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The present article reads

"If the index finger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed at a right angle to both, the three fingers indicate the relative orientation of the x-, y-, and z-axes in a right-handed system. The thumb indicates the x-axis, the index finger the y-axis and the middle finger the z-axis."

The "Right-hand rule" article reads

"For right-handed coordinates the right thumb points along the z axis in the positive direction and the curling motion of the fingers of the right hand represents a motion from the first or x axis to the second or y axis."

George Rodney Maruri Game (talk) 05:14, 24 December 2022 (UTC)[reply]

The two descriptions define the same orientation, as differing by a circular permutation of x, y, z. D.Lazard (talk) 10:43, 24 December 2022 (UTC)[reply]

Famously, though I don't know how verified this is, Descartes came up with the coordinate system watching a fly in the corner of his house: https://wild.maths.org/ren%C3%A9-descartes-and-fly-ceiling

Would this be worthwhile including in the article? It seems to me, as a non mathey person, that additional historical context is often useful for better understanding concepts. Even when they don't have direct bearing on the thing, it gives color and texture to things, helps me understand the problems that individuals were trying to solve when they came up with their solutions. Thedonquixotic (talk) 21:16, 24 March 2023 (UTC)[reply]

Moved from Talk:D.Lazard

To let you know, coordinate systems in general, and Cartesian coordinate systems in particular, are topics in Analytic Geometry. Just like any other mathematical topic, they are used in other branches of mathematics and have applications in many other fields of science. You have stated that “Cartesian coordinates are used in almost all parts of geometry and its applications”. However, this does not mean that “Cartesian coordinates” do not belong to a more specific branch of mathematics. Belonging to a specific branch of mathematics does not restrict the application of a mathematical topic in other fields. Coordinate systems are, in fact, a topic in Analytic Geometry, as it combines geometry with algebra. By the way, coordinate systems are NOT used in plane geometry. That’s another reason why coordinate systems are not generally a topic in “geometry” but are actually a topic in “analytic geometry”. Persianwise (talk) 10:05, 28 August 2023 (UTC)[reply]

Thank for this course on a topic that I know well since many years. I know that analytic geometry is a part of geometry, although, nowadays, it is more the name of courses in colleges, than that of a specific area of geometry; this is not the problem here. The problem is that the first phrase "In geometry" is not aimed to categorize the subject (this is the object of the categories appearing at the end of the article). It is aimed to tell to readers whether they could be interested to the article. Here, everybody knows geometry, but people who know analytic geometry, know generally what are Cartesian coordinates, and thus your first phrase is not useful for them. On the other hand, many people encounter Cartesian coordinates in their studies or their work, without knowing that their use is called analytic geometry. This is the case, for example, of students in physics. This is for such people that the first phrase is intended. D.Lazard (talk) 10:33, 28 August 2023 (UTC)[reply]

You have stated: "... nowadays, it [analytic geometry] is more the name of courses in colleges, than that of a specific area of geometry."

However, not just analytic geometry is the name of a course. Algebra, calculus, topology, complex analysis, and almost every other mathematical subject are also the names of courses, and this has always been the case.

Your other statements are subjective, chaotic, and do not aim to make the article accurate.

For your information, although analytic geometry combines geometry and algebra, it is neither a branch of geometry nor algebra. If you were correct in saying that analytic geometry is a branch of geometry, then you could also say it is a branch of algebra too, which is not the case.

Furthermore, Wikipedia articles are written for all audiences, not just for physics students, and should be as accurate as possible for all readers.

That's why "geometry" should be changed to "analytic geometry" in order to start the article with correct phrasing. Persianwise (talk) 13:58, 28 August 2023 (UTC)[reply]

End of the moved discussion. D.Lazard (talk) 11:25, 31 August 2023 (UTC)[reply]

By "[analytic geometry] is more the name of courses in colleges, than that of a specific area of geometry", I meant that, nowadays, the phrase "analytic geometry" is rarely used outside college studies. In fact, it has been proved (See Cantor–Dedekind axiom) that the synthetic geometry (the geometry defined by geometric axioms) is strictly equivalent to analytic geometry, in the sense that every theorem of synthetic geometry can be proved by means of analytic geometry, and vice-versa.

