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Brief subsection for mathematicians as a subculture / in pop-culture?
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The Definition of Mathematics in the First Sentence Doesn't Define Mathematics
Latest comment: 2 months ago28 comments11 people in discussion
This article used to contain a more accurate and simplified definition of mathematics in the summary (the first sentence). Checking Merriam-Webster, Cambridge, Dictionary.com and Britannica.com will show that these (among other) generally accepted lexicographers all define mathematics as something very similar to "the scientific study of space, structure, quantity and change", or words very similar to those. This is also what Wikipedia used to define it as, but since I last consulted the article, it has changed the definition to "a field of study that discovers and organizes methods, theories and theorems for various sciences . . ." This is too abstract; it doesn't even describe what types of methods or theorems mathematics discovers or organizes.
At the very least, any definition of the term should include the fact that the subject involves quantities and the relationships among them. I'm not saying the definition is completely incorrect, but it isn't specific enough to describe what mathematics is, and rather only describes a part of what abstract or pure mathematicians do. TheLibrivore (talk) 11:33, 17 November 2025 (UTC)Reply
The problem with attempting to define mathematics is that no one in the history of the world has been successful at this. Why would the first sentence of the article succeed where everyone else has failed? Tito Omburo (talk) 11:36, 17 November 2025 (UTC)Reply
In my opinion, in this definition:
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.
"that are developed and proved" can be omitted, which yields:
Mathematics is a field of study that discovers and organizes methods, theories, and theorems for the needs of empirical sciences and mathematics itself.
A lot of mathematics is developing methods and proving (or disproving) conjectures. So, the definition should keep proved or say something about proofs. EulerianTrail (talk) 15:15, 17 November 2025 (UTC)Reply
How is this one?
Mathematics is a field of study that discovers, develops, organizes and proves methods, theories, and theorems for the needs of empirical sciences and mathematics itself.
A side from that, I think this definition lacks "Concepts". For example, "number" and "shape", "circle", "rectangle" etc. are concepts, neither method, not theory, nor theorem. But they are concepts. So I propose:
Mathematics is a field of study that discovers, develops, organizes and proves concepts, methods, theories, and theorems for the needs of empirical sciences and mathematics itself.
The first sentence of the article is not intended to be a definition. We have a Proposed Definitions section later.
We have argued about the first sentence incessantly for years on this talk page. Writing an opening sentence that is verifiable, correct, concise, and thorough is extremely difficult. For example, I object that the current opening sentence (and the amended version proposed by Hooman Mallahzadeh) also applies to computer science and statistics. Mgnbar (talk) 15:41, 17 November 2025 (UTC)Reply
I have linked "proved" to mathematical proof. So, the opening sentence applies to computer science and statistics only when mathematical proofs are involved. A work in computer science or statistics that involves mathematical proofs belongs to both mathematics and computer science or mathematics. By the way, I'll link theorem also. D.Lazard (talk) 19:14, 17 November 2025 (UTC)Reply
Links are not supposed to change or enhance the meaning of the text. You provide the links as a courtesy for people who want to know more, not to clarify what the words mean. You always have to assume that the links will not be followed. --Trovatore (talk) 19:54, 17 November 2025 (UTC)Reply
Computer science and statistics are both branches of mathematics. (Especially those parts of them involved with defining and proving theorems about abstract objects.) They are split into separate academic departments for practical reasons: they are large, have wide practical application, and their students are not expected to have general expertise in all areas of pure mathematics to the level of "pure mathematics" students. –jacobolus(t)20:27, 17 November 2025 (UTC)Reply
I think more to the point, the lead sentence and lead paragraph are meant to describe mathematics, not to demarcate it from everything that is not mathematics. We don't have to take a position on whether computer science is or is not mathematics. --Trovatore (talk) 20:39, 17 November 2025 (UTC)Reply
In my opinion, this definition lacks the adjective "abstract", and I propose this definition:
Mathematics is a field of study that discovers and organizes abstract methods, theories, and theorems that are developed and proved for the needs of empirical sciences and mathematics itself.
Thanks for your effort, but no, this is a non-starter. Look back at the older discussions to understand why. We aren't going to write a lead sentence that appears to be a definition of mathematics. --Trovatore (talk) 18:43, 21 May 2026 (UTC)Reply
The opening sentence is also discussed in § Redundant opening sentence, and it seems that a consensus is reached (or near to be reached) at the end of this section. Instrad of trying to write from scratch a new first sentence, you would have better to give there your opinion. If you have objections or ideas for improving the suggested new first paragraph, please, explain them there. D.Lazard (talk) 20:10, 21 May 2026 (UTC)Reply
@D.Lazard: Hi, and sorry for pinging you. In fact, much of mathematics is about mathematical objects (which is an umbrella term for numbers, expressions, shapes, functions, and sets etc.) and their properties. But we see nothing about that in this definition. I really propose to include that. Thanks, Hooman Mallahzadeh (talk) 14:59, 19 November 2025 (UTC)Reply
@Jacobolus Yes, it is mentioned there. But we need that in definition. I personally would define mathematics as:
Mathematics is a set of primitive notions and derived objects that have properties that are proved by methods, theories and theorems from primitive ones.
@Hooman Mallahzadeh Before continuing, I would recommend you spend some time reading all of the discussions about the lead section of this article from the past few years. If you want, you can also read discussions going back further than that. –jacobolus(t)19:20, 19 November 2025 (UTC)Reply
I agree with Jacobulus. The current first sentence is a compromise resulting from many discussions on this talk page, and I guess that nobody is fully satisfied with it. Personally, I would suggest
Such a first sentence has the advantage of getting rid of WP:weasel words ("field of study", "organizes methods"), of not excluding any area of mathematics, and of distinguishing mathematics from other sciences and fields of study.
I have a little hope that this formulation could get a consensus, but, if a discussion on the first sentence is opened again, this suggestion must be taken into account. D.Lazard (talk) 15:55, 20 November 2025 (UTC)Reply
My view continues to be: The opening sentence is not a definition. We should improve it, to make it clear that it is not even trying to be a definition. Mgnbar (talk) 18:37, 20 November 2025 (UTC)Reply
Proposed Definition of Mathematics for Your Consideration
Mathematics is the science of measurement applied to both abstract and concrete entities through the establishment of standardized units by which measurable aspects of such entities may be analyzed and compared and, through the use of such units, by which entities with new measurements may be created.
Of course the test of any definition is whether it includes all and only those things necessary to identify what one is defining, so I invite you to identify any mathematics that this definition would miss or any non-mathematics it includes. -----RalphBlanchette (talk) 22:26, 20 May 2026 (UTC)Reply
Sorry, but I don't think that this is better than our numerous other imperfect candidates. For starters, the science of measurement is metrology. For another, I'm surprised at the mention of units. For another, it emphasizes measurement (loosely, real numbers) but doesn't mention counting (loosely, integers). Some editors here will insist that the definition mention logic and proof, which yours doesn't. Regards, Mgnbar (talk) 22:37, 20 May 2026 (UTC)Reply
Thank you for the criticisms and I will try to take account of them in a revised definition. I think metrology is the wrong word for comparing abstract entities. I may have trouble finding a better word than "unit" to identify even non-numerical entities that are "measured", which is to say compared and contrasted, i.e. topological properties, for which each instance of a homeomorphic space is a "unit" of that topological space irrespective of their dimensions, the species of spaces of which I think must have only integral numbers. I use "measurement" more broadly than you do here and do not see why it excludes measuring in integer units. For me logic and proof are required in order to do valid measuring, but I appreciate the issues you raise. RalphBlanchette (talk) 23:11, 20 May 2026 (UTC)Reply
Ralph, I really think you need to take on board that the lead sentence is not intended to define mathematics, and we aren't going to write a lead sentence that appears to be a definition of mathematics. That's based on years of discussion. The lead sentence is going to describe mathematics, but not demarcate mathematics from that which is not mathematics. Any work you do trying to come up with a lead sentence that appears to be a definition of mathematics is not going anywhere. --Trovatore (talk) 18:46, 21 May 2026 (UTC)Reply
The opening sentence is also discussed in § Redundant opening sentence, and it seems that a consensus is reached (or near to be reached) at the end of this section. Instrad of trying to write from scratch a new first sentence, you would have better to give there your opinion. If you have objections or ideas for improving the suggested new first paragraph, please, explain them there. D.Lazard (talk) 20:17, 21 May 2026 (UTC)Reply
Lead Image
Latest comment: 25 days ago70 comments15 people in discussion
As significant as mathematics is, there should be an image in the introduction of the article to represent it. Are there any thoughts as to what it should be? I am thinking a visual proof, or a fractal, or something else that is simple but still is representative of mathematics. EulerianTrail (talk) 22:33, 26 November 2025 (UTC)Reply
No image exists which meets this criterion. For pure flair, an image could be added, say, of Archimedes pondering his orbs, but I struggle to see the relevance of such an image. 23:53, 26 November 2025 (UTC)— Preceding unsigned comment added by Tito Omburo (talk • contribs) 00:53, 27 November 2025 (UTC)Reply
As said below, the removed image meets perfectly this criterion. Moreover, this image contains both formulas and geometric figures, which is much better than an image containing only figures or only formulas. D.Lazard (talk) 15:57, 27 November 2025 (UTC)Reply
You removed an image of a mathematician reasoning about formulas on a blackboard, because you believe that it illustrates "the Galton–Watson process of probability theory". I congratulate you to be able to recognize this, but I think that that this will not be the case for most readers. Persomally, I was only able to guess that it was about probabilities, because of the notation , but I was not sure, because this notation has other uses in mathematics.
