Talk:Natural number

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There are not[edit]

  • 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. -- (talk) 04:51, 27 July 2014 (UTC)

Implementation of whole number to redirect here[edit]

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. (talk) 05:48, 1 September 2014 (UTC)

I've implemented the change. (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. (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? (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. (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. (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. (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. (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 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)

Sentence in lead about zero and textbooks[edit]

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. -- (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)

Modern Convention[edit]

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 (talkcontribs) 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 (talkcontribs) 14:32, 3 October 2014 (UTC)

— Preceding unsigned comment added by (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? (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" [1] 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 \{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.
-- (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. -- (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. -- (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.) -- (talk) 06:17, 5 October 2014 (UTC)

Discussion of lead[edit]

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:

Natural numbers can be used for counting (one apple, two apples, three apples, ...)
"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.
-- (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. (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? -- (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. -- (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. -- (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. -- (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? -- (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.


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. (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:


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.


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. (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.


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. (talk) 05:37, 17 October 2014 (UTC)

von Neumann's definition in lede?[edit]

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." -- (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. -- (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) -- (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. -- (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 (talkcontribs) 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. Thomas Walker Lynch (talk) 16:16, 6 October 2014 (UTC)

Ok building from "Special:Contributions/" 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, and, 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? -- (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. -- (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 f the counting function and denotes the codomain by \mathbb{N}_0. (p. 3) Can you figure out where he defines \mathbb{N}_0? -- (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. \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) -- (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"). -- (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 (talkcontribs) 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? -- (talk) 08:20, 7 October 2014 (UTC)
The article on Pierre Dugac lists a German translation: Geschichte der Mathematik 1700-1900. Ein Abriß. (1985) -- (talk) 09:37, 7 October 2014 (UTC) 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. -- (talk) 19:56, 7 October 2014 (UTC)
I have placed the pages from Diudonné on one of my webservers, see 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 \mathbb{N}_0 and \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) -- (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)

writing zero out, not working with other pages[edit]

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)

Origin of "Natural Numbers"[edit]

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
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 (talkcontribs) 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 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)

Why does positive integers redirect here? Whole numbers not related to integers??[edit]

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

Editing Etiquette[edit]

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. -- (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. -- (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? -- (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? -- (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)

the current article appears to be confusing counting numbers with natural numbers[edit]

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)

translation of a quote from Abrégé d'histoire des mathématiques by Jean Dieudonné[edit]

[Comments copied from User_talk:D.Lazard by (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.

-- (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) -- (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.
-- (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 His five axioms are here. Interestingly, Peano calls them "propositions primitives".
-- (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. -- (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. -- (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. -- (talk) 14:56, 10 October 2014 (UTC)
To 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. -- (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)

current redirects that land here, and proposed changes[edit]

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


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. -- (talk) 18:20, 12 October 2014 (UTC)
The links to Whole number (disambiguation) have been changed to Whole number. (1, 2) -- (talk) 18:35, 12 October 2014 (UTC)

Halmos on the Peano axioms[edit]

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: "0 = \varnothing" and "n^+ = n \cup \{ n \}". Is Halmos using the term "Peano Axioms" in a significantly different sense than the article? -- (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? -- (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. -- (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?
-- (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." -- (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)

Use of "whole number" in lede of integer, as a synonym[edit]

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. (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? (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) 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.
-- (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) -- (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. -- (talk) 19:04, 13 October 2014 (UTC)

Title of the article[edit]

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)

are whole numbers, natural numbers and counting numbers sets identical, what is the relationship?[edit]

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. -- (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": 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: . 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
  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)

History and etymology of the terminology[edit]

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. -- (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. -- (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. -- (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.

-- (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".
-- (talk) 18:30, 14 October 2014 (UTC)
  • "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."
-- (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. -- (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. -- (talk) 11:11, 15 October 2014 (UTC)
Would it be better if it said: "the positive integers and the non-negative integers"? -- (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. (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?
-- (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. -- (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."
-- (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".
-- (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)

a hold on counting numbers[edit]

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". (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. -- (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.
-- (talk) 10:26, 15 October 2014 (UTC)
Here are two more sources for "counting number":
The problem is that I can't add them to the article until you revert yourself.
-- (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
At what educational level is the term "counting numbers" replaced with another term?
Sourcing ideas include books on pre-algebra and mathematics curriculum standards.
-- (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. (talk) 16:04, 15 October 2014 (UTC)
Thanks. The sentence now reads:
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.)
-- (talk) 17:44, 15 October 2014 (UTC)
The Random House Dictionary dates "counting number" to 1960-65. -- (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) -- (talk) 18:46, 17 October 2014 (UTC)
A Google ngram that includes "whole numbers" shows that it outnumbers both "natural numbers" and "counting numbers". -- (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?
-- (talk) 10:39, 18 October 2014 (UTC)

isomorphism of ordered sets[edit]

I've tagged this as needing a citation:

  • "... there is a unique isomorphism of ordered sets between them."