It is not accurate to say that analytic geometry "combines" geometry and algebra. The correct statement is that analytic geometry is the use of algebra in geometry (or, more precisely, the use of real numbers and their properties). So analytic geometry is definitively a part of geometry.

About the first phrase: "In analytic geometry" is not wrong, but it may be confusing to many readers, who are not supposed to know what is analytic geometry: for knowing analytic geometry, one must know Cartesion coordinates, and this article if for people who do not know what are Cartesian coordinates (otherwise, they do not need this article). Also, people who have never heard of analytic geometry may have encountered Cartesian coordinates in other contexts, and they may understand the phrase "In analytic geometry" as an indication that the article is not for them. D.Lazard (talk) 17:24, 1 September 2023 (UTC)[reply]

I reverted this edit of Persianwise, with the edit summary Axes and number lines are two different sorts of specific lines. In turn, he reverted my revert with the edit summary Notoce: "axis" is a term that is commonly used for a number line in mathematics, and posted Axes are number lines. Are you familiar with the x-axis and y-axis? These are number lines that are denoted by the letters x and y, respectively. Please stop preventing the article from being improved. on my talk.

After that, I tried to get a compromise, but I was reverted again. Thinking about that, I got convinced that the use of (coordinate) axis for a number line is very uncommon. So, I'll revert the section § One dimensional to the old version where the word "axis" did not appear.

By the way, I'll revert the other edits of the same user for reasons given in the preceding thread and in my edit summaies. D.Lazard (talk) 14:29, 2 September 2023 (UTC)[reply]

In the same paragraph, it is stated: ‘A line with a chosen Cartesian system is called a number line.’ It should be clarified that any coordinate system, including Cartesian coordinate systems, have axes. A number line is essentially a one-dimensional coordinate system. If the number line isn’t considered an axis, then where would the axis be located? If you answer this question correctly, you would understand that number lines and axes are two different names referring to the same concept.

DO not make any unwarranted revertions.

Persianwise (talk) Persianwise (talk) 23:20, 17 September 2023 (UTC)[reply]

You have not provided any reliable source that uses "axis" to refer to a numbeer line. So, your edit is a personal opinion. This is called WP:original research in Wikipedia, and not allowed in articles per policy WP:NOR. D.Lazard (talk) 11:20, 18 September 2023 (UTC)[reply]

Every axis is a number line but not every number line is an axis; the two are not synonyms, and your edit incorrectly proposed that they are.

Almost every part of D.Lazard's most recent edit was an improvement. In my opinion, there are two exceptions: (1) the cn tag needs double curly braces instead of single curly-braces, and (2) I think the spelling out of the meaning of bijection (that each point corresponds to a real number and each real number corresponds to a point) was a less technical way of conveying the point, so is likely worth preserving. 100.36.106.199 (talk) 13:22, 18 September 2023 (UTC)[reply]

Analytic Geometry is the branch of mathematics that introduced the concept of Coordinate Systems. Euclidean or non-Euclidean Geometries, by themselves, do NOT use Coordinate Systems. Therefore, it would be more accurate to change ‘geometry’ to ‘analytic geometry’, as this is what analytic geometry entails.

D.Lazard's comment:

"analytic geometry is much too restrictive, as it may prevent people interested in computational geometry, physics, mechanics, astronomy, etc. to read this article (all know geometry, but many may not know analytic geometry."

is subjective and lacks meaningful content. It is, in fact, a fallacy.

He has been warned before on his talk page, and he is hereby warned again to cease any attempts to make unwarranted reversions and to stop introducing inaccuracies due to insufficient mathematical knowledge. Persianwise (talk) 22:47, 17 September 2023 (UTC)[reply]

It is not analytic geometry that introduced coordinates systems, but it is René Descartes, a long time before the term analytic geometry was coined. Moreover, it is not geometries but geometers that uses coordinates systems. This is documented in the article Geometry, where coordinates systems and analytic geometry are clearly mentioned as parts of geometry.