Since reasonning on formulas is the core of mathematics, I consider that this image illustrates well the whole mathematics and I strongly suggest to restore the image. D.Lazard (talk) 15:42, 27 November 2025 (UTC)Reply
Mathematics Barnstar
As you stated in the changelog "the first image must not be specific to an area of mathematics" which the image you are referring to was added recently by @Fgnievinski and is in probability theory. Thus, that image is not representative of the whole of mathematics and it is unfamiliar to most readers including yourself as you mentioned. Which you stated in the changelog as a reason not to include an image in the lead when I added a geometric visualization of Euler's formula to which you stated many readers are not familiar with complex numbers.
A possible image could be a photomontage of various math symbols and objects just like the math barnstar. I would be willing to do this and include a variety of symbols and objects. EulerianTrail (talk) 16:47, 27 November 2025 (UTC)Reply
I think this is all missing the point. An image is supposed to convey something encyclopedic. It is not decoration. An image is not required, and no image adequately conveys (or sheds any light on) the subject of the article. Tito Omburo (talk) 17:47, 27 November 2025 (UTC)Reply
The main purpose of the lead image is to "allow readers to quickly assess if they have arrived at the right page" MOS:LEADIMAGE and to be representative of the subject. It may very well be that for this article no image is better since any individual image becomes too specific, but that is why I suggested a photomontage. EulerianTrail (talk) 18:02, 27 November 2025 (UTC)Reply
I really don't think we need an image for that purpose at the "mathematics" article. I strongly prefer continuing not to have a lead image. --Trovatore (talk) 18:21, 27 November 2025 (UTC)Reply
@Trovatore We should consider the needs of the audience over the preference of editors, especially if the editors are experts and the audience is mainly non-expert. (Experts in math education excepted.)
The proposed image, of a person writing mathematics on a chalkboard, is representative of the common practice of mathematics. Most readers won't understand the content on the chalkboard, unless we choose a really simple expression, such as "1+2". But that is unnecessary as the content is irrelevant, since the image seeks to illustrate the process of doing math, not its result (which wouldn't fit on a chalkboard anyways). fgnievinski (talk) 13:43, 1 December 2025 (UTC)Reply
I don't think a lead image is necessary per se, but the chalkboard image does suggest the good point that we should discuss in much more detail about the basic tools and methods used for mathematics, perhaps under § Symbolic notation and terminology or § Training and practice, or in a new section, where it can be mentioned that e.g. mathematics routinely involves graphical manipulation of symbolic notation and/or diagrams (typically with pen and paper or a chalkboard), used to represent abstract objects; that the historical format for presenting mathematical knowledge for over 2 millennia has been lectures with supporting diagrams and notation, including not only school lectures but also lectures targeted to an expert audience (e.g. today at conferences); that work is also shared through research papers, monographs, and textbooks; that mathematical training and practice throughout history has involved posing and solving of contrived problems, ranging from easy problems given as schoolwork to very difficult problems posed as a challenge to the top experts.
Our current discussion of all of these topics is non-existent or limited, with some brief mentions in passing that assume that readers already basically know what mathematics is like, as a practice, and that do a very poor job conveying it to an unfamiliar audience. We only have such examples as "In mathematics, the experimentation may consist of computation on selected examples or of the study of figures or other representations of mathematical objects (often mind representations without physical support)", "Euclid organized mathematical knowledge by way of postulates and first principles, which evolved into the axiomatic method that is used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements, is widely considered the most successful and influential textbook of all time", "many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process" but there is no further context or elaboration about any of these. The word "diagram" is not mentioned except without explanation in a couple of captions (and no synonyms like "figure" or "drawing" are used anywhere). The only mention of graphing functions comes in "Analytic geometry allows the study of curves unrelated to circles and lines. Such curves can be defined as the graph of functions, the study of which led to differential geometry. They can also be defined as implicit equations, often polynomial equations (which spawned algebraic geometry). Analytic geometry also makes it possible to consider Euclidean spaces of higher than three dimensions."
It would be appropriate to include quite a few relevant images of mathematicians doing work or of mathematical work itself in such sections, whether or not an image goes in the lead. –jacobolus(t)21:36, 27 November 2025 (UTC)Reply
The image that was removed seems to meet the requirements of MOS:LEADIMAGE. It gives the reader an indication that they landed in the right place, it's neutral and not shocking. It also humanizes a very abstract subject. Perhaps there are better lead images, but this one is at worst harmless and does not violate any policy or guideline. It should It should not have been removed without discussion. I am restoring it pending a consensus for change.--agr (talk) 22:05, 1 December 2025 (UTC)Reply
I think the concern whether a reader has "landed in the right place" seems unserious. This is the article on mathematics. Any (hypothetical, possibly non-existent) reader unsure about this is unlikely to be satisfied by an abstract martingale calculation at a blackboard. So this invocation of LEADIMAGE strikes me as unserious.
I think the concern being raised is not literally whether a reader knows they have “landed in the right place,” but whether the lead image meets the requirement in MOS:LEADIMAGE that it be representative of the subject as a whole. That is the actual guideline at issue.
For a topic as broad and abstract as mathematics, a photograph of a single person performing a very specific, concrete activity does not satisfy that requirement. Mathematics comprises logic, proof, abstraction, structure, quantity, geometry, analysis, algebra, and many other domains that have no visual resemblance to a chalkboard computation. Using an image of one practitioner performing one narrow task inevitably privileges that activity as emblematic of the field, which is precisely what LEADIMAGE cautions against.
So the invocation of MOS:LEADIMAGE here is not about hypothetical confused readers; it is about representational scope. On that criterion, the image does not meet the guideline. Tito Omburo (talk) 23:20, 1 December 2025 (UTC)Reply
I would add, to Arnold's point about not removing the image without discussion, that the image was a recent addition. It's not as though someone modified the article's stable version by removing a longstanding image; it's the reverse. --Trovatore (talk) 21:52, 2 December 2025 (UTC)Reply
I searched Wikimedia Commons for "Mathematics" and there are a few images that might work. The first one that jumped out to me was the women teaching geometry image. Failing this, maybe a famous figure from a math text.
Likely represents Sophia/logos (red robe with purple high status) Illumination of P Pythagoria Pi and phi - geometrical shapes set within circle bounded in green representing material worldGeogSage (⚔Chat?⚔) 22:50, 3 April 2026 (UTC)Reply
I am not sure why we are having this edit war. We seem to be divided about fifty fifty. Maybe we need to give more people a chance to weigh in. I note that the articles computer science and statistics both have images at the top of the page, I suspect most articles on major academic subjects do. I favor adding an image at the top of the page, and am at a loss to understand why some people object so strongly. Rick Norwood (talk) 20:38, 3 December 2025 (UTC)Reply
My view is that there are probably a lot of articles that have lead images "just because", and for which a lead image is not really useful. I don't think we should follow that trend without having a better reason than that.
A couple of other cases of similar breadth: Philosophy has a lead image but it's a statue of a thinker. I don't think that's really particularly relevant to philosophy, and I can imagine it being seen as "problematic" on cultural grounds, not that I personally pay a lot of attention to that stuff but it's not a fight I want to pick. Science has no lead image per se, though there's a stylized picture of an atom in the navbox for the series on science. I would get rid of that one too, as not sufficiently representative of science in general. --Trovatore (talk) 21:56, 3 December 2025 (UTC)Reply
There were historically two older images that were lead images being The School of Athens painting zoomed in on Euclid and a math formula (formulas are bad images since they can just be type for accessibility). EulerianTrail (talk) 04:24, 4 December 2025 (UTC)Reply
I was looking at the article and other level-1 vital topics for another discussion below. What puzzled me for a while was an image of a globe reading a newspaper which appeared at top right (pictured). This seems incongruous and I wondered whether it was vandalism or an April Fool. Clicking on it didn't seem to do anything or provide a clue what it was. After some research, my impression is that it's the baby globe mascot created for the 25th anniversary of Wikipedia and that it's some kind of easter egg. I'd like to get rid of it but I'm not sure how.
The absence of a lead image for the Mathematics article makes the matter worse visually as it then appears that the baby globe is the lead image, which looks weird.
Checking the equivalent articles in other languages such as French, German and Spanish, I find that they all have proper lead images. The English language version thus seems perverse and persnickety. Tsk.
