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== Number theory topline definition ==

{{Reply to|D.Lazard}} Daniel: 90% of number theory is about the properties of algebraic numbers, and saying "numbers" in general is very misleading, since most study of real numbers occurs in analysis. "Whole numbers and fractions" are the main interest of number theory, and algebraic numbers appear as generalizations of them.

I wouldn't use "algebraic number" in the lead, but do you really think "whole number" and "fraction" are too technical, that they will confuse readers who are curious about mathematics, but do not know what whole numbers are? 

[[User:Magyar25|Magyar25]] ([[User talk:Magyar25|talk]]) 21:50, 5 November 2024 (UTC)
:My take:  If we need to gloss the term "number theory" at all, I would prefer "the theory of the natural numbers", which is accurate in spite of the fact that arithmeticians consider other sorts of numbers.  Rationals are ratios of natural numbers.  Algebraic numbers are algebraic ''over'' the natural numbers.  Et cetera.
:As to the term {{!xt|whole number}}, my preference would be that we should never use (as opposed to mention) it at all, especially in a math article.  Mathematicians essentially never use the term. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 22:18, 5 November 2024 (UTC)
:The lead of the article [[Mathematics]] is not the place for an accurate definition of number theory. This sentence is here to explain what mathematics is about, and the way readers understand "number" does not really matter. Moreover, restricting number theory to some sort of numbers would go against a common consensus: the facts "every real number has an infinite decimal expansion" and "{{pi}} is not an algebraic number" are properties of real numbers that everyone considers as belonging to number theory. If there is something that is misleading in this sentence it is the definition of analysis as "the study of continuous changes", since the phrase in rarely used in analysis, except for explaining one of several motivations of analysis. Nevertheless, after many discussions on this talk page, nobody has found a better phrase. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 01:49, 6 November 2024 (UTC)
::Sorry, I ''completely'' disagree with the claim ''the facts "every real number has an infinite decimal expansion" and "{{pi}} is not an algebraic number" are properties of real numbers that everyone considers as belonging to number theory''.  I think that's just absolutely wrong.  The first one is ''definitely'' not part of number theory.  The one about &pi; is a little closer but I still think it's unlikely to be called number theory. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 19:01, 6 November 2024 (UTC)
:::Transcendental number theory and diophantine approximation are both part of number theory, fwiw. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 19:26, 6 November 2024 (UTC)
::::But &pi; is not. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 19:31, 6 November 2024 (UTC)
::::I mean, of course you ''use'' &pi; in number theory, to give approximations and so forth.  But you don't really ''study'' &pi;. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 19:32, 6 November 2024 (UTC)
:::::: The Borweins would disagree. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 19:34, 6 November 2024 (UTC)
::::::: Ref? --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 19:42, 6 November 2024 (UTC)
:::::::: Pi and the AGM: a study in analytic number theory and computational complexity, Jonathan and Peter Borwein, 1987. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 19:45, 6 November 2024 (UTC)
::::::::: Well.  Categorizing branches is always fraught.  I did say &pi; was "a little closer".  My general take is that nothing that involves the completed infinite is part of the ''subject matter'' of number theory, though it might be part of the ''methods''.
::::::::: Anyway the best solution might be just not to gloss "number theory" at all in the lead.  I don't see that a gloss saying it's the "theory of numbers" adds anything at all; it just sounds like the natural meaning of the words.  If we are to have a gloss I continue to think the "theory of natural numbers" is better wording. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 20:12, 6 November 2024 (UTC)
:::::::::: p-adic numbers and adeles are unambiguously a part of number theory, and certainly involve completion. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 20:39, 6 November 2024 (UTC)
::::::::::: OK.  I was never a number theorist, and maybe the field has moved on since I took my one class in it as an undergrad (we used Apostol's ''Introduction to Analytic Number Theory'').  I still don't find the current gloss useful.  (Note that p-adic numbers and adele rings are not likely to be evoked by the phrase "the study of numbers".)  Do you agree with just removing the gloss? --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 21:29, 6 November 2024 (UTC)
::::::::::{{ec}}The gloss for number theory is here for the balance of the sentence. If you can propose a better gloss, please do.
::::::::::The first sentence of [[Transcendental number theory]] is "'''Transcendental number theory''' is a branch of [[number theory]] that investigates [[Transcendental number|transcendental numbers]]". If you read the article, you will learn that a major result of this branch of number theory is [[Gelfond–Schneider theorem]], which implies that {{tmath|e^\pi}} is trancendental, and that a major open question is whether {{tmath|e+\pi}} is transcendental. 
::::::::::About "completed infinity" in number theory: I never saw anybody writing that [[Fermat's Last Theorem]] and [[Wiles's proof of Fermat's Last Theorem]] do not belong to number theory, although the proof makes a fundamental implicit use of the [[axiom of infinity]], and even (in the original proof) of a much stronger axiom. So, your opinion on the subject matter of number theory goes against a consensus of number theorists. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 21:35, 6 November 2024 (UTC)
::::::::::: On the other hand, the first sentence of [[number theory]] says that it's about natural numbers and arithmetic functions.  So there's a bit of a conflict there.  Myself, I would not have counted transcendental number theory as part of number theory, but I don't know how workers in the field think about it.
::::::::::: My proposal is simply to have no gloss at all.
::::::::::: As to your second paragraph, you're talking about the proofs, not the subject matter.  Fermat's last theorem is ''about'' natural numbers.  Its ''proof'' uses completed infinite objects, but that is not what it is [[aboutness|about]]. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 22:30, 6 November 2024 (UTC)
:I see that this conversation is moribund but I would definitely second this -- characterizing number theory as the "study of numbers" is extremely misleading. It would be preferable to list it without the parenthetical if we don't feel that it can be easily summarized. 
:[[User:Lauciusa|Lauciusa]] ([[User talk:Lauciusa|talk]]) 17:58, 25 March 2025 (UTC)

