Talk:Fermat's Last Theorem/Archive 7

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Request for comment (RfC) on inclusion of Infobox mathematical statement

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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Should Template:Infobox mathematical statement be applied to the article Fermat's Last Theorem in any form? The proposed implementation can be seen here and here.

For background and previous input, please see the discussion above at Talk:Fermat's Last Theorem#Inclusion of Infobox mathematical statement. An overview of associated discussions is listed at User:Worm That Turned/FLT. 07:24, 19 March 2019 (UTC)

• Oppose as above. The sort of thing that can go into an infobox is too superficial and distorting (too much making mathematics into a competitive sport of who scores first rather than conveying any understanding of the actual mathematics involved and the cooperation required to achieve that understanding) to justify the waste of article space and reader attention. —David Eppstein (talk) 07:38, 19 March 2019 (UTC)
  • I, as a reader, can't get from the prose that the article is related to number theory, - just one example. The infobox should not present mathematical undestanding, imho, but tell a reader the context of this article, in history and mathematics. The prose begins for experts only, not for me, a random reader. --Gerda Arendt (talk) 09:54, 20 March 2019 (UTC)
    • I do not see why you do not get from the prose that it is related to number theory when the first sentence starts with "In number theory"! PJTraill (talk) 11:12, 20 March 2019 (UTC)
      • Because I am a superficial reader, who began reading at the bolded item. My mistake. Readers are not perfect, that's my point. I vote for giving something to the not so good readers also. It doesn't take away from those who can safely ignore it. --Gerda Arendt (talk) 11:42, 20 March 2019 (UTC)
• Support: To avoid wasting space here, I will refer to the arguments others and I made in the main body and the post-TfD part of the previous discussion. Briefest of summaries: accessibility for casual + knowledgeable readers; logical connections to other conjectures/theorems for knowledgeable + expert readers; valuable third-party information; summarizes (already explicitly stated) info. — MarkH21 (talk) 07:52, 19 March 2019 (UTC)
• Support: an infobox makes the article much more accessible to the occasional reader unfamiliar with the topic. Needless to say, with only confirmed facts. --Gerda Arendt (talk) 08:13, 19 March 2019 (UTC)
• Oppose per the good points made by David Eppstein. I would also add that the presence of the infobox and the trivial (as in trivia) manner in which it presents information misleading implies that there was no progress for hundreds of years until a solution was instantly produced. The article should reflect that the history and development of this idea is more complicated. I might support a sidebar timeline that shows progress and advancement. While early sources and the news media tend to over-hype the myth of the lone genius, the overwhelming trend in modern scholarship is to provide a more nuanced historiography. --mikeu ^talk 12:20, 19 March 2019 (UTC)
• Comment I kind of like the idea of a sidebar timeline. As for the infamous infobox itself, I'm finding it hard to summon real passion either for or against. This isn't some attempt to claim a moral high ground by being above the fray (this week, I just seem to be drained in general), but both the benefits and the harms have come across to me as a touch oversold. If the infobox is just repeating data given in the prose, well, people can read that. On the other hand, it does give fairly prominent visibility to the idea of connections among mathematical topics, which might (I don't know for sure) actually help promote the sense that mathematics is collaborative. Sorry for bringing fizzle instead of fireworks, but that's where I am. XOR'easter (talk) 16:51, 19 March 2019 (UTC)
• Oppose per arguments given by David Eppstein. While I have a mild dislike of infoboxes in general, this feeling is much stronger when the infobox concerns mathematical statements such as this one. If the purpose of an infobox is to make the subject accessible to the casual reader, then the infobox should say something about the subject! Outside of the re-statement of the theorem, there is nothing here which gives any insight into the theorem. Statements as to why it was so hard to prove and where does it fit within the structure of mathematical theory are not to be found here. Instead we are treated to auxiliary factoids; who conjectured it, who proved it, etc. The reason for this is clear, any substantial remarks about the theorem can't be squeezed into the infobox format where only sound-bites fit.--Bill Cherowitzo (talk) 19:30, 19 March 2019 (UTC)
• Support The Infobox succinctly communicates a number of key points of information which the lead does not so readily communicate at all. It summarises enough of those key points of information to serve a purpose - the most important being the amazing time delay between its conjecture in 1637 and its proof in 1995. To me, that is a stunningly significant point to communicate, and the use of an Infobox here does so far more effectively (to a mathematical ignoramus like me, who didn't even understand the first sentence of the lead, yet has long been aware of the existence of FLT) than does wading through the lead to try to comprehend its significance. This is what Infoboxes are intended to do, and this one does it well in my view. What a shame it requires an RfC to decide upon its merits. Haven't you all got better things to do than fuss over such trivial matters of presentation? Nick Moyes (talk) 23:27, 19 March 2019 (UTC)
  • Just in case this helps, I'll add some context to my reply above by explaining that I have not read any prior arguments for or against the use of this template in this article, nor, if I'm honest, do I have any interest in wading through them. I simply arrived here following an attempt to assist an editor on Infobox use at the WP:TH. My comments above are a simple !vote in response to seeing an RfC request, and are based upon my lifelong involvement in helping uninformed users of any resource to quickly understand key elements of complicated concepts. If just one of them goes away knowing a tiny amount more than they did when they arrived, then that didactic process has, in part, been successful. Nick Moyes (talk) 23:43, 19 March 2019 (UTC)
• Support It gives valuable information about the topic in a succinct form, which is exactly what a good infobox should do. A lot of the arguments against including it strike me as stuffy gatekeeping. e.g. (from the discussion about creating a mathematical statement infobox): Infoboxes are inane. They're fine for filling out the details of inane subjects, like the teams a footballer has played on, but they are inherently a way of reducing material to a 5-second soundbite for readers who don't even have the attention span for a single full sentence at the start of the article.. I couldn't disagree with this more strongly. For example, I would hardly call Lawrence v. Texas an inane subject, and yet the SCOTUS case infobox there serves a very useful purpose in serving up much of the most salient information about the case in a structured format that's easily scanned. The arguments that purport to be against the suitability of the infobox in this particular case (rather than arguing against the acceptability of infoboxes in general) seem pretty thin to me. e.g. the the idea that a reader seeing that FLT was conjectured in 1637 and proven in 1995 would assume that no progress was made in between strikes me as specious. I simply don't believe that's how a normal human would react to that information. (Especially when a glance at the scrollbar shows me that this is an awfully long article, and a glance at the ToC shows that there's a long history section). Colin M (talk) 03:30, 20 March 2019 (UTC)