The appearance menu (accessed from the spectacles icon or sidebar) allows the baby globe to be turned off. Having dispensed with that, I plan to try some more images. I found some good inspiration at Portal:Mathematics/Featured picture archive and mathematics and art. As a benchmark, let's start by displaying the image that was the lead picture from 2006 to 2021 – Euclid by Raphael. Andrew🐉(talk) 18:09, 3 April 2026 (UTC)Reply
Comment. This discussion seems a little unfocused, having gone through several waves of debate fairly haphazardly. I would strongly support the image File:Woman_teaching_geometry.jpg suggested above as ideal for the lede. Several reasons that come to mind are:
it shows mathematical shapes and implements that will be recognizable without overwhelming the viewer with specificity;
the activities shown are more recognizably schematic rather than narrowly specific, as the image of the blackboard filled with specialized notation was;
it presents mathematics as a human activity (particularly a social and pedagogical on) rather than a specific object, aligning with the broad scope of the article and topic;
it avoids the strong stylistic framing of Renaissance or allegorical paintings, where mathematics is often secondary to artistic or symbolic concerns;
the allegorical aspect of the illumination is philosophically relevant to the topic (in this case the Platonic interpretation) but restrained, and does not force that ontology onto the viewer;
it shows the role of teaching and transmission in mathematics;
it includes representation (in this case women in mathematics) without making that the focus;
it does not show recognizable historical figures (like Archimedes) that would unduly narrow the scope;
I went ahead and added the image suggested above. While there wasn't anything like consensus for any particular image, there does seem broadly to be consensus for an image, so I thought it useful to boldly cut the gordian knot. Sławomir Biały (talk) 07:09, 4 April 2026 (UTC)Reply
I don't find that image beautiful but it is a reasonable choice. Additional reasons include:
It comes from an edition of Euclid's Elements which is a famous foundational work of mathematics
It is antiquated and so shows the deep roots and tradition of mathematics as a formal topic
It is a featured picture on Commons and so is technically approved as excellent
Why do we need a lead image in the first place? It's not like someone will visit the page and not know it's about the field of mathematics unless there is a lead image. Again, any lead image, especially in an article about a field as broad as mathematics, will appear to heavily skew the lead to one area of mathematics Mechanikin (talk) 05:04, 8 April 2026 (UTC)Reply
This image was selected for a variety of reasons, enumerated above. Several of the reasons: it does not unduly bias to one area of mathematics, it is not overtly allegorical (and the allegory it does present subtly is one of the major philosophical views of mathematics), it includes representation in mathematics (women in mathematics), it shows mathematics as a human activity and includes one such activity prototypical of mathematical practice (teaching, but only in a fairly coded way), it is a featured image, it has historical depth. Other proposed lede images, by contrast, are either very allegorical, illustrate recognizable historical figures, or show very specialized activities such as specific calculations in probability theory. While the image is grounded in geometry, that is historically the most accessible and widely recognizable form of mathematics, and it is presented here in a schematic way rather than as a specialized subfield. This makes it less narrowly scoped than images tied to modern technical notation. Per MOS:LEADIMAGE, the lead image should be representative of the topic as a whole and not overly specific. For a subject as broad as mathematics, perfect representativeness is not achievable, so the relevant standard is whether the image is less specific and less misleading than the alternatives, and how much of the overall picture it conveys. This image ticks a lot of boxes. Sławomir Biały (talk) 05:17, 8 April 2026 (UTC)Reply
It is impossible for one to accurately generalize the intent of one who visits the article, even with specificity as low as the broad region of mathematics that is being looked for, and thus how the lead image, if it exists, will affect one's judgement on what the field of mathematics is about is and should be, in my honest opinion, of relatively low importance (accounting for the fact that any visitor of this page has, within their capacity, at least a rough idea of what mathematics is, and will not judge the field solely based on one image), thus, the excessive effort in deciding, and discussion about whether or not there should be a lead image, what the lead image should be, and the specific image(s) chosen that depict humans engaging in mathematics rather than the concept of mathematics itself (which, from a modern point of view, would be sufficed for by depicting the process of deductive reasoning and the axiomatic foundations of set theory, which all of mathematics is based on), is a signal of the article being treated as a museum specimen or a fine art piece rather than an encyclopedic article with the sole purpose of conveying academic information to those willing to learn. Mechanikin (talk) 07:40, 8 April 2026 (UTC)Reply
The image does in fact cover several of the aspects being discussed. The history of mathematics is alluded through reference to Euclid's Elements. The teaching of mathematics is depicted. Philosophical interpretations of mathematics are suggested through the allegorical elements (Sophia inscribing mathematics in the world). Foundational and exploratory aspects of mathematics are both represented through Euclidean construction and geometric reasoning. (It also avoids the familiar pattern of representing mathematics only through named male figures that is common in Renaissance art for example.) I would also dispute the characterization of "the concept of mathematics" as essentially deductive reasoning grounded in set theory. That is one important modern perspective, but it is much narrower than the scope of the article, which explicitly includes history, pedagogy, social and cultural aspects, philosophy, and a wide range of mathematical practices (including diagrammatic and geometric reasoning). A lead image should not privilege one such perspective to the exclusion of the others. Mathematics is also a human activity, and the image makes that visible without tying it to any one modern specialty. No single image can represent the whole field, but this one captures multiple dimensions of it in a way that is broader and less specialized than the alternatives discussed above. Sławomir Biały (talk) 08:34, 8 April 2026 (UTC)Reply
My feeling is that we need an image that clearly describes how mathematicians perform research and this image characterizes it better than any other. But it is not available in Commons, so I think we should keep what you already placed there. Yesterday, all my dreams... (talk) 08:47, 8 April 2026 (UTC)Reply
Lol. I like the one with "and a miracle happens", with the other mathematician saying "you might want to be more explicit in step 3". Reminds me of an anecdote a Romanian friend told me: two mathematicians are discussing a proof step; one says its trivial; the other goes off and sits in their office alone for a few hours, comes back, and says "yes, it's trivial". Sławomir Biały (talk) 09:03, 8 April 2026 (UTC)Reply
I believe this image of mathematical symbols is broad enough to function as a lead image. If we wanted to add additional symbols, then I would be willing to do so.MathematicsEulerianTrail (talk) 13:05, 8 April 2026 (UTC)Reply
Likewise, mathematics is not an Illumination P. The current image is heavily biased towards geometry. Also, if you want a women teaching mathematics and wanted to show diversity then an image a women teacher and children of both genders is better than a group of monks with a personification of a subject as a fictional women. EulerianTrail (talk) 13:27, 8 April 2026 (UTC)Reply
The point is that the lead image should reflect, as far as possible, the scope and breadth of the subject. For an article as broad as mathematics, that includes not only particular subfields, but also history, pedagogy, philosophy, cultural transmission, visual and diagrammatic practice, exploration, and deductive reasoning. No single image can literally depict all of that, so the relevant question is comparative: which image is less narrow and less misleading as a representation of the topic as a whole? On that standard, a collage of symbols smashed into a pane does not seem adequate. I'm also happy to consider your alternative image proposal. It shows a woman teaching grammar in front of a class of young children. I do not think this is a good image, because it does not illustrate the history, philosophy, cultural transmission, visual practice, or deductive method associated with mathematics. Perhaps you feel it is more suitable than it seems to me, but I think you should clarify the reasons. Sławomir Biały (talk) 13:42, 8 April 2026 (UTC)Reply
First of all, that is not clear from the image. There is a drawing of a house on the board and what appear to be letters. Secondly, that is responsive to "history, philosophy, cultural transmission, visual practice, or deductive method"... how? Sławomir Biały (talk) 13:48, 8 April 2026 (UTC)Reply
I would like to be provided an outline of the disadvantages of not having a lead image in this article. It seems that the inclusion of any lead image is the cause of much disagreement Mechanikin (talk) 13:54, 8 April 2026 (UTC)Reply
If a lead image is needed. I think it is best to give much discussion before plopping an image into the article. The current image is extremely biased to a 14th century view and geometry. The image might be useful for a history of geometry article but not of mathematics. I only shared the teaching image since it seemed to fit better your goals of a lead image than the one you placed in the article.
Would it be nice to have a lead image? Probably, but is it necessary. I think the current consensus is leaning toward image, but we have not completed this discussion.
Assume we reach consensus to have a lead image, what should it be? This discussion needs to take place, where we find an image that is both representative of mathematics and is not confusing to a reader upon landing on the page. The 14th century image is like using the lizard image for the page dinosaur. Hylaeosaurus
I think the page Dinosaur might actually have the solution we need for the mathematics article. Instead of a lead image, they have a collection of images in the infobox showing different dinosaurs. I think we should do something similar and have different mathematical topics be represented as images in the infobox: teaching, history (The School of Athens is a good choice), topology, graph theory, probability, number theory, geometry, algebra. EulerianTrail (talk) 14:22, 8 April 2026 (UTC)Reply
This is a good idea. It is also what the wikipedia page Physics does for its lead. It's not a single image, but different images showing different areas of physics. I think that without adhering to a purist view of mathematics (that is, it's nothing except deductive reasoning from axioms), deciding what a lead image should be is messy.