Halmos and Hamilton don't use the term "isomorphism". Morash lists it in the index, but the Google books snippet doesn't show the indexed page.

What can be used to source the isomorphism?

-- (talk) 14:07, 15 October 2014 (UTC)

The term "isomorphism" is elementary. It refers to a one-to-one map of one set onto another that preserves certain properties. A group isomorphism preserves group properties. An order isomorphism preserves order properties. In this article, the word is used twice. The fact that it is an "order isomorphism" is stated in one case and implied (but should be stated) in the other. Both of these paragraphs are unclear and need a rewrite. If no one else wants to do it, I'll try to do it.

Here is a reference. The Encyclopedic Dictionary of Mathematics, 2nd edition, MIT Press, 1993, ISBN 0262590204, p. 1169 "A mapping \phi: A \rightarrow A' of an ordered set A into an ordered set A' is called an order-preserving mapping (monotone mapping or order homomorphism) if a \le b always implies \phi(a) \le \phi(b). Moreover, if \phi is bijective and \phi inverse is also an order-preserving mapping from A' onto A, then \phi is called an order isomorphism.— Preceding unsigned comment added by Rick Norwood (talkcontribs) 16:19, 15 October 2014‎

Thanks for your explanation, the reference, and the extended quote. There is an article on Order isomorphism, and it could use some simplification along the same lines. Anyway, I was actually asking for a source that proves the existence and uniqueness of the isomorphism in the context of the natural numbers. I have modified the tag to make that clear. If the proof is so elementary that it is assigned as an exercise, the exercise would probably be a satisfactory source. -- (talk) 17:06, 15 October 2014 (UTC)
Mendelson (1973) states and proves:
Number Systems and the Foundations of Analysis By Elliott Mendelson
-- (talk) 18:26, 15 October 2014 (UTC)
Warner (1965) says: "We may now prove that every naturally ordered semigroup is isomorphic to ( \mathbb{N}, +, \le )." (p. 129) He then states and proves Theorem 16.15.
Modern Algebra By Seth Warner
-- (talk) 19:35, 15 October 2014 (UTC)

──────────────────────────────────────────────────────────────────────────────────────────────────── Can these theorems be used to source the statement that says, in part: "... there is a unique order isomorphism between them"? Or is the terminology too different? -- (talk) 18:37, 16 October 2014 (UTC)

The isomorphism in the Seth Warner quote is an order isomorphism with the additional property that it preserves + (that is f(a+b)=f(a)+f(b). A naturally ordered semigroup is a set which has an identity (usually denoted 0) and an associative binary operation (usually denoted +). It is naturally ordered if for every element n, if a > b then a + n > b + n. So, the isomorphism Warner is talking about is an order isomorphism with some additional properties. Rick Norwood (talk) 20:48, 16 October 2014 (UTC)

Proposal to rewrite lede, part 1[edit]

I propose the following:

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").


In mathematics, the natural numbers or whole numbers are those used for counting ("there are six coins on the table") and ordering ("this is the third largest city in the country").


The term whole number is also used to refer to the natural numbers, with or without zero. Whole number is sometimes used to refer to any integer, whether positive, zero, or negative. In non-mathematical contexts, typically in education, natural numbers may be referred to as counting numbers to distinguishing them from the decimal numbers which serve for measuring and often contain a decimal mark.