== High compliments ==

The first paragraph of the lede here is excellent. Extreme kudos to everyone who contributed. [[User_talk:HiDrNick|<span style="color:#CC3300">Hi</span><span style="color:#0088FF"><b>DrNick</b></span>]]! 22:51, 22 December 2024 (UTC)

== Jain mathematics ==

Two editors attempted to add a long papagraph on Jain mathematics. This is misplaced here and gives an [[WP:UNDUE]] weight to this part of the history of mathematics. 

Indeed the {{tl|main article}} template at the beginning of {{alink|History}} say that this section is a summary of [[History of mathematics]]. This article does not mention Jain mathematics, but has a section on Indian mathematics with a template {{tlx|main article|Indian mathematics}}. [[Indian mathematics]] has several sections including a section on [[Janism]]. If more must be added on Janism, this must be done first in these "main articles".

Important contributions of Indian mathematics are already mentioned at the end of {{alink|Ancient}}. I have no opinion whether other important contributions must be mentioned there, but, if this requires more than a few words to mention them, this must be discussed first in this talk page. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 12:03, 14 January 2025 (UTC)

== Add reference to Albert Einstein in Mathematics#Reality section ==

I found the issue by reading an excerpt of this article in the Philosophy of Mathematics article. It looks a bit inconsistent, as all other enlisted scientists are referenced properly. [[User:Sunlamb|Sunlamb]] ([[User talk:Sunlamb|talk]]) 12:55, 25 January 2025 (UTC)

:I guess that what you call a bad reference is the fact the mention of Einstein is not wikilinked to the article [[Albert Einstein]]. It was intensional, since the link appears already in two other sections. However, this excerpt has been inseterd more recently in [[Philosophy of mathematics]]. Also, readers are not supposed to read sections in their occurring order. So, I'll add the link. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:02, 25 January 2025 (UTC)
::Thank you [[User:Sunlamb|Sunlamb]] ([[User talk:Sunlamb|talk]]) 21:34, 22 March 2025 (UTC)

== Definition and Introduction Proposal ==

The current definition reads 
 Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.

This is awful, not least because it is circular. An operational definition of mathematics is basically impossible because of the huge variety of the field (there's even a section of the page on this), and what I would propose is skirting the issue by not proposing a definition.

 <del>Mathematics is a large and varied field of study that has its origins in the analysis and manipulation of numbers and shapes</del>.

 Mathematics is a large and varied field of study that has its origins in the analysis and manipulation of quantities and shapes.

And then continuing from there, with some examples of fields of study.

[[User:Lauciusa|Lauciusa]] ([[User talk:Lauciusa|talk]]) 17:54, 25 March 2025 (UTC)

:I recommend searching the archives for "first sentence".  It's written how it is for rather good reasons. <span style="border-radius:2px;padding:3px;background:#1E816F">[[User:Remsense|<span style="color:#fff">'''Remsense'''</span>]]<span style="color:#fff">&nbsp;‥&nbsp;</span>[[User talk:Remsense|<span lang="zh" style="color:#fff">'''论'''</span>]]</span> 17:59, 25 March 2025 (UTC)