Either we have a lead with multiple images showing different areas of mathematics, or we decide not to have a lead image. I am thoroughly unconvinced of the idea that the current lead image (the woman teaching geometry) is in any way better than not having a lead image at all. Again, this article is an encyclopedic text, not a cultural artefact. Mechanikin (talk) 14:38, 8 April 2026 (UTC)Reply
I think the first two images would be more suited for a 'history of mathematics' and a 'mathematics pedagogy' page than this. The last image you provided is merely a symbol. Similar to the physics page (where the order of the images is from classical mechanics, to thermodynamics, to electricity, then modern physics etc. in the order of better theories), a good lead image of mathematics would start from set theory, to order theory, to abstract algebra, to mathematical analysis, to general topology, to measure theory, to probability theory, and maybe applied mathematics; in that general direction. A historical depiction of geometry loses focus of what mathematics is, and a classroom teaching mathematics focuses a bit too much on pedagogy and less on the logical hierarchy of mathematics Mechanikin (talk) 15:00, 8 April 2026 (UTC)Reply
There is mathematics education that stands as its own field in mathematics. Likewise, history of mathematics is a common subject covered in many schools, and it would be neglectful to leave out some representation for it. I really like the magma to group diagram, die, and topology mug images as recognizable and representative images. For the others, I am sure there are better images, I just put this together as a starting point. EulerianTrail (talk) 15:26, 8 April 2026 (UTC)Reply
Yeah, I don't like this idea of showing multiple images. Someone suggested an image, fine. But if there isn't a single image that gets consensus, I don't see how 5 images will: it just creates needless visual bloat. Most readers will expect an image of something. It might as well be an image that shows multiple aspects of the subject: mathematics in art, history, pedagogy, experimentation, philosophy, and reasoning. Jacobolus's image below is interesting, in that it shows mathematics in commerce, science, and the Arab world, all of which are worthy things of an image. But I think the mathematical content is too hard to discern, without very close examination. I think a wider RfC might be useful. Mechanikin's suggestion that perhaps having no image is the best solution, could also be an option there (I realize they may not hold this view, but did put it out as a suggestion). Sławomir Biały (talk) 13:24, 9 April 2026 (UTC)Reply
Interesting item about math instruments. Thanks for pointing it out. I had forgotten about it altogether. The German article mentions calculators and computers as such. Are they? Supercomputers? I will add a few "further reading" items for now given that excellent sources exist. Yesterday, all my dreams... (talk) 13:07, 9 April 2026 (UTC)Reply
I would leave out any significant discussion of calculators and computers and focus on "analog" instruments, since by the time of computers I don't think the term had much of any use. Among more recently used tools, could perhaps include nomograms, slide rules, the stereographic Wulff net, etc. But for many centuries there were competing approaches to solving many practical problems: an arithmetical approach involving pen-and-paper calculations with heavy use of table lookups and an "instrumental" approach involving measuring with dividers on various scales, etc. The "instrumental mathematics" has been substantially erased from historical summary of mathematics even though it was probably more practically important at the time and was tremendously influential on more "theoretical" mathematicians. –jacobolus(t)19:52, 9 April 2026 (UTC)Reply
Given that you obviously know more about the history of those, you should fix that page rather than instructing me how to do it. But I don't agree with the computer issue. Even recently Wolfram was pushing for that and part of the negative response to him was due to the new kind of arrogance in the presentation. And the overlap was closer than you suggest. When l was taking my 2nd year undergrad classes in math (before most of you were born) we were not allowed to use handheld calculators in exams. In fact I had never used a calculator because I could do a lot in my head in those days. But a professor got me an account on the newly installed supercomputer to test his ideas about prime numbers. So we were using supercomputer but not calculators. The computer was an instrument. I continued to use supercomputers for a few more decades and saw them as tools. Yesterday, all my dreams... (talk) 20:48, 9 April 2026 (UTC)Reply
Sorry, I'm not trying to tell anyone what to do. I do have a medium/long-term goal to improve that page and other related pages. –jacobolus(t)22:40, 9 April 2026 (UTC)Reply
16th century "mathematicians" and "astronomers" were basically the same group. Earlier as well: you can feel free to call Eudoxus, Eratosthenes, Euclid, Apollonius, Archimedes, Hipparchus, Theodosius, Menelaus, Ptolemy, etc. either "astronomers" or "mathematicians" based on personal preference, both descriptions are accurate. And for that matter this was still true to a substantial degree for a few centuries afterward as well. –jacobolus(t)19:34, 9 April 2026 (UTC)Reply
Thus, I propose to use LLMs and maybe more accurate prompts to get a better leading image. We can do minor edits on the main idea of LLMs to bypass its licences on Wikipedia. Thanks, Rais-Ali Delvari (talk) 12:18, 13 July 2026 (UTC)Reply
Can we get only a main idea from AI and then create a fully human made version of that main idea and upload and use this human version on Wikipedia? Rais-Ali Delvari (talk) 12:32, 13 July 2026 (UTC)Reply
Wikipedia's editor community has rejected the use of AI for such purposes. There really aren't any technicalities or loopholes - we should not do it. MrOllie (talk) 12:39, 13 July 2026 (UTC)Reply
Because it is fully AI generated I can not upload that (it has license). Please apply the above prompt yourself on ChatGPT or other LLMs, and wait for the results. No way! These AI-generated images are useful for acquiring some ideas to make some human-made image for the "Lead image" of this article. Rais-Ali Delvari (talk) 12:52, 13 July 2026 (UTC)Reply
This is the result of the above idea: AI created and human modified version.
The image is nice and ingenious, but not convenient. Inddeed, the first image of the article must be understandable at first glance, which is not the case here. Moreover, the first image appears often in search results in a (very) small format. The image is definitively not suited for this usage. D.Lazard (talk) 09:44, 15 July 2026 (UTC)Reply
I also don't love the idea of a lead image presenting an academic taxonomy of mathematics, and perhaps inaccurately. Why is art not included? Why is applied mathematics only a small part of it, when almost all mathematics in the world is "applied"? Why is linear algebra presented as a main branch of mathematics? Sławomir Biały (talk) 10:09, 15 July 2026 (UTC)Reply
Redundant opening sentence
Latest comment: 16 days ago35 comments8 people in discussion
The opening sentence is a little redundant :
"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."
The word isn't really defined, and to make matters worse, the sentence ends by saying that mathematics serves mathematics. I don't pretend to know better, but I think an opening sentence should put more effort into defining the word itself, and it definitely shouldn't include the word itself in its attempt at a definition.
Perhaps a little more attention needs to be devoted to this first part of the article, which is the most crucial in providing the essential information a reader wants to know. Sure, everybody knows what mathematics is, but its a hard thing to define and anyone looking for a formal definition will be disappointed in this article. Endangeredreader (talk) 19:07, 22 December 2025 (UTC)Reply
I don't really disagree with you, but you have to understand that massive effort has been put into the opening sentence. You can read about it on this talk page and its archives. Agreeing on a definition of mathematics is difficult, and people have a lot of conflicting goals for the opening sentence, and what you're currently seeing is the result of painful compromise.
I agree that this first sentence is poor. Its reference to empirical science seems to be recentism. The basics of mathematics are grounded in everyday life -- measuring and counting for practical purposes such as administration, business, crafts, trade, navigation and so forth. Let's see how it compares with the versions in other languages:
French: Les mathématiques (ou la mathématique) sont un ensemble de connaissances abstraites résultant de raisonnements logiques appliqués à des objets divers tels que les ensembles mathématiques, les nombres, les formes, les structures, les transformations, etc. ; ainsi qu'aux relations et opérations mathématiques qui existent entre ces objets.
(Mathematics (or mathematical theory) is a body of abstract knowledge resulting from logical reasoning applied to various objects such as mathematical sets, numbers, shapes, structures, transformations, etc.; as well as to the mathematical relations and operations that exist between these objects.)
German: Die Mathematik (bundesdeutsches Hochdeutsch: [matemaˈtiːk], [matemaˈtik]; österreichisches Hochdeutsch: [mateˈmaːtik];[1] von altgriechisch μαθηματικὴ [τέχνη] mathēmatikē [téchnē] „Kunst des Lernens“) ist eine Formalwissenschaft, die aus der Untersuchung von geometrischen Figuren und dem Rechnen mit Zahlen entstand.
(Mathematics (... from Ancient Greek μαθηματικὴ [τέχνη] mathēmatikē [téchnē] “art of learning”) is a formal science that arose from the study of geometric figures and calculation with numbers.)
Both seem better than the current English version. The French version lists major components while the German gives its historical origin.
I don't think the reference to empirical science is "recentist". Historically, mathematics developed in close interaction with astronomy, mechanics, physics, and other sciences, just as it also developed from practical activities such as counting, measurement, administration, and trade. So I would not want the lead to suggest that mathematics arose only from geometry and arithmetic in a narrow sense, as the foreign wikis do. That said, I agree that the present opening sentence is not ideal. I also think "formal science", while defensible, has been contentious here and is probably not the best place to start. Likewise, I do not like defining mathematics as a mere "body of abstract knowledge": mathematics is not only a collection of results, but also an activity of (among other things) reasoning, abstraction, and proof. Sławomir Biały (talk) 11:43, 4 April 2026 (UTC)Reply
A few years ago, the opening sentence of this article was more like the French one you cite, in that it listed some vague topics: numbers, shapes, etc. Mgnbar (talk) 12:20, 4 April 2026 (UTC)Reply
I’ve drafted an alternative version of the lead sentence that attempts to resolve the circularity and structural issues discussed above, and also introduces the topic more accurately:
"Mathematics is a field of study that uses logic to develop, organize, and prove the structures, relationships, and patterns of abstract objects, and uses such findings to answer abstract and practical questions."
It explains the method, the activities, the objects, the findings (structures, relationships, patterns) and the fact that these findings can be used to answer questions (which addresses Andrew's point about mathematics being grounded in everyday life).
It makes generalizations like "logic" for the method or "develop, organize, and prove" for the activities which aren't perfectly precise but are mostly accurate and do a good job of introducing the essence of the article.
I think it is certainly better than the current one, which has a circular definition and also explains mathematics using empirical sciences, which math predates.