In non-mathematical contexts, typically in education, natural numbers may be referred to as counting numbers to distinguishing them from the decimal numbers which serve for measuring and often contain a decimal mark. (talk) 15:54, 15 October 2014 (UTC)

In the current version, "or" should be "and", because the natural numbers and the whole numbers are used for counting. Also, "or" suggests that the two terms are synonymous:
  • "In mathematics, the natural numbers or and the whole numbers are those used for counting ..."
-- (talk) 04:57, 17 October 2014 (UTC)

Proposal to rewrite lede, part 2[edit]

My next proposal is to delete this part of ==History==:

The term counting number is also used to refer to the natural numbers,[citation needed] with or without zero, though in modern usage it is convenient to use this term to refer to the case where zero is excluded. Some authors use the term whole number to mean a natural number while others use whole number to mean counting number; while still others use whole number to refer to any integer, whether positive, zero, or negative. (talk) 16:01, 15 October 2014 (UTC)

Could you put <nowiki></nowiki> tags around the citations, so they don't get expanded on the talk page? -- (talk) 16:14, 15 October 2014 (UTC)
The citations were being expanded at the bottom of the talk page, which causes confusion and clutter, so I have commented them out by putting them inside wiki markup comments (<!-- xxx -->). This doesn't affect the proposed text. -- (talk) 06:38, 16 October 2014 (UTC)
Hi User: I saw your request here but was heading to bed so I did not have time to respond, but thanks for doing it for me. Just to explain why I've proposed these changes is because that the text is being restated, and the later restatement is in ==History== which is seems less germane than if it was in lede. (talk) 11:59, 16 October 2014 (UTC)

Proposal of New Lead[edit]

I present this for integration so that we don't have edit conflicts on the same section. As has shown in the edit history for the last few days, I have proposed a new lead and there has been some discussion of it in the Discussion of Lead section, but not be the editors here. I am moving the discussion here to this newer section as the editors in this section have been proposing parallel edits. The statements in this proposed lead have been substantiated with citations and discussed in these talk pages. If someone has a argument with citations against a statement in this lead, I hope you will provide a link to that here.

The goal of this proposed lead is to address the successor function issue and construction of arithmetic, i.e. it isn't just for counting, as mentioned in my hatnote comment above, and other threads of discussion on these talk pages. Note comments about counting in the history and etymology of the terminology section as shown there the term counting is ancient beginning with things called counting stones and counting sticks with a continuous history to the present day and should not be written out.

<--> 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.

<--> Thomas Walker Lynch (talk) 06:06, 17 October 2014 (UTC)

I don't think Peano axioms need to be in the lead, and I think the words "wide variety of sets" are misleading. Certainly not every set that obeys the Peano axioms is called the set of natural numbers, that phrase only applies to 0, 1, 2, 3, ... and 1, 2, 3, ... . There are various ways of defining 0 and 1, but that is entirely different from having "a wide variety of sets".

The arithmetic structure of the Natural numbers against a successor function is what they are about. Without this structure you have counting numbers. Peano's axioms are number, and order is required to state them. This shift from the naturalist view of counting to the formal view of counting is described in ["History and Philosophy of Modern Mathematics" Minnesota Studies in the Philosophy of Science, Abrégé d'histoire des mathématiques by Jean Dieudonné, and in "Philosophy of Mathematics and Logic" ed. Shapiro where Arithmetic is referred to as theory of natural numbers]. It is part of the constructions that appeared in many parts of mathematics then or shortly after.
The arithmetic structure of natural number sets is the basis of computation theory, where natural number sets are defined such as N ={ "s", "ss", "sss"...} (abstraction based on a first number of "s" and a successor function of appending "s"to a string. [See Papadimitriou "Automata Theory, Languages, and Computation"] Note also the set theoretic N={ {}, {{}}, {{{}} with the first number of {}, ad the successor function of appending another level of nesting. This latter set was important in proving such important things as omega being in omega. Neither of these latter two sets are {0,1,2..}
It is a mathematically incorrect to say that the first number in a natural number set is always "0", and the second "1" because these numbers have algebraic properties that are not required in arithmetic. E.g. The additive identity property of zero is not referenced in the Peano Axioms, and it is not referenced in Arithmetic. It only occurs two levels up, when one defines an algebraic structure. I am perfectly fine in defining addition and subtraction on a set that starts with the number 12, for example. As another example, there is no effort made to establish that "{}" has the properties of an additive identity.
People do often leave the arithmetic structure implied when using natural number because they don't need to explicitly state it successor, as is done when constructing arithmetic. Though there are many times where this structure is important.
I did point out earlier when the convention of zero as a natural number was written out of this article that it was important to mention this convention because 0 is an additive identity for algebraic structures. It does't have to be there, as just discussed, it is just that if algebraic structures do come about it is convenient, see Modern Convention
I think these points, and their citations, thoroughly address the questions you have raised. If not, what remains? (talk) 08:25, 18 October 2014 (UTC)

The next paragraph seems much too complicated for the lead.

I'll attempt to simplify it.