  • Question Would it be possible to include information about the progress made during those intermediate centuries, say in a field called "notable advances" or "partial results" or the like? XOR'easter (talk) 16:48, 20 March 2019 (UTC)
• Support The proposed format seems reasonably calculated to present key information; the fact that it does not articulate every aspect of the sequence of work on the the theorem is not in itself compelling reason that an infobox is detrimental to the aritcle: that is both a false choice and a non sequitur. The infobox can and should be improved with time and it's function is clearly not just to to tell the internal story of the progress towards a proof but also to contextualize the topic within the broader framework of mathematics. Furthermore, I don't see why the assertion that the infobox listing only a few key dates implies that this is all there was to the story, any more than presenting a written summary in the lead suggests a lack of fuller context to be explored lower in the article; I just read the infobox for the very first time (thanks to the links in the RfC prompt) immediately before reading the above !votes, and the infobox left me with no such impression. All told, the infobox as proposed seems like a perfectly reasonable and well-approached addition to the article. I'm sure it can be improved upon with time, but I see no particularly compelling argument for omitting the element entirely simply because it can't tell the whole story in itself. Snow ^{let's rap} 07:10, 20 March 2019 (UTC)
• Oppose as presented. Very little of the information proposed for the infobox is accurate, sourcable, and helpful. Generally opposed to the use of this infobox, but could be convinced if entries were accurate, sourcable (including avoiding WP:SYNTHESIS), and helpful. — Arthur Rubin (talk) 09:08, 20 March 2019 (UTC)
• Support I do not find infoboxes clutter, as I find it trivial to ignore them (on a large screen, I admit). PJTraill (talk) 11:12, 20 March 2019 (UTC)
• Comment. Does nobody else care that the oversimplifications made in condensing information down to an infobox make it factually incorrect? It is false that the abc conjecture implies Fermat's last theorem. It is also false that the Fermat–Catalan conjecture implies the theorem. Both of those things imply that there are at most finitely many solutions to the Fermat equation, but the theorem states instead that there are no solutions at all. Also what is the difference between "implied by" and "generalizations" supposed to mean? Because if it doesn't imply the theorem, how can it be a generalization? To be more nitpicky, it is also incorrect that the theorem was posed in 1637 (we only know an approximate date) or that it was proved in 1995 (that's when the proof was published, not when it was found). —David Eppstein (talk) 16:58, 20 March 2019 (UTC)
  • I'm still not sold on the idea of the infobox in the first place, but on the other hand, those problems look fixable, e.g., by changing "proof" to "proof published", adding a "circa" to the 1637, and removing the items like "abc conjecture". XOR'easter (talk) 18:25, 20 March 2019 (UTC)
    • Or we could, you know, write in full sentences and paragraphs that are capable of conveying nuance, rather than trying to condense everything down to inaccurate "descriptor: answer" fragments. —David Eppstein (talk) 21:15, 20 March 2019 (UTC)
  • I'll respond to the concerns about 1) Implication vs generalization, 2) Fermat–Catalan conjecture, and 3) the abc conjecture:
    1. Regarding implication vs generalization, the difference is that an implication has a logical meaning (if X then Y) and a generalization is taking something specific and applying it more broadly. From the previous discussion above: For an example of the distinction between "implied by" and "generalizations": The Generalized Poincaré conjecture is a generalization of the Poincaré conjecture as it extends the statement to broader settings. The geometrization conjecture implies the Poincaré conjecture but is not a generalization as it concerns a different kind of problem that has an application to the setting of the Poincaré conjecture. Generalizations of X always imply X, but the converse is not necessarily true!
    2. The Fermat-Catalan conjecture broadens the context of Fermat's Last Theorem and is often called a generalization of Fermat's Last Theorem in the literature. There is no claim that it implies Fermat's Last Theorem here.
    3. Regarding the abc conjecture, it is true that the versions given at abc conjecture only implies Fermat's Last Theorem for sufficiently large exponents, but there are several variations of the conjecture (as can be seen from the formulations already in the article) and several effective versions of the abc conjecture do imply Fermat's Last Theorem. In fact, one of them is described here. If others find that we should remove the abc conjecture from the "Implied by" list (and therefore remove the modified Szpiro conjecture as well), I am fine with that. We could also modify it to read effective abc conjecture.
       I hope that addresses some of the concerns here. — MarkH21 (talk) 21:25, 20 March 2019 (UTC)
    • Details on the abc conjecture, Beal conjecture, and Fermat-Catalan conjecture have now been added to the article with appropriate references. — MarkH21 (talk) 02:41, 22 March 2019 (UTC)
      • This is the second time you've explained implication vs. generalization to a PhD-holding professional mathematician. I think you should consider that the reason PhD-holding professional mathematicians have raised the issue twice independently is that the distinction you are making is highly debatable in practice: the two categories are not clearly separated, as you suggest, and the classification of a statement into exactly one of them is not typically going to be straightforward or uncontentious. --JBL (talk) 14:29, 22 March 2019 (UTC)
        • I mean sure, one or both of the categories can certainly be omitted if/when this is contentious. — MarkH21 (talk) 14:48, 22 March 2019 (UTC)
  • Infoboxes are a companion to the text of an article, not a replacement for it. If someone wants a short answer to the question "When was Fermat's Last Theorem proven?", "1995" is the most reasonable answer you could give them. This NYT piece, for example, briefly mentions that Andrew Wiles "finally proved the theorem in 1995". If someone had more time to listen, you could expand on that answer by talking about the flawed proof that was released in 1993, how Wiles eventually realized how to repair the proof in September 1994, and how a final proof was submitted in manuscript form in October of 1994, underwent peer review, and finally was published in May of 1995. The infobox is the place for the short answer. The prose of the article is the place for the long answer.