Thanks for trying. It's pretty good. I'm not sure that it's better than what we have. It could be read as saying that abstract objects are particular to math (whereas many subjects study abstract objects), but maybe I'm alone in that complaint. It would be better with some citations to reliable sources. (Yes, I know that the current opening paragraph has no citations either.) Mgnbar (talk) 23:40, 17 May 2026 (UTC)Reply
Thanks for that, EulerianTrail. I'd forgotten. Still, if you're going to convince editors on this talk page that your opening sentence better summarizes math than others do, it is helpful to be able to say, "See, this is how math is summarized here and here and here." Mgnbar (talk) 23:57, 17 May 2026 (UTC)Reply
Thanks for your feedback, Mgnbar. That's a valid concern. It could be tweaked slightly so it doesn't suggest abstract objects are particular to math. Maybe something like:
"Mathematics is a field of study that uses logic to develop, organize, and prove the structures, relationships, and patterns of numbers, shapes, and other abstract objects, and uses such findings to answer abstract and practical questions."
It is a little more wordy but it might address that concern. What do you think? Also, what are your thoughts on the essence of the sentence, do you think it is in the right direction or should we look for a different path? Cicada419 (talk) 00:33, 18 May 2026 (UTC)Reply
I think this is a step in the right direction, but still tries to do too much at once. I think a split into two sentences, something like this:
"Mathematics is a field of study that uses abstraction, logic, and proof to investigate patterns, structures, and relationships. Its questions may arise from numbers, shapes, space, change, or from practical problems in the empirical sciences and everyday life."
I am strongly against these sentences: "pattern" is almost never used in mathematics, and does not appear in § Areas of mathematics; Newton's and Leibniz' infinitesimal calculus is a fundamental area of mathematics that does not fit in this definition; etc. IMO, this is a wrong direction, since it is a tentative to give a definition for which there is no consensus neither in Wikipedia nor outside. So, no such suggestion can satisfy WP:NPOV. D.Lazard (talk) 10:55, 18 May 2026 (UTC)Reply
I am a major contributor of this first sentence. So, I am rather satisfied by it. Nevertheless, as every compromise it could be improved.
I think that replacing "field of study" by "field of knowledge" would be an improvement, since mathematics must be learnt, and as for every knowledge, learning mathematics is difficult.
Replacing "empirical science" by "empirical practice" could be discussed for including problems of everyday life.
Also I would like for more emphasis on theorems anf proofs. Indeed it results from my experience as a referee for journals and conferences, that, when there is a doubt whether an article or a contrinution is mathematics, the main criterion is the existence of theorems and proofs. For example, the logic of mathematics became a part of mathematics, called mathematical logic when its axiomatization allowed the proof of theorems about logic (for example Gödel's incompleteness theorems). D.Lazard (talk) 11:33, 18 May 2026 (UTC)Reply
The definition in the lead is for general readers and so should use familiar, general language rather than mathematical jargon. Per MOS:INTRO, Make the lead section accessible to as broad an audience as possible.Andrew🐉(talk) 11:09, 18 May 2026 (UTC)Reply
I do not se any mathematical jargon in the current first sentence, but I would agree with linking theories and theorem. On the other hand, in the suggested sentences, "abstraction", "logic", and "structure" are unneeded technical terms, and "pattern" and "relationship" are misleading or even wrong (it is truethat mathematics studies relations, but mathematical relations are not rlationships). It is not an improper use of familiar language that makes things more accessible. D.Lazard (talk) 11:55, 18 May 2026 (UTC)Reply
Some of these suggestions are OK. But I want to address the supposed "circularity" claim. The existing first sentence is not a circular definition, because it's not a definition at all. And it isn't supposed to be a definition. Cicada419, I really want you to understand that point; it's really really important. We must never allow the first sentence to appear to be a definition of mathematics. --Trovatore (talk) 17:40, 18 May 2026 (UTC)Reply
I understand it is not a definition, but it serves a similar role as one since its goal is to introduce the topic. I think "circular dependence" would have been a better term. Regardless, the current lead sentence describes what motivates the development of mathematics which I think does not belong to the lead sentence. I'm thinking that something simpler like this may work:
"Mathematics is a field of knowledge that uses logic and formal proof to develop and study abstract structures, such as numbers, shapes, space, and change. Its findings are applied to solve practical problems in everyday life and the empirical sciences."
Or simply:
"Mathematics is a field of knowledge that uses logic and formal proof to develop and study abstract structures, such as numbers, shapes, space, and change."
I think something in those lines might be a good introduction to the topic even if it is not perfectly accurate. Thoughts? Cicada419 (talk) 17:50, 18 May 2026 (UTC)Reply
This seems better than the current lead sentence. It introduces the article better than the current one and appears to have few objectional parts. I will replace the lead sentence since this one is better.
I think it should say pattern instead of space. It is very nice that it says mathematics uses somethings and gives examples of abstract structure. It is not too narrow nor too broad. EulerianTrail (talk) 18:22, 18 May 2026 (UTC)Reply
The proposed sentence nay be a good start, but is still far from a consensus. So, I have reverted it from the article. My concerns: "formal proof" is wrong since most mathematical proofs are not fully formalized; "logic and formal proof(s)" seems redundant; "structure" seems too technical and too narrow (for example, neither probability theory nor infinitesimal calculus are the study of structures; "space" is ambiguous, since, for most readers, this refers to the physical space; etc.
I suggest the following instead: Mathematics is a field of knowledge where abstract concepts are formally defined, such as numbers, geometrical shapes, infinite sets and probabilities. The properties of these abstract concepts are studied and proved with logical reasoning, and stated as theorems.
Just for the record, I was surprised when the first sentence was changed with a claim of talk page consensus. Like D.Lazard, I didn't really think that we had consensus yet.
I tend to favor the "let's give examples" version of the opening sentence, which list some mathematical topics such as numbers, shapes, etc. In this way, I prefer D.Lazard's proposal. My only complaint is that a reader might reasonably infer that other fields don't prove theorems logically, which isn't right (CS, statistics, the more mathematical parts of physics). Mgnbar (talk) 22:01, 18 May 2026 (UTC)Reply
@D.Lazard thank you for your feedback and for taking the time to propose an alternative. I like it. Especially the terms “abstract concepts,” “properties,” "logical reasoning," and the examples you have given.
I think it could be made a little more concise and readable though. Also, I am not a fan of introducing theorems in the second sentence, as they are fundamental but need explanation. I think the second paragraph does a good job of introducing them.
I’ve made an alternative that may be an improvement:
Mathematics is a field of knowledge that uses logical reasoning to study and prove the properties of abstract concepts such as numbers, geometrical shapes, infinite sets, and probabilities.
Lazard's proposal above is not bad, but gives undue weight to what I might call research mathematics. Most mathematics in the world in not research mathematics, probably not even what most mathematicians would call applied mathematics. I would propose something like this Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometrical shapes, sets and probabilities. These concepts are defined, calculated with, modeled, compared, and studied using logical reasoning, symbolic notation, and proof. Mathematical results are often stated as theorems, while mathematical methods are used to model and solve problems in the natural sciences, engineering, economics, technology, and everyday life.Sławomir Biały (talk) 05:30, 19 May 2026 (UTC)Reply
I think this is a step in the right direction. I like the idea of limiting the first sentence to introducing what mathematics is concerned with. I also think symbolic notation is important. I would make some refinements to improve conciseness, readability, and also explain that mathematics is concerned with the properties and structures of abstract concepts and not only the concepts themselves. Here is my revised version:
"Mathematics is a field of study that is concerned with abstract concepts such as numbers, geometrical shapes, sets, functions, and probabilities. It explores their structures and properties using logical reasoning, symbolic notation, and proof. Its findings are essential to the empirical sciences, engineering, and technology, and useful in everyday life."
Except for the examples, the beginning applies to other fields such as philosophy and physics, where logical reasoning is applied to abstract concepts. So, one must emphasize on aspects that distinguish mathematics, which are formal definitions, proofs (plural not singular) and theorems.
I would like to mention functions in the exemples, but I did not, because I do not know how doing this is a non-technical and non-confusing way ("function" has many meanings, and the mathemaical one is probably not the one that most readers have in mind).
"Structure" is redundant, since, in this sentence "structure refers to some kind of poperty.
"Symbolic notation" seems too technical. "Formulas and equations" would be clearer to most readers, but are not easy to include in the first sentence. Howeverm, in my proposal, one can replace "stated as theorems" with "stated as theorems, formulas and equations".
IMO, the last sentence would be very good, with "its finding are" replaced with "mathematics is". So, my new proposal is Mathematics is a field of knowledge where abstract concepts are formally defined, such as numbers, geometrical shapes, infinite sets and probabilities. The properties of these abstract concepts are studied and proved with logical reasoning, and stated as theorems, formulas and equations. Mathematics is essential to the empirical sciences, engineering, and technology, and useful in everyday life.D.Lazard (talk) 10:29, 19 May 2026 (UTC)Reply
I don't think "function" is unduly technical, any more than "set" or "probability" is, and it certainly belongs here. Second, I think the emphasis is still too much on theorems and proofs. These are one very important mode of mathematics, but they leave out a lot of mathematical practice. I also don't think "defined" is really the right verb for the first sentence; "concerned with" seems fine to me. Regarding physics and the other sciences, mathematics is certainly used there, and I don't think we should exclude mathematical methods in the sciences just because they also belong to other fields: they are still mathematics. Saying that mathematics is "essential" in other fields also elides an important aspect of mathematics: it is used for modeling and problem-solving (these constitute a huge proportion of actual use-cases of mathematics in the world). So I would prefer something more like this: Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometrical shapes, sets, functions, and probabilities. These objects and concepts are defined, calculated with, and studied using logical reasoning and proof. Mathematical results are often stated as theorems, formulas, or equations. Mathematical methods are used to model and solve problems in the natural sciences, engineering, economics, technology, and everyday life.Sławomir Biały (talk) 11:29, 19 May 2026 (UTC)Reply
I like Slawomir's version of the first sentence. Given the years of debate I think it is better to focus on neutrality over precision. It doesn't exclude philosophy and physics but the examples orient the reader in the right direction.