And the final paragraph does not capture how arithmetic is actually constructed. Arithmetic precedes the formalization by centuries. Rick Norwood (talk) 12:03, 17 October 2014 (UTC)

Yes, I would like to add a section with a summary of how arithmetic is constructed from naturals and refer the reader to the arithmetic wiki.
No, it doesn't precede. Arithmetic is formally constructed from the first number and successor function of the Peano Axioms, i.e. upon the Natural Number abstraction. One typically starts the construction by adopting a definition of N= {"s", "ss"...} (quotation marks denoting a subset), then places the successor function in correspondence with the symbol s , - then you can say that repeated successor application is addition. It goes forward from there. See [See Papadimitriou "Automata Theory, Languages, and Computation"] for a step by step example of this. See Shapiro for a universal statement of this, [Shapiro "Philosophy of Mathematics and Logic" page 8], where he notes that arithmetic is the study of the natural numbers.
Though modern Arithmetic is constructed from the natural number abstraction (first number and a successor function), yes people have been doing Arithmetic in an informal, and occasionally incorrect, manner since the beginning. The bone in the picture on our article is the earliest known example of math by humands, and believed to have been used for arithmetic as well as counting. You appeal here to the Naturalists Philosophy of Mathematics, the one that was transformed by Peano's "coup" of providing a construction. This construction is of central importance to the definition of Natural numbers, but not to counting numbers (or counting sticks, or counting stones, etc.) see also History and Etymology and Origin sections on this talk page. The observations on other parts of the talk page that counting number is more organic, more human, useful for teaching, only confirms that "counting numbers" is the correct terminology for describing the set Peano worked from when he formalized and abstracted "it". (talk) 08:25, 18 October 2014 (UTC)

  • Oppose the proposal, and in general I oppose anything that gives more prominence to the term whole numbers, which is for the most part not used by mathematicians. We have to accommodate people who use whole number as a search term, but that's all we have to do. --Trovatore (talk) 18:57, 17 October 2014 (UTC)
I agree with you. I wish the whole numbers went to the Integer page. I see it as a digression and artificial appendage here, but there is a long talk section on why they redirect here, and as the redirection is here, whole redirect here I tried to add an explanation. I did my best to find a short definition, and that "no fractional part" observation is ubiquitous and bared mentioning. There could be shorter sentence and a section I suppose. Anway, I don't have strong opinions apart from the one that as the redirect comes here now that something should be said.
Notice that reference 8 relative to math education thing and counting numbers is wrong, and should be removed, all Eric said on that page was that it was preferable to talk about integers, than to talk about any of these sets, Counting, Natural, or Whole. If we are to depend on this reference in the lead, it seems all this stuff should redirect to integer. (talk) 08:25, 18 October 2014 (UTC)

Quotes from pre-algebra books[edit]

Pre-algebra is taught in middle school (US grades 6, 7, 8).

  • "Numbers make up the foundation of mathematics. The first numbers people used were the natural or counting numbers, consisting of 1, 2, 3, .... When 0 is added to the set of natural numbers, the set is called the whole numbers." (Chapter 1: Whole Numbers, p. 1)
Pre-Algebra DeMYSTiFieD, Second Edition By Allan Bluman (2010)
  • "The most basic collection of numbers is called the natural numbers. The first numbers you learned were probably the natural numbers, those that describe how many objects you can have starting at 1: 1, 2, 3, .... You can have two hands, ten fingers, a dozen cupcakes, one million dollars. All of these quantities are part of the collection of natural numbers. Another important collection of numbers is the whole numbers, the natural numbers together with zero. There are no negatives in the collection of whole numbers." (Chapter 1: The Whole Story, p. 4)
The Complete Idiot's Guide to Pre-algebra By Amy F. Szczepanski, Andrew P. Kositsky (2008)

-- (talk) 10:52, 16 October 2014 (UTC)