::As Remsense alluded to, this sentence is highly contentious. And the supposed circularity doesn't bother me at all.
::That said, I think that Lauciusa's solution is pretty good, and better than what we have. Lauciusa, your case would be strengthened if you could cite reliable sources (possibly tweaking the text according to them). [[User:Mgnbar|Mgnbar]] ([[User talk:Mgnbar|talk]]) 18:07, 25 March 2025 (UTC)
:::Cards on the table, I wouldn't agree. The present opening sentence is ponderous, but at least each word is considerably precise and meaningful. I'm skeptical of {{xt|manipulation}}, which seems highly impressionistic—is tax collection in ancient Babylon "manipulation of numbers"? <span style="border-radius:2px;padding:3px;background:#1E816F">[[User:Remsense|<span style="color:#fff">'''Remsense'''</span>]]<span style="color:#fff">&nbsp;‥&nbsp;</span>[[User talk:Remsense|<span lang="zh" style="color:#fff">'''论'''</span>]]</span> 18:15, 25 March 2025 (UTC)
::::I don't feel that the use of "manipulation" here is less precise and meaningful than the "needs of ... mathematics itself". There is literally nothing in the universe of objects and concepts that cannot be fit into this definition because there is no base case for the recursion (this is pedantic, I know, but we're editing a wikipedia article on mathematics so I think pedantic is where we are).
::::Is tax collection in ancient Babylon mathematics? I mean, I would argue that in terms of the use of numbers for record keeping and analysis, it is, yes. Certainly this plays into the origins of mathematics -- the early use of numbers and sums and products led to discoveries about those systems.
::::[[User:Lauciusa|Lauciusa]] ([[User talk:Lauciusa|talk]]) 14:16, 26 March 2025 (UTC)
:::::That's going to be an argument one can make for every noun and verb we could use here. Ultimately, I'm pretty convinced we cannot write a more elegant, "non-circular" definition of what mathematics is because it simply would not reflect reality. Others may agree with me here—as human endeavors go (this characterization applies to some extent according to pretty much everyone, though some may quibble on margins for invention vs. discovery reasons?) mathematics is uniquely abstract and not always commensurable with "independent terms" (the connection to empirical science may be the best we can do), so the "feedback loop" you seem particularly bothered by, is to me a plain reflection of what mathematics has meant historically. I hope that makes sense. <span style="border-radius:2px;padding:3px;background:#1E816F">[[User:Remsense|<span style="color:#fff">'''Remsense'''</span>]]<span style="color:#fff">&nbsp;‥&nbsp;</span>[[User talk:Remsense|<span lang="zh" style="color:#fff">'''论'''</span>]]</span> 14:35, 26 March 2025 (UTC)
::::::In a larger sense, I am pretty convinced that we cannot write a concise definition of mathematics at all. Gun to my head, I would probably write a checklist of 20 features of mathematical study, and 20 anti-features of mathematical study, and if a field scored more than +9 then it would be considered within "mathematics". 
::::::But my proposal here is to precisely avoid getting pulled into a game of trying to create a definition at all. We give a general idea of the historical motivations for the study (which I think is unambiguously factual) and then in the next sentence we give some examples of modern mathematical subfields of study that have emerged from that.
::::::I think where we disagree is that the existing definition is at all satisfactory. I feel that it is not. It is so broadly phrased (in the recursiveness) that it fails to exclude anything, but so restrictive in the remainder (specifically "the needs of empirical sciences") that it excludes the overwhelming majority of fields of mathematical endeavor.
::::::There's a section of the article on the dispute around the definition and scope of mathematics; it is a field of study in itself (although more a field of semantics in my mind than anything prescriptive) and not worth trying to engage with in the first sentence.
::::::[[User:Lauciusa|Lauciusa]] ([[User talk:Lauciusa|talk]]) 14:45, 26 March 2025 (UTC)
:::::::See, I agree we're not really defining mathematics in the traditional sense either way, but rather we are identifying it by its fruits. Yours is simply less helpful in doing so, though—{{xt|large and varied field}} communicates very little at all, and {{xt|manipulation of numbers and shapes}} is still offputtingly concrete in its vagueness.
:::::::Right now I do not see how any point in the present lead paragraph can actually be argued to be false, misleading, or even unhelpful to readers. It's not a good solution to answer the anxiety about lacking an airtight definition by resolving to say as little as possible instead. We can clearly say more of substance, and the present lead does. <span style="border-radius:2px;padding:3px;background:#1E816F">[[User:Remsense|<span style="color:#fff">'''Remsense'''</span>]]<span style="color:#fff">&nbsp;‥&nbsp;</span>[[User talk:Remsense|<span lang="zh" style="color:#fff">'''论'''</span>]]</span> 14:55, 26 March 2025 (UTC)
::::::::I don't really disagree with your post, Remsense, but it exemplifies what I said earlier, about how editors disagree in their priorities. Your post is focused on factual content, rather than on making that content understandable to the reader.
::::::::In my opinion, the current opening sentence is unhelpful to readers because its structure is too complicated, especially for non-native speakers of English. The complication is what gives a lot of readers a knee-jerk "I don't like this" response. In general, we ask too much of the opening sentence. [[User:Mgnbar|Mgnbar]] ([[User talk:Mgnbar|talk]]) 15:02, 26 March 2025 (UTC)
:::::::::I agree in principle also—its diction may well be worth tinkering with. As you say, if we're doing either that or a more drastic downsizing (which I'm not presently in favor of) we should be clear about what issues we're trying to solve. <span style="border-radius:2px;padding:3px;background:#1E816F">[[User:Remsense|<span style="color:#fff">'''Remsense'''</span>]]<span style="color:#fff">&nbsp;‥&nbsp;</span>[[User talk:Remsense|<span lang="zh" style="color:#fff">'''论'''</span>]]</span> 15:07, 26 March 2025 (UTC)
:::This seems like a good avenue; I'll see what I can dig up. The "large and diverse field" I can probably cite much of the same material as in [[Definition of mathematics]] since that article directly addresses the controversy in defining the field.
:::For the historical origins of mathematics (i.e. numbers & shapes as I have proposed here) it shouldn't be hard to find supporting material in the introduction of any survey of mathematics. If you have suggestions I'm open to them, otherwise I'll dig in further when I can.
:::[[User:Lauciusa|Lauciusa]] ([[User talk:Lauciusa|talk]]) 15:26, 26 March 2025 (UTC)
:The present first sentence is a compromise that could certainly be improved. Nevertheless, Lauciusa's proposal is much worse, as it hides the most fundamental features of mathematics: All areas of mathematics involve theories, theorems and proofs and the relatioship between mathematics and empirical sciences is fundamental. So, I am strongly against Lauciusa's proposal. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 18:32, 25 March 2025 (UTC)