I also like Lazard's emphasis on Mathematics being interested in the properties of abstract concepts rather than simply the concepts themselves. I am thinking that something in these lines may balance everything well:
Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometrical shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties. Its results are often stated as theorems, formulas, and equations. Mathematics is used to model and solve problems in the natural sciences, engineering, technology, economics, and everyday life.
I think it balances accuracy and clarity while remaining neutral.
Yes that seems pretty good. Assuming this get's general "pretty good" assent, I would go forward with this. I think it could replace the first sentence, keeping the next sentence to cover the main areas. Sławomir Biały (talk) 04:33, 20 May 2026 (UTC)Reply
I agree with this version. However, I suggest to merge the second and third sentences into:It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. D.Lazard (talk) 10:14, 20 May 2026 (UTC)Reply
With or without this change, I would propose first to start a clean thread at the bottom as a more targeted discussion/informal rfc to gauge local consensus. Sławomir Biały (talk) 10:39, 20 May 2026 (UTC)Reply
This one is nicer to read. It attaches theorems/formulas/equations to 'properties' though. I am not sure if attaching them to 'results' would be more or less accurate. The four sentence version has the benefit of being easier to tweak by adding nouns or verbs, if future editors want that. Any thoughts on that?
These are all minor details though. The important thing is that we are starting to reach consensus. If no further comments are made I will make a new thread asking for opinions, like Slawomir suggested. Cicada419 (talk) 21:21, 20 May 2026 (UTC)Reply
There is a clear consensus that we have now a version of the beginning of the lead that is much better than the current one. So, it is time to close this long thread, implement this new version, and open a new thread for further discussions. So, please, add new comments on the beginning of the lead in § New first sentence(s). D.Lazard (talk) 10:09, 22 May 2026 (UTC)Reply
Lead paragraph is awful
Latest comment: 1 month ago6 comments5 people in discussion
Im a seasoned mathematician. I have absolutely no idea what mathematics is. That paragraph feels like it was written by a math professional and is overly precise ~2026-24027-33 (talk) 23:14, 18 April 2026 (UTC)Reply
Yep, there has been a lot of argumentation over the first paragraph, and what you're currently seeing is the meta-stable result of compromise. Feel free to propose a better paragraph on this talk page, supporting your text with Wikipedia:Reliable sources. Regards, Mgnbar (talk) 00:00, 19 April 2026 (UTC)Reply
I also wonder whether the parenthesis in the second paragraph can be removed. Mainly, because there is already a section for each of them under the "Areas of Mathematics".Fishes3 (talk) 20:21, 22 June 2026 (UTC)Reply
Please, read WP:EXPLAINLEAD: In short the lead must be accessible to the most general audience and should, as far as possible, be understandable without following links or access to sections. A short definition of the main areas of mathematics is therefore useful in the lead, at least for people with a low knowledge of mathematics. D.Lazard (talk) 08:03, 23 June 2026 (UTC)Reply
Mathematics is a field of study that discovers and organizes methods, theories, and theorems that are developed and proved either in response to the needs of empirical sciences or the needs of mathematics itself.
with
Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometrical shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in the natural sciences, engineering, technology, economics, and everyday life.
While implementing this new beginning, I remarked that there is some redundancy with what follows. IMO, this is not a problem, since these new first sentences may be viewed as a less technical versionsummary of the remainder of the lead. D.Lazard (talk) 11:01, 22 May 2026 (UTC)Reply
Latest comment: 2 months ago19 comments5 people in discussion
The use of "geometrical" here seems redundant: all shapes are geometric by definition. It comes off as unnecessary padding: "Mathematics is a field of knowledge concerned with abstract concepts such as numeric numbers, geometrical shapes, collective sets, mapping functions, and statistical probabilities." Or, even there are somehow shapes not considered "geometric", then it's still unnecessary qualification, since any definable shape can be studied in mathematics.
I didn't see a problem removing geometrical, but can see a case for leaving it in. It emphasizes the geometry of shapes, rather than just shapes as things in their own right. Sławomir Biały (talk) 16:05, 22 May 2026 (UTC)Reply
I'm so confused... This is a discussion about the redundancy of an individual word. I'm trying to be productive here. Is there a better way I could have asked that question? – Farkle Griffen (talk)16:21, 22 May 2026 (UTC)Reply
I am not terribly interested in pursuing pedantry here, so if you feel inclined to respond in a similar argumentative vein, please withhold. Shape may or may not refer to geometry. Children play with shapes that also have color. In ordinary usage, shape and mean visual form, outline, physical configuration, or metaphorical structure. The point is that "geometrical" is useful as emphasis, even if it might be redundant in some opinions. Sławomir Biały (talk) 16:23, 22 May 2026 (UTC)Reply
As far as I understand, a shape is a property of a physical object or of its graphical representation (see Shape), and a geometrical shape is the mathematical abstraction of this empirical concept. A circle is a geometricalal shape, but is not the shape of any physical object. Since this is an article about mathematics, "geometrical" is useful for disambiguating between the empirical and the mathematical meanings, which are distinct, although strongly related. D.Lazard (talk) 17:21, 22 May 2026 (UTC)Reply
I suppose that's fair in terms of the distinction, but I'm not sure the qualification is helpful. Nearly all of the objects mentioned have more distinct counterparts. For example, functions in computer science are not the same as functions in mathematics, and "functions" is also used to mean "the roles of an object or process" (see Function (biology)). We don't correct this by attaching qualifiers to each term, nor do I think we should. It would only add bloating with very minimal added clarity. Same reasoning for "geometrical". – Farkle Griffen (talk)17:33, 22 May 2026 (UTC)Reply
It is the reason for which, in my first version of the sentence, I omitted "function", and I was talking of "infinite sets" rather that "sets". I accepted the change because it is important to mention functions even if seems impossble to clarify the meaning without destroying the flow of the sentence. Similarly, there is few harm with the ambiguity of the word "set", while readers can be confused by the question 'why "infinite sets" and not simply "sets"'?. On the other hand, "geometric shape" is a vey commom phrase, andI'll change "geometrical shape" into "geometric shape". D.Lazard (talk) 18:32, 22 May 2026 (UTC)Reply
I don't think this really addressed my point. Why is the ambiguity of the words "set" and "function" okay, but not "shape"? I think the reader can reasonably assume what we mean. The relative popularity of the term is not really my issue. – Farkle Griffen (talk)18:57, 22 May 2026 (UTC)Reply
It wasn't meant to be a discussion, it was a minor copyedit to reduce redundancy. It was @EulerianTrail who insisted it needed a discussion (who also hasn't contributed to the discussion themselves).
@Farkle Griffen I am not on wikipedia 24/7, so I can only contribute when I go on the site. Sorry for any inconvenience.
A quick google search shows a few meanings of non-geometric shapes [1][2]; although I do not think there is a standard definition. Looking at the wiki page for shape, chirality, orientation, and size are independent of a geometric shape, but most lay people would probably say a diamond vs square, left handprint vs right handprint, and similar examples are different shapes. So, geometric adds some value of clarity.
In doing the revert, I was pointing out that consensus was just made on this sentence and there is already a talk section above titled "New first sentence(s)" that is for discussing improving the wording. If it was in the main body, I would not be so worried about a "copy edit" but since it is changing the lead sentence without consensus, right after consensus on the new version was made, I reverted it. I am not too attached to the wording, but I think it is wise to discuss changes to significant parts of the article on the talk page first. EulerianTrail (talk) 02:52, 23 May 2026 (UTC)Reply
I find reasonable Farkle Griffen's argument that the word "shape" alone should be clear from context, given that any "set" and "function" alone are unambiguous in this context. I will remove "geometric" as there's not enough people seeming to agree that it needs to be there Mechanikin (talk) 17:56, 23 May 2026 (UTC)Reply
"Geometric shape" is a very common phrase (more thn 6 million hits in Scholar Google), that is correctly understood by almost everybody. On the other hand I am not sure that most people understand well whether a shape is the same thing as a geometric shape (this is shown by the above discussion). The first paragraph is intended for the most general audience, and "geometric shape" is definitively more adapted for this audience. D.Lazard (talk) 19:07, 23 May 2026 (UTC)Reply
Should "the natural sciences" in the first paragraph be replaced by "science"? You reverted my edit for that, but I don't see why it should be restricted to natural science. Social and behavioural science also use mathematics Mechanikin (talk) 19:24, 23 May 2026 (UTC)Reply
Latest comment: 2 months ago2 comments2 people in discussion
There were two wikilinks in he first paragraph that duplicated links of the third paragraph. I removed them since things are more detailed there and links are more useful there. However this is a personal choice, and further discussion may be useful. D.Lazard (talk) 19:47, 23 May 2026 (UTC)Reply
Latest comment: 2 months ago4 comments4 people in discussion
There is already a section on the talk page for lead image, but it's getting too clogged for certain questions to be adequately noticed and addressed.