Fantastic sources User: We should be using these as citations for when "counting number" appears. (talk) 12:37, 16 October 2014 (UTC)
I have added both books to the References section. You may cite them as you like. I have been using {{harvtxt}} to generate citations linked to the References. -- (talk) 18:22, 16 October 2014 (UTC)
So "whole number" = "natural number" = "counting number". Are there other synonyms? (talk) 12:52, 16 October 2014 (UTC)
Both quotes above say that:
  1. The "natural numbers" are the set { 1, 2, 3, … } .
  2. The "whole numbers" are the set { 0, 1, 2, … } .
  1. Szczepanski & Kositsky use the term "collection" instead of "set" in the quote above.
  2. Bluman uses "counting number" once in his book, and that is in the quote above. (per Google books and Amazon searches)
  3. Szczepanski & Kositsky say this in their summary of Chapter 1: "Natural numbers are the counting numbers starting with 1 and continuing forever." (p. 17)
  4. In their glossary (pp. 313-319), Szczepanski & Kositsky list:
  • "integers All of the natural numbers, their negatives, and zero."
  • "natural numbers ... These are sometimes called the counting numbers: 1, 2, 3 ...."
  • "whole numbers The natural numbers together with zero."
They do not list "counting numbers".
The term "counting numbers" is obsolescent in mathematics education. (Disclaimer: This is original research.)
-- (talk) 16:41, 16 October 2014 (UTC)

unsourced paragraph on notation that was removed from the article[edit]

[This unsourced paragraph on notation was removed from the article by D.Lazard here.]

(Sometimes, an index or superscript "+" is added to signify "positive". However, this is often used for "nonnegative" in other cases, as R+ = [0,∞) and Z+ = { 0, 1, 2, ... }, but rarely in European scientific journals. The notation "*", however, is standard for nonzero, or rather, invertible elements. The notation \mathbb{N}^0 could also mean the empty direct product \prod_{i=1}^k \mathbb{N} resp. the empty direct sum \bigoplus_{i=1}^k \mathbb{N} in the case k=0.)

-- (talk) 13:11, 19 October 2014 (UTC)


The "formal definition" section reads as follows: The various definitions of the natural numbers are equivalent in the following sense: these sets are naturally ordered and there is a unique order isomorphism[citation needed] between them. Therefore, when using or studying natural numbers, it is not necessary to take care of the particular method that has been used to construct them. The first sentence is possibly defensible so long as one sticks to the definition of "equivalence" in terms of the order. Even then one wonders whether this is not merely an equivalence of the ordinal natural numbers. At any rate the second sentence is misleading because it is apparently based on a broader interpretation of equivalence. It can only be claimed that it is unnecessary to pay attention ot the particular method, etc., if the structures are truly equivalent. If there is an aspect which is not equivalent, one cannot make such a sweeping claim of independence of the particular method of definition. For example, Goodstein's theorem is not provable in PA. This is a more careful version of an earlier post that I deleted. Tkuvho (talk) 13:46, 19 October 2014 (UTC)

I agree. I've attempted to address this concern. It would be good if you added something about the sense in which PA and set theory are and are not equivalent and mentioned Goodstein's theorem. Rick Norwood (talk) 14:14, 19 October 2014 (UTC)
The Formal definitions section no longer mentions any isomorphism theorems. Why did you remove that term? The section says "The two approaches have been proven to be equivalent." That statement will need to be sourced. And the way to say that they are "equivalent" is to cite specific isomorphism theorems. -- (talk) 16:51, 19 October 2014 (UTC)
As I mentioned above, the two approaches cannot be said to be equivalent since there are theorems about the number-theoretic integers that cannot be proved from PA. The most we can affirm here is that certain structures are isomorphic, e.g., the order type as in an earlier version. Misleading blanket statements about the equivalence of the approaches should be avoided. Tkuvho (talk) 17:17, 19 October 2014 (UTC)
I cited two isomorphism theorems in the section #isomorphism of ordered sets (one from Mendelson and one from Warner). Could you comment on their relevance to this "equivalence" problem? -- (talk) 17:26, 19 October 2014 (UTC)

Tkuvho, are you able to fill in the senses in which the two approaches are equivalent, and the ways in which they are not equivalent? Rick Norwood (talk) 00:11, 20 October 2014 (UTC)