Please, stop using accusations ("fallacy" and "insufficient mathematical knowledge") for trying to support your personal views. Use WP:reliable sources, if you can. D.Lazard (talk) 11:11, 18 September 2023 (UTC)[reply]

I have fully protected the article for a short period so the edit warring can be resolved on this talk page. Please review WP:DR and seek assistance at a relevant wikiproject. If necessary, and per WP:DR, use an WP:RFC if necessary. I will watch this page for a while. Feel free to ping me or any administrator if the article is ready to be unprotected due to consensus here (remembering that WP:CONSENSUS is not unanimity). Johnuniq (talk) 09:43, 18 September 2023 (UTC)[reply]

It should be mentioned that the page Cartesian coordinate system is about a subject of mathematics. To edit a mathematical subject, like any other scientific field, one must have the proper knowledge. Without the proper knowledge, any attempt to edit or make changes leads to inaccuracies that would confuse and misinform the readers. Notice that the user D.Lazard, due to lack of sufficient mathematical knowledge, is haphazardly changing sentences and publishing:

Cartesian coordinates are named for René Descartes, whose invention of them in the 17th century revolutionized mathematics by providing the first systematic link between geometry and algebra.

The correct and precise statement should be:

Cartesian coordinates are named for René Descartes, whose invention of them in the 17th century revolutionized mathematics by combining algebra and geometry.

The explanation for this was presented on his talk page: User_talk:D.Lazard.

But then he replaced it with:

Cartesian coordinates are named for René Descartes, whose invention of them in the 17th century revolutionized mathematics by allowing the expression of every problem of geometry in terms of algebra and calculus.

This is wrong because calculus was not even invented at that time.

(Analytic gemetry invented by René Descartes in 1637] , but Calculaus was invented in 1665 by Isaac Newton.

The user is evidently not just initiating the edit warring but also is actively engaged in sabotaging the article and preventing further improvement to the article. He should be blocked from editing mathematical related pages.

More information regarding his other unwarranted reversions will be given. Persianwise (talk) 17:03, 18 September 2023 (UTC)[reply]

The talk page of an article is not the correct place to discuss other editors. The correct page is WP:ANI but please examine WP:BOOMERANG before thinking about that. It is very common for disagreements to occur at Wikipedia and participants must follow standard procedures to avoid sanctions. Handling disputes is covered at WP:DR. Do not mention other editors or their alleged shortcomings on this page. If you want independent opinions on the accuracy of what I have written, ask at WP:Teahouse. The point is that editors must focus on content not on contributors. Have you asked for opinions at the wikiproject (see headings at top of this page)? I have not done more than glance at the issue and will take no part in discussing what should be done. However, you might bear in mind that articles generally focus on current practice, that is, what do people mean when they talk about the Cartesian coordinate system now? Please do not reply to this: I just raised an issue for you to consider. I suggest starting a new section with a brief outline of what problem you see in the current article and what should be done to fix it, with a brief explanation. Then try to get other opinions per WP:DR. Johnuniq (talk) 04:41, 19 September 2023 (UTC)[reply]

"Allowing the expression of every problem of geometry in terms of algebra and calculus" may perhaps not be ideal wording, but the essential point is not invalidated by the fact that calculus had not yet been invented, any more than "the invention of the wheel has had revolutionary effects on transport, such as enabling the manufacture of motor cars" is invalidated by the fact that the motor car was not invented until long after the invention of the wheel. JBW (talk) 20:33, 20 September 2023 (UTC)[reply]

I find both first and second versions acceptable, but the third... It's too strong to assume that "every" problem of geometry is reduced to algebra and calculus. 慈居 (talk) 08:49, 21 September 2023 (UTC)[reply]

@慈居: Well, in the context of Cartesian coordinate "geometry" clearly means Euclidean geometry, and Euclidean space is isomorphic to linear algebra in a vector space over the real numbers, and therefore every proposition about geometry in Euclidean space can indeed be expressed as an equivalent proposition in real algebra and calculus That is what was said in the quoted passage, which refers to "the expression of every problem of geometry in terms of algebra and calculus" (my emphasis) which is certainly possible. That is true whatever you may have in mind when you refer to "reducing" every problem of geometry to algebra and calculus, and whether or not such "reduction" in your intended sense is possible. JBW (talk) 17:07, 21 September 2023 (UTC)[reply]

