2, … } , but the hatnote does not use the word "set".

BTW, the Simple English version of the article might offer some inspiration: simple:Natural number.

--50.53.39.150 (talk) 16:07, 14 October 2014 (UTC)

I was not fully satisfied by this edit, although preceding hatnote was worse. I'll try something else. D.Lazard (talk) 17:10, 14 October 2014 (UTC)

Thanks. That is much better:

• "This article is about positive and nonnegative integers. For properties involving negative numbers, see Integer."

1. When I first read the hatnote, it seemed to be saying that the integers are the negative numbers.
2. The article on the integers is about more than their properties. In particular, it has sections on their construction and their use in computing. The article doesn't have a history section, but if it did, I would note that too. :-)
3. The lead has a dash in "non-negative".

--50.53.39.150 (talk) 18:30, 14 October 2014 (UTC)

Thanks:

• "This article is about positive and non-negative integers. For the whole set {..., -2, -1, 0, 1, 2, ...}, see Integer."

1. The word "whole" momentarily caused me to think the hatnote was saying something about whole numbers.
2. The hatnote is mixing prose and set notation. Here is what it would look like if it consistently used set notation:

• "This article is about the sets { 1, 2, 3, … } and { 0, 1, 2, … } . For the set { …, -2, -1, 0, 1, 2, … } , see Integer."

--192.183.212.87 (talk) 11:01, 15 October 2014 (UTC) (Sorry about the very different IP address.)

The word "elementary" has two meanings. It can mean "for beginners" as in "elementary school". It can also mean fundamental, as in the name of Euclid's book, "Elements". The Peano Axioms are elementary in the second sense, as are the natural numbers. "Elementary, my dear Watson."Rick Norwood (talk) 21:46, 14 October 2014 (UTC)

Thanks for pointing that out. "Elementary" is ambiguous. --50.53.39.150 (talk) 22:05, 14 October 2014 (UTC)

I think the article currently reads very well. Any thoughts on upgrading it to at least a B class? Rick Norwood (talk) 22:24, 14 October 2014 (UTC)

To a reader not already familiar with the issues that have been discussed here, "positive and nonnegative integers" reads oddly. It sounds as if the article is about the positive integers but excluding those which are also negative. Maproom (talk) 07:16, 15 October 2014 (UTC)

Unfortunately, the only well-defined terms for the two sets this article is discussing are "positive integers" and "non-negative integers". MathWorld has the recommended terminology here. --192.183.212.87 (talk) 11:11, 15 October 2014 (UTC)

Would it be better if it said: "the positive integers and the non-negative integers"? --192.183.212.87 (talk) 11:24, 15 October 2014 (UTC)

I prefer "This article is about the sets { 1, 2, 3, … } and { 0, 1, 2, … } . For the set { …, -2, -1, 0, 1, 2, … } , see Integer." if we are going to use set notation in the latter part of the hatnote. The reason is that it is much more elegant than to have one half of a hat note being in prose, and the latter in mathematical notation.174.3.125.23 (talk) 15:23, 15 October 2014 (UTC)

MOS:MATH#Article introduction contains the sentence "specialized terminology and symbols should be avoided as much as possible". This applies to hatnotes. I have not followed this guideline because "for the whole set of integers, see Integer" seems awkward. On the other hand there is no reason, except some editor's preference, to not follow the guideline for the first part of the hatnote. There is a stronger reason for keeping the mention of positive integers and non-negative integers in the hatnote: these are redirects to this article, and a reader looking for them may be confused, as it is not immediately clear, from the lead that these topics are the subject of this article. Citing them in the hatnote is therefore useful. On the other hand the choice between "about positive and non-negative integers" and "about positive integers and non-negative integers" seems a question of preference or of linguistic tradition. I have chosen to avoid the repetition of "integers" because the repetition seems unnecessary for avoiding ambiguity. D.Lazard (talk) 12:58, 16 October 2014 (UTC)

This avoids specialized terminology:

• "This article is about the numbers used for counting. For the numbers used for [what?], see Integer."

What are integers used for?

--50.53.36.23 (talk) 19:09, 16 October 2014 (UTC)

Szczepanski & Kositsky have a nice section on The Number Line and Absolute Value. They number houses along a street with positive numbers to the right of the house at 0 and negative numbers to the left of the house at 0. (pp. 13-14) (NB: Google books doesn't show these pages.) Books on pre-algebra and elementary mathematics are readily available at libraries and bookstores. --50.53.36.23 (talk) 19:43, 16 October 2014 (UTC)

The negative integers are used for opposites. If a natural number indicates movement to the right, a negative number can be used to indicate movement to the left. Positive numbers up? Then negative means down. Positive numbers a profit? Then a negative number represents a loss. And so on.