::Lauciusa, these responses give you an idea of what you're up against. People have high standards for the opening sentence: verifiability, accuracy, clarity, completeness, etc. We have not found a sentence that achieves all of these goals, and we disagree about which goals to abandon first, etc. (I hope that I am not mis-representing any of my fellow editors here. I write this post with sincerity and empathy.) Regards, [[User:Mgnbar|Mgnbar]] ([[User talk:Mgnbar|talk]]) 19:13, 25 March 2025 (UTC)
:::I can live with both the current first sentence and with Lauciusa's proposal, and don't have a strong preference between them.
:::I will say that one clear advantage of Lauciusa's proposal is that it at least says something about the objects of study of mathematics, which the current first sentence does not.
:::As an aside, I don't think I ''completely'' agree with Prof Lazard's claim that "all" areas of mathematics involve theorems and proofs; that seems shading towards a Euclidean POV.  It's true as a practical matter for research mathematics today, but there's lots of stuff that I would consider "mathematics" for which it's a problematic claim. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 20:03, 25 March 2025 (UTC)
::::I think the lack of "study objects" in the first sentence is nicely made moot by the following two sentences—sometimes I feel I drill down too fractally to the level of the sentence when judging how material is weighed in the lead. <span style="border-radius:2px;padding:3px;background:#1E816F">[[User:Remsense|<span style="color:#fff">'''Remsense'''</span>]]<span style="color:#fff">&nbsp;‥&nbsp;</span>[[User talk:Remsense|<span lang="zh" style="color:#fff">'''论'''</span>]]</span> 20:06, 25 March 2025 (UTC)
::::Can you give an example for something you'd consider "mathematics" which does not involve theorems? –[[user:jacobolus|jacobolus]] [[user_talk:jacobolus|(t)]] 20:38, 25 March 2025 (UTC)
:::::Well, its ancient origins, for example, before the notion of "theorem" was really formulated.  Or similar explorations even today (for example in [[recreational mathematics]] or [[experimental mathematics]]). --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 20:42, 25 March 2025 (UTC)
::Let me start by saying that my proposed change does not contradict any of this, but is simply a factual statement about the origin of mathematics as a field, and is non-controversial as far as I'm aware (although also admittedly not comprehensive by design)
::The idea that all areas of mathematics involves theories, theorems, and proofs does not seem accurate to me. Calculation and arithmetic, while distasteful to the practicing mathematician, are clearly part of mathematics, and arguably are the foundation upon which all of mathematics is motivated.
::In terms of the relationship of mathematics and empirical sciences, I would broadly say an even stronger converse -- mathematics is more accurately the study of things that cannot be contradicted by empirical methods!
::[[User:Lauciusa|Lauciusa]] ([[User talk:Lauciusa|talk]]) 14:03, 26 March 2025 (UTC)
:Most mathematics concerns the sciences, not "numbers and shapes". Try again. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 21:46, 25 March 2025 (UTC)
::First, I want to say that my proposed opening does not contradict this. Even if we grant that mathematics concerns the sciences, its origins in the study of arithmetic and geometry are hard to dispute.
::But as a direct response, I don't think this is true; mathematics in most cases distinguishes itself from the sciences based almost entirely on the fact that it is not an empirical examination; it cannot be contradicted by empirical study.
::[[User:Lauciusa|Lauciusa]] ([[User talk:Lauciusa|talk]]) 14:07, 26 March 2025 (UTC)
:::This is exacly what is said in the current first sentence: mathematics is not an empirical science but most mathematics were developped as tools for empirical sciences and for mathematics itself. Also, it is arguable wether the empirical rules developed before [[Greek mathematics]] were mathematics. In any case, this is Greek mathematicians who coined the word "mathematics" and elaborated the concept of proof. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 15:08, 26 March 2025 (UTC)
:::::See, I think this is a bit POV, specifically a Euclidean-foundationalist POV.  In my view, mathematics is distinguished by its subject matter (mathematical objects) more than by its methods (proof being the method currently under scrutiny, though not the only method used in mathematics).  There used to be a first sentence that was more explicitly about the subject matter (in various versions it had a list of things {{tq|...including quantity, space, pattern, and change...}} or something like that).  It was less elegant than the current first sentence, but it did have the advantage of bringing the subject matter front and center. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 16:53, 26 March 2025 (UTC)
::::::Of course, and at the same time it's nontrivial to really discern what the ideal, modern, internally consistent notion of a concept (here, {{linktext|lang=grc|μάθημα}}) should be when {{scareq|you}} were the ones with which the word historically emerged to refer to a particular historically contingent but certainly antecedent sense of what it means now.
::::::(I'm currently doing a lot of research for [[Logic in China]], which seems a bit more cut and dry as being logic unless we really think a tradition doesn't count unless it assigns special symbols while analyzing methods of reasoning and argumentation.) <span style="border-radius:2px;padding:3px;background:#1E816F">[[User:Remsense|<span style="color:#fff">'''Remsense'''</span>]]<span style="color:#fff">&nbsp;‥&nbsp;</span>[[User talk:Remsense|<span lang="zh" style="color:#fff">'''论'''</span>]]</span> 17:56, 26 March 2025 (UTC)
::::As I've said before, the current opening is both overly expansive and overly restrictive in ways that only make sense if you already have an intuition of what mathematics is. Microscopes were developed as tools for empirical sciences. Why is optics not part of mathematics? Note that I am not saying that the definition implies the inverse, that all tools for empirical sciences are mathematics; just that it fails to add clarity on what exactly distinguishes mathematics from other tools.
::::I don't think we can provide a clear definition here, and I don't think we should try to create one. The definition provided is bad and inaccurate.
::::I do not propose an alternate definition, very deliberately. Instead we open with historical context and follow up with broad strokes on where it lies today.
::::If you want to say that arithmetic and the development of counting are not part of mathematics, then I most strenuously disagree, and I think any realistic look at primary and secondary mathematical education means that I am not alone in this disagreement.
::::[[User:Lauciusa|Lauciusa]] ([[User talk:Lauciusa|talk]]) 15:20, 26 March 2025 (UTC)