I think you made a fair point. Instead of a boring argument, I'd like to mention the Jordan curve theorem, which says that every closed curve partitions the whole plane into two parts. This looks trivial (despite its lengthy complicated proof), but it yields nontrival conclusions when it's applied to some crazy curves like the Hilbert curve. This is how the word "every" is powerful. Even for classical Euclidean geometry, there was a famous (meta-)problem asking whether there is a hyperbolic geometry, and certainly this is not something expressable only with algebra and calculus (at least with the method of coordinate geometry). So in my opinion, both three sentences make sense (as you have pointed out for the third one), but I'd pick the first two of them as the better choices. 慈居 (talk) 20:11, 21 September 2023 (UTC)[reply]

OK, prompted by your comments, 慈居, I have thought some more about this. What do we mean by algebra? My post above in effect treats "algebra" as including anything which can be expressed in terms of an algebraic structure (in this case a finite dimensional real linear algebra) but that is a dubious interpretation; for example the metric and topological properties of such a space are not usually regarded as "algebraic". One could argue about exactly what features are added to the situation by the words "and calculus", but again topological properties are not, I think, generally regarded as being part of calculus. Above, I took issue with your statement on the basis that the quote attributed above to D.Lazard (but I haven't checked that attribution) refers not to reducing every problem in geometry to algebra and calculus, but to expressing it as such, but on reflection I think I was wrong: that was based on an interpretation of "algebra" as "any property of an algebraic space", but that is an unhelpfully broad interpretation of the word. Indeed, your example is highly pertinent: it would be difficult to express the Jordan curve theorem in terms which could reasonably be regarded as belonging to algebra and calculus. In the context in which this appears in the article, it is probably reasonably true that the kinds of geometric properties considered at the time when Cartesian coordinates were introduced can be expressed in terms of algebra and calculus, but since the statement in the form in which it is given is not justifiable, on reflection I think probably the best thing to do is to remove the word "every". I therefore intend to remove that word once I have posted this message, but of course I will be willing and interested to read any arguments in favour of its retention. (Well, any arguments by anyone who comes anywhere near to knowing what they are talking about.) JBW (talk) 21:15, 21 September 2023 (UTC)[reply]

About "calculus": I used "calculus" instead of analysis, because it is known by a wider audience. However "analysis" would be a proper term since, AFAIK, it denoted initially Descartes's method of coordinates and led to the term Analytic geometry, before becoming an alternative name of calculus.

About Jordan curve theorem: This is not a good example, as it involves the concept of "continuous curve" which cannot be defined in terms of pure Euclidean geometry. I do not imagine a way to define continuous curves without using coordinates, at least implicitly.

Nevertheless, I agree with the removal of "every", which would require either a source or an accurate definition of "algebra and calculus". D.Lazard (talk) 09:54, 22 September 2023 (UTC)[reply]

I also agree with the removal of "every". By the way @D.Lazard, the Jordan curve theorem was not at all intended to be a good example, or even an example. It was to illustrate how an assumption can be unnecessarily strong with the word "every". 慈居 (talk) 10:36, 22 September 2023 (UTC)[reply]

Moved from Talk:Persianwise

You are starting an WP:edit war for trying to impose the idea that "number line" is a synonymous of "axis". It is true that the axes of a coordinate system are number lines, but this does not mean that the words are synonyms. Otherwise, this would be said in number line. In any case, Wikipedia rule are that any new assertion like yours must be supported by a reliable source (in this case of elementary mathematics, a widely used textbook). Otherwise, it is forbidden to add the assertion, per the policy WP:OR.

Please, remember also that edit warring may lead to an edit block (see WP:Edit warring). D.Lazard (talk) 18:00, 30 August 2023 (UTC)[reply]

In mathematics, a number line and an axis are equivalent concepts. They are often used interchangeably and can be regarded as mathematically synonymous, but they are not considered synonyms in the dictionary.

You stated that

“It is true that the axes of a coordinate system are number lines.”

However, on your own talk page, you stated that

"A number line and an axis are two different things."

This is contradictory to mathematical facts. Your actions evidently constitute Edit warring, as you have repeatedly reverted my correct assertion that a number line and an axis are the same thing. If you continue to engage in this behavior, I will report you as a Disruptive user. Persianwise (talk) 21:42, 30 August 2023 (UTC)[reply]

I don't see the logic of what you say. A cat is an animal, but "cat" and "animal" are not the same thing. There is no contradiction there; nor is there any contradiction in what you quote D.Lazard as saying. JBW (talk) 15:52, 20 September 2023 (UTC)[reply]

Your use of analogy is not valid here. The fact that "not all animals are cats" is because the set of all cats is a proper subset of the set of all animals.