In all my years of teaching, essentially all of my students have been taught negative numbers in school, and essentially none of them have been taught what negative numbers are used for. Rick Norwood (talk) 20:44, 16 October 2014 (UTC)

Thanks. The hatnote could then read:

• "This article is about the numbers used for counting. For the numbers used with their opposites, see Integer."
• "This article is about the numbers used for counting. For the numbers that have opposites, see Integer."
• "This article is about the numbers used for counting. For the numbers that have negatives, see Integer."
• "This article is about the numbers used for counting. For the numbers that have additive inverses, see Integer."

--50.53.36.23 (talk) 21:54, 16 October 2014 (UTC)

The natural numbers are used for counting. There are also used to define all other kinds of numbers, and to build all mathematics. I do not know of any part of mathematics, which does not use natural numbers, directly or indirectly. Do you have a source for asserting that the counting use is more important than the others? The hatnotes also must have a neutral point of view. D.Lazard (talk) 22:42, 16 October 2014 (UTC)

A hatnote is for disambiguation. Here, there are two alternatives: the current article and the article on the integers. The hatnote only needs to provide enough information for the user to make a decision about which article to read. A reader who is not certain about the difference between the natural numbers and the integers is unlikely to be helped by the information that this article is about the numbers that are the foundation of mathematics. Could you please clarify your position? Are you satisfied with the current hatnote or not?

BTW, the lead should say that the natural numbers are "used to define all other kinds of numbers, and to build all mathematics". That would be far more informative than the current drivel about "linguistic notions".

--50.53.36.23 (talk) 04:04, 17 October 2014 (UTC)

We should not forget that the Peano Axioms, i.e. Natural Numbers, also give us an abstracted successor function. Yes, this is about counting, but not necessarily about counting by one, as the term implies to some, but possibly counting by anything that maintains the properties specified by the axioms. Von Neumann for example used set nesting [set theoretic wiki]. Also, Natural number sets are used for more than making correspondences to other sets to form counts or to give order to other sets and term them into sequences. As one example, the abstract successor function is what arithmetic is built from. Computation theory works such as that by Papadimitriou [in his Automata Theory book] used string concatenation as a successor function to build arithmetic. Etc. As another example of alternative use, sometimes we perform proofs on the set of natural numbers rather than placing the set's elements into correspondence with the elements in another set.Thomas Walker Lynch (talk) 04:18, 17 October 2014 (UTC)

I checked the 29 sept version of the page, which is the last version before the most recent round of intensive editing. I noticed that the lede did not mention the term "counting number". I think this is appropriate because the term is not in common usage at the level this article is aiming at. I will therefore delete the recent additions of counting numbers to the lede. Editors wishing to argue for their inclusion need to provide better reasons than the fact that counting numbers have been around for thousands of years, more specifically including reliable sources. Tkuvho (talk) 08:59, 15 October 2014 (UTC)

I am unsure why you made the edits you did: tagged a phrase with {{cn}} and removed "counting number" here. Mathworld, as a widely used reference/citation on wikipedia, has specifically an entry on "counting number".174.3.125.23 (talk) 09:29, 15 October 2014 (UTC)

Counting numbers should be discussed in a later section rather than the lede. That's the appropriate place to provide a reference. Tkuvho (talk) 10:12, 15 October 2014 (UTC)

I have asked Tkuvho to revert himself here. --192.183.212.87 (talk) 12:21, 15 October 2014 (UTC)

Tkuvho said: "Editors wishing to argue for their inclusion need to provide better reasons than the fact that counting numbers have been around for thousands of years, more specifically including reliable sources."

1. The article is citing MathWorld for both "counting number" and "whole number".
2. Counting number redirects to Natural number, so counting number should appear in boldface in the lead per WP:MOSBOLD.

--192.183.212.87 (talk) 10:26, 15 October 2014 (UTC)

Here are two more sources for "counting number":

• Merriam-webster.com (note the 1965 date)
• James&James

The problem is that I can't add them to the article until you revert yourself.

--50.53.39.110 (talk) 13:32, 15 October 2014 (UTC)

I agree with Tkuvho that "counting number" is not used in mathematics and therefore must not be mentioned as the same level as "whole number". I agree with IP users that "counting number" deserve to appear in the lead. This apparent contradiction may be solved by a specific paragraph at the end of the lead, which could be

In non-mathematical contexts, typically in education, natural numbers may be referred to as counting numbers, for distinguishing them from the other kind of numbers that everybody knows, the decimal numbers, which serve for measuring and often contain a decimal mark..

Per WP:BRD, I'll add this sentence to the end of the lead. D.Lazard (talk) 14:17, 15 October 2014 (UTC)

Thanks, a separate sentence is an excellent idea, but we will need reliable sources for it. Rather, than tag bomb the sentence, I'll do it here:

• "typically in education"^{[citation needed]}
• "everybody knows"^{[clarification needed]} (students are included in "everybody", but they don't know "the other kind of numbers")
• "natural numbers may be referred to as^[clarify] counting numbers" (natural numbers with or without zero?)