This is a very common pattern in articles. e.g. "How big is the Atlantic Ocean?" Short answer (infobox): 106,460,000 km2. Long answer (prose): well, different authorities define the oceanic boundaries differently, and they've changed over time, and it depends whether you include its marginal seas, so it might be X, or it might be Y, or... Colin M (talk) 18:28, 21 March 2019 (UTC)

References

• Support for the reasons I gave above. Andy Mabbett (Pigsonthewing); Talk to Andy; Andy's edits 23:45, 20 March 2019 (UTC)
• Oppose, for all arguments against including this IB are only met with intimidation, condescendence, emphasized "I-say-so", repeated false claims of transporting "key facts", where there is mostly irrelevance heaped upon misguidance, and appealing to common sense, identified with the own opinion, even denying to responsibly measure IB's (non-)triviality. The hope for IBs would help the cursory, superficial readers is also in vain for this article (and most mathematical articles), because of the mentioned lack of relevance for the info transferable in IBs. Additionally, one should have in mind the many articles about Mathematical statements, now unspoiled, that have to expect being plastered with this template soon. Principiis obsta. Even when the article were deteriorated only a minuscule amount by adding this IB, I would not support the addressed "third party" interests, and also not the personal gain for superficial readers at the expense of genuine readers with a mature interest in the topic, deserving not to be distracted by pretending-only IBs. To sum it up, the IB here is confined to mostly trivia, missing the inherent values of this article, by presenting rubbish "executive briefs" in a show case. Purgy (talk) 18:17, 21 March 2019 (UTC)
• Oppose. Infobox of this discussion, i.e. executive summary for the cursory reader: those who contribute to the maths articles consider this kind of oversimplifying infobox as only misleading, while the infoboxers are more than often people that don't contribute to this kind of articles (or even don't read them). But it could be useful to write the first lead sentence in blinking orange in order to help the cursory passer-by to detect that Fermat's Theorem is a Theorem that belongs to Number Theory and is named by reference to a guy who, marvelous coincidence, was himself named Fermat. Pldx1 (talk) 23:09, 21 March 2019 (UTC)

• Fermat's Last Theorem

  Field	Number theory
  Conjectured by	Pierre de Fermat
  Conjectured in	1637
  First proof by	Andrew Wiles
  First proof in	1995
  Implied by	Modularity theorem

  Support inclusion of an infobox. I believe that the infobox should, for now, contain these fields in addition to the lead image:
  Contrary to Arthur Rubin's assertion, each of those facts is obviously substantially sourced in the article. They are accurate, uncontroversial and helpful to a casual reader who wants to find a simple fact, as well as to a reader whose first language is not English (and therefore may have more difficulty in finding those key facts in the article). Using that infobox also presents each of those facts as structured data that is easily accessible by third-party re-users, an accepted part of the mission of Wikipedia.
  I reject the contention that those facts are "too superficial and distorting". They are patently not.
  I reject the contention that the "trivial (as in trivia) manner in which it presents information misleading implies that there was no progress for hundreds of years" The dates demarcate the start point and the end point of the proof, there is clearly no implication beyond that. Who would draw the conclusion that nobody worked on the proof for hundreds of years and then it was suddenly proved overnight? It's not a reasonable objection.
  I reject the implied accusation that the "infobox [does not] say something about the subject!" It tells the reader who formulated the theorem; who proved it; when those events occurred; and what field of mathematics the theorem belongs to. That's a perfectly reasonable use of an infobox here.
  I also object strongly to Purgy Purgatorio's continued personalisation of this debate. I would like to know if an uninvolved admin would find the their diatribe above, aimed at those who disagree with them, to be compatible with the discretionary sanctions applicable here, particularly ArbComs injunction: "All editors are reminded to maintain decorum and civility when engaged in discussions about infoboxes, and to not turn discussions about a single article's infobox into a discussion about infoboxes in general." --RexxS (talk) 00:14, 22 March 2019 (UTC)
  • Thanks for those links - that was an interesting rabbit hole to climb down. I had no idea that the inclusion of infoboxes had a history of provoking such ferocious debate (apparently going back at least as far as 2013). It's helpful in contextualizing this discussion. Colin M (talk) 17:44, 22 March 2019 (UTC)

• Cantor-Bernstein-Schroder theorem

  two injections imply a bijection
  Field	set theory
  Conjectured by	Cantor
  Conjectured in	1895
  First proof by	Schroder
  First proof in	1898
  Implied by	Axiom of choice

