Talk:Arithmetic/GA2

GA Review

Article (edit | visual edit | history) · Article talk (edit | history) · Watch

Reviewer: Dedhert.Jr (talk · contribs) 07:53, 21 February 2024 (UTC)

I do not know if I'm able to review this article in which the content is surprisingly written, cited in bunch of sources. In other words, I could possibly ask for a second opinion here. Will try my best. Dedhert.Jr (talk) 07:53, 21 February 2024 (UTC)

Hello Dedhert.Jr and thanks for doing this review. I'll try to help with any questions as best as I can, and asking for second opinion can be a good way to address uncertainties. Phlsph7 (talk) 08:53, 21 February 2024 (UTC)

Okay. At least this will take a long time to review and write the comments as well. Please give more times. Dedhert.Jr (talk) 08:56, 21 February 2024 (UTC)

This is a big topic so please take the time you need. Phlsph7 (talk) 09:03, 21 February 2024 (UTC)

Initial comments:

Before I began to write the list of comments, I was confused about why the sections "kinds of numbers" and "types of arithmetic" have the similarity content? They explain the definition, properties, and examples of kinds of numbers. The difference, however, is that the latter section explains how are these numbers calculated. Any reason not to merge them into one section? Also, the ordinal numbers and cardinal numbers are supposed to be suitable in other various fields? Also, again, I have never heard the terms such as "integer arithmetic", "rational arithmetic", and others; are these terms officially used? Dedhert.Jr (talk) 04:15, 22 February 2024 (UTC)

There are different ways to arrange the topics into sections and they all have different advantages and disadvantages. The idea behind the current approach is the following: the subsection "Kinds" discusses the different types of numbers while the first subsections of "Types of arithmetic" discuss specific techniques or algorithms of how to perform calculations with them. For example, the subsection "Integer arithmetic" starts by explaining how to perform operations on one-digit integers and then discusses algorithms for how operations on integers with several digits can be calculated using a series of one-digit operations. The subsection "Kinds" only defines and compares different types of numbers without introducing any algorithms for how calculations on them can be performed. This approach is also found in reliable sources like Khattar 2010. Since other sections, like "Axiomatic foundations" and "History", also rely on the distinction between the different kinds of numbers, I thought it best to have them in their own subsection.

In regard to ordinal numbers and cardinal numbers, I assume you mean that they are primarily relevant to natural and whole numbers. I moved that paragraph up and slightly adjusted the text. Please let me know if you had a different idea in mind.

I've added sources for the terms of the different types of arithmetic. The term "rational arithmetic" is also sometimes used but "rational number arithmetic" seems to be more common and is also clearer since it avoids the danger of misreading the "rational" in the sense of being based on reason. Phlsph7 (talk) 09:12, 22 February 2024 (UTC)

I would expect that the combined section I referred to can have its advantages. With this section, it is described the system of numbers, and how they perform their calculation: if the operation does have restriction, meaning that the results after some operations are not in the same kind of numbers, the latter paragraph or section could be the next hierarchy numbers with a different definition. What I am trying to say, is if natural numbers and integers could be defined in the operation of addition and multiplication, but division is not, then the latter paragraph contains the definition of rational numbers along with the examples and how they perform their calculation. This could be repeated up to irrational, real, and complex numbers.

Re: the ordinal numbers and cardinal numbers. I thought you would put them in the education in the last section "In various fields", because both of these numbers are related to linguistics, but yeah, I do not have a particular reason for not recommending you to move them out. Dedhert.Jr (talk) 11:34, 22 February 2024 (UTC)

Also, yeah. I would expect again that the combined section would give the consequence of refactoring the other section, which is putting the operations first, followed by the kinds of numbers, although this necessarily makes up less WP:TECHNICAL, more specifically WP:ONEDOWN, meaning that this topic could be targeted by all readers, especially for graders in elementary school who wants to study even more. However, I would consider that there are other options despite of being repeatedly explaining the same topic in different sections. By the way, I have seen the discussion between you and @Jacobolus on the talk page, so I think I could hear an opinion on the case of targeting the audience. Dedhert.Jr (talk) 12:01, 22 February 2024 (UTC)

I'm not sure how exactly you envision your suggestions but, from what I can tell, they seem to involve several fundamental changes that include removing some sections, refactoring them into others, introducing the different sections and their relations in new ways, and changing the order of the remaining sections.