It would be more appropriate to speak about order-isomorphism (which is sourced, as user pointed out) and avoid making blanket claims about "equivalence", which are not only unsourced but also incorrect, as I already mentioned above (e.g., Goodstein's theorem). Tkuvho (talk) 11:39, 20 October 2014 (UTC)
I have edited this paragraph in order to clarify this equivalence. I believe that, now, it may be sourced. I have also edited the whole paragraph for having less ideas in the same sentence, and a clearer explanation (I hope). If this formulation is kept, I am not sure if the sentence between dashes should been kept in the body of the article or moved in a footnote. D.Lazard (talk) 13:39, 20 October 2014 (UTC)
There is something odd about claiming a purported "equivalence" between an axiomatic approach, on the one hand, and a specific model constructed by set-theoretic means, on the other. The page should stick to a statement of an order isomorphism as sourced by user and avoid philosophical commitments inherent in such "equivalence" claims. Tkuvho (talk) 06:51, 21 October 2014 (UTC)
There is a more immediate problem with the equivalence claim in that it is unsourced and therefore constitutes WP:OR. In fact, I doubt it we can source it since it is incorrect in view of the existence of Goodstein's theorem. Tkuvho (talk) 08:36, 21 October 2014 (UTC)
D.Lazard: "I believe that, now, it may be sourced."
You are going about that backwards. You should be starting with some sources and writing the paragraph based on those sources. -- (talk) 09:12, 21 October 2014 (UTC)
The comments about "truth" in the context of the Peano axioms are really not clear. I think these comments may be confusing the so-called intended interpretation with the Peano axioms. Tkuvho (talk) 10:29, 21 October 2014 (UTC)
IMO, the important fact on which we must emphasize is that the various formalizations are essentially equivalent, sufficiently for allowing the mathematicians, which are not concerned by proof theory, to consider that there only one notion of natural numbers. I am not fully satisfy by the present formalization, because the distinction between true and provable is too technical here, and also controversial. Maybe we could write something like Although the various formalizations are not equivalent from the point of view of proof theory, they are sufficiently close to be equivalent to allow the mathematicians that are not concerned by proof theory of considering that there is only one notion of natural number and not taking care of the chosen formalization. What do you think of this formulation? Can you propose something better? D.Lazard (talk) 13:29, 21 October 2014 (UTC)
What sources are you consulting? -- (talk) 15:23, 21 October 2014 (UTC)
Daniel, reassuring mathematicians that "there is only one notion of natural numbers" may be a worthy goal if it is accurate and sourceable. Meanwhile, at Peano axioms we find the following comment: "Peano arithmetic is equiconsistent with several weak systems of set theory.[12] One such system is ZFC with the axiom of infinity replaced by its negation." Tkuvho (talk) 15:45, 21 October 2014 (UTC)

But did you look at Equivalent definitions of mathematical structures? I think, it could be relevant. Boris Tsirelson (talk) 16:31, 21 October 2014 (UTC)

Is "notion" ambiguous?[edit]

A IP user has tagged "notion" as needing clarification, saying that the word is ambiguous. Sure, it may be ambiguous as are almost every English words. Sure "notion of number" could be replaced by "concept of number", but I am not sure that this would clarify anything. Therefore, I'll remove this tag until this IP user will explain which misunderstanding could occur of propose a better term. D.Lazard (talk) 17:04, 19 October 2014 (UTC)

"Notion" is used by philosophers, not mathematicians. Can't you find some standard mathematical terminology to use? How about "set"? BTW, do you know about this sense of "notion" in English: Notions (sewing)? See an English-language dictionary for the term "notion". -- (talk) 17:19, 19 October 2014 (UTC)
This footnote has the solution: Say "number systems" instead of "notions of number". The source is Elliott Mendelson. -- (talk) 17:45, 19 October 2014 (UTC)
"Notion" is widely used in mathematics, in the same meaning as in philosophy and the primary meaning of the dictionaries. See, for example Kernel (algebra), Erlangen program, Symmetry, Compact space, Cancellation property, Line (geometry), Projective object, General position, Permutation, ..., where the word appears with this meaning. For more examples, search "notion of" in Wikipedia. "Numbers system" has another meaning, complex number may hardly qualifies as a number system. D.Lazard (talk) 18:45, 19 October 2014 (UTC)
All your examples need to be copy-edited. The word "notion" could and should be removed from all of them. The word "notion" is an example of a weasel word. Anyway, a reliable source and a redirect are sufficient reason to use "number systems" instead of "notions of number". -- (talk) 19:06, 19 October 2014 (UTC)
You have just started an edit war. Please admit that you are wrong, and revert. -- (talk) 19:09, 19 October 2014 (UTC)
Bourbaki in English uses "notion" multiple times and "notion of number" three times. (Elements of Mathematics - Algebra part 1) Striking myself. -- (talk) 04:19, 20 October 2014 (UTC)
The French Bourbaki uses "notion" too, so that explains the use in the English Bourbaki: Algèbre: Chapitre 8 -- (talk) 05:59, 20 October 2014 (UTC)

50.5355.68: You are being silly, and remind me of another editor of this article. Please stop. Rick Norwood (talk) 00:12, 20 October 2014 (UTC)