After reading through most of the replies, there are a few questions that I don't see being appropriately addressed.
In particular, does there need to be a lead image at all?
And why the current lead image? Some people say it covers different areas of mathematics, but the point is that it is not at all clear from the image. The image caption says "Geometry is shown personified as a woman", but all I see is a woman holding a compass and a straightedge and drawing some shapes, which would be great for portraying geometric constructions, but not geometric proofs, nor any process used in modern mathematics beyond geometry.
For reference, Sławomir Biały gave the following reasons for this lead image:
— — —
"it shows mathematical shapes and implements that will be recognizable without overwhelming the viewer with specificity;
the activities shown are more recognizably schematic rather than narrowly specific, as the image of the blackboardfill;
it presents mathematics as a human activity (particularly a social and pedagogical on) rather than a specific object, aligning with the broad scope of the article and topic;
it avoids the strong stylistic framing of Renaissance or allegorical paintings, where mathematics is often secondary to artistic or symbolic concerns;
the allegorical aspect of the illumination is philosophically relevant to the topic (in this case the Platonic interpretation) but restrained, and does not force that ontology onto the viewer;
it shows the role of teaching and transmission in mathematics;
it includes representation (in this case women in mathematics) without making that the focus;
it does not show recognizable historical figures (like Archimedes) that would unduly narrow the scope;
it is beautiful without being distractingly so."
— — —
I would address some of these points the following way:
"it shows mathematical shapes and implements that will be recognizable without overwhelming the viewer with specificity"—The only mathematical thing I see in the image is geometry, which, as portrayed here, is highly specific.
"it avoids the strong stylistic framing of Renaissance or allegorical paintings, where mathematics is often secondary to artistic or symbolic concerns"—It has its own stylistic framing, that is far from what one would think of as modern mathematics.
"it shows the role of teaching and transmission in mathematics"—It doesn't. It only shows students looking, with awe, at a woman drawing shapes. This is what an interested classroom may look like when a teacher is teaching, but it does not necessarily show the role of teaching.
"it is beautiful without being distractingly so"—This is subjective, but to my eyes, it is distracting to use a mediaeval-style painting of people wearing medieval clothes that only portrays mathematics restricted to drawing shapes. I'm also not sure what one would mean by saying the image is "beautiful". Why does it need to be beautiful?
— — —
I don't know of any lead image that addresses all these points and addresses all the points that the current lead image addresses, but I think it's better to not have any lead image at all than to have the current lead image. Consensus is needed to remove the current lead image. Mechanikin (talk) 13:24, 24 May 2026 (UTC)Reply
Agree, I don't think any 1 image could represent mathematics because it's a very broad field. And a symbol wouldn't be helpful for readers. UTC16:07, 24 May 2026 (UTC)Reply
Physics lead image is actually a collection of images, which appears to be the best way to satisfy the lead image for mathematics. I believe that there should be a lead image as it aids in showing different math concepts from the start of the article as well as aids in showing immediately what article you land on. EulerianTrail (talk) 16:38, 24 May 2026 (UTC)Reply
It seems to me that these are all arguments against a lead image at all. An image may not show transmission of mathematics, may show a different area of mathematics, may show historical figures rather than mathematics per se, may not have the Platonistic allegorical content of Sophia inscribing mathematics into the world, may not be from the most famous mathematical work in existence, or may not be a featured image. And any one of these would be a reason to prefer this image to those. I am strongly opposed to a mashup-style image as proposed above. I think the best thing would be for an RfC. Present a few options, including no image, this image, and whatever other images have been proposed. Sławomir Biały (talk) 16:52, 24 May 2026 (UTC)Reply
Subfields???
Latest comment: 2 months ago6 comments4 people in discussion
One sentence begins as follows:
"Calculus, consisting of the two subfields differential calculus and integral calculus ...".
But to label "differential calculus" and "integral calculus" as "subfields of calculus" is just like labeling "history of even-numbered years" and "history of odd-numbered years" as "subfields of history". ~2026-28336-51 (talk) 17:01, 31 May 2026 (UTC)Reply
Differential calculus and integral calculus are entirely distinct fields. Differential calculus is about rates of change, integral calculus is about accumulation. There is a theorem, called the fundamental theorem of calculus, which links the two.
The division of calculus into differential calculus and integral calculus is not even remotely arbitrary, as the two fields are completely independent of each other. I personally don't think they should belong under the common umbrella of "calculus". However, the division of history into "history of even-numbered years" and "history of odd-numbered years" is exceptionally arbitrary. Mechanikin (talk) 18:54, 31 May 2026 (UTC)Reply
I'm sorry, Mehcanikin, but I think that's a pretty silly take. When you're using calculus in any way other than just answering questions like "differentiate this" or "integrate that", you're mixing them up all the time. Once you get to the point that calculus is actually good for something other than passing a class, there is no separation. I agree with the temp account; they are not "subfields". --Trovatore (talk) 19:19, 31 May 2026 (UTC)Reply
Differential calculus: derivatives, partial derivatives, directional derivatives, gradient, differential equations; basically anything relating to instantaneous rate of change.
Integral calculus: Riemann integral, line integral, surface integral, volume integral, vector field flux; basically anything relating to accumulation, that is, the continuous analog of summation.
These are different tools that are almost always used together. Knowing one without the other is of very little value. Basically it's all one subject. --Trovatore (talk) 19:33, 31 May 2026 (UTC)Reply
Latest comment: 2 months ago2 comments2 people in discussion
The subfield of geometry listed in the article as "Differential Geometry" (the study of differentiable curves and surfaces, etc.) should instead be correctly labeled as "Differential Topology".
That whole list is a mess. For example, differential topology (and geometry) deal with manifolds, which listed separately. And projective geometry is super-common in algebraic geometry, which is listed separately. Mgnbar (talk) 20:48, 31 May 2026 (UTC)Reply
Sentence regarding deduction vs experiment
Latest comment: 1 month ago10 comments3 people in discussion
The recent sentence in dispute:
Although mathematics is widely used to model empirical phenomena, its results are established by deductive proof rather than by experiment.
I understand the point this is trying to make, but I think the way it is expressed is not fully neutral without qualification. Proof is absolutely a key tool in the establishment of mathematical results but it is not necessarily the only one. Axioms in particular are not necessarily arbitrary and their choice arguably has an empirical component.
I think this sentence, read literally and uncritically, is a bit biased towards Euclidean foundationalism and even formalism, and we should fix that. I haven't figured out the optimal fix. I would be happy enough just removing the sentence but there is probably some version of it that I would agree had a net positive value. --Trovatore (talk) 22:30, 13 June 2026 (UTC)Reply
I put that in because the prior edit removed an earlier, even stronger sentence, without discussing the removal. Personally, I find the concern that it is biased towards foundationalism a bit overwrought. It is just saying that mathematical results come from deductive proof rather than experiment, and I think that is a pretty uncontroversial view regardless of where one stands on foundations: it specifies the activity that leads to mathematical results, without taking a position on their truth as such. But these two sentences on foundations feel a little out of place in context, and like unnecessary nit-picking there. It might be better to end the preceding paragraph with a sentence about empiricism versus mathematical "truth", assuming that can be done neutrally. Sławomir Biały (talk) 10:52, 14 June 2026 (UTC)Reply
At a quick glance I think you might be onto something in terms of the discussion living more naturally in the third paragraph than the fourth. I wouldn't put it as empiricism versus mathematical truth, though, since my point is that empiricism is sometimes in fact used to discover mathematical truth. --Trovatore (talk) 16:46, 14 June 2026 (UTC)Reply
For arguing that the sentence is not correct, one must provide a single mathematical result that was not obtained by a proof (axioms and conjectures are not results). However, I would not oppose to add a sentence such as However, empirical and mental experiments are widely used in mathematics for suggesting conjectures, finding methods of proof, choosing useful axioms, etc.
I would also suggest to replace "deductive proofs" with "logical proofs". Indeed, I would not call a proof by contradiction a "deductive proof". D.Lazard (talk) 12:05, 14 June 2026 (UTC)Reply
The parallel postulate has been "postulated" or chosen because, with it, geometry is a good abstraction of the perceveid reality. It is not true on the surface of Earth, nor in Universe, as shown by general relativity.
See, here you're coming at me with an explicitly Euclidean-foundationalist view. It's a common one, but it's not the only one.
The way I would put it is that the axioms of infinity and choice are true, understood in the structure in which they are meant to be interpreted, which is the von Neumann universe. You can get models of their negations, and sometimes those models are of great interest, but they are not the von Neumann universe, which is well-specified up to a canonical isomorphism, and which can be described in an intuitively clear way without starting with axioms. --Trovatore (talk) 18:39, 14 June 2026 (UTC)Reply
Continuing discussion
There has been a recent edit to the lead, which I do not think is an improvement. I have reverted that edit, but it does expose a real issue with the old lead that the paragraph on applications is already a bit redundant with a sentence of the first paragraph and, following recent edits, is a bit muddled in scope now. It seems to do no real work and I think could safely be removed from the lead. The only question to my mind is whether the pair of sentences on mathematics vs empiricism vs reality should be added to the end of the third paragraph. Sławomir Biały (talk) 06:19, 5 July 2026 (UTC)Reply
I would agree to remove the sentence "Mathematics is essential in the natural sciences, engineering, medicine, finance, computer science, and the social sciences" from the 4th paragraph, since it is essentially the same as the last sentence of the first paragaph, except for a MOS:SEAOFBLUE. My opinionis that the remainder of the 4th paragraph should not be merged with the 3rd one: the 3rd paragraph is about the internal "behaviour" of mathematics, while the 4th one is about the relationship between mathematics and the external world. D.Lazard (talk) 10:43, 5 July 2026 (UTC)Reply
"Statistics and probability" as areas of mathematics
Latest comment: 1 month ago7 comments7 people in discussion
There are many areas of mathematics, including number theory (the study of integers and their properties), algebra (the study of operations and the structures they form), geometry (the study of shapes and spaces that contain them), analysis (the study of approximating continuous changes), and set theory (presently used as a foundation for all mathematics).
