Neither mathematicians nor college textbooks use the phrase "whole number" very often. "no fractional part" is a bad definition, since it assumes the reader already knows the difference between a whole number and a fraction. I would prefer something along the lines of "A whole number is a counting number, such as "1, 2, 3, ...". Some include 0 and, if the students have been introduced to negative number, include -1, -2, -3, ... . Professionals usually use the more technical terms [[natural number]] or [[integer]]." But this is off the top of my head. I'm not out at school right now, and reference books are not easily to hand, so I leave this edit to others who are interested. [[User:Rick Norwood|Rick Norwood]] ([[User talk:Rick Norwood|talk]]) 14:40, 13 October 2014 (UTC)
Neither mathematicians nor college textbooks use the phrase "whole number" very often. "no fractional part" is a bad definition, since it assumes the reader already knows the difference between a whole number and a fraction. I would prefer something along the lines of "A whole number is a counting number, such as "1, 2, 3, ...". Some include 0 and, if the students have been introduced to negative number, include -1, -2, -3, ... . Professionals usually use the more technical terms [[natural number]] or [[integer]]." But this is off the top of my head. I'm not out at school right now, and reference books are not easily to hand, so I leave this edit to others who are interested. [[User:Rick Norwood|Rick Norwood]] ([[User talk:Rick Norwood|talk]]) 14:40, 13 October 2014 (UTC)
:If the phrase {{tq|"Professionals usually use ..."}} appears in the article, I will immediately [[WP:TAGGING|tag]] it with {{tl|pov-inline}}, because anyone can seek to use precise terminology. Sources that explain <u>why</u> a particular definition was chosen, would be very interesting, though. --[[Special:Contributions/50.53.41.238|50.53.41.238]] ([[User talk:50.53.41.238|talk]]) 19:04, 13 October 2014 (UTC)
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of mathematics on Wikipedia. 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 articles
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
Math Pedagogy has special challenges not found in the rest of mathematics. E.g. They go in a special order. So a teacher talking about whole numbers as not having a fractional part might define the set depending on whether the concepts of negative integers, zero have been introduced. If one were to describe whole as an adjective meaning the number has no fractional part, it skirts the issue as to which set is chosen and legitimizes the definition relative to the universe of discourse currently defined. I.e. an explanation of the use of whole numbers as a didactic tool would help readers understand why there are multiple conventions for the definition of the set of whole numbers. Thomas Walker Lynch (talk) 07:23, 12 October 2014 (UTC)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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" [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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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.
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
One of the things that has come to light in the discussion below is that there are many redirects to this page. Does anyone know how to list them all instead of just running into them?
I've gone over that, the title fits the guidelines of "short, natural, and recognizable" quite well, indeed, we have "countable, whole, and natural", which is about the same thing ;-)Thomas Walker Lynch (talk) 14:58, 12 October 2014 (UTC)[reply]
Ok, Dr. 50.53, I read that \{\{DISPLAYTITLE:Counting, Whole, and Natural Numbers \}\} (do not include backslashes .. and why do they show ???) would change the title heading on the article, but it does not. Is that formulated incorrectly? What is the interaction between the page name (appears in the link), the article title that displays, and the search box?Thomas Walker Lynch (talk) 15:21, 13 October 2014 (UTC)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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. rather it must be understood from the notation.Thomas Walker Lynch (talk) 18:18, 7 October 2014 (UTC)[reply]
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)[reply]
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)[reply]
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.
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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)[reply]
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:
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.
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.
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. MjolnirPantsTell me all about it.13:41, 7 October 2014 (UTC)[reply]
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)[reply]
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.
Towards the end of the 19th century there was a raging debate in Europe between the mathematical Naturalists and the Logicians. The Naturalists believed that numbers stemmed from the human mind. The Logicians believed they came from logic. (With this in mind we can see the irony of the Peano Axioms for arithmetic).
This comes from the book "History and Philosophy of Modern Mathematics":
[naturalism philosophy in mathematics]
Minnesota Studies in the Philosophy of Science;Volume XI;Copyright 1988 by the University of Minnesota; William Aspray and Philip Pitcher, editors
p32
Pincaré criticized the logicist definitions of the numerals on the grounds they were ultimately circular, and he contended that that the proper resolution of the set-theoretic pradoxes should proceed by honoring the vicious circle principle. Goldfarb argues that the former criticism is not an elementary logical blunder, but the product of Poincareé's insistence that legitimate definitions must trace the obscure to th eclear, where the notions of clarity and obscurity are understood psychologically.
.. Ultimately, then the difference between Poincaré and his opponents comes down to a deep divergence in agendas for the philosophy of mathematics. Where Frege and later logicists saw the task of finding foundations as one of the showing how mathematics results from the most general conditions on rational thought, Poincareé saw mathematics as the product of natural objects – human beings – so that the task of finding foundations is intimately linked to bringing clarity (judged by the standards appropriate for such beings) to areas that are currently obscure (again, jugded by the standards appropriate for such beings). As Goldfarb hints, this contrast between Poincaré and the definders of the logicist program is not only useful for throsing into relief the central tenets of logicism, but it also enables us to see interesting parallels between the early criticisms of logicism and contemporary naturalistic approaches to the philosophy of mathematics.
Now in "über den Zhlbergriff" Kronecker writes
The difference in principles between geometry and the mechanics on th eone hand and the remaining mathematical disciplines, here comprised under the designation “arithmetic,” consists according to the Gauss in this, that the object of the latter, Number, is solely the product of our mind, whereas Space as well as Tim have also a reality, outside our mind, whose laws are unable to prescribe completelly a priori. [
Kronecker, Werke, vol 3, 1st half-volume, ed K. Hensel (Leipzig: Teubner, 1899), p 253 (emphasis original; Kronecker quotes, in a footnote a letter fom Gauss to Bessel, 9 April 1830).]
And of course we have Kronecker's quote:
God made the integers, all the rest is the work of man.
Quoted in "Philosophies of Mathematics" - Page 13 - by Alexander George, Daniel J. Velleman - Philosophy - 2002
Kronecker gives credit to the natural numbers to God, but this is also a naturalists quote, as this is the point that man works from. We also have the other Kronecker writing above to confirm this.
So yes, Mr. Norwood, that crazy Mathematician who coined this term was certainly thinking about the mindset of men independent of mathematics when he coined the term "natural numbers". You can set aside your doubts.
Thomas Walker Lynch (talk) 15:42, 6 October 2014 (UTC)[reply]
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)[reply]
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)[reply]
now it seems half of it has been put back .. 18:19, 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
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. MjolnirPantsTell me all about it.20:14, 8 October 2014 (UTC)[reply]
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. MjolnirPantsTell me all about it.20:49, 8 October 2014 (UTC)[reply]
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)[reply]
No, I didn't ask. I explained wikipedia policy, and I explained what I intended to do in accordance with that policy. Why don't you go back to that link you threw up earlier (Right Here) and quote me the part where it says one must poll all participants in a discussion for permission before moving the discussion? Wait, don't bother. I've got the entirety of that page's (note that it's not a policy) treatment of the subject of moving right here:
Material can be userfied or moved to a different page where it is more appropriate. If the refactoring is later reverted, the moved material should be deleted on the pages it was moved to prevent proliferation of the text.
So let's break this down: You are making the highly contentious assertion that I didn't follow the how to guide for moving talk page content based on your own interpretation of how you think it should be done, whereas I took steps to enforce a wikipedia policy after explaining that I would be taking these steps and waiting several hours for any objections (of which I received none). MjolnirPantsTell me all about it.21:31, 8 October 2014 (UTC)[reply]
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 additive identity,
or have a distance metric that includes a single point, then we had better put zero
into our set of counting numbers. Zero will the be the 'first number' and the counting
numbers with zero will still be natural. Note all these reasons for using zero are relative
latecomers in the history of mathematics.
Whole numbers have no fractional part. Natural numbers may or may not be whole. The
counting numbers are whole numbers and natural numbers.
I think this is a good summary of the literature, is mathematically precise, and ironically it appears to
take something from all the editors who have commented here, even the b******s who
keep deleting my comments ;-)Thomas Walker Lynch (talk) 21:14, 8 October 2014 (UTC)[reply]
Can you share some diffs showing where someone else deleted your comments? I've seen where you were reverted for deleting some of mine, and if someone were doing it to you previously, that might excuse it somewhat. Either way, if someone is deleting your comments it needs to stop right away.
see your own comments on my talk page for a description of some of them. If anyone can tell me how this is relevant to the subject of this section please do so now, otherwise, please move this back to the talk page, the etiquette section, or redact it. Of course I will also redact the sarcastic smiley remark, the humor of which apparently isn't universally accepted. Thomas Walker Lynch (talk) 10:34, 9 October 2014 (UTC)[reply]
note the integers are also natural numbers.
set 0 to be the first number
for a successor function use: if n > 0 then m = -n else m = -n + 1 ; n is the number operated on, m is the result
The rationals are natural also, have a two dimensional array of points (p,q), and run a spiral out from the
You are correct that the sets you list obey Peano Axioms (though not Peano's Axioms) but no mathematician calls them "natural numbers". They are in one-to-one correspondence with the natural numbers, but not isomorphic to the natural numbers. The natural numbers starting with zero are a semi-group under addition, but not a group. In any case, whether we begin with 0 or 1 is arbitrary, and all major sources agree that there is, currently, no bull goose mathematician who can make all mathematicians accept that one or the other is right.
All sets built against the Peano Axioms will have isomorphic arithmetic. If one brings in algebraic structures as an supplemental constraint, then an additive identity is needed. The additive identity is not important for finding a set as being natural or not, but rather to qualify it for an algebraic structure. Such a requirement would eliminate the set {1, 2, ..} as well as the sets above.
better not to leave this dangling as declaration for those who might come later. To see this is true, first note that we can count a-priori, second note that the Peano axioms place the numbers that go into our N_s in a sequence n_0, n_1, n_2 .. , so place these numbers into correspondence with their count in the sequence n_i <=> i, now note that S(n_i) = n_(i+1) make the correspondence S(n_i) <=> S(i). Now any arithmetic built with S on N_a will necessarily be isomorphic to every other built on N_b. Now yes, wish there was a reference for it, common knowledge perhaps?Thomas Walker Lynch (talk) 07:59, 12 October 2014 (UTC)[reply]
Von Neuman's natural number definition is an example natural number set that is not based on whole numbers or even +1 as the successor, yet the arithmetic is isomorphic. An additive identity can be defined, but this is not made use of in proofs such as omega containing itself.
I believe it is clear from the mathematics philosophy citings I have provided, two of which I have uploaded for this group to see, that what is natural or not depends upon context. That context is namely the choice of a first number and of a successor function. We can not say generally that '23 is natural' rather we can only say it is natural relative to, say, a first number of 12 and a successor function of 1. Note the description in Stewart Shapiro's Philosophy of Mathematics and Logic 2005, where he calls arithmetic the study of natural numbers. That shows that what is natural depends on context.
Anyway, that is the implications of the foundation work, but if I'm misreading that I very much look forward to learning more as it has implications in computation theory. I.e. where is this lack of isomorphism between the arithmetics over differing sets chosen from Peano axioms. Showing one example would suffice to show a failure in the generalization.Thomas Walker Lynch (talk) 10:04, 9 October 2014 (UTC)[reply]
Accordingly a lead consistent with definition, given all the redirects to this page, would be titled "Whole, Counting, and Natural Numbers". And continue to say, the set of counting numbers are those used for counting objects. Whole numbers are those with no fractional part, and natural numbers are defined by the context of a base number and a successor function.
The first sentence is justified by the school of naturalism. This sentence would correspond to a section on counting numbers, most of which is already there. The beginning of that section would mention naturalism and provide references.
The second sentence is a bit more problematic as it refers to a concept not yet defined, and the only way to do that is to again appeal to naturalism and that of wholeness. Another section would do that. This section would also include a discussion of whole numbers as a didactic tool in math pedagogy. Math pedagogy has some special burdens that bare mentioning.
The third sentence would correspond to a modern discussion starting with a discussion of the school of Logicism and the 'coup' described by Dieudonné. Thus provide the Peano axioms, describe arithmetic, set theory, and number theory relevance. There would also be a discussion of conventions used. This would be an expansion of the current formal definitions section. Thomas Walker Lynch (talk) 10:27, 9 October 2014 (UTC)[reply]
translation of a quote from Abrégé d'histoire des mathématiques by Jean Dieudonné, now relevance of history discussion for this article, now more on prior section confusion between counting and natural numbers
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..."
"From volume II of his Formulaire ([214], 1897-1899) on, Peano substitutes the set N of all natural integers to the set N* of the nonzero positive integers, [and] the zero to the 1, ..."
The normal translation of "celui" would be "that", but I am not sure that "that N* of the positive integers" would be correct English. Adding "and" is required because of the truncation of the citation. D.Lazard (talk) 08:33, 9 October 2014 (UTC)[reply]
Note also the following: the fact that in French the set of natural integers contains zero does not imply that the same convention applies in English. The sentence contains an example where the English and French conventions differs: "nonzero positive integers" is redundant in English, not in French, where "positif" means "nonnegative". D.Lazard (talk) 08:48, 9 October 2014 (UTC)[reply]
thank you very much. I have received email back from a colleague expert in arithmetic at Lip-ens, you surely know him, and he throws this into the mix <<:
It seems to me that there is a slight inversion: "Peano substitue l'ensemble N de tous les entiers naturels à celui N* des entiers positif non nuls » means « Peano replaces the set N* of the positive nonzero integers by set set N of all natural numbers » and he also replaces the one by the zero in the writing of axioms.
I have used "substitute ... to" and not "substitute ... by". Thus, there is no inversion. By the way, it is impossible to use French texts for differentiating between "integer", "whole number" and "natural number". In fact, the literal translation of "entier naturel" is "natural whole", as "entier" is an adjective meaning "whole", which becomes a noun only in mathematics, where it means "integer". I suspect that, initially, "integer" and "whole number" were synonyms, one being a direct translation from Latin, and the other being a translation from the French translation of the same Latin word. By the way again, the term "natural number" seems much older than 19th century. It dates from the time where the negative numbers were not fully accepted in mathematics and were considered as "unnatural". The opposition between "natural" and "negative" is very similar to that of "real" and "imaginary" numbers. D.Lazard (talk) 13:40, 9 October 2014 (UTC)[reply]
The inversion the French professor of arithmetic is referring to is in the translation to English of the original French sentence. I.e. he is giving me advice relative to the translation. That is his email response verbatim after the top '<<' he is not speaking to you in that quote, but rather to me. He has only the original French text and my proposed translation, nothing more, none of this wikipage. He has not seen your entries here, nor mine or anyone else's. There is not bias. I can share the email correspondence if pressed because one finds that important, but he might be unhappy if I pulled him into the fray here.Thomas Walker Lynch (talk)
p39 "N_0 == <<nombre, (entier, positif ou nul)>>" =english=> " N_0 = number (whole, positive, or zero). Can entier be translated as integer?
Age of the natural number term - great. Can you point me at a source for that information? In the discussion I've been able to find on natural numbers, [afore mentioned texts Abrégé d'histoire des mathematiques, History and Philosophy of Modern Mathematics, and Philosophy of Mathematics and Logic] all first discuss natural numbers by name at the end of the 19th century either in the context of Poincaré and naturalism, or as a formalization by Peano. These two are placed in juxtaposition. I have gone so far as to write Dr. Shapiro and point blank ask him if the term was used sooner that that, and what for. If you have a reference for an earlier use, that would be wonderful. Though even if there is an earlier reference, the 'coup' of Peano is dominate in all the references above, he has provided the modern definition. Shapiro's discussion appears particularly poignant when he calls arithmetic the study of natural numbers.
When people equate whole numbers, counting numbers, and natural numbers, as this article appears to have originally done, one has great difficulty in making sense of the history of the terms and tracing etymology and evolution. It turns out to be important to place these terms into their appropriate schools of mathematical philosophy in order to see a continuity in going back and to have an understanding of what they are. When doing this, the above texts appear to run into a wall for natural numbers with Peano. In the lead sentence I am proposing, I have divided the terms by whether they are a property of a set, or a set itself, and placed them into their appropriate schools of thought: counting numbers goes to naturalism and is a unique set, whole numbers goes to math pedagogy and naturalism and as a didactic tool can take multiple forms (as noted in the current lede), and natural numbers goes to Logicism and Peano. One must have context to say a number is or isn't natural. That context is a first number and successor function with special qualities. Again this is described in the above texts, references linkedin into the wikipage for Peano Axioms, and via common usage.Thomas Walker Lynch (talk) 15:41, 9 October 2014 (UTC)[reply]
This page is not a forum to discuss the history of mathematics. The history of terminology is out of scope here. A section on the history of terminology for the integers would certainly be interesting in Wikipedia, but not here. The best place would be in Integer. However, to write such an history, it is required, by Wikipedia policy WP:OR, that a published text written by a specialist discusses this point. It is useless to discuss these questions here until some reliable source is found. As it is, the article is convenient, as it reports the various modern uses of these terms. It may certainly be improved, but nothing in your lengthy posts allows to improve it. Thus, again stop to use this page as a forum for your questions and thoughts about history and philosophy of mathematics. D.Lazard (talk) 16:36, 9 October 2014 (UTC)[reply]
The "history of numbers" page redirects here since 2005. There is a section in this article you are 'talking' about. Are you proposing to change this now? What will your edits be?
The definition of the natural numbers is very much part of the philosophy of mathematics even today. The Shapiro reference, "Philosophy of Mathematics and Logic" is from 2005, and he still publishes (perhaps you didn't see this? I have left a copy of the first chapter on my server, see the link above). It is a pretty good reference. The fact that there is a continuous understanding of its meaning going back to ~1900 is highly relevant, not a reason to dismiss the literature or be dismissive of my bringing this to light.
The lede section is much improved since I arrived here and criticized it for being circular and only providing {1,2,3, .. } in the lead sentence. Thank you for the positive feedback on this.Thomas Walker Lynch (talk) 19:10, 9 October 2014 (UTC)[reply]
The purpose of a Wikipedia talk page (accessible via the talk or discussion tab) is to provide space for editors to discuss changes to its associated article or project page. Article talk pages should not be used by editors as platforms for their personal views on a subject.
Yes, this page is trying to be too much perhaps. The history of numbers section in this natural numbers article is well intended, and echos the mathematical school of naturalism discussions of the late nineteenth century [see "History and Philosophy of Modern Mathematics" book (discussed in the 'Origin of Natural Numbers" section on this talk page, "Philosophy of Mathematics and Logic" book full reference and link to first chapter given further up on this talk page, and the "Abégé d'Histore des Mathematiques" book discussed here and "von Neumann's definition in lede?" section on this talk page. (I introduced each of these references further above in the discussion)] I suggest the redirect be changed as you propose, and that the current history of numbers discussion on this page be converted to a short introduction to Logicism and Naturalism views on natural numbers as a summary of the reference material. Some of the material already there could be kept to buttress the naturalists position. This would lead nicely to the Peano axioms. Thomas Walker Lynch (talk) 08:32, 10 October 2014 (UTC)[reply]
The redirect from "history of numbers" does not promise the reader any history of natural numbers so it would be safe to change that now. The redirect here is probably based on a technically incorrect presumption of counting numbers being identical to natural numbers ["History and Philosophy .., "Philosophy of Mathematics ..", "Abégé d'Histore..", and the newer Peano citings added]. Accordingly, the history of numbers section in our article here should start out "the counting numbers" instead of the "the natural numbers", and needs to discuss Peano's coup [This is a quote from Dieudonné, see "Abégé d'Histore.."] in defining arithmetic when introducing the term natural numbers to be consistent with the references cited just above. (If someone has a reference to the term used before that, we would love to see it.) Seems we should update the history section here before adding something to the history of numbers page. Does someone have a reference to the 'dot' as an origin to mathematics? I added the picture of the counting stick, though the corresponding text was deleted. Can we change 'dot' to 'mark or stone in a pile', this is more accurate based on the citings we have, and it causes the bone picture to make sense. This bone is billed as the earliest example of mechanized counting (as opposed to using one's fingers which we suppose is much older) in discussion of it, there is a wiki page for it. Of course piles of stones don't survive time as do bones.Thomas Walker Lynch (talk) 14:33, 10 October 2014 (UTC)[reply]
The redirect from "history of numbers" does not promise the reader any history of natural numbers so it would be safe to change that now. — User:Thomas Walker Lynch
To editor Thomas Walker Lynch: The distinction that you make between various kinds of numbers is wrong from a mathematical point of view as well from an historical point of view. Historically, until 15th century, there were only (whole) numbers and fractions. The fact of considering zero, negative numbers and real numbers as true numbers is thus relatively recent. The use of "whole" or "integer" to distinguish "natural" numbers from fraction does not implies any philosophical distinctions among integers. Your insistence of making conceptual differences is your own idea, which is not based on any reliable source. All the citations that you provide support only the fact that the name given to the positive integers or to the non-negative integers vary with the authors. You cite also Frege and Peano as if they have a different philosophical conception of the integers. This is possible but of low interest. The fact is that both were concerned by the need of a formal definition of the integers that is logically solid. They gave different formalisms that have been proved equivalent. This implies that for modern mathematicians as well as for ancient ones there is only one notion of natural numbers (positive integers). Thus your sentence "a technically incorrect presumption of counting numbers being identical to natural numbers" is completely wrong. Both are and have always been the same thing. Therefore, there is no valid reason to modify the article as you suggest.
It seems that your concern is the philosophical problem of the relationship between the mathematical and abstract concept of number and the real world. This a wide question, which is a special instance of the relationship between mathematics and the real world. This is completely out of the scope of this article. Moreover, very few may be written in an encyclopedia about this because of the lack of any consensus, except that the answer to this question is of low importance for mathematicians. For the question itself, you may hardly find two mathematicians or philosophers that have the same opinion. D.Lazard (talk) 15:30, 10 October 2014 (UTC)[reply]
D.Lazard just a few corrections here, I have not made any statements about zero, negatives, or reals being older that the 15th century. Though I did ask you for a reference or source material using the term natural number before the Peano Axioms as you said the term was used earlier. Do you know of such a reference??
As far as the integer bit I gave one citation from Dieudoné saying that in Peano's volume II, he placed 0 in N. The rest of the conversation on that was driven by you and and other editor, I only replied to a request for further translation. You do know that the 'history of numbers' comes to this article right?
I have not used 'whole' or 'integer'to distinguish natural from fraction. Again, I don't know where you are getting that, perhaps it is because of my remarks on whole numbers. You might not realize, but 'whole numbers' is also comes to this article. My input on whole numbers is now in the article is cited.
Yes I did summarize p32 from "History and Philosophy of Modern Mathematics" I'm sorry you find that of low interest. Gosh I find it fascinating, as apparently did the authors who took the time to have researched it and write about it.
The conceptual difference I am grappling with is between 'the set of counting numbers' and 'a set of natural numbers'. The references show that a definition of natural number has context of a starting number and a successor function (and that set choices such as {1,2,...} are conventions). How do we get from there to "It is positive integers." Here are a couple of questions I hope you will answer to help us understand that:
Are we to dismiss the set-theoretic definition of natural numbers as natural numbers. It is based on sets not counting numbers. I see they can be placed in correspondence, but that isn't equivalence because the successor functions are defined differently.
If the counting numbers and the natural numbers are identical, how is it that the Peano Axioms are numbered?
I saw in this article a few problems, people gave me a hard time here, but the article is much improved. There are still some issues with the history, and unresolved differences between your description of natural numbers and that in some of the source material. I'm trying to reconcile these sources to understand if you are describing convention or a profound equivalence, i.e. that arithmetic is no richer than counting. Don't you think that an opening sentence for this article that says that 'counting numbers are used for counting' would be better than 'natural numbers are used for counting'?Thomas Walker Lynch (talk) 04:25, 11 October 2014 (UTC)[reply]
DLazard, your indented comments here do not speak to the text above on the subject of changing the history redirect. Please take note of WP:INDENT. There is a section on this topic above.
DLazard, I went to the trouble of creating a talk section on history so that we could discuss your comments that history should not be discussed here, and adjust the "history of numbers" redirection accordingly. You have deleted that section thus leaving the proposal buried here were some may not have seen it. Can you please put it back?Thomas Walker Lynch (talk) 21:19, 11 October 2014 (UTC)[reply]
current redirects that land here, and proposed changes
Counting number
Counting numbers
History of numbers
ℕ
Natural integer
Natural number (transclusion)
Natural Numbers
Natural numbers
Non-negative integer
Nonnegative integer
OTTFFSSENT
Positive integer
Positive integers
Unnatural number
Unnatural numbers
Von Neumann natural number
Von Neumann natural numbers
Whole number
Whole number (disambiguation)
Whole Numbers
Whole numbers
What is a transclusion?
Is it proper to have a Whole number dsimabiguation and a direction? Is that redundant or does it help in some way?
IMHO:
Will delete the capitalization variations as they are unnecessary.
The history of numbers and Von Neumann natural number to be moved to respective articles.
From math is fun: "Question : What are Unnatural Numbers ? Answer :There is no such term called unnatural numbers." Nor is this 'term' explained on the page, will delete this redirect.
OTTFFSSENT? One Two Three... should be put on the acronyms list and redirect there. Probably predates the acronyms list. I will remove redirect.
The integer terms to be forwarded to the integer article.
Natural Integer removed so it goes to disambiguation. How can we assure that the natural number page and the integer page both appear on that disambiguation page?
Thomas Walker Lynch (talk) 15:35, 12 October 2014 (UTC)[reply]
Thomas Walker Lynch asked: "Is it proper to have a Whole number dsimabiguation and a direction? Is that redundant or does it help in some way?"
There have been three aspects people have spoken from: a) mathematical definition/understanding b) convention c) math pedagogy. When I first saw this page I was concerned upon finding the 0 based convention was written out.
Rick's comments have been insightful, and I appreciate his patience. Thank you Rick. Particularly two comments A. he doubted a mathematician was thinking of archaeology when he coined natural numbers. B. That the first axiom should be defined in this article in terms of a 'first number'.
At the time when Rick said 'A' I understood as DLazard, that natural numbers we considered 'natural' exactly because they were what people originally used (no negatives etc.). But the math sources told a more sophisticated story, yes there were the naturalists (dignity saved..), but now 'arithmetic is the theory of natural numbers' [Shapiro p8], and Dieudonné called it Peano Axioms a 'coup', (among Peanos many coups) [citing goes here ;-)].
Rick's 'B' allows for a unified presentation of Von Neumann's definition with a first number of'{}'. Please forgive me Rick, but in trying to address the convention issue, I asked you to update the Peano Axioms with a zero as the first number, (a common convention, but not a full definition). Can you please put back the 'a first number' version back? We can then state Von Neumann's definition in terms of the Peano Axioms.
Rick you also made the argument about algebraic structure, the same one I had made earlier and discussed with another editor at length. Zero provides for algebraic structure (which goes beyond arithmetic structure) this is stated on the Peano Axioms page and we should mention it too.
Blame Google Books and the publisher. They want you to buy a copy, not read it online. I find that counterproductive. Anyway, a large public library or an academic library would probably have it. You might be able see more if you can guess a search term: "peano axioms" or "natural numbers", say. --50.53.37.49 (talk) 00:06, 13 October 2014 (UTC)[reply]
In the conventional phrasing of the Peano Axioms, the context consists of a first number and a successor function. One can build many arithmetic systems by changing the parameters. In Halmos's definition both first number and the successor function are embedded in the definition of natural number. There are no free parameters, so we get exactly one arithmetic. One could call this a 'base' arithmetic, or some such, and then say any system that is isomorphic to it is 'representative'. The symbols in a the representative system 'represent' numbers, etc. But here is the key difference, Halmos's system does not automatically give us the isomorphism, rather we have to go find it. With the conventional statement of the Peano Axioms, if we use two different first numbers, those are in correspondence. If we use two different successor functions, those are in correspondence. It seems that Halmos's system is less rich. Said richness is what is added to counting numbers with the Peano Axioms, so daresay, Halmos's definition appears at first blush to be an identity function on the counting numbers and not a set of Peano Axioms.Thomas Walker Lynch (talk) 20:57, 12 October 2014 (UTC)[reply]
I don't recall saying that this article should state the first axiom in terms of "first number". I do recall saying that some books state the axiom that way. For this article, which should be readable by non-mathematicians, think Axiom One should just state "Zero is a natural number." After the axioms, we should mention that some authors start with 1. Whether we need to get into "first number" at all at this level of exposition I doubt. As for the Halmos quote. Halmos defined "0" as "{}", so his version starts with 0. Rick Norwood (talk) 22:44, 12 October 2014 (UTC)[reply]
I'm not going to misrepresent the source in the Notes. Could you please answer the question:
Is Halmos using the term "Peano Axioms" in a significantly different sense than the article?