Insisting on accurate and sourced technical terminology is not being "silly". -- (talk) 00:32, 20 October 2014 (UTC)
For the record, James&James say:
-- (talk) 00:50, 20 October 2014 (UTC)
Firstly there is no reason for insisting to substitute a technical word to a word which is clearly intended as non-technical. Secondly, "number system" does not have any definition that is widely accepted in mathematics (you may hardly find any textbook of algebra or number theory that define and uses this term). Also this term is ambiguous, as almost everybody confuse it with "numeral system". Thirdly "number system", when used, refers to an explicit structured set of number, that is a syntactic object; this is not intended here, where one has to emphasize on the concept (a synonymous of "notion", which IMO would be too pedantic here). D.Lazard (talk) 09:46, 20 October 2014 (UTC)
"Thirdly "number system", when used, refers to an explicit structured set of number, that is a syntactic object;"
That is essentially what James&James say in sense (2) above. Sense (1) of "number system", which I omitted, appears to be defining what you are calling a "numeral system". There is no separate entry for "numeral system" in James&James. On WP, disambiguation can be done with links, and since there are different articles on the two senses, the ambiguity can be resolved by linking to "number system". NB: I am referring to my copy of the fourth edition of James&James. Do you have a copy of James&James? -- (talk) 14:22, 20 October 2014 (UTC)

I am going to assume good faith, and explain to why he is wasting our time. Hint,, it is not because you insist on accurate and sourced technical terminology, it is because you are nit-picking, and do not understand the sources. To object to "notion" because it is also used in sewing is like objecting to "bat" because it can mean either a baseball bat or a furry mammal. Which meaning is intended is clear from context. You seem not to understand the difference between a number system and a numeral system. A number system is a set of numbers with certain properties, e.g. a group, a ring, or a field. A numeral system is a way of writing numbers, e.g. the decimal system (base 10). You should only edit articles you understand.Rick Norwood (talk) 15:09, 20 October 2014 (UTC)

Did you even read my comment? I linked to numeral system and number system, and I referenced both senses as given in James&James. -- (talk) 16:28, 20 October 2014 (UTC)

Is there a standard definition of "whole number"?[edit]

When some books say one thing and other books say something else, we must report that fact.Rick Norwood (talk) 17:39, 20 October 2014 (UTC)

I have added two curriculum standards that define "whole number". The two books and the Ontario source give definitions of "natural number" and "whole number". They happen to be consistent, but I have been careful not to generalize from them. AFAICT, the Common Core State Standards do not mention "natural numbers" or "counting numbers". If you find some reasonably recent pre-algebra books or comprehensive curriculum standards that say something different, please say so. You should be able to find pre-algebra books at a library or a bookstore.
Also, note that Szczepanski & Kositsky (2008) say on the inside front cover: "We based this book on the state standards for pre-algebra in California, Florida, New York, and Texas, ..." What curriculum standards are used in Tennessee?
-- (talk) 18:59, 20 October 2014 (UTC)
Szczepanski "is a member of the Department of Mathematics at the University of Tennessee". (back cover)
Bluman "taught mathematics and statistics in high school, college, and graduate school for 39 years. He received his doctor's degree from the University of Pittsburgh." (p. vii)
-- (talk) 19:46, 20 October 2014 (UTC)
Here is a third pre-algebra book that defines the "counting numbers" and the "natural numbers" to be 1, 2, 3, .... The "whole numbers" are defined to be the counting numbers with 0 added to the set. See Chapter 25: Ten Important Number Sets to Know, Counting on Counting (or Natural) Numbers, p. 340:
-- (talk) 19:17, 20 October 2014 (UTC)

I would like it if it were standard for the natural numbers to begin with 1 and the whole numbers to begin with 0. Unfortunately, many professional mathematicians disagree and disparage the attempt by educators, few of whom are working research mathematicians, to tell research mathematicians what to do. For more information on this subject, see math wars. Rick Norwood (talk) 21:36, 20 October 2014 (UTC)

How did you get from complaining about the sources, to explicating a theory about the oppression of research mathematicians by educators? Could you please just comment on the sources per WP:RS? -- (talk) 22:55, 20 October 2014 (UTC)

Basic Math and Pre-Algebra For Dummies is not as respected a source as, say, Naïve Set Theory by Paul Halmos, but my point is that since sources disagree, and we all accept that sources disagree, adding a large number of sources based on US education "standards" does not change the simple fact that sources disagree. I suppose it doesn't do any harm to expand the list of references, but it doesn't do any good, either. Rick Norwood (talk) 14:05, 21 October 2014 (UTC)