I do agree that we should make some effort, to have the Mathematics article and the Math topics TOC template agree.
Any claim that statistics is an area of mathematics is highly contentious. In my experience, statisticians strongly disagree, and mathematicians somewhat disagree.
Probability is certainly regarded as an area of mathematics. In the past century or so it has become a branch of analysis, although this view de-emphasizes the combinatorial aspects. This always happens when we try to categorize topics that overlap and interact.
I would be inclielned to include at least probability, and lean against statistics. (Fwiw, I think the math topics template serves no useful purpose, certainly not in this discussion.) I do not think the distinction between pure and applied mathematics is worth making in the lead. Most areas of mathematics are both "pure" and "applied", the question being one of purpose, not subject-matter, and the lead already does foreground applications of mathematics to the sciences and other areas. I don't pure/applied is a useful taxonomy, having more to do with the institutionalization of mathematics than with a lead-worthy distinction. Sławomir Biały (talk) 13:25, 24 June 2026 (UTC)Reply
"This sentence ... lacks many important areas." True and explicitly said by the sentence itself: There are many areas of mathematics, including ... (emphasis is mine). So the problem is : are there other mathematical areas that are sufficiently important for being listed here? My personal opinion is no, but, in any case, if any other area should be listed in this sentece, it should appear as one of the main subsections of § Area of mathematics. This is not the case, although § Statistics and other decision sciences appears as a subsection of § Specific sciences, and probability theory is cited as a subarea partly of § Discrete mathematics and partly of § Probability and analysis.
Also, presenting in the lead the classification into pure and applied mathematics would contradicts the opinion of most present mathematicians, which is summarized in § Pure and applied mathematics as In the present day, the distinction between pure and applied mathematics is more a question of personal research aim of mathematicians than a division of mathematics into broad areas. D.Lazard (talk) 13:52, 24 June 2026 (UTC)Reply
I don't agree that separating pure and applied mathematics in the lead is a good idea, though it is good that these common terms are explained later in the article. Regarding the sentence in question, "the study of approximating continuous changes" is a terrible summary of analysis. Only a fraction of analysis matches that. It would be better to have something like the study of functions, feel free to suggest something better. McKay (talk) 05:19, 1 July 2026 (UTC)Reply
I think it's hard to argue that statistics isn't at least an area of applied mathematics. Very very roughly probability:statistics :: pure:applied. It's absolutely true that statisticians deal with questions that are not typical concerns of pure mathematicians. Just as an off-the-cuff example, they might learn by experience that when comparing two drugs to see which one is better at prolonging survival, it works best if you apply these experimental designs and statistical tests rather than those other ones rather than trying to formulate that as a theorem, and pure mathematicians don't think that way, or at least they tell themselves they don't. But applied mathematicians unabashedly do that sort of thing all the time, and this is not the pure mathematics article. So I think stats deserves a reasonably prominent mention. But probably not in the first paragraph, if that's all we're discussing here (not sure I quite understood exactly what we are discussing). --Trovatore (talk) 05:53, 1 July 2026 (UTC)Reply
The intersection of statistics and mathematics is nonempty but it does not include all of statistics. (The same is true for computer science and physics, although the overlap may be greater for statistics.) ~2026-28744-62 (talk) 14:11, 3 July 2026 (UTC)Reply
Hyperlinks missed in the first paragraph
Latest comment: 1 month ago7 comments5 people in discussion
Hi, in my opinon, in this sentence:
Mathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
these items need hyperlink:
abstract concepts
numbers
geometric shapes
sets
functions
probabilities
logical reasoning
proof
theorems
formulas
equations
model
science
engineering
technology
economics
Please note that we want to introduce the concept of «mathematics» to someone who knows nothing about it; thus, all the concepts mentioned above might have hyperlinks. Thanks, Rais-Ali Delvari (talk) 09:42, 1 July 2026 (UTC)Reply
I agree with some of these, but I think we should be selective. I would remove the link to knowledge, and probably not link some of these (e.g., abstract concepts and logical reasoning), because too many links hinders readability. Sławomir Biały (talk) 09:59, 1 July 2026 (UTC)Reply
Right. Linking all of these would produce a sea of blue, in which it was difficult to tell where one link ended and another began. For additional guidance see MOS:OVERLINK (although it does not forbid most of these proposed links). Mgnbar (talk) 12:14, 1 July 2026 (UTC)Reply
Do you really someone who knows nothing of mathematics and is able to access to this article and follow links in it? I doubt that such people exist. More seriously, MOS:OL says A good question to ask yourself is whether reading the article you're about to link to would help someone understand the article you are linking from. Most of the suggested links would not be helful, either, such as "number", because people know sifficiently of them for understanding the sentence, or because the target is much too technical for the first sentence of this article; see the first sentences of concept and function). The few terms that could need explanation are better explained in the other paragraphs of the lead.
The Manual of Style consists only of guidelines, that reflect consensus of Wikipedia editors. If there is a consensus for not following them in a specific case, this is this consensus that prevails (see also WP:IAR). Here there is a consensus of three aditors against your request. D.Lazard (talk) 11:27, 2 July 2026 (UTC)Reply
Latest comment: 1 month ago4 comments4 people in discussion
Hi, @D.Lazard @Trovatore. To provide a good definition for mathematics, I propose to exclude some areas of knowledge that are not mathematics. For example a sentence like this:
Mathematics is in contrast to sciences which deal with concrete objects.
or
Mathematics do not deal with human emotion or language.
is really helpful for introducing mathematics.
Please see
Just no. A long long time ago there used to be a "what mathematics is not" section. Luckily it's long gone. It makes me too upset to think about it but if you like you can probably find my comments on that in the talk archives. --Trovatore (talk) 16:34, 1 July 2026 (UTC)Reply
Semi-protected edit request on 5 July 2026
Latest comment: 1 month ago2 comments2 people in discussion
This edit request has been answered. Set the |answered= parameter to no to reactivate your request.
Not done:MOS:OVERLINK. There have been many discussions about wikilinking in the first paragraph (1, 2, 3, 4) and this seems to be a controversial topic. Currently, it seems like most editors agree on unlinking 'knowledge'. jolielover♥talk04:09, 5 July 2026 (UTC)Reply
Greek origin of the word "Mathematics"
Latest comment: 14 days ago5 comments3 people in discussion
Hi, I propose to promote the first line by this footnote:
From the Greek μάθημα (máthema), meaning "science, knowledge, or learning"
Yielding:
Mathematics[1] is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities.
@Jacobolus Hi and thanks. In my opinion an abstract "footnote version" is really helpful for the first line. In simple English article here, it is correctly mentioned, making the article more understandable. Rais-Ali Delvari (talk) 07:29, 26 July 2026 (UTC)Reply
I agree with Jacobolus: We may safely suppose that most readers have already heard of mathematics, and come here for knowing more on the topic. Since the meaning of the word is not exactly the same as in Ancient Greece, etymology is a minor aspect of the topic that needs not to be emphasized in the lead when there is no place in the lead for much more important aspects. D.Lazard (talk) 10:51, 26 July 2026 (UTC)Reply
Pedagogy began with pedagogy teaching of Jesuits
Latest comment: 2 days ago2 comments2 people in discussion
@D.Lazard you removed part of a sentence stating that the formal instruction in pedagogy began with members of the Jesuits. You state that you did this since it was unsourced in your edit summary. But it was sourced, the source in reference is currently labeled 201: Jones, Phillip S. (1967). "The History of Mathematical Education". The American Mathematical Monthly. 74 (1). Taylor & Francis, Ltd.: 38–55. doi:10.2307/2314867. JSTOR 2314867. EulerianTrail (talk) 17:16, 7 August 2026 (UTC)Reply
Yes, Jesuits seem to be the first to teach pedagogy of mathematics. But this section is about mathematical education, not pedagocical education. SO, the sentence about Jesuits is confusing and out of scope. IMO, it should be removed. Nevertheless, I'll not do this, and, as a compromise, I'll edit the sentence for making clear that it is about the formation of future teachers. D.Lazard (talk) 21:03, 7 August 2026 (UTC)Reply
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EulerianTrail (talk) 14:45, 8 April 2026 (UTC)
.jpg)
Agree, I don't think any 1 image could represent mathematics because it's a very broad field. And a symbol wouldn't be helpful for readers. UTC 16:07, 24 May 2026 (UTC)
Mathematics Portal
Not done: MOS:OVERLINK. There have been many discussions about wikilinking in the first paragraph (1, 2, 3, 4) and this seems to be a controversial topic. Currently, it seems like most editors agree on unlinking 'knowledge'. jolielover♥talk 04:09, 5 July 2026 (UTC)