Why was the term "counting numbers" introduced "circa 1965"? (per Merriam-webster.com)

At what educational level is the term "counting numbers" replaced with another term?

Sourcing ideas include books on pre-algebra and mathematics curriculum standards.

--50.53.39.110 (talk) 15:52, 15 October 2014 (UTC)

I've actually made significant changes to User:D.Lazard's addition, please peruse and change as you see fit.174.3.125.23 (talk) 16:04, 15 October 2014 (UTC)

Thanks. The sentence now reads:

• "In non-mathematical contexts, typically in education, natural numbers may be referred to as counting numbers to distinguish them from the decimal numbers which serve for measuring and often contain a decimal mark."

The sentence suggests that there only two kinds of numbers: counting numbers and decimal numbers. Is that really so, even in elementary education? (NB: Curricula are graded, so more kinds of numbers (e.g. negative numbers) may be introduced in higher grades. The sentence should reflect that. Some sources would help here.)

--50.53.39.110 (talk) 17:44, 15 October 2014 (UTC)

The Random House Dictionary dates "counting number" to 1960-65. --50.53.36.23 (talk) 01:42, 16 October 2014 (UTC)

Hold on here,

1. Curious as to why this conversation of etymology did not continue from History and Etymology But to reiterate from that section, counting numbers in various forms are among the oldest known mathematics, counting stones, counting sticks, etc. You do not find natural stones, natural sticks, etc. All these have citations if you need those copied here let me know.
2. If the term "counting numbers" was not used, it is because that is mostly what all numbers were for. Notice that the dictionary also says that the word "air" comes from circa 1350 -- but air certainly existed much before that. Note the James&James above does not give an etymology, and the other dictionary entry mentioned is probably not independent. Also note, that the term counting number exists in other forms and languages, it is not clear what the dictionary is actually referring to.
3. You don't have a citation saying that the Peano axioms do not abstract the set, indeed you have examples of set abstractions, for example the so called on this wiki, set theoretic natural numbers. The abstraction is needed to construct arithmetics and should be part of the article. (Note The Halmos article proves Axioms IV and V, making it a three axiom system, where those three axioms are the same as the first three Peano Axioms. Thus, it is a different system, related, but missing the abstraction provided by those other two axioms. Cherry picking references from a vast literature base with many vagaries could lead to a tedious discussion. Halmos is far outnumbered.)
4. most importantly the term "counting number" redirects to this page and it is recognized today and does appear in many scholarly works. It is not just for the classroom. Do a google search on: "counting number" "journal of" -game -child -education and you will find many examples, if you need me to cut and past from that list I can. Here let me mention one, "Discrete Mathematics with Applications" By Susanna Epp.
5. Note also the current tone of this wiki article, with its history section. It all starts with counting. Is the history section to be deleted? People count, they don't natural. The term natural surely comes from the school of naturalism in mathematical philosophy, this is discussed in two of the cited works see the Origin of Natural Numbers discussion and the citations there. It is hard to prove a vacuum, but DLazard nor anyone else came up with reference to this term before that use, and it hasn't been for lack of searching for one. Dieudoné called Peano's work a coup as it provided a construction for these numbers. So even if they were called natural before that, they changed in character signficantly due to the formalization and abstraction, and thus it is very appropriate to call the set prior to that "counting numbers", as its modern definition matches the concept of the set before Peano's "coup".

--> don't take counting numbers out

Thomas Walker Lynch (talk) 05:36, 17 October 2014 (UTC)

According to this ngram comparison the term counting number is outnumbered tenfold by natural number. It should not be given much prominence. −Woodstone (talk) 17:15, 17 October 2014 (UTC)

Thanks. That's very interesting. Can you think of a reason for the spike centered around 1965? Two dictionaries cite that time interval, but they don't say why.(1, 2) --50.53.38.50 (talk) 18:46, 17 October 2014 (UTC)

A Google ngram that includes "whole numbers" shows that it outnumbers both "natural numbers" and "counting numbers". --50.53.38.50 (talk) 19:28, 17 October 2014 (UTC)

Counting and natural numbers are more or less synonyms, whereas whole number is more likely to include negatives. I think the spike in the late 60s is the advent of the computer, when the naming and distinction of the various number classes became relevant to more people. −Woodstone (talk) 05:38, 18 October 2014 (UTC)

"the spike in the late 60s is the advent of the computer"

The term "personal computer" does not correlate with the "numbers" terms. "personal computer" peaks at 1988. Can you suggest another term?

--50.53.55.68 (talk) 10:39, 18 October 2014 (UTC)