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)