Talk:Natural number

Template:WP1.0

There are not

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

Implementation of whole number to redirect here

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Sentence in lead about zero and textbooks

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)

Modern Convention

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

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

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

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

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

---

>

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

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

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

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

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

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

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

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

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

You should sign your posts with four tildes.

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

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

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

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

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

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

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

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

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

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

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

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

I agree that anyone looking for counting number, natural number or whole number should quickly get a clear statement of what the phrase means. Before 14 September, there was a "disambiguation page" [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.

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

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

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

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

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

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

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

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

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

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

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

Discussion of lead

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.

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

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

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

218.187.84.185 (talk) 07:46, 5 October 2014

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

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

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

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

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

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

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

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

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

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

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

von Neumann's definition in lede?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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"

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 Moder Mathematics": [naturalism philosophy in mathematics]

p32

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

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

Now in "über den Zhlbergriff" Kronecker writes

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

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

And of course we have Kronecker's quote:

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

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

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

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

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

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

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

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

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

Editing Etiquette

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

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

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

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

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

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

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

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

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

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

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

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

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

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

the current article appears to be confusing counting numbers with natural numbers

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 arithmetic of zero, 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)