Talk:Mathematics/Archive 17

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The Definition of Mathematics in the First Sentence Doesn't Define Mathematics

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)

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)

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.

we have "discovers and organizes" so no need for "developed and proved". Hooman Mallahzadeh (talk) 14:05, 17 November 2025 (UTC)

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)

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.

Hooman Mallahzadeh (talk) 15:40, 17 November 2025 (UTC)

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)

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)

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)

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)

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)

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, Hooman Mallahzadeh (talk) 11:55, 18 November 2025 (UTC)

Mathematics is the study of patterns and the relations within and between them.

I propose this as a first sentence. ~2026-30641-45 (talk) 14:03, 21 May 2026 (UTC)

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)

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)

Including mathematical object in the definition

@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)

It's mentioned and wikilinked from the beginning of the second paragraph. –jacobolus (t) 15:42, 19 November 2025 (UTC)

@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.

What is your opinion? Is that good? Hooman Mallahzadeh (talk) 16:41, 19 November 2025 (UTC)

No, this is very bad. Mathematics isn't a formal system. Tito Omburo (talk) 17:22, 19 November 2025 (UTC)

@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)

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

Mathematics consists mainly of discovering and proving by pure reasoning properties (theorems) of abstractions of reality, of models elaborated by empirical sciences, and of previously established mathematical theories.

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)

This formulation is fine with me, maybe modulo swapping out "elaborated by" for a more suitable construction. Tito Omburo (talk) 17:26, 20 November 2025 (UTC)

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)

I can see what this is going for, but I don't think it succeeds. "Field of study" isn't a weasel word, but "mainly" is; the parenthetical "(theorems)" is awkward. Stepwise Continuous Dysfunction (talk) 21:18, 20 November 2025 (UTC)

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)

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)

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)

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)

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)

Wikilinks in the first paragraph

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)

I think the MOS recommendation is to link the first occurrence but not the second. I myself am neutral on the question here. Sławomir Biały (talk) 19:50, 23 May 2026 (UTC)

"Geometrical shapes"

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.

@EulerianTrail, you asserted "there are a few different types of non-geometric shapes" in your revert. Do you mind clarifying that statement? – Farkle Griffen (talk) 16:02, 22 May 2026 (UTC)

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)

"It emphasizes the geometry of shapes..." Do shapes have properties outside of their geometry? – Farkle Griffen (talk) 16:12, 22 May 2026 (UTC)

I am not interested in pedantry. Thanks! Sławomir Biały (talk) 16:18, 22 May 2026 (UTC)

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)

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)

@Sławomir Biały: please see your talk page – Farkle Griffen (talk) 17:13, 22 May 2026 (UTC)

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)

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)

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)

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)

For what it's worth, I don't think it's worth anyone litigating over "geometric(al) shape". It's just too marginal. Who cares? Sławomir Biały (talk) 19:00, 22 May 2026 (UTC)

Me neither.

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).

But this is Wikipedia. And dealing with bureaucracy is part of what we signed up for. – Farkle Griffen (talk) 19:07, 22 May 2026 (UTC)

@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 ; 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)

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)

"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)

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)

That change seems good (and I think Lazard agrees). Sławomir Biały (talk) 19:35, 23 May 2026 (UTC)

I reverted my revert of this particular change. D.Lazard (talk) 19:51, 23 May 2026 (UTC)

Woman teaching geometry?

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)

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)

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)

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)

Subfields???

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)

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)

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)

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.

I don't see how there is zero separation between them. Mechanikin (talk) 19:30, 31 May 2026 (UTC)

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)

Yes, describing them as two distinct "subfields" is strange. I would change the text right now, if I had a better wording that still managed to mention Differential calculus and Integral calculus. Mgnbar (talk) 20:47, 31 May 2026 (UTC)

Subfields of geometry

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".

The current listing of "Riemannian Geometry" itself covers most of the field of Differential Geometry, in any case. ~2026-28336-51 (talk) 17:07, 31 May 2026 (UTC)

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)

Lead paragraph is awful

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)

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)

Are the parentheticals even necessary? I feel like that's the point of the links to those fields. ~ oklopfer (💬) 00:11, 19 April 2026 (UTC)

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)

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)

Thanks, that now makes perfect sense. Fishes3 (talk) 19:15, 23 June 2026 (UTC)

New first sentence(s)

There are several recent sections about tthe first line of the lead: § The Definition of Mathematics in the First Sentence Doesn't Define Mathematics, with two subsections, and § Redundant opening sentence. At the end of the latter, a consensus emerged to replace

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.