:The first sentence is not a definition of mathematics. I'm unclear why your proposal is less of a definition. The rest of the first paragraph provides added context. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 15:34, 26 March 2025 (UTC)
::I'm not sure I understand. The existing first sentence looks like definition to me. It defines what "mathematics" is. Is there something subtle I'm missing? It attempts to be an exhaustive description of what mathematics does and does not study.
::My proposed sentence on the other hand is not a definition -- is says that mathematics is a "large and varied field of study" which is descriptive, not definitional, because it says nothing about what it is studying or not studying: is it numbers? Frogs? Zeus? From there it just talks about the historical context to give direction, and relies on the rest of the introduction to give enough examples of mathematical areas of interest to hopefully provide a fuller context.
::[[User:Lauciusa|Lauciusa]] ([[User talk:Lauciusa|talk]]) 16:47, 27 March 2025 (UTC)
:::The current first sentence is descriptive. That's the point. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 17:25, 27 March 2025 (UTC)
::::The reason why I say prescriptive rather than descriptive is that the existing opening rules out some things. If a subject is not developed or proved for the empirical sciences nor for mathematics itself, then it is not part of mathematics. So methods and calculations designed for record keeping or actuarial uses are not mathematics. This is simply not true; arguably these are the origin of the field and the way that it is taught in primary and secondary school to this day.
::::On the other hand, my introduction draws a very big circle of things that it could study, and gives two members of that set; members that are, in particular, the origins of the entire field of study.
::::[[User:Lauciusa|Lauciusa]] ([[User talk:Lauciusa|talk]]) 18:34, 28 March 2025 (UTC)
:::{{br}}Lauciusa, I'm really having trouble following you here.  It seems to me your proposal says more about the objects of study, at least originally ("numbers and shapes"), whereas the current first sentence says absolutely nothing about the objects of study, and instead describes the methodology.  That was what I actually ''liked'' better about your proposal.  (But as Remsense correctly points out, this is less of an issue with the current lead if you also consider the second sentence.)
:::In any case, neither version seems to be a definition.  --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 06:01, 28 March 2025 (UTC)
::::See my reply to [[User:Tito Omburo|Tito Omburo]] just above this. My problem with the current definition is that it is inaccurate. My alternative is admittedly overly broad, but is not incorrect, and at least gives a motivation for the field in historical terms.
::::[[User:Lauciusa|Lauciusa]] ([[User talk:Lauciusa|talk]]) 18:42, 28 March 2025 (UTC)