Your main reason for this proposal seems to be that, according to you, the current version is too technical (GA criterion 1a) and has problems with redundancies (GA criterion 3b?). My suggestion would be to first try to tackle these potential problems with smaller adjustments, and see if that works. For example, which passages in the current version do you think are too difficult to understand? I'm all for presenting the ideas in a simple way but I don't think that elementary school students are the key audience of Wikipedia articles. If they are our key audience then, I agree, the article fails. We could approach the issue of redundancy the same way. For example, which passages in the subsection "Integer arithmetic" merely repeat claims from the subsection "Kinds"? I had a look now but I did not spot any, but maybe you are seeing something that I don't. Phlsph7 (talk) 12:59, 22 February 2024 (UTC)

I would say that the current version is fine and it's not very technical at all. The main reason I provide my opinion on that is I was trying to find a way of how to remove the redundancies of repeated topics over and over again in different topics. After I read carefully again, it seems that everything is fine. My apologies for misunderstanding and missing the spot, which I have to retract my comment. Dedhert.Jr (talk) 13:32, 22 February 2024 (UTC)

I think it's a good idea to consider different perspectives to probe the current version. There is no one correct way to organize the topics and it's quite possible that your way of dividing them into sections would work fine as well. The danger when implementing big changes to address smaller issues is that various new issues may arise that one did not anticipate before. Phlsph7 (talk) 14:52, 22 February 2024 (UTC)

First reading

Following comments:

• Definition, etymology, and related fields: The first paragraph explains the definition of arithmetic and the origin of its name. In the first sentence, Merriam-Webster cites the meaning of arithmetic and the operations, but I'm surprised there is the phrase "staffs" by the author. It seems that I cannot access the "Romanowski"-named source, seeming to solely redirect to the encyclopedia.com site without explicitly saying the pages here. Ditto for the citation in the second paragraph, the second sentence, and possibly in other citations including some of them. I do think both sentences have the same citation, so why are they numbered differently in the citation list? Note that I'm skeptical about the EoM source and whether is reliable or not, although it is supported by Springer and EMS. Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  The problem in relation to the "staffs" is that the sfn citation format needs an author but the citation suggestion at [1] does not include an author. I've followed the "staffs"-approach so far in other articles without problems so far, including FA articles like Philosophy. One alternative I could think of would be to use the publisher in the author field. Do you think we should use that instead?

  The very first article at [2] is Romanowski 2008. It's a digitalized version that lacks page numbers but I thought it might be better to have this one than nothing. The article starts on page 302 and page 303 ends with "Like addition, the operation of multiplication has three axioms related to it. There is the commutative law of multiplication stated by the equation a× b=b× a."

  This and the following reference to Romanowski 2008 are numbered differently because they are bundled together with other references using the template "multiref". They only get the same number if all references in a bundle are identical.

  I think EoM qualifies as a reliable source. If you feel that it is used somewhere to support a controversial claim without any other sources to back it up then please let me know and I'll check what other sources say. Phlsph7 (talk) 09:07, 28 February 2024 (UTC)

• Definition, etymology, and related fields: Some definitions restrict arithmetic to the field of numerical calculations. I wonder what is the field of numerical calculations you referring to? Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  What I was trying to express in this and the following sentence is that there are different definitions of arithmetic that vary in scope. Numerical calculations would be how to add numbers, multiply numbers, etc. For example, various parts of number theory would be excluded from that definition while other definitions see number theory as a part of arithmetic. Phlsph7 (talk) 09:16, 28 February 2024 (UTC)

• Definition, etymology, and related fields: The last paragraph contains the relation of arithmetic in many branches of mathematics. Musser, Peterson & Burger (2013) describe the role of manipulating equations in algebