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Frequently asked questions (FAQ)
Q: Are you positive that 0.999... equals 1 exactly, not approximately?
A: In the set of real numbers, yes. This is covered in the article. If you still have doubts, you can discuss it at Talk:0.999.../Arguments. However, please note that original research should never be added to a Wikipedia article, and original arguments and research in the talk pages will not change the content of the article—only reputable secondary and tertiary sources can do so.

Q: Can't "1 - 0.999..." be expressed as "0.000...1"?
A: No. The string "0.000...1" is not a meaningful real decimal because, although a decimal representation of a real number has a potentially infinite number of decimal places, each of the decimal places is a finite distance from the decimal point; the meaning of digit d being k places past the decimal point is that the digit contributes d · 10-k toward the value of the number represented. It may help to ask yourself how many places past the decimal point the "1" is. It cannot be an infinite number of real decimal places, because all real places must be finite. Also ask yourself what the value of would be. Those proposing this argument generally believe the answer to be 0.000...1, but, basic algebra shows that, if a real number divided by 10 is itself, then that number must be 0.

Q: The highest number in 0.999... is 0.999...9, with a last '9' after an infinite number of 9s, so isn't it smaller than 1?
A: If you have a number like 0.999...9, it is not the last number in the sequence (0.9, 0.99, ...); you can always create 0.999...99, which is a higher number. The limit is not defined as the highest number in the sequence, but as the smallest number that is higher than any number in the sequence. In the reals, that smallest number is the number 1.

Q: 0.9 < 1, 0.99 < 1, and so forth. Therefore it's obvious that 0.999... < 1.
A: No. By this logic, 0.9<0.999...; 0.99<0.999... and so forth. Therefore 0.999...<0.999..., which is absurd.
Something that holds for various values need not hold for the limit of those values. For example, f (x)=x 3/x is positive (>0) for all values in its implied domain (x ≠ 0). However, the limit as x goes to 0 is 0, which is not positive. This is an important consideration in proving inequalities based on limits. Moreover, although you may have been taught that must be less than for any values, this is not an axiom of decimal representation, but rather a property for terminating decimals that can be derived from the definition of decimals and the axioms of the real numbers. Systems of numbers have axioms; representations of numbers do not. To emphasize: Decimal representation, being only a representation, has no associated axioms or other special significance over any other numerical representation.

Q: 0.999... is written differently from 1, so it can't be equal.
A: 1 can be written many ways: 1/1, 2/2, cos 0, ln e, i 4, 2 - 1, 1e0, 12, and so forth. Another way of writing it is 0.999...; contrary to the intuition of many people, decimal notation is not a bijection from decimal representations to real numbers.

Q: Is it possible to create a new number system other than the reals in which 0.999... < 1, the difference being an infinitesimal amount?
A: Yes, although such systems are neither as used nor as useful as the real numbers, lacking properties such as the ability to take limits (which defines the real numbers), to divide (which defines the rational numbers, and thus applies to real numbers), or to add and subtract (which defines the integers, and thus applies to real numbers). Furthermore, we must define what we mean by "an infinitesimal amount." There is no nonzero constant infinitesimal in the real numbers; quantities generally thought of informally as "infinitesimal" include ε, which is not a fixed constant; differentials, which are not numbers at all; differential forms, which are not real numbers and have anticommutativity; 0+, which is not a number, but rather part of the expression , the right limit of x (which can also be expressed without the "+" as ); and values in number systems such as dual numbers and hyperreals. In these systems, 0.999... = 1 still holds due to real numbers being a subfield. As detailed in the main article, there are systems for which 0.999... and 1 are distinct, systems that have both alternative means of notation and alternative properties, and systems for which subtraction no longer holds. These, however, are rarely used and possess little to no practical application.

Q: Are you sure 0.999... equals 1 in hyperreals?
A: If notation '0.999...' means anything useful in hyperreals, it still means number 1. There are several ways to define hyperreal numbers, but if we use the construction given here, the problem is that almost same sequences give different hyperreal numbers, , and even the '()' notation doesn't represent all hyperreals. The correct notation is (0.9; 0.99; 0,999; ...).

Q: If it is possible to construct number systems in which 0.999... is less than 1, shouldn't we be talking about those instead of focusing so much on the real numbers? Aren't people justified in believing that 0.999... is less than one when other number systems can show this explicitly?
A: At the expense of abandoning many familiar features of mathematics, it is possible to construct a system of notation in which the string of symbols "0.999..." is different than the number 1. This object would represent a different number than the topic of this article, and this notation has no use in applied mathematics. Moreover, it does not change the fact that 0.999... = 1 in the real number system. The fact that 0.999... = 1 is not a "problem" with the real number system and is not something that other number systems "fix". Absent a WP:POV desire to cling to intuitive misconceptions about real numbers, there is little incentive to use a different system.
Q: The initial proofs don't seem formal and the later proofs don't seem understandable. Are you sure you proved this? I'm an intelligent person, but this doesn't seem right.
A: Yes. The initial proofs are necessarily somewhat informal so as to be understandable by novices. The later proofs are formal, but more difficult to understand. If you haven't completed a course on real analysis, it shouldn't be surprising that you find difficulty understanding some of the proofs, and, indeed, might have some skepticism that 0.999... = 1; this isn't a sign of inferior intelligence. Hopefully the informal arguments can give you a flavor of why 0.999... = 1. If you want to formally understand 0.999..., however, you'd be best to study real analysis. If you're getting a college degree in engineering, mathematics, statistics, computer science, or a natural science, it would probably help you in the future anyway.

Q: But I still think I'm right! Shouldn't both sides of the debate be discussed in the article?
A: The criteria for inclusion in Wikipedia is for information to be attributable to a reliable published source, not an editor's opinion. Regardless of how confident you may be, at least one published, reliable source is needed to warrant space in the article. Until such a document is provided, including such material would violate Wikipedia policy. Arguments posted on the Talk:0.999.../Arguments page are disqualified, as their inclusion would violate Wikipedia policy on original research.
Featured article 0.999... is a featured article; it (or a previous version of it) has been identified as one of the best articles produced by the Wikipedia community. Even so, if you can update or improve it, please do so.
Main Page trophy This article appeared on Wikipedia's Main Page as Today's featured article on October 25, 2006.
Article milestones
Date Process Result
May 5, 2006 Articles for deletion Kept
October 10, 2006 Featured article candidate Promoted
August 31, 2010 Featured article review Kept
Current status: Featured article

On secondary sources[edit]

There seems to be some confusion here about what a secondary source is in the context of the In popular culture section. One cited source is Straight Dope, a secondary source about several discussions that took place on a message board. Those discussions would be primary sources. Likewise, the Blizzard Entertainment press release was a secondary source concerning discussions that took place on the discussion forum. Those discussions themselves are primary sources, but the press release is a third party secondary source.

I believe that the content here is important for establishing the wider cultural context for the subject of this article. It is not an indiscriminate collection at all, but indeed is very discriminate. It was suggested that editors here study the essay WP:IPC. I think that summarizes nicely why sections like this are an important part of Wikipedia. Sławomir Biały (talk) 12:02, 27 May 2016 (UTC)

Per this RFC, content in IPC sections requires reliable sourcing that indicates the significance of the reference to the topic. In this particular case, The Straight Dope and Blizzard Entertainment can't demonstrate their own significance. You would need much better sourcing to support an argument that a noticeboard post is significant enough to warrant discussion here. Nikkimaria (talk) 12:06, 27 May 2016 (UTC)
Well, we disagree about that. So does User:JohnBlackburne, apparently. Perhaps some other editors watching this page, who aren't just drive-by taggers, might care to weigh in? The consensus at the RfC is that secondary sources are required, pure and simple. We have that. If the secondary sources are adequate is a matter for local consensus, not administrative fiat. Sławomir Biały (talk) 12:31, 27 May 2016 (UTC)
To quote from the RFC close, "The source(s) cited should not only establish the verifiability of the pop culture reference, but also its significance". These sources don't do that. Nikkimaria (talk) 12:44, 27 May 2016 (UTC)
The sources do establish significance. They are secondary sources about message board discussions, indicating that the discussions had a broad impact upon that online community. That is clear, and direct, "significance", even on a very strict reading of the term "significance" in the closer's rationale at the RfC you cited. Also, typically on Wikipedia secondary sources are ipso facto evidence of significance. See, for example, our general notability guideline. I'm not seeing any evidence in the comments put forward at the RfC that sources like this would not be considered acceptable, the closer's comment notwithstanding. Sławomir Biały (talk) 12:52, 27 May 2016 (UTC)
A company remarking on its own noticeboards doesn't mean that the posts are significant to anyone but the company - it certainly doesn't mean that the posts are significant to an encyclopedic understanding of the topic. Nikkimaria (talk) 16:45, 27 May 2016 (UTC)
No, it doesn't necessarily mean this. That's why we have editors to decide whether something is noteworthy enough to go in the article. Obviously not everything is necessarily worth including, but something that indicates the cultural impact of the topic outside of mathematics is definitely germane and important information worthy of inclusion in the article. Sławomir Biały (talk) 16:51, 27 May 2016 (UTC)
It would be if there were actually sourcing to demonstrate broad cultural impact and not just "someone on our noticeboard thought this was cool". Nikkimaria (talk) 17:05, 27 May 2016 (UTC)
Well, I happen to think that the Cecil Adams piece does establish broad cultural impact. In fact, he addresses precisely that question in a fairly direct manner. Sławomir Biały (talk) 17:31, 27 May 2016 (UTC)
To expand on my edit summary, WP:IPC is just an essay, but even looking at it I cannot see anything in the section that is problematic. It is not an indiscriminate unsourced list. It does not contain only meaningless irrelevant mentions. The RFC requires sources and that they establish the significance of the entries, but I think they do that – they all relate to how 0.999... is perceived in popular culture. It is a very unusual thing in mathematics, elementary enough to be widely 'understood', but also easily misunderstood because it involves subtleties which might not be apparent on first encountering it. So it has an outsize presence in popular culture, and this is worth documenting.--JohnBlackburnewordsdeeds 12:55, 27 May 2016 (UTC)
It's worth documenting, but it isn't worth overemphasizing the impact of a couple of noticeboard posts on cultural understanding of the topic. Nikkimaria (talk) 16:45, 27 May 2016 (UTC)
If you have other sources to bring to bear, now would be a good time to bring them up. Here "overemphasizing" seems to mean "at the exclusion of other, better sources on the cultural understanding of the topic". If you have such sources, I'm sure that the article would benefit from their inclusion. Sławomir Biały (talk) 16:51, 27 May 2016 (UTC)
"Overemphasizing" means presenting as "important information worthy of inclusion", something that really isn't absent better sourcing. I don't have sources to suggest these noticeboard posts are significant; unless you do, they shouldn't be here. Nikkimaria (talk) 17:05, 27 May 2016 (UTC)
It seems like you're alone in that view. WP:CON. Sławomir Biały (talk) 17:21, 27 May 2016 (UTC)
WP:LOCALCON. Nikkimaria (talk) 17:46, 27 May 2016 (UTC)
Right. But you're also the only editor who thinks that these sources aren't acceptable under that RfC and the essay you referred to. As far as this article goes, the local consensus and community consensus apparently agree, unless you can make some convincing case, which so far you havent. Indeed, your eagerness to disregard local consensus smacks of bureaucratic I-know-best cabalism. I think we're done here, unless you have anything of substance to say. If all you can do is refer to alphabet soup that's not what Wikipedia is about. Sławomir
18:05, 27 May 2016 (UTC)
Let's keep the discussion focused on content and not each other, shall we? Nikkimaria (talk) 00:49, 28 May 2016 (UTC)
Lovely. Let me know when you're ready to start! Sławomir
00:58, 28 May 2016 (UTC)
A video game reliable source search shows 3 foreign references (which does not mean they should be disincluded) to the Blizzard April Fools' day joke. I have not assessed the sources to see whether they are passing mentions or basically "echoing" the original press release. --Izno (talk) 17:17, 27 May 2016 (UTC)
It's great to look for additional sources, but let's not lose sight of the big picture. The subject of this article is one of perennial confusion in lots of online message boards. The secondary sources that we have gathered amply show this I think. While it might be reasonable to question, for instance, the Blizzard source (which is by far the weakest), it too is a secondary source attesting a similar kind of cultural impact that the others do. So we have here a plurality of seconary sources showing a similar kind of cultural impact in different online communities. They fit together. Sławomir
18:27, 27 May 2016 (UTC)
Are there any sources that say anything beyond "people on noticeboards are confused"? We would expect "cultural impact" to extend past that. Nikkimaria (talk) 00:49, 28 May 2016 (UTC)
Go, find the sources about the kind of "cultural impact" that you feel is more momentous and significant. It seems like you have something very specific in mind, and if you can produce sources to match your high expectations, I'm sure we'd love to discuss them. But I've already given reasons why the content is reasonable for the article. So has another editor. As far as I can tell, that point still stands. Sławomir
01:07, 28 May 2016 (UTC)
I don't think those sources exist, which is why I don't think this content should be included. Nothing you've said on this talk page explains why even minor changes are unacceptable. Nikkimaria (talk) 01:19, 28 May 2016 (UTC)
Those changes weren't minor. Sławomir
01:20, 28 May 2016 (UTC)
Why is it important to say what else the FAQ covers, when those details have absolutely nothing to do with this article? Why must we go down the rabbit-hole that some unknown noticeboard about video games discussed something that was then discussed on another noticeboard which was then discussed in a column? Why do we have to have a quote that says that a proof was offered, and then say again that a proof was offered? It seems like all possible changes are being rejected. Nikkimaria (talk) 01:40, 28 May 2016 (UTC)
I've numbered your questions. 1. They clearly do have to do with this article. This shows that you've not read the article, and one wonders if this is the best position from which to be making such "minor" edits. 2. Those paragraphs in the section concern the impact of the identity 0.999...=1 in those online communities, so mentioning the online communities seems germane. 3. There are various proofs covered in this article, and it is worthwhile pointing out that the proofs discussed in the press release are related to other parts of the article. 4. I realize this is not a "question" per se, but if you do not make good edits, I do not know why you should expect those edits to stay. You're certainly free to propose edits here that you think might be good. That would certainly prevent bad edits from being reverted, and might actually lead somewhere constructive. Sławomir
01:50, 28 May 2016 (UTC)
You've stated that the point of overemphasizing these noticeboard posts is to demonstrate the cultural impact of the concept; we don't need these details to do that. Nikkimaria (talk) 14:41, 28 May 2016 (UTC)
I have replied in good faith to all of your questions. I think we are done here. Come back when and if you have something constructive to say, like additional content-building sources. Sławomir Biały (talk) 14:47, 28 May 2016 (UTC)

Interpretation within the ultrafinitistic framework[edit]

Exposition moved to Arguments page. A suggested change to the article related to this argument is possible, but this is not it. --Trovatore (talk) 00:49, 14 July 2016 (UTC)

I moved this back from the /Arguments page. I think this is a notable perspective that should be mentioned somewhere in the article. Sławomir Biały (talk) 23:58, 13 July 2016 (UTC)
I didn't say it wasn't. Just the same, the contribution itself didn't suggest anything like that; it just attempted an exposition of a POV. I think those should get moved automatically to Arguments. If Iblis wants to suggest putting something in the article, that request would take a considerably different form, and also a considerably different tone. --Trovatore (talk) 00:47, 14 July 2016 (UTC)
After Sławomir's edit and Hawkeye's reversion, I suppose we'd better talk about this.
I agree that it's a point of view that should probably be treated. But a few lines of get-off-my-lawn blogging from Zeilberger don't strike me as rising to the level of something that we should virtually copy into the article. (By the way, I didn't notice that Iblis's post was essentially a direct quote of Zeilberger's proto-tweet.)
Surely there must be something better somewhere. Could even be from Zeilberger, if he's written about it more seriously somewhere. --Trovatore (talk) 23:23, 14 July 2016 (UTC)

I agree with the removal. as it was it was sourced only to a blog post which is far from a reliable source. It’s hard to extract any useful information from that reference, it certainly does not clearly reference ultrafinitistism. Absent proper sources it does not belong in the article.--JohnBlackburnewordsdeeds 00:17, 15 July 2016 (UTC)

I also agree with Hawkeye7's action, but I think Sławomir is probably right that the POV deserves some mention. Needs a higher-quality source, and also better exposition (in particular, what Zeilberger means by "symbolically" is not well explained). --Trovatore (talk) 00:46, 15 July 2016 (UTC)
I think the solution is not removal, much less under WP:FRINGE, but to improve the content. Sławomir Biały (talk) 11:22, 15 July 2016 (UTC)
But perhaps you could get it right, before re-inserting. such as the summation of infinitely many decimal digits implicit in the notation is wrong William M. Connolley (talk) 13:44, 15 July 2016 (UTC)
Thanks, but the notation 0.999... by definition is , which is an example of an infinite series. It is a theorem of mathematical analysis that this series is equal to one. And, since your reasoning that it's "wrong" is puely ad hominem ("It's wrong because I said so..."), perhaps you at least feel that your mathematical bona fides are relevant to the discussion? For instance: How many post-graduate degrees in mathematics you have? How many years of experience do you have teaching mathematical analysis at an undergraduate/graduate level? Sławomir Biały (talk) 14:46, 15 July 2016 (UTC)
I'm going to guess that WMC's point is that is not a "digit", which I think you could argue either way, but doesn't strike me as ideal wording in any case (precisely because you could argue it either way).
What does "some particular number (like 64)" mean? I'm afraid people will read this as something special about 64, which I don't think is what's intended.
By the way, people mix up "fringe" and "crackpot". Fringe views are not necessarily wrong. Ultrafinitism is definitely fringe, but that doesn't mean it shouldn't be covered, just that it should be given due weight. I'm still a bit uncomfortable that we have a reference to one of Zeilberger's belly grumbles. --Trovatore (talk) 18:04, 15 July 2016 (UTC)
Perhaps it would be best working to improve the ultrafinitism article first. Its been marked as no-decent-refs since 2008 William M. Connolley (talk) 14:26, 15 July 2016 (UTC)
Yes, and I think Zeilberger wrote his Ph.D. thesis on this subject, so there likely do exist some good sources for us to work with. But simply put, it's about replacing infinite sets by another formalism. Compare this to we not using use infinitesimals like Newton did and instead using a rigorous limit procedure. Or instead of using distributions non rigorously like is often done on physics where they are treated as if they are ordinary functions, one can set up a rigorous mathematical framework where they are functionals on a suitably defined space. Count Iblis (talk) 18:11, 15 July 2016 (UTC)

such as the summation of infinitely many decimal digits implicit in the notation is wrong William M. Connolley (talk) 21:55, 18 July 2016 (UTC)

You've already expressed that opinion, thanks. I assert that it is correct. Your basis for making this proclamation is, apparently, "Wrong because I said so." Obviously, if we are arguing purely ad hominem, then I win the argument. (I sincerely doubt you boast similar qualifications in mathematical analysis to my own.) If you wish to give reasons for your opinion, those are welcome. Anyway, assuming you are no happier with this ad hominem reasoning than I am with yours, you are certainly welcome to look at the article decimal representation that discusses what the decimal representation of a number actually means. Sławomir Biały (talk) 22:06, 18 July 2016 (UTC)
While I wouldn't have noticed a problem if WMC hadn't pointed it out, our numerical digit article says that a digit is a numeric symbol (such as "2" or "5") used in combinations (such as "25") to represent numbers (such as the number 25) in positional numeral systems.
The "numeric symbols" here are all 9s (well, also some 0s, to the left of the decimal point). We aren't adding up infinitely many 9s (ignoring the question of whether you can even add 9s as numeric symbols as opposed to numbers).
So I don't know how much to make of this; I think the meaning is pretty clear, but the statement in the article is not literally correct. Could be "positional values of decimal digits" or some such, maybe. --Trovatore (talk) 07:22, 19 July 2016 (UTC)
I think this is splitting hairs well beyond the point of reasonableness. And Connolley's attitude of "It's wrong, delete it" and, when asked for an explanation merely repeats "It's wrong", does not strike me as constructive. It seems like borderline trolling. Sławomir Biały (talk) 11:13, 19 July 2016 (UTC)

I'm moving this to the arguments subpage. KSFTC 21:37, 20 July 2016 (UTC)

@Sławomir Biały You are the one trolling here. Sometimes you have to let someone else manage the problem. And the guy who wrote this "ultrafinitism section" is a troll not only on wikipedia but also on different forums. If you'd understand what it is about, you'd agree. @William M. Connolley: Please remove it, it has nothing to do on this article. (talk) 03:04, 14 November 2016 (UTC)

Horizontal ellipsis (U+2026) vs. three periods[edit]

Considering that the title uses "..." as a single character that represents infinite nines, I think it would be more semantic if we called this article 0.999 instead and replaced all instances of "0.999..." in the article with "0.999…". Then a redirect could be set up. Please advise. lol md4 U|T 00:32, 10 November 2016 (UTC)

Don't do it. See MOS:ELLIPSES for the rationale. Hawkeye7 (talk) 06:31, 14 November 2016 (UTC)

Induction proofs and 0.999.....[edit]

This is primarily a humerous arguement against those that think that because for any finite amount of 9s then 0.999...<1.

Start with the empty set, it's cardinality is clearly 0. Now the induction step is assume that for a set with a cardinality of n, finitely such, then attatching another element gives anotehr finite set with n+1 in cardinality. As this is always true, therefore natural numbers has finite cardinality.

Yes I know this is faulty and it is meant to be to show the errors of this arguement. TheZelos (talk) 13:02, 14 February 2017 (UTC)

I see the point that you're trying to show here. Obviously, the flaw of the argument as a whole is that no single finite set/sequence represents the complete infinite set/sequence represented by 0.999... It's equivalent to the argument that the limit of the sequence {0.9, 0.99, 0.999, ...} (which are all finite strings of digits) is 0.999..., which is not itself a member of the sequence. But more to the point, I'm not sure if using such a (purposely flawed) example argument for cardinality would be sufficiently instructive for readers of the article. Specifically, I'm not sure naive readers who are trying to grasp the equivalence of 0.999... and 1 are going to understand the concept of cardinality to begin with. — Loadmaster (talk) 18:30, 14 February 2017 (UTC)
They do not need to know very advanced cardinality such as bijections and stuff, all they need to know is that natural numbers are not finite, their arguement means they are, the contradiction is reached. TheZelos (talk) 07:49, 17 February 2017 (UTC)

Ultrafinitism removal[edit]

Regarding this, I do not see how removing the ultrafinitist/finitist perspective from the "In alternative number systems" is constructive. Why is it that a group of things that includes "infinitesimals", "hackenbush", "revisiting subtraction", and "p-adic numbers", ultrafinitism is somehow ruled as the FRIGE perspective? I think it is just as relevant as these other strange mathematical objects, in the sense that I don't think the idea of a finite number system is any more "fringe mathematics" than branches of mathematics that include infinitesimals. (Consider Edward Nelson, for instance, who worked in fundamental questions regarding both such kinds of number systems.)

As an aside, looking at some of Trovatore's remarks earlier to Hawkeye, I think ultrafinitism rejects that the conventional interpretation of "0.999.." as an "infinite string" is meaningful. Sp one side of the equation "0.999... = 1" is meaningless under the conventional interpretation, in their view. In fact, I suspect that the ultrafinitist interpretation more accurately reflects Haweye's own perspective on this equality, when he writes "0.999... is defined as 1". The ultrafinitist would be perfectly happy adjoining the eight character string "0.999..." as a term, and adjoining the axiom "0.999... = 1". All conceptual problems, magically swept away ;-P Sławomir Biały (talk) 21:40, 16 February 2017 (UTC)

The paragraph was incompletely sourced, which is not permissible in a Featured Article. I thought it had been given a reasonable amount of time for this to be rectified. I've had a go at correcting it myself. I haven't changed the text, just the referencing. Assuming that the two of us have understood it correctly, the statement would be covered by Katz and Katz, who basically examine the notion that if the notation is what is confusing people, then maybe the notation could be changed to make the mathematics easier to teach. Face-confused.svg Hawkeye7 (talk) 22:18, 16 February 2017 (UTC)
I think we should get rid of the link to Zeilberger's poorly explained rant. It's a very low-quality source, possibly sufficient to establish that this is Zeilberger's opinion, but not remotely good enough to make it clear why the reader ought to care. --Trovatore (talk) 23:21, 16 February 2017 (UTC)
What do you mean "completely unsourced"? It had two quality sources. One of which is (apparently) an important one in ultrafinitist circles, which also explicitly discusses the subject of this article, and the other is on the subject of this article, which mentions ultrafinitism. Both are in peer reviewed journals. I think these sources easily establish due weight for the present context of "Alternative number systems". Sławomir Biały (talk) 12:15, 17 February 2017 (UTC)
It should also be added that ultrafinitism IS a fringe in mathematics, mathematics works on the axiomatic method which means you accept the axioms at hand to make claims about propositions within the system at hand. But ultrafinitists, like Zeilberger, rejects all axioms that does not fit his personal taste, that makes it a fringe and him a crank. TheZelos (talk) 07:53, 17 February 2017 (UTC)
Can't say I like it either.I took it out again William M. Connolley (talk) 08:05, 17 February 2017 (UTC)
We do say that it is not widely accepted mathematics. Ultrafinistists, like Zeilberger and Nelson are definitely not cranks. Also, WMC's reason for removal is that he "doesn't like it". This is not the first time that WMC has put his authoritarian and uninformed opinion ahead of actual reasonable discussion on this matter. Sławomir Biały (talk) 11:12, 17 February 2017 (UTC)
When Zeilberger deviate from the axiomatic method, the foundation of mathematics, to only play in his own prefered realm while rejecting findings in other axiomatic systems, then he most definately is a crank. TheZelos (talk) 14:01, 17 February 2017 (UTC)
Where did you get the idea that Doron Zeilberger chose to deviate from the axiomatic method? He is actually a considerable expert on formal systems, as a number of famous theorems bearing his name (and, most significantly, their proofs) show. His work on formal systems in fact earned him an Euler Medal. He earned a Leroy P. Steele Prize for a famous paper with Herbert Wilf that rational functions can computably verify certain combinatorial identities (see Wilf–Zeilberger pair). He was recently made a fellow of the American Mathematical Society and, this past year, won the David P. Robbins Prize for a formalized proof of the q-TSPP conjecture. The idea that he is a crank, or indeed is not a leading expert on formal systems, is simply laughable. He does have ideas that are outside the mainstream, but we are here including ultrafinitism along with other alternatives to the standard interpretations, certainly not giving it equal validity or suggesting that it is the correct interpretation. But it is notable, there are sources, and a number of high profile proponents such that I believe it deserves mention. Sławomir Biały (talk) 14:28, 17 February 2017 (UTC)
I don't get that either in my understanding (without being particularly familiar with it) ultrafinitism is legitimate math following an axiomatic method, just that its set of axioms is (unnecessarily) more restrictive than that what most mathematician would consider as appropriate. This might make it fringe as not that many mathematicians work with it, but it is still legitimate math and there is no "deviation from axiomatic method" I'm aware of, just different axioms. So as long as it is properly sourced (textbook or journal articles) I don't really see any problem with including it as a notable minority viewpoint.--Kmhkmh (talk) 14:46, 17 February 2017 (UTC)
Ultrafinitism is not a mathematical methodology, but rather a philosophical position. It is the position that methods that invoke completed infinite objects (or sometimes even arbitrarily large finite ones) are illegitimate or meaningless.
As such, it is a sharply minority viewpoint outside the mainstream. That's what "fringe" means. "Fringe" does not mean "wrong"; plate tectonics, for example, was fringe, but is now considered proven. I think the problem is that some people use "fringe" as a euphemism, to avoid saying what they really mean, which is "crackpot". That's unfortunate, but it shouldn't prevent us from using the word correctly. Ultrafinitism is not crackpot. But it is fringe.
I have no objection to covering an ultrafinitist take on 0.999.... I do have an objection to the link to Zeilberger's ill-explained opinion. That is not a serious source; it's just venting. Zeilberger is a serious mathematician and could have created a serious source if he wanted to. Maybe he even has, for all I know. But it's not that one. --Trovatore (talk) 18:39, 17 February 2017 (UTC)
I have removed the Zeilberger reference. Sławomir Biały (talk) 19:04, 17 February 2017 (UTC)
Fine with me.--Kmhkmh (talk) 19:24, 17 February 2017 (UTC)
And me too. Hawkeye7 (talk) 19:53, 17 February 2017 (UTC)
Does the article go too far in stating that "the philosophy lacks a generally agreed-upon formal mathematical foundation"? Hawkeye7 (talk) 21:25, 17 February 2017 (UTC)
I think it's not wrong, but it's misleading and poorly worded. There is no reason to expect a philosophical position to have a formal mathematical foundation.
What is probably meant is that (as is common with ontologically restrictive viewpoints) there is no clear agreement among its proponents about where to draw the line between the real stuff and the meaningless. For example, some accept arbitrarily large natural numbers and some do not (the former group are sometimes just called "finitists" without the "ultra-"). Among those that accept that every natural number has a successor, there is disagreement about whether we can justifiably assert that for every n there's a 2n. That sort of thing. --Trovatore (talk) 22:01, 17 February 2017 (UTC)
Where does he deviate? The instant he rejects infinity due to personal taste and claim all propositions based on it is wrong/not-even-wrong due to including that one axiom he doesn't like it. TheZelos (talk) 06:43, 20 February 2017 (UTC)
One can reject the axiom of infinity, yet still employ the axiomatic method, just as there are mathematicians who reject the axiom of choice. For example, intuitionistic mathematics typically rejects the latter and sometimes the former, while finitism rejects both. The mathematicians holding these perspectives are most definitely not cranks, but rather share a different idea about what constitutes a "reasonable" set of axioms. Recall that hyperbolic geometry was discovered by rejecting Euclid's fifth postulate. This was not a rejection of the axiomatic method as such, but a rejection of one of the postulates. Sławomir Biały (talk) 11:25, 20 February 2017 (UTC)
Absolutely. And if you believe that the total amount of computation possible in the lifespan of the universe is finite (see here for a paper that considers that hypothesis), so will be the number of theorems deducible from any finite axiom system. (Note however that if that axiom system allows infinite sets, it will allow construction of theorems about infinite sets: but only a finite number of such theorems. And of course Gödelization allows the generation of theorems about infinite sets of theorems, and so on... but the "top level" set of theorems must be finite. Or am I missing something Permutation City-like here?) -- The Anome (talk) 12:31, 20 February 2017 (UTC)
Not really relevant to this page, but the fact that there is an upper bound to the information density of the universe, a la Bekenstein, seems to be a strong reason against believing in the reasonableness of the axiom of infinity. (Bracing myself for the onslaught of Platonic idealists, who believe that infinite sets are more real than reality. That's also a possibility, but to me requires something like faith in the infinite.) Sławomir Biały (talk) 13:19, 20 February 2017 (UTC)
Actually if you reject it and then proclaim the findings of it is false because you reject it, you've violated it. The axiomatic method means you accept the axioms at hand and then go where they lead, regardless of personal preferences. While one can question the utility, that is no relevance to axiomatic method. TheZelos (talk) 13:47, 20 February 2017 (UTC)
Once again, at least crudely, they are rejecting the axioms, not the method. There is a difference. There is perhaps another sense, in which some mathematical theorems are regarded not as describing "reality", despite being derivable from the axioms. It seems to me that this is a matter of interpretation: one is under no philosophical obligation to regard the outcome of logic games as describing reality any more than one is obliged to regard a chess game as describing reality. Yet we can still agree that the moves proceeded according to the established rules. Indeed, many chess players do not regard the game as descriptive of reality. That would be silly. Sławomir Biały (talk) 14:12, 20 February 2017 (UTC)
I never claimed axioms have to do with reality. I know it doens't and don't care fori t really and when you reject axioms like that you are rejecting the methodology. By the methodology you accept the axiom at hand and move on so rejecting an axiom means rejecting the methodology. TheZelos (talk) 07:23, 21 February 2017 (UTC)
Hi TheZelos; we're getting off-track here. This is not the place to argue foundational philosophy. Just very quickly, you are advocating a version of formalism, whereas at least some ultrafinitists (and in particular Zeilberger) are realists, albeit very ontologically restrictive realists. Realism is a reasonably common position; it's the restrictive version of it that is not so common. Whether it is correct or not is irrelevant. However, if you think that it's the consensus position among mathematicians that axioms are arbitrary, then you're just wrong. --Trovatore (talk) 09:15, 21 February 2017 (UTC)
"By the methodology you accept the axiom at hand and move on so rejecting an axiom means rejecting the methodology." No, there is substantial disagreement about what constitutes a reasonable set of axioms. Ultrafinitists, and indeed mathematicians of any stripe, are likely to agree on whether a proposition is derivable from a finite number of axioms, using a finite sequence of rules for inference (which also satisfy a finite set of axioms). Now, if a theorem derived from such a set of axioms and rules has no "meaning" (as per your view that axioms have nothing to do with reality), possibly apart from some formal interpretation, then it does not matter if a mathematician views the result as "meaningless". It is meaningless in the same way that a chess game is meaningless: it does not correspond to reality. He does not claim that the theorem is not a theorem, or that the theorem is not "true" (i.e., its truth value that is determined in a given interpretation of the formal system), merely that it has no ontological meaning to him, in the sense of corresponding to a capital-T Truth of the "real world". Just as he agrees that the rules of chess were observed throughout the chess game. When asked "What is the meaning of the chess game?", I think most of us would probably agree that it has no meaning apart from the game itself.
For the ultrafinitist, in the real world there is no such thing as an infinite set, so of course conclusions about infinite sets are as meaningful as conclusions about invisible pink unicorns. We can agree from the "rules" of English predicates that invisible pink unicorns are invisible, that they are pink, and that they are unicorns. But these are not actually meaningful statements about the world, because when we attempt to assign meanings to these predicates in the real world, we obtain a manifest inconsistency. (Unicorns are entirely fictitious, and moreover cannot be both invisible and pink.) So even though we are able to apply formal reasoning to the English compound noun "invisible pink unicorn", the results of the analysis are meaningless. Sławomir Biały (talk) 18:47, 21 February 2017 (UTC)