  Comment. Adding the following desinfobox to the Cantor-Bernstein-Schroder theorem page would have the great merit of provoking ferocious debates where a casual passer-by could take a better part. No need to understand what the theorem is about to discuss the infobox at infinity about when, who, was it a proof, ", is a blog a sufficient source to back-up 1898, who ever said that your source is reliable, has the image the right size, and so on. On the contrary, discussing what happens when excluding the 'excluded middle' would be too horrible for the casual passer-by. On the other hand, a math article should perhaps be centered about mathematics and cosmetics should only be used with an actual purpose. Pldx1 (talk) 18:43, 22 March 2019 (UTC)
  • Better yet, why not put that on the Straw man page where it would be a fine example. This RfC is about the article Fermat's Last Theorem. The date that Fermat's Last Theorem was first expounded and by whom are both significant pieces of information, which are verifiable by reliable sources and are accurate, not misleading, summaries of content already in the article. Exactly the same applies to the date of and author of the first proof. And to the field of mathematics that Fermat's Last Theorem belongs to. This RfC does not suggest that the infobox should debate "who said it was a proof", and there is no reliance on a blog for either of the dates. They are already reliably sourced in the article. The image is a standard size in an infobox, so there's no debate to be had there. Not one of the issues you raise address the value of adding an infobox (with content to be decided by consensus) to Fermat's Last Theorem. Perhaps you need me to remind you to not turn discussions about a single article's infobox into a discussion about infoboxes in general. --RexxS (talk) 22:07, 22 March 2019 (UTC)
    • discussion about infoboxes in general is a straw elephant. It is clear that nobody discusses here if articles about Korean painters should have infoboxen or not. Paintings and theorems are two different kinds of animals, aren't they ? But the very name of the template {{Infobox mathematical statement}} indicates that it has not been written to be used only at FLT. The intent seems to experiment with one article, and then, in case of success, invade all articles containing mathematical statements, i.e. all the non empty math articles. Anders gesagt: behind the straw-elephant comes the elephant-in-the-room ! Pldx1 (talk) 00:26, 23 March 2019 (UTC)
      • You lost that argument when the attempt to delete the infobox template failed. You need to stop taking a battlefield approach – "invaded", indeed! – to the topic of infoboxes or it will end badly for you. Concentrate on the arguments for and against an infobox in this article. Here's a cue sheet for you to learn the proper decorum from: User:RexxS/Infobox factors. --RexxS (talk) 10:13, 23 March 2019 (UTC)
        • It doesn't come across as particularly honest to start a comment with "you lost" and then admonish the other person for taking a battleground approach. —David Eppstein (talk) 12:50, 23 March 2019 (UTC)
          • Then obviously your standards for usage of English are far different from mine. There's a world of difference between using the commonplace phrase "lost an argument" and the phrase "invade all articles containing mathematical statements". I find the former a rather neutral use of the verb 'to lose', and the latter to be a rather pointed hyperbole indicative of a battlefield mentality. Calling a fellow editor dishonest on those sort of grounds really should be sanctionable in a place where WP:AC/DS apply. Don't you agree? --RexxS (talk) 16:12, 23 March 2019 (UTC)
            • Don't I agree that your use of battleground win-lose terminology is perfectly civil disagreement while those you oppose in the disagreement are exhibiting a battleground mentality? Don't I agree that anyone who points out your double standard is being uncivil to you and should be sanctioned while your classifying others' remarks as "pointed hyperbole" and calling for sanctions on them is again perfectly civil disagreement? No and no. —David Eppstein (talk) 19:41, 24 March 2019 (UTC)
              • So you think my use of "lose an argument" is sanctionable, and your pal's use of "invade all articles containing mathematical statements" is perfectly acceptable? There's only one of us being partisan here, and it's not me. Your personalisation of other editors' disagreement with your view is what is sanctionable; and if it continues it will only be a matter of time before an uninvolved admin spots that and takes action. --RexxS (talk) 21:51, 24 March 2019 (UTC)
  • In response to the comment (rather than the preceding discussion of it), surely it is not hard to add optional parameters to cope with controversial (abc) proofs and those that depend on particular axiomatisations or systems of logic, not to mention status of conjectures? PJTraill (talk) 12:34, 25 March 2019 (UTC)
• Support Part of the population of readers need overarching information followed by information specifics to understand and learn. Others do better dealing with specifics followed by the overarching or can do with out the overarching altogether. This infobox, in a topic area that requires some speciality in education provides that kind of overarching information which makes information more accessible to some people. Those of us who teach know there is no one way of presenting information that works for everyone; we learn to inform all types. By extension as an encyclopedia whose mission is to present information, to teach, there is no argument that allows us to exclude one kind of reader, ever. Littleolive oil (talk) 17:08, 23 March 2019 (UTC)

• Support - adding Wikipedia:Tarage's Law. The example RexxS provided was helpful. I also find series infoboxes quite helpful. ^Atsme _Talk 📧 17:14, 23 March 2019 (UTC)

• Support Useful for many readers and very easy to maintain. Jrheller1 (talk) 18:42, 23 March 2019 (UTC)

• Support for reasons I outlined in the prior discussion. Mathematicians should guard against the notion that our field is a special snowflake whose depths resist concise summary. Lagrange613 19:11, 23 March 2019 (UTC)

• Comment. Answering to Concentrate on the arguments for and against an infobox in this article by pointing to the logical fallacy of following this very sentence by a link to a catechism for infobox defenders would be too easy. Let us concentrate on the teaching argument. Indeed, teaching of mathematics should be improved. Too many lectures follow the line of
  Lecture_1: God said to Abraham: “there aren't genuine $x^{n}+y^{n}=z^{n}$ when $n>2$”. But
  Lecture_2: God said to Abraham: “I got a proof and a large margin” is not better.
  Maths are starting when the believer turns into a student and asks: “hey, you, let's look at your proof!”. When the angry consumer comes and shouts “Be damned with your large margins. Give me something to believe and sufficiently narrowed be written in an infobox”, this turns into:
  ---  Well Abe says, “Where you say this proof been done?”
  ---   God says, “Out on Highway Sixty-One”.
  And the angry consumer is satisfied, repeating 1995, 1995 while God runs Her hand through Her long hairs and says to Herself, mezza-voce : it only takes a week to create a world, but 10 semesters to teach maths. If there were miracles, I would already know ! Pldx1 (talk) 13:00, 24 March 2019 (UTC)

  Why not concentrate on the arguments for and against an infobox in this article? It's not that difficult if you have genuine grounds to argue from. --RexxS (talk) 21:55, 24 March 2019 (UTC)

  A more explicit statement of the same arguments. Doing maths is (1) finding results ; (2) finding proofs ; (3) checking if a given proof really proves what the proof is intended to prove. Everyone involved in maths knows that steps (1) and (2) are the more difficult ones and the more time-consuming. And the ones that should be targeted when teaching maths. The key fact here, if any, is that step (2) taked so long a time, and resulted into the discovery of a large list of results... that were irrelevant to the final proof of the FLT, but were nevertheless fundamental results for maths as a general topic. By the way, saying that 1995 is overarching here, i.e. an all-embracing fact, looks as a joke intended to bad-mouth the believers of this specific infobox. Lets do a grep on overarching to detect who dared to use that word ! Pldx1 (talk) 08:41, 26 March 2019 (UTC)

• Support. An infobox such as the one included in this discussion section will help highlight key facts for those who are looking for exactly that and doesn't in anyway hinder those who are looking for more in-depth information that cannot be quickly summarised. While its possible that there are other things someone could be looking for in this manner which cannot be succinctly summarised, they are not present in the proposed infobox and are not a barrier to the inclusion of things that can be so condensed. Thryduulf (talk) 13:43, 24 March 2019 (UTC)
• Oppose While there are decent arguments being made by both sides (quite a rarity on this site), I'm swayed by the points David Eppstein and Arthur Rubin have raised. Joefromrandb (talk) 06:32, 26 March 2019 (UTC)
• Keep or improve the template or its form or content in context Having checked the "Templates for discussion" log, I consider the objections to templates and denial of their value to users to be mistaken, much like the rwars against links and technical terms in article titles. It is quite possible to use them constructively, and as is true for any useful tool, to misapply them uselessly or harmfully or pointlessly. The fact that certain classes of user, typically professionals or others advanced in the field, might consider them as useless, redundant, misleading, or aesthetically displeasing, is beside the point. Those are not the only users. Many readers (most, I think) are either ill-equipped to go beyond factoids, or uninterested; even a professional might well look up a date or a name for constructive purposes. If anyone has objections, the appropriate course is to improve the template, not gratify a point of personal distaste. JonRichfield (talk) 18:42, 26 March 2019 (UTC)
• Support: there are two separate issues: the accuracy of the infobox and the merit of having one at all. There are many math articles in which infoboxes are probably useless; e.g., the box might say the theorem has various formulations proved by various people. The accurate box would just then be the reproduction of the intro. On the other hand, some article that is of interest to the general public can benefit from having an infobox. I assume it is not necessary the common knowledge that Fermat conjectured but didn't prove (or widely assumed so) the theorem, which is not obvious from the article title; many theorems are named after those who proved them. This one is an exception and the info box efficiently conveys such information. Some delicate information (e.g., connection to abc conjecture) can be omitted or vaguely mentioned. -- Taku (talk) 23:26, 27 March 2019 (UTC)