:How about:
 Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of analysis and manipulation of numbers, shapes, and other kinds of appropriate objects.

:[[User:Paul August|Paul August]] [[User_talk:Paul August|&#9742;]] 13:43, 28 March 2025 (UTC)
::I guess I don't hate it, but would prefer "quantities" to "numbers". [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 15:36, 28 March 2025 (UTC)
:::What "appropriate object" is intended to mean, and for which audience this phrase is meaningful?
:::For avoiding apparent circularity, I suggest to replace "mathematics itself" with "previously established parts of mathematics". This would give the first sentence {{tqq|Mathematics is a field of study that discovers and organizes methods, theories and theorems that are developed and proved for the needs of empirical sciences and previously established parts of mathematics}}. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:45, 28 March 2025 (UTC)
::::Thinking again, I do not like either "field of study that discovers and organizes" (almost each term is more or less controversial). So, I suggest: {{tqq|Mathematics deals with abstract methods, theories and theorems that are developed and proved for the needs of empirical sciences and previously established parts of mathematics.}} [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:54, 28 March 2025 (UTC)
:::::I like the "previously established"; it's overly formal but does address the recursiveness. I also like the "deals with" which is less limiting than the current language. If we adopted this phrasing it would be a strict improvement.
:::::I feel like the connection to the empirical sciences is a relatively recent development in the history of mathematics and does not deserve to be in the opening. The answer to what particular aspects of the real world led to the development of mathematics is unaddressed while I feel like my opening does at least indicate some intuition about where mathematics is used.
:::::[[User:Lauciusa|Lauciusa]] ([[User talk:Lauciusa|talk]]) 18:46, 28 March 2025 (UTC)
:::::: The earliest mathematics was used in commerce and architecture by the Babylonian civilization, primarily for reckoning lengths and weights. That is, it was used as a tool to describe the world. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 19:46, 28 March 2025 (UTC)
:::I am not a huge fan of this phrasing because I feel it is too limiting, but I love the idea of using "quantities" instead of "numbers". [[User:Lauciusa|Lauciusa]] ([[User talk:Lauciusa|talk]]) 18:16, 28 March 2025 (UTC)
::::There is only so much one can do in 80 words: I would prioritize providing meaningful description over that which endeavors to completely circumscribe at the expense of being overly vague. Really, the description is the entire article. We should favor providing as much "survey coverage" as quickly as possible in the lead. <span style="border-radius:2px;padding:3px;background:#1E816F">[[User:Remsense|<span style="color:#fff">'''Remsense'''</span>]]<span style="color:#fff">&nbsp;‥&nbsp;</span>[[User talk:Remsense|<span lang="zh" style="color:#fff">'''论'''</span>]]</span> 18:27, 28 March 2025 (UTC)

== Introduction counter-proposal ==

We argue incessantly (actually: periodically with high frequency) about the opening sentence. It's so over-constrained that no solution is ever entirely satisfying. The most recent proposal by Paul August is admirable but typical: It attempts to say everything important, and thus becomes a complicated nest of multiple dependent clauses. I believe that such sentences are off-putting to many native speakers and incomprehensible to many non-native speakers. The opening sentence often resembles a definition, which invites further complaints, even though we disavow it as a definition.

In the interest of breaking this cycle, here is my radical proposal: We make the opening sentence so short, that it can't possibly be taken as a definition or summary of math. Rather, it is obviously just the start of a paragraph. The paragraph has enough room, to hold everything that we wanted to put in the opening sentence.

So, for those of you willing to consider this alternative approach, here is a proposed opening sentence:
 Mathematics is a field of study.
[[User:Mgnbar|Mgnbar]] ([[User talk:Mgnbar|talk]]) 21:14, 28 March 2025 (UTC)

: Thanks, but I hate it. ;-) [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 21:22, 28 March 2025 (UTC)
:I totally disagree, since mathematics is also a field of knowledge, and knowledge is not included in study, even if study is required to obtain knowledge. This is one of the things I had in mind by saying that almost all words that I suggest to replace with "deals with" are controversial. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 21:40, 28 March 2025 (UTC)

::I just took "field of study" from the current version at [[Mathematics]]. Other, similarly short formulations fit my proposal equally well. [[User:Mgnbar|Mgnbar]] ([[User talk:Mgnbar|talk]]) 22:15, 28 March 2025 (UTC)