Halmos is a reliable source, but you don't need to be told that, right? Both Szczepanski and Bluman have advanced degrees, and their books are published by established publishers. What more can you add about Halmos that makes him more "respected"? Also, both books are based on education standards. The Ontario source is Canadian, not US. Did you even bother to look at it? Would you like me to add some French sources?Face-smile.svg -- (talk) 20:32, 21 October 2014 (UTC)

While there is no point in loading the article with a large number of non-noteworthy sources, I'm not the one who reverted your edit. Incidentally, you should know that Paul Halmos is one of the most respected mathematicians of the 20th century. Rick Norwood (talk) 21:41, 21 October 2014 (UTC)

A search for "halmos whole number" led to this quote. I hope you like it.Face-smile.svg
  • "As mentioned earlier, the study of the set of whole numbers, W = {0, 1, 2, 3, 4, ...}, is the foundation of elementary school mathematics." (p. 60)
  • Halmos is quoted on p. 145.
Mathematics for elementary teachers: a contemporary approach
Gary L. Musser, William F. Burger, Blake E. Peterson
J. Wiley, Jan 4, 2005 - Education - 1042 pages
-- (talk) 20:07, 22 October 2014 (UTC)
The 10th edition of Musser, et al (2013), references both NCTM and Common Core. (The 10th edition doesn't appear to quote Halmos, but it does reference Pólya several times. Is Pólya respectable enough for you?Face-smile.svg) -- (talk) 20:27, 22 October 2014 (UTC)
Pólya is good.Rick Norwood (talk) 23:07, 22 October 2014 (UTC)

Bluman's preface[edit]

"In his preface, Bluman says that his book is "closely linked to the standard high school and college curricula"

"closely linked" is the problem here. There's a pov problem because Bluman should not be assessing whether or not his book is closely linked or not, since it would simply be subjective. (talk) 09:11, 22 October 2014 (UTC)

This is a bit of a problem but since this occurs inside a footnote this may be tolerable. Other editors are invited to comment. Tkuvho (talk) 09:38, 22 October 2014 (UTC)
I wouldn't consider it tolerable. 3 editors have removed it, you and me included. (talk) 09:45, 22 October 2014 (UTC)
I was mostly bothered by the registered trademark, which has been removed by now. Certainly if this comment bothers more than one editor it should be re-removed. Tkuvho (talk) 09:48, 22 October 2014 (UTC)
The Bluman quote establishes the reliability of the source by connecting the citation to curriculum standards. There is a similar quote from Szczepanski & Kositsky that establishes their reliability. NB: I removed the publisher's trademarked name, because the publisher (McGraw-Hill Professional) is identified in the References, and the publisher also establishes the reliability of the source. -- (talk) 10:35, 22 October 2014 (UTC)
This Bluman quote is quite different from Szczepanski because Bluman is explaining that his book is "closely linked" rather than citing them directly. That's the source of the subjectivity/bias. (talk) 11:46, 22 October 2014 (UTC)
The phrase "closely linked" is very vague, but it shows that the author and publisher are aware of curriculum standards, and that is sufficient to show that the book seeks to be mainstream, and that it is not fringe. Can you cite something from WP:RS that you believe applies? -- (talk) 12:02, 22 October 2014 (UTC)
I emailed Judy Bass at McGraw-Hill Professional asking what "closely linked" means and for sources (web site or book reviews). -- (talk) 12:54, 22 October 2014 (UTC)

This was hardly necessary. When an author says (as many do) that his book is "closely linked" with the standards, he means he wrote the book with the standards in front of him and followed them closely.Rick Norwood (talk) 13:44, 22 October 2014 (UTC)

This is different from stating "Roger said ...". Bluman's case is "I said, but paraphrased ...". Additionally, he mentions that they are paraphrased from "... standard high school and college curricula ..." which is impossible if they change every year. (talk) 13:58, 22 October 2014 (UTC)

In this context, I think "standard" means NCTM standard, but you are right, for this to be included, it should specify NCTM standard. Rick Norwood (talk) 17:58, 22 October 2014 (UTC)

If he did, Allan Bluman should say that. He didn't. (talk) 04:37, 23 October 2014 (UTC)
I have added the textbook by Musser, et al, which explicitly references both the NCTM and the Common Core standards. IMO, Bluman could be removed as a source, although he is being quoted here. -- (talk) 09:06, 23 October 2014 (UTC)