I'll install this new beginning, and I suggest to add here any new comment on the beginning of the article. D.Lazard (talk) 10:37, 22 May 2026 (UTC)

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)

Looks much better than the earlier version. Sławomir Biały (talk) 11:15, 22 May 2026 (UTC)

Thank you so much for pushing this over the finish line! Count me among those who found the old lead paragraph to be terrible and regretted not being able to get any traction. Moving from a definitional perspective to a descriptive perspective and grounding in reality is so much better. Lauciusa (talk) 19:41, 30 June 2026 (UTC)

What is not mathematics?

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

1. https://cs.stanford.edu/people/zjl/pdf/science.pdf
2. This qurey

Thanks, Rais-Ali Delvari (talk) 10:33, 1 July 2026 (UTC)

This idea is too vague to be actionable at best. The contrast with empirical sciences is already controversial as it is. Sławomir Biały (talk) 10:52, 1 July 2026 (UTC)

Mathematics can also be consider an empirical science. Similiar to the applied versus pure, it depends on the individual researcher. EulerianTrail (talk) 16:13, 1 July 2026 (UTC)

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)

Hyperlinks missed in the first paragraph

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:

1. abstract concepts
2. numbers
3. geometric shapes
4. sets
5. functions
6. probabilities
7. logical reasoning
8. proof
9. theorems
10. formulas
11. equations
12. model
13. science
14. engineering
15. technology
16. 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)

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)

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)

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.

For these reasons, I agree with the suggestion of Sławomir Biały to unkink "knowledge". D.Lazard (talk) 13:23, 1 July 2026 (UTC)

Linking at the first paragraph or other paragraphs

Please see Wikipedia:Manual of Style/Linking:

Consider including links where readers might want to use them;

But as a rule of thumb, link only the first occurrence of a term in both the lead and body of the article

Many of the above-mentioned words are linked in other paragraphs, and this is not true. Only the "first" occurrence should have a hyperlink:

1. hyperlink for "abstraction/theorem/proof/reason" is in the third paragraph
2. hyperlink for "science/engineering" is in the fourth paragraph
3. hyperlink for the "number" is in the ninth paragraph.
4. hyperlink for the "formulas" is in the second paragraph of #Algebra

and so on. All these hyperlinks can be moved to the first paragraph because it is the "first occurrence." Rais-Ali Delvari (talk) 11:17, 2 July 2026 (UTC)

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)

In this specific case, only modest linking should be done in the first paragraph to avoid MOS:SEAOFBLUE and MOS:OVERLINK. EulerianTrail (talk) 13:41, 2 July 2026 (UTC)

"Statistics and probability" as areas of mathematics

Hi, @D.Lazard Please see

1. https://byjus.com/maths/branches-of-mathematics/
2. https://www.reddit.com/r/learnmath/comments/mvlb2s/general_list_of_areas_of_mathematics/
3. https://www.geeksforgeeks.org/maths/branches-of-mathematics/

And many other sources. And also this template in Wikipedia:

Thus, I think this sentence:

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).

lacks many important areas.

I also propose a separation of "Applied" and "Pure" mathematics to make it more understandable and applicable for users. Thanks, Hooman Mallahzadeh (talk) 11:19, 24 June 2026 (UTC)

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.

By the way, the sources that you've cited are not Wikipedia:Reliable sources. Cheers, Mgnbar (talk) 11:54, 24 June 2026 (UTC)

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)

"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)

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)

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)

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)

Semi-protected edit request on 5 July 2026

Link the word knowledge in the first paragraph to the Wikipedia article knowledge. Helpful editor 3 (talk) 03:47, 5 July 2026 (UTC)

 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)

Sentence regarding deduction vs experiment

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)

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)

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)

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)

I dispute the claim that axioms and conjectures are not results. --Trovatore (talk) 16:36, 14 June 2026 (UTC)

I also object to the implication that axioms are always "chosen" to be "useful" rather than discovered to be true. --Trovatore (talk) 16:39, 14 June 2026 (UTC)

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.

The axiom of infinity and the axiom of choice may be chosen to be true or false. They are chosen to be true by 99,9% of the mathematicians, because they make all mathematics much easier (or much less difficult). D.Lazard (talk) 18:15, 14 June 2026 (UTC)

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)

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)

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)