== Inclusion of abbreviated term "math" in lead sentence ==

Would like to gain a consensus and gather more opinions regarding the following. I believe the simplified word "math" should be included in the lead, since it is commonly called such and was mentioned often throughout the article (until a user completely removed each of the words "math" which have been featured on here for a while until I made my edits), and is a standard practice to include common names on this encyclopedia. For example: "Mathematics (simply referred to as "math") ..." etc. [[User:Tinton5|Tinton5]] ([[User talk:Tinton5|talk]]) 00:12, 12 April 2025 (UTC)
:I feel very strongly that we should not include either "math" or "maths" in the lead section.  They are colloquialisms and this is an encyclopedia article.  The mention in the etymology section is more than enough.
:The other point to consider here is that, if we include one, we also have to include the other, and then we have to pick which one goes first.  I would really rather not.  The term "maths" is a little nails-on-chalkboard to me and I would rather not see it at the start of the article.  I imagine that our Commonwealth friends feel the same about "math". --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 00:28, 12 April 2025 (UTC)
:<small>By the way, it is not true that "math" was "mentioned often throughout the article".  [https://en.wikipedia.org/w/index.php?title=Mathematics&diff=1285131399&oldid=1285130105 My cleanup edit] was limited to the "Education" section, and changed only five instances in two paragraphs.  That took care of all the uses (as opposed to mentions).  Of course I did not change any usages in external publication titles, names of organizations, or URLs. --[[User:Trovatore|Trovatore]] ([[User talk:Trovatore|talk]]) 00:38, 12 April 2025 (UTC) </small>
:We argue about the opening sentence and paragraph interminably. (And we have argued about this specific point in the past, although I haven't found it in the talk page archives.) The opening sentence tends to be overly long and complicated. I support leaving this minor point to Etymology. [[User:Mgnbar|Mgnbar]] ([[User talk:Mgnbar|talk]]) 12:25, 12 April 2025 (UTC)
:Leave it to etymology. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 18:19, 12 April 2025 (UTC)

== Description of Algebra as the study of formulas ==

I believe we should try and reach a new consensus on how best to describe algebra. The article makes statements such as:

Algebra is "the study of formulas and related structures"

"Roughly speaking, [algebra] consists of the study and manipulation of formulas"

"Algebra is the art of manipulation equations and formulas"

I feel that this seriously misrepresents algebra, especially how its been studied in the last 500 years. The Wikipedia page for algebra itself is much better, saying 

"Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems."

I think it's best to have a discussion about the best ways to succinctly describe algebra in an accessible way which represents it more accurately. [[User:Finitefields|Finitefields]] ([[User talk:Finitefields|talk]]) 04:25, 18 May 2025 (UTC)

:What do you propose?
:Your quotation of article [[Algebra]] "misrepresents algebra, especially how its [sic] been studied in the last 500 years", since algebraic structures were introduced less than 150 years ago.
:Here are some facts that support the current presentation of algebra in [[Mathematics]]:
:*The word "algebra" is derived from the title of an [[Al-Khwarizmi]]'s book, where it refers to a method for manipulating equations.
:*[[François Viète]] introduced variables that allow manipulating (computing) with unspecified quanities as if there were numbers.
:*Until the 19th century, algebra was essentially the [[theory of equations]], that is methods for transforming equations for getting their solutions.
:*The axioms of an algebraic structure are rules allowing to transform [[expression (mathematics)|expression]]s in view of obtaining formulas that are true in the structure.
:Again, can you propose something that is better than the current state of the article. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 09:14, 18 May 2025 (UTC)
::I agree that some of these formulations could be improved.
::* {{green|Algebra is "the study of formulas and related structures"}}
::*: Depending on how we interpret "related structures", this could be fine as is.
::* {{green|"Roughly speaking, [algebra] consists of the study and manipulation of formulas"}}
::*: Could be improved by talking about "classical algebra" instead of "algebra" in general
::*{{green|Algebra is the art of manipulating equations and formulas}}
::*: Could be improved by talking about "classical algebra" instead of "algebra" in general
::[[User:Phlsph7|Phlsph7]] ([[User talk:Phlsph7|talk]]) 09:56, 18 May 2025 (UTC)