So, we have the Taku Conjecture: "it could exist readers that are genuinely thinking that all this hype of nowadays around Wiles could be about a theorem already proven by Fermat some (large) times ago", Conjectured In: 2019. At first sight, it seems difficult to write a counter-proof in only a small margin. But in the frame of Constructive Mathematics, it seems equally difficult to construct and exhibit a living reader fulfilling this conjecture. Pldx1 (talk) 12:11, 29 March 2019 (UTC)

Do you really consider your weak sarcasm directed at another editor to be helpful to the debate? If you've nothing constructive to say, why not stay silent? This sort of sniping at other editors that you don't agree with is disruptive. --RexxS (talk) 17:32, 29 March 2019 (UTC)

Agree with RexxS that these comments are getting old. Your case would be better served if you would state your disagreements plainly, rather than making the reader unwrap layers of caustic irony and sarcasm. Colin M (talk) 18:53, 29 March 2019 (UTC)

• support: not all readers are here to fully understand the intricacies of the topic. while those intricacies are certainly important, we should not withold information from others under the guise of protecting them from misunderstanding it. a person who cares to understand it fully, perhaps a graduate level mathematics student, will take the necessary time to read the article and look at the context. a person who wants to know about the topic will almost certainly walk away with misunderstandings of its implications no matter how carefully the article is written. this type of reader will skim through and pull whatever information seems relevant to their reasons for being here. Many of the primary objections I've read come from well meaning editors who clearly love mathematics in the way I do (or more) and hope that everyone else will love it and want to know it deeply. Many readers of Wikipedia don't care to know mathematics in that way. Many readers want "trivia". we cannot protect them from their own misunderstandings of the context and implications, and the presence or absence of an infobox doesn't necessarily correlate with readers having better or poorer understanding of the topic, as has been implied by some vocal editors in this debate. the infobox is more helpful than harmful.Cliff (talk) 07:18, 11 April 2019 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

The triplet (a,b,c)=(0,0,0) wants to have a word or two

The article states that there is no triplet (a,b,c) in N^3 where a^n+b^n=c^n where n>=3. But 0 is a part of N (it isn't N* where 0 is excluded) so... Serouj2000 (talk) 23:43, 4 May 2019 (UTC)

Where exactly does it say this? Because in the lead it says a,b,c are positive. "Equivalent statements of the theorem" repeatedly uses "non-trivial" to exclude this sort of solution (and the ones where only one of the three is 0). And the word "triplet" doesn't appear very often in the article. I did find and fix one instance buried deep within the article, in the "Other exponents" section, where it neglected to say "nontrivial" or "positive"; was that the one you meant? —David Eppstein (talk) 00:03, 5 May 2019 (UTC)

In the overview section, it says this:"Original statement. With n, x, y, z ∈ N (meaning n, x, y, z are all positive whole numbers) and n > 2 the equation x^n + y^n = z^n has no solutions".When it said for all x,y,z in N, it meant that they CAN be 0. If it were " With n, x, y, z ∈ N* (meaning n, x, y, z are all positive nonzero whole numbers) and n > 2 the equation x^n + y^n = z^n has no solutions", it would have made more sense.Also, sure. The word triplet isn't in here, but the meaning is still the same. 20:44, 5 May 2019 (UTC)Serouj2000 (talk)

I think it's the usual difference in notation of whether to include 0 in the natural numbers. Here, of course, 0 is excluded from N. It can certainly be reworded to remove this ambiguity though. — MarkH21 (talk) 01:29, 6 May 2019 (UTC)

Ah, I see. The issue is whether the "natural numbers" $\mathbb {N}$ include 0. I don't think there is universal agreement on which way to go. As Mark says, it is a difference of notation. —David Eppstein (talk) 01:37, 6 May 2019 (UTC)

In every math book I've had in secondary class,N did include 0 while the 0-less set is N*. Idk if it is otherwise in other books. 17:30, 6 May 2019 (UTC)Serouj2000 (talk)

And even if N didn't include 0 (ie the original statement was true), the equivalent ones (at least the first two) would be false, because Z and Q being a ring and a field, respectively, they both include 0. 17:40, 6 May 2019 (UTC)

Historically, the natural numbers did not include 0 and many texts still exclude 0 today (especially those from the United States) — see Natural number#Modern definitions. On the FLT article, The equivalent statements say "non-trivial" and the original statement says "(meaning n, x, y, z are all positive whole numbers)", so none of them are incorrect. — MarkH21 (talk) 19:32, 6 May 2019 (UTC)

"Conjecture"

It seems appropriate to open the discussion of "conjecture" in connection with FLT in a new section.

I think the article opens badly. Fermat did not conjecture FLT in the margin of a book; he asserted it. He believed it to be true. Later he changed his mind about having a proof, or so the evidence seems to suggest (he forgot to tell us). I think a more accurate statement would be that "Fermat proposed" FLT; see my next paragraph.

