Article snapshot taken from Wikipedia with creative commons attribution-sharealike license.
Give it a read and then ask your questions in the chat.
We can research this topic together.
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:
I think for the discussion of the lead we should start further back, before all this started. At the lead looked like this:
{{quote|], two apples, three apples, ...)]]{{Redirect|Whole number|other uses of the term|Integer}}
{{quote|], two apples, three apples, ...)]]{{Redirect|Whole number|other uses of the term|Integer}}
In ], the '''natural numbers''' are those used for ] ("there are six coins on the table") and ] ("this is the third largest city in the country"). These purposes are related to the linguistic notions of ] and ], respectively. A later notion is that of a ], which is used only for naming.
In ], the '''natural numbers''' are those used for ] ("there are six coins on the table") and ] ("this is the third largest city in the country"). These purposes are related to the linguistic notions of ] and ], respectively. A later notion is that of a ], which is used only for naming.
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Misplaced Pages. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.MathematicsWikipedia:WikiProject MathematicsTemplate:WikiProject Mathematicsmathematics
Currently the image of apples that is used to illustrate counting has two problems: The apples look too identical, so one could say it is a picture of one apple; but . There are six apples in the picture and they could be grouped by the eye in different ways, not only the intended one. The intended way of grouping the apples as (1 single apple, a pair of 2 apples, a row of 3 apples) could be highlighted by connecting them in a colored rectangular background or other helpful way.
I thought the same thing; their (apparently exact) similarity hides the issue of "differences among identical objects", which gets into some issues about the Peano axioms and the reflexive definition of equality: "For every natural number x, x = x. That is, equality is reflexive." http://en.wikipedia.org/Peano_axioms Does 1 apple = 1 apple? What if the apples are of different size? Or of different type? Then 1 apple might not equal 1 apple ... Bruce Schuman (talk) 14:28, 3 August 2013 (UTC)
I have replaced the image by a version, helpfully created by MjolnirPants, in which the apples within each row differ. I hope this deals with the identicality issue. Maproom (talk) 06:14, 13 June 2014 (UTC)
I don't know. To me, the caption says "one apple, two apples, three apples", but the picture just shows six apples, arranged in a triangle. The idea that the one, two, three are supposed to correspond to rows of the triangle is not really obvious.
Maybe if the rows were spaced farther apart? Or if we made them different things (one apple, two cats, three waterfalls)?
But I have to say I'm a little bit skeptical as to whether this notion is really well served by an image of this sort. Our readers pretty much know what it means to count; I don't know whether they really need a picture of counted things. --06:33, 13 June 2014 (UTC)
I agree that the image does nothing to help anyone understand the article – it just makes the article look prettier. But at least it is now technically correct. Maproom (talk) 08:36, 13 June 2014 (UTC)
I can make further changes to the image if anyone thinks it would be helpful. The image is being used to illustrates sets, right? I might be able to pull off some baskets to put the apples in. MjolnirPantsTell me all about it.12:33, 13 June 2014 (UTC)
Is there agreement - at least in America - about whether the natural numbers include zero? (I don't care about university-level math in this context: I want to know what to tell my students so they'll "get the question right" on the high-stakes statewide tests.) --Uncle Ed (talk) 12:12, 25 April 2013 (UTC)
I don't know about America but in Australia the students get taught that natural numbers don't include zero, because whole numbers are natural numbers INCLUDING the number zero. — Preceding unsigned comment added by 121.45.228.1 (talk) 11:23, 5 February 2014 (UTC)
No test should ever rely on this; in most countries (including the US) different people use it differently. If the test in question has ever used the concept of "natural numbers" without defining them, you should raise an official complaint.
128.112.17.131 (talk) 19:09, 11 June 2014 (UTC)
There are not
Strictly speaking, what is the subtraction of natural numbers. Types of division. What is Euclidean division.
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)
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'. MjolnirPantsTell 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. MjolnirPantsTell 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)
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)
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 positiveintegers {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)
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.
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 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" 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.
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.
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)
Much of the above is beside the point (following numbering):
the natural numbers are a group nor a ring
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.
0 to n-1 as result of modulo division is an arbitrary software implementation choice (see line above)
many other mathematicians exclude 0
cartesian coordinates: this is rather more about real numbers, not naturals
empty set has no members
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.
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)
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 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.
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.
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.
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.
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 positiveintegers {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.
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.
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 Neuman'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)
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)
. There are six apples in the picture and they could be grouped by the eye in different ways, not only the intended one. The intended way of grouping the apples as (1 single apple, a pair of 2 apples, a row of 3 apples) could be highlighted by connecting them in a colored rectangular background or other helpful way.
Done 
are always considered natural numbers, so the first sentence should say at least that much. And the quibble about zero should be supported by .svg)