However, my assertion is that the set of all number-lines and the set of all one-dimensional coordinate systems are identical.

This mathematical fact is even visually evident if you draw a number-line and a coordinate system in one dimension, both on the same geometric straight line, on a piece of paper. You will notice that a number-line is exactly a one-dimensional coordinate system because the axis in the one-dimensional coordinate system coincides with the number-line.

So, there is a bijection between the set of all points on the number-line and the set of all points on the axis; hence, they are the same.

Otherwise, where do you think the axis and the number-line are located relative to the aforementioned geometric straight line? Persianwise (talk) 18:19, 20 September 2023 (UTC)[reply]

That is completely missing the point. You said that there was a contradiction between the statement that axes of a coordinate system are number lines and the statement that a number line and an axis are two different things. Whether it is or is not true that a number line and an axis are both the same thing does not have any bearing on whether or not the statement (whether true or false) that they aren't is inconsistent with the statement (whether true or false) that the axes of a coordinate system are number lines. And that is quite apart from the fact that there being a bijection between the elements of two sets is not at all the same thing as their being the same thing. If you really can't see those two points then you should probably not be trying to improve the logic of the treatment of mathematical topics in articles. JBW (talk) 19:15, 20 September 2023 (UTC)[reply]

For clarification, it should be noted that the set of real numbers is a Field. It is obvious that the structure of a number-line defined by the same operations of addition and multiplication makes the two isomorphic - See Isomorphism. To mathematicians, two isomorphic fields (or any other relevant structures) are the same thing.

Analogously, the same thing can be said about the axis (more precisely, the set of points on the axis, with the familiar operations).

So all three are isomorphic, there are no differences between them, except the naming. They are all the same thing! Persianwise (talk) 20:48, 20 September 2023 (UTC)[reply]

NOTE: By “This is contradictory”, I meant “This [the second statement] is contradictory [to the established mathematical facts] (and not to the first statement). Persianwise (talk) 21:07, 20 September 2023 (UTC)[reply]

Oh, Christ. if you are going to say things and then say you meant something completely different than what any reasonable person would have thought you meant, then I may as well give up even trying. JBW (talk) 21:18, 20 September 2023 (UTC)[reply]

“If you noticed, I had stated ‘a number line and an axis are the same thing’, but the other user had said ‘A number line and an axis are two different things.’ which was contradictory to my assertion, and to mathematical facts. When ambiguity arises, why not explain things further to make things precise to get rid of ambiguity? Isn’t that the right thing to do?” Persianwise (talk) 22:01, 20 September 2023 (UTC)[reply]

You are saying that the x-axis and the y-axis of a Cartesian coordinate system are the "same thing" since they are both isomorphic to the field of the real numbers, and thus they are isomorphic. However, these two axes are different, and thus they cannot be the same thing. D.Lazard (talk) 22:08, 20 September 2023 (UTC)[reply]

The main meaning of isomorphism is the assertion that two seemingly different things are actually the same thing, except possibly for the names of their elements and the way the operations defined on them work. This is an advanced topic, hard to grasp for those who lack in-depth mathematical knowledge. I suggest you read the topic on this page that I posted at 18:19, 20 September 2023 (UTC). Anyone can see how to visualize why a number line and an axis (in a coordinate system) coincide, both representing the real number system. If you cannot see it, it is then crucial to answer the question that was asked there:

“Otherwise, where do you think the axis and the number line are located relative to the aforementioned geometric straight line?” Persianwise (talk) 22:46, 20 September 2023 (UTC)[reply]

How kind of you to help Daniel Lazard by explaining to him ideas which you understand but which are beyond him because of his being one of "those who lack in-depth mathematical knowledge". What a pity that nobody thought to teach him some elementary mathematics before he started his career. Hmm. Or... JBW (talk) 13:36, 21 September 2023 (UTC)[reply]

Your response is not pertinent to a mathematical discussion. He was asked this question twice:

"Otherwise, where do you think the axis and the number line are located relative to the aforementioned geometric straight line?”

and you know he failed to answer it.

Can you supply an answer to the aforementioned question yourself?

One does not even have to be a mathematician to answer it.

What is your answer? Persianwise (talk) 20:46, 24 September 2023 (UTC)[reply]