As for "conjecture", I am a working mathematician and have had and read many discussions about the appropriate use of the word in math. Merriam-Webster is good in general, but they are too definite about this (def. 1c). Some mathematicians agree with their version and some disagree. Many (I gather) would say a true conjecture is something proposed as true but without proof, thereby differing from a question. Others would say a proposition proposed but not necessarily as true qualifies as a conjecture. Some things called X's conjecture were never proposed even as an open problem by X (I believe the "Collatz conjecture" is one such). There is opinion but no consensus. Since the only evidence we have is that Fermat asserted FLT as true and later appears to have realized his proof was incorrect, I think "conjecture" has the wrong flavor. Hence my suggestion of "proposed", which could mean he asserted it, as he did, or he only put it out there to be considered, which may (or may not) have been his final opinion. Zaslav (talk) 05:19, 21 May 2019 (UTC)

Attribution

1. https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem, article Fermat's Last Theorem.
2. cc by-sa 3.0
3. English Wikipedia
4. Authors can be found at https://en.wikipedia.org/w/index.php?title=Fermat%27s_Last_Theorem&action=history.
5. Used with changes.

That's the attribution for my edit if I need to make that attribution. ParticipantOfTheEncyclopedia (talk) 17:32, 18 December 2019 (UTC)

I am not sure what you are trying to do here: are you trying to tell your fellow editors something, or do you intend to use this elsewhere? If the latter, is this only for your personal convenience? What effect does this attribution have on whom or what? Moreover, your formatting and links seem not to be what you presumably want:

• Putting numbers on new lines in the wiki source does not create a numbered list. (I have amended that, which I hope meets your approval.)
• To link a specific edit, use the link under "diff" in the history, presumably (where you say "my edit") you want this.

PJTraill (talk) 17:53, 18 December 2019 (UTC)

Falsehoods in infobox

I removed the infobox today with the edit summary "It was not conjectured by Fermat; he stated it as a theorem. The date is not known to be 1637; that is guesswork. The proof combined work by many mathematicians, not just Wiles. And as a proven statement of mathematics it is implied by any statement (true or false), not just the ones listed." User:MarkH21 reverted, restoring these falsehoods to the article and also without comment reverting my simultaneous changes to the lead of the article. I happen to think that in this sort of article the best an infobox can be is a waste of space and a distraction, but when it is also disinforming people it needs to go. But I welcome discussion here of whether it is possible to make an infobox that accurately summarizes the important information about this theorem without so many distortions, or whether we are better off without it altogether. —David Eppstein (talk) 21:38, 4 October 2020 (UTC)

You're edt-warring against the existing consensus at Talk:Fermat's Last Theorem#Request for comment (RfC) on inclusion of Infobox mathematical statement (which proposed this version of the infobox and was clearly closed by DannyS712 with consensus in favor of including an infobox) by removing the entire infobox.

Which details are in the infobox can be discussed and modified, but the consensus was that the given version of the infobox should be included.

I didn't object to the other edits, but you reverted my edit in less than one minute, before I even had a chance to re-add them. — MarkH₂₁^talk 21:42, 4 October 2020 (UTC)

When all of the details are wrong, arguing that these are mere details and that the overall infobox is good comes across as disingenuous. My point is that not only are all of the details wrong, but that it is the nature of infoboxes to oversummarize and make things wrong. —David Eppstein (talk) 21:45, 4 October 2020 (UTC)

Your argument of as a proven statement of mathematics it is implied by any statement (true or false), not just the ones listed advocates for removing every mathematical consequence listed on Wikipedia, including in article bodies. That is also not the language/viewpoint used by most published reliable sources. There are dozens of published reliable academic sources that regularly state that, for instance, "Fermat's Last Theorem for n ≥ 6 follows immediately [from the abc conjecture]" (Notices of the AMS by Andrew Granville and Thomas J. Tucker), or that "the abc conjecture immediately solves the Fermat–Catalan problem" (Computational Number Theory by Carl Pomerance). The infobox also does claim to be a complete list, just as a person is not only known for the listed items in the infobox of a biographical article.

Of course the proof of many theorems comes about as the result of several peoples' work. This just gives you the end date and what is the simply stated by RSes, e.g. Fermat's Last Theorem "had remained unproved for three-and-a-half centuries until Andrew Wiles published a proof in 1995" (same Pomerance source).

But either way, your outright removal of the infobox is editing against the consensus of the RfC where you presented exactly the same points, regardless of whether you think you are right. If you just remove the entire infobox, you're disruptively editing against consensus.

Believing that you have a valid point does not confer upon you the right to act as though your point must be accepted by the community when you have been told that it is not accepted.

— WP:IDHT

You need to self-revert your removal of the infobox, and we can build consensus on each part of the infobox. — MarkH₂₁^talk 22:02, 4 October 2020 (UTC)

I am not going to re-insert falsehoods into the article by self-reverting. If you think the encyclopedia is improved by re-inserting falsehoods, that's not my problem. A quiz for you: When Fermat stated this, he did so as a conjecture, not a theorem: true or false? It is known that the exact year he did so was 1637: true or false? If you can answer these questions honestly, you should be able to conclude from there whether it is appropriate to reinsert these claims. —David Eppstein (talk) 22:10, 4 October 2020 (UTC)

Then you are blatantly and consciously editing against existing consensus, which is disruptive editing. You know that editors must abide by consensus, even if they think the consensus is factually incorrect.

Sure, those can be made more precise by adding two parameters that say "First stated by"/"First stated in" instead of "Conjectured by"/"Conjectured in", and using {{circa}} for "1637". — MarkH₂₁^talk 22:22, 4 October 2020 (UTC)

Further improvements

On the current points (current version):

1. "Field": The entry Number theory reflects what the article says, Number theory.
2. "First stated by": The replacement of "Conjectured" with "First stated" more precisely represents the historical scenario (that Fermat claimed a proof and did not literally state it as an unknown/question). As the article says: The proposition was first stated as a theorem by Pierre de Fermat around 1637
3. "First stated in"The addition of {{circa}} should resolve the issue of the year not being exactly 1637. It reflects what the article says: Around 1637.
4. "First proof by": The entry Andrew Wiles reflects what the article and dozens of RSes cite as the person providing the first complete proof, e.g. the first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995. Of course, it was the result of the cumulative work of dozens of mathematicians before him, but Wiles providing the complete proof is one of the key points of the article.
5. "First proof in": The entry says 1995 because that is the publication date of the final proof. As academic publications generally cite the publication date of research papers rather than its formulation (which is also the case for FLT), this is probably preferable to 1994.
6. "Implied by": All of the listed entries also reflect the RS-cited WP-notable implications from the article itself. Infobox parameters never claim to be complete, whether they're implications here, "Awards" in {{Infobox scientist}}, or "Known for" in {{Infobox person}}.