:::All of math deals with abstract systems. So that part of the [[Algebra]] page should not be emulated, unless it's all clarified. For example, it could be argued that algebra is the abstraction of ''number'' systems while topology is the abstraction of ''spatial'' systems. [[User:Mgnbar|Mgnbar]] ([[User talk:Mgnbar|talk]]) 12:12, 18 May 2025 (UTC)
::::You all make good points. As [[User:Phlsph7|Phlsph7]] suggested, we should try to emphasize the difference between modern and classical algebra. I do acknowledge the continued importance of formulas in algebra today, such as in [[Galois theory]], but I feel that emphasizing formulas first in the definition of algebra gives an incorrect impression of how the field is approached by modern mathematicians, especially in topics such as [[Group theory]].
::::''Algebra is "the study of formulas and related structures"''
::::As this is the first time algebra is described in the article, I'd propose a slightly more modern definition. Maybe just '''"the study of formulas and algebraic structures"''' or '''"the study of algebraic structures and related formulas"'''. [[User:Mgnbar|Mgnbar]] is correct that simply saying "structures" would not be specific to algebra.
::::''"Roughly speaking, [algebra] consists of the study and manipulation of formulas"''
::::This sentence comes from a section about the Renaissance. I think the sentence should be changed as such: '''During the Renaissance, two more areas appeared. [[Mathematical notation]] led to [[Algebra]], ''as it allowed for broader study and manipulation of formulas''.'''
::::''"Algebra is the art of manipulating equations and formulas"''
::::This is the first sentence of the algebra section, and should be changed, perhaps in a similar fashion to my proposal for the first sentence. [[User:Finitefields|Finitefields]] ([[User talk:Finitefields|talk]]) 13:14, 18 May 2025 (UTC)
:::First of all, the only definition of algebra given in this article is the whole section {{alink|Algebra}}. A single sentence cannot define such a wide area. Before changing a sentence, the place where it appears in the article must be considered for deciding whether a better formulation can be found.
:::* The first above quotation is not a definition but a parenthetical explanation of what algebra is, intended for people who have no idea of that. It has to be compared with the corresponding parenthetical phrase for geometry (the study of shapes and spaces that contain them). As they are intended for non mathematicians, they have to avoid every technical terms such as "algebraic structure" that is nonsensical for people who know nothing of algebra. Also, using "algebraic" in an explanation of what algebra is is a sort of circular reasoning, and must definitively be avoided. 
:::* The complete second notation is {{tqq|[During Renaissance,] [[Mathematical notation]] led to [[algebra]] which, roughly speaking, consists of the study and the manipulation of formulas}}. It is the only sentence about algebra in this section. The phrase "roughly speaking" establishes clearly that the sentence is not intended to be an accurate definition. I do not see above anything suggesting that this is a wrong description of algebra (at least at that time). On the oposite, "as it allowed for broader study and manipulation of formulas" is an opinion on the reasons of the rise of algebra, that has not its place in the article.
:::* {{tqq|Algebra is the art of manipulating equations and formulas}}. Nothing is wrong in this sentence, since algebraic structures consist essentially of frameworks for the study of formulas subject to some manipulation rules (axioms of the structure). This sentence is useful for clarifying why [[Diophantus]] and [[al-Khwarismi]] were precursors of algebra.
:::[[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 14:12, 18 May 2025 (UTC)

== Timeline of origins of algebra ==
The beginning of the first section says "Before the Renaissance, mathematics was divided into two main areas. During the Renaissance, two more areas appeared. Mathematical notation led to algebra..." [[Renaissance]] is 15th and 16th centuries. But the [[algebra]] article says that it originated with a body of work called [[Al-Jabr]] from the year 820. I am confused, which is correct? [[Special:Contributions/96.250.205.153|96.250.205.153]] ([[User talk:96.250.205.153|talk]])  <!--Template:Undated--><small class="autosigned">—&nbsp;Preceding [[Wikipedia:Signatures|undated]] comment added 15:51, 26 May 2025 (UTC)</small> <!--Autosigned by SineBot-->

:Good point. The whole notion of the Renaissance (at least, the one we're talking about here) is Euro-centric. [[User:Mgnbar|Mgnbar]] ([[User talk:Mgnbar|talk]]) 16:15, 26 May 2025 (UTC)
:The quoted sentences are developed and explained in section {{alink|Algebra}} that says {{tqq|[[Diophantus]] (3rd century) and [[al-Khwarizmi]] (9th century) were the two main precursors of algebra. [...] Algebra became an area in '''its own right''' only with [[François Viète]] (1540–1603), [...]}} (bolding is mine). So, there is no contradiction between the two articles. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 16:26, 26 May 2025 (UTC)

::Is there a consensus among historians of mathematics that the discipline of algebra began with Viete? My understanding is that while Viete invented symbolic algebra, algebra as an independent field of study precedes him. The work in [[Al-Jabr]], for instance, appears to be quite systematic, general and methodological. Even texts like [[Ars Magna (Cardano book)|Ars Magna]], which is known to many modern algebra students today from the story of the insolvability of the quintic, precedes Viete. [[Special:Contributions/96.250.205.153|96.250.205.153]] ([[User talk:96.250.205.153|talk]]) 22:54, 26 May 2025 (UTC)
:::My own hot take is that it is dubious to assign historical priority to such matters as "when did the subject of algebra first appear". It's like asking "when did the subject of biology appear", illegible in these epistemes. Perhaps Viete first recognized there was a subject, which could be called algebra, at that time. [[User:Tito Omburo|Tito Omburo]] ([[User talk:Tito Omburo|talk]]) 00:03, 27 May 2025 (UTC)
::::I mostly agree. We should probably make François Viète less prominent by replacing "with François  Viète" by "during the 16th century" and adapting the sentence for including both ''Ars Magna'' and Viète. Indeed, as far as I know (I did not read ''Ars Magna''), Cardano did not considered equations with unspecified coefficients, and the "rules of algebras" mentioned in his subtitle should probably be interpreted as "algorithmic methods in arithmetic". 
::::It would be interesting to know when the word "algebra" was first used in a sense close to that given by Cardano, whichever it is. My guess is that it is not before the 14th or 15th century. [[User:D.Lazard|D.Lazard]] ([[User talk:D.Lazard|talk]]) 01:34, 27 May 2025 (UTC)