Answer: No. I thought I made that clear. {} is the symbol Halmos uses for 0. The main difference in the Halmos version of the Peano Axioms is that he uses the language of set theory instead of the language of arithmetic. Rick Norwood (talk) 00:17, 13 October 2014 (UTC)[reply]
Thanks. Your wording is exactly what I was looking for: "[Halmos] uses the language of set theory instead of the language of arithmetic." --50.53.37.49 (talk) 00:42, 13 October 2014 (UTC)[reply]
What of the fixed successor function? Halmos has shown counting numbers have recursive structure i.e. are natural. Other sets which the Peano Axioms could be used to show have recursive structure this set of axioms does not speak to. It would be analogous to defining a group on a specific set and saying there is only one group in the universe, rather than specifying the properties of a group - when talking about algebraic structure. A piece of abstraction is missing.Thomas Walker Lynch (talk) 07:59, 13 October 2014 (UTC)[reply]
Rick, I understand from WP:IDENT that a new subject in a section starts at the bottom outdented, so let me continue the above other subject on the formal section and first number rather than mix is in above. Just a question. Wouldn't it be better to give the definition you gave with the 'there is a first number' and then to start the Von Neumann section with '{} is taken as the first number' rather than starting with the demonstrably false, and very confusing to a younger reader, "0 = {}"? I do know people often write it this way, but that doesn't make it a requirement to do so here. Also I do understand the intended meaning is that the two entities are to be placed into correspondence, but to a person seeing this the first time with no prior context such would be found in a book .. just asking.Thomas Walker Lynch (talk) 08:35, 13 October 2014 (UTC)[reply]
Use of "whole number" in lede of integer, as a synonym
As an issue that has been raised, integer uses "whole number" as an synonym to integer. Since it is relevant to this page as this "whole number" redirects to natural number, I am posting this notice that I will remove it as a synonym on integer. Additionally, the use of "whole number" as a synonym to "integer" is not cited.174.3.125.23 (talk) 08:58, 13 October 2014 (UTC)[reply]
Here are a few things to consider, whatever you do:
The definition of whole number is consistently given as "number without fraction". However, the domain of discourse varies, so sometimes 'the set of whole numbers' ends up being the integers, the non-negatives, or the positives. The citing to that definition of 'set of whole numbers' is in the lede here, Weisstein, Eric W., "Counting Number", and "Whole Number", MathWorld. This talk page has a section where the original redirect was discussed. It looks like you were involved in that.
The definition "number without fraction" is a bit problematic, as it requires knowing what a fraction is, a higher level construct. You could instead define a whole number as a number reachable with a successor function of ++1 or --1 from zero via the Peano Axioms. (or variations leaving out --1 or starting at 1). Such a definition would fit better on this page, though currently it is not given this way. I noticed that the integer construction discussion defines integers on top of naturals via the spiral over a plane. Hence, if you define wholes on integers, you get essentially the same definition as this one.
As a final note, as was discussed earlier though deleted, it seems the set of whole numbers is not commonly relied upon in serious works, but it is common in math textbooks. There is a section on this talk page requesting a math textbook sentence about wholes in the article here.
It is not necessary to always specifically cite everything. I'd have thought that knowing people sometimes refer to them as whole numbers was a sort of the sky is blue sort of thing. Dmcq (talk) 14:58, 13 October 2014 (UTC)[reply]
Neither mathematicians nor college textbooks use the phrase "whole number" very often. "no fractional part" is a bad definition, since it assumes the reader already knows the difference between a whole number and a fraction. I would prefer something along the lines of "A whole number is a counting number, such as "1, 2, 3, ...". Some include 0 and, if the students have been introduced to negative number, include -1, -2, -3, ... . Professionals usually use the more technical terms natural number or integer." But this is off the top of my head. I'm not out at school right now, and reference books are not easily to hand, so I leave this edit to others who are interested. Rick Norwood (talk) 14:40, 13 October 2014 (UTC)[reply]
If the phrase "Professionals usually use ..." appears in the article, I will immediately tag it with {{pov-inline}}, because anyone can seek to use precise terminology. Sources that explain why a particular definition was chosen, would be very interesting, though. --50.53.41.238 (talk) 19:04, 13 October 2014 (UTC)[reply]
.svg)