Are there further adjustments that need to be made, whether on these existing parameters or new parameters? In line with MOS:INFOBOXPURPOSE, the listed points in the infobox should only contain key facts (that already in the article itself) in short form.

The infobox previously (original proposed version from the RfC) included the Beal conjecture and Fermat-Catalan conjecture as generalizations. They're both described in Fermat's Last Theorem#Generalized Fermat equation as generalizations. Although it might be confusing to include both "Generalizations" and "Implied by" as separate parameters because their entries would be largely overlapping. — MarkH₂₁^talk 23:19, 4 October 2020 (UTC)

• This infobox should be removed. It shouldn't be used anywhere, in fact. Infoboxes are good when there's basic, standard data to summarize, like birth dates, etc. The reality around things like theorems are messsy and aren't suited to an infobox. –Deacon Vorbis (carbon • videos) 23:53, 4 October 2020 (UTC)

  The point of this isn't to relitigate the RfC that closed for inclusion of the infobox. There was consensus to include the infobox, but we can still improve what is there regardless of agreement/disagreement with the RfC result. — MarkH₂₁^talk 00:23, 5 October 2020 (UTC)

  It was a bad close. It's a bad use of an infobox. There are no good uses for this particular infobox. It should be removed here. It should be removed everywhere. –Deacon Vorbis (carbon • videos) 01:48, 5 October 2020 (UTC)

• Although I continue to prefer total removal, in its current state it's less objectionable than it was, merely wasting space and drawing reader attention to unimportant things rather than also misinforming. However, there are points in MH21's latest response that I continue to disagree with. When we have two conflicting pieces of information, we should not be deciding that it is preferable to omit one and imply that the other one is the only truth; we should state both. The proof was released in 1994 and published in 1995. This is stated in some detail within the article, and summarized with both dates in the lead of the article. If we're going to provide the date of first proof in the infobox, we should state both dates, with a description of what each date means, rather than continuing to oversimplify. As for the "implied by", in the current state of mathematics, 1+1=2 implies FLT. 1+1=3 implies FLT as well. Every well-formulated statement in mathematics, true or false, implies FLT. Some of those implications were unknown prior to the proof of FLT, but so what? What does the chronology of knowledge of the implications vis-a-vis the proof date have to do with anything? —David Eppstein (talk) 23:58, 4 October 2020 (UTC)
  • I have no qualms with including both years in the "First proof in" parameter, that seems reasonable.
    These implications highlight what RSes say are relevant logical connections between various conjectures, e.g. that Fermat's Last Theorem inspired / is generalized by / is a case of the Beal conjecture. Meanwhile, your argument regarding "Implied by" is one for removing all mentions of "A implies B" in all mathematical articles (not just this infobox) even if they are cited to published books and research papers that say things like "A implies B". It is arguing against the convention for "implies" that is used by published sources about Fermat's Last Theorem. The article (under "Relationship to other problems and generalizations") already says outside of the infobox what RSes state (emphasis mine):

    The abc conjecture implies what we shall call the Asymptotic Fermat Theorem
    — Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001 p. 196.

    "Fermat's Last Theorem for n ≥ 6 follows immediately [from the abc conjecture]"
    — Article in Notices of the AMS by Andrew Granville and Thomas J. Tucker)

    In 1991 Elkies showed that using an explicit version of the abc-conjecture [...] one can deduce an explicit version of Faltings' Theorem.
    — Granville & Tucker

    "Generalizations" may be the better word in some cases (e.g. Fermat-Catalan conjecture) to use instead of "Implied by", but these are connections that are already in the article and directly stated by RSes. — MarkH₂₁^talk 00:22, 5 October 2020 (UTC)
    • Even Noam Elkies's aforementioned 1991 paper is titled ABC implies Mordell (emphasis mine) even though the Mordell conjecture was already proved 8 years prior, and says (emphasis mine):

      An effective version of the ABC conjecture would imply Mordell's conjecture with an effective height bound [...]

      Our proof generalizes the known implication "effective ABC -> eventual Fermat" which was the original motivation for the ABC conjecture
      — ABC implies Mordell by Noam Elkies

      You might avoid the word "implies", but that is exactly the terminology in the relevant mathematical publications. — MarkH₂₁^talk 00:47, 5 October 2020 (UTC)
      • Is there still a stance against using "implied by", despite it being in the actual article and it being the standard terminology among the cited papers? — MarkH₂₁^talk 04:32, 7 October 2020 (UTC)
        • Brought it back then. Let me know if further discussion is desired. — MarkH₂₁^talk 02:19, 9 October 2020 (UTC)
          • An infobox about a topic that fails to even identify the topic that the article is about is a disgrace, and you should be ashamed for continuing to perpetrate it. Unlike many, this is not a topic whose statement is difficult to state. I looked in vain for a statement= parameter in the infobox in which to put the statement of the theorem. If we're going to distract people with this predigested garbage, we can at least make an attempt at informing them of the important parts of the subject while doing so. Instead, you are continuing to dance around the subject, putting stuff into the infobox because it's easy to summarize rather than because it's an important aspect of the topic. —David Eppstein (talk) 02:27, 9 October 2020 (UTC)
            • Let's be civil as we work towards improving it.
              If you think that it would be useful, we can easily add a |statement= parameter to the template code. — MarkH₂₁^talk 02:58, 9 October 2020 (UTC)
• Here is a sample one: User:MarkH21/Fermat's Last Theorem sandbox. Let me know if there’s some potential tweaking for the statement or if it’s okay. Or perhaps if the parameter should generally be above "Field"? — MarkH₂₁^talk 03:10, 9 October 2020 (UTC)

Infoboxes as style elements

Maybe you're thinking about infoboxes the wrong way. They are stylistic elements meant to be soothing to they eye, and to perk up the attention of the distracted reader -- something to idly stare at when overwhelmed by the wall of text -- something to return to when lost in thought -- something to grab hold of while researching some high-school term paper. Remember those? They were hard. I suffered. (I excelled at math; I was doing math in my spare time as a pre-teen. But having to write a term paper was unbearable torture; infoboxes were the life-preserver to the drowning man.) Decades later I learned about marketing; they have concepts like "proof points" and "talking points" and similar which are techniques for encapsulating key ideas into a small number of words, into bulleted lists. Once the customer/interested party has gotten the gist of a point or two, they are much more willing to dig deeper into the supporting material. They are a guide, a map to the territory, a quick-reference to key claims. ...

Back to the math infobox. Maybe scratch the date, scratch the names. Replace them with key concepts instead. The infobox for the poincare conjecture looks great - it talks about homotopy and supplies a pretty picture. So for the infobox here, maybe talk about finite function fields and how they provide the "atomic structure" of addition and multiplication. I kid you not: David Goss says as much in the preface to one of his books... Here, I found it: Basic Structures of Function Field Arithmetic Springer 1998:

"By abusing language, and perhaps reality, we like to say that in the theory of function fields, number theoretic objects break up into little "arithmetic quarks". While the reader may not consider this analogy to be particularly apt, the reader will see this splitting up phenomenon again and again throughout the book."

He was aiming those sentences at working mathematicians, he wrote them after 4 pages of Drinfeld-this and shtuka-that. I don't know how to explain a drinfeld module in 20 words or less, but maybe 20 words on elliptic functions, or maybe just polynomials and fields is enough. And if you insist on having names, then Goss puts it like this, I quote: "the well known solution due to G. Frey, J.-P. Serre, K. Ribet R. Taylor and most importantly A. Wiles" There, that's an authoritative, quotable secondary source. Seriously -- make the infoboxes as visually pretty, as visually soothing and pleasing as possible, and for the actual info, sketch the key ideas in 2-5 bullet points. Don't sweat the dates. Mathematics is not about Hollywood red-carpets. I agree with David Eppstein, infoboxes should NOT be National-Enquirer-style tabloids. That's not what math is about. 67.198.37.16 (talk) 03:45, 5 October 2020 (UTC)

Visual prettiness

The 3-adic integers, with selected corresponding characters on their Pontryagin dual group

One more idea: it is hard to make arithmetic visually pretty and stimulating. But it can be done. This article cites: Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001). "On the modularity of elliptic curves over Q: wild 3-adic exercises". Journal of the American Mathematical Society. and if I click over to p-adic number I see the picture at right. So, OK, its kind of a difficult route to get from 3-adic anything to this article, but it can be done, and a picture like this can be intoxicating to a certain class of readers. Arithmetic does not have to be portrayed as dusty old tomes from 4 centuries ago! It can be made thrilling! It can be better than some crappy sci-fi movie! No disrespect to Medieval scholars; dusty old tomes can be exciting too, Buffy the Vampire Slayer proved that ... but ... really... colors! fractal-like things! 67.198.37.16 (talk) 04:31, 5 October 2020 (UTC)

Fractals are one of the most common modern mathematical objects in pop-culture for a good reason - they're extremely appealing to look at! I agree, although if not 3 adics because of their tenuous albeit existing connection with the theorem, I definitely advocate for something similar. _{Integral Python} ^{click here to argue with me} 11:14, 5 October 2020 (UTC)

@67.198.37.16: Yes, it would be nice to have a visually interesting image like the dual coloring of the 3-adics. In fact, the p-adic numbers are fundamentally used in Wiles's proof of FLT (through l-adic representations associated to modular forms ) and are much more related than you suggest. However, this article does not go anywhere near that level of detail and an image only about p-adic numbers would not be directly relevant to the article about FLT as a whole.

I'm not sure if there are any visually striking images that are more directly connected to FLT. Perhaps there is something related to the Fermat curve, I'm not sure. — MarkH₂₁^talk 06:23, 6 October 2020 (UTC)

Well, that would be a reasonable graphic, actually. One could draw a square lattice and the n=3,4,5 curves on it, and have a caption that reads "these curves will never hit any pair of rationals (a/b, c/d), which is remarkable and perhaps counter-intuitive, given that the rationals are dense in the reals". Or something like that. It instills a sense of wonder that is otherwise absent in the arithmetic statement. I mean, how the heck can a curve avoid bumping into a pair of rationals? Perhaps some animated zooming in with the fermat curve flying past a filigree of rationals that are trying to "get it" like some 2D video arcade game... Isn't there somewhere on WP where you can make a "request for graphic art"?

A variant would be the fermat curve wound around a torus. The point being it never hits itself. The caption could mumble something about moduli spaces or teichmuller characters or something along those lines.67.198.37.16 (talk) 04:53, 7 October 2020 (UTC)

I agree it’s a reasonable idea. I’m not sure where one would ask though. Wikipedia:Reference desk/Mathematics or Commons:Help desk are the closest things I can think of. — MarkH₂₁^talk 02:23, 9 October 2020 (UTC)

A reminder: "Images must be significant and relevant in the topic's context, not primarily decorative". There is, possibly, some relevance in Category:Fermat cubic surface, but they're not really that interesting as images. —David Eppstein (talk) 05:24, 9 October 2020 (UTC)

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Equivalent statement

The article says:

• Equivalent statement 4 – connection to elliptic curves: If a, b, c is a non-trivial solution to xp + yp = zp, p odd prime, then y2 = x(x − ap)(x + bp) (Frey curve) will be an elliptic curve[15].

Is this really an equivalent statement of the theorem (except in the trivial sense that, since we now know there are no such solutions, then any statement of the form "If ... is a solution then ..." must be true)? W. P. Uzer (talk) 17:43, 18 February 2021 (UTC)

capitalization

Doctrines, ideologies, philosophies, theologies, theories, movements, methods, processes, systems or "schools" of thought and practice, and fields of academic study or professional practice are not capitalized, unless the name derives from a proper name. Very difficult to understand how this article warrants going against such clear guidance. If it's good enough for the theory of relativity it's surely good enough for this. Primergrey (talk) 15:01, 2 August 2021 (UTC)

The theory of relativity is a theory of relativity. Fermat's Last Theorem wasn't a theorem of Fermat, and even had it been, it wouldn't (I guess) have been his last. So it doesn't really make sense to use the words uncapitalized (a bit like writing 'holy Roman empire', perhaps). Anyway, theorems are not on the list you quote, so the guidance doesn't seem entirely applicable. But if it turns out that good sources more commonly use the phrase uncapitalized, then I won't object to doing likewise. W. P. Uzer (talk) 15:37, 2 August 2021 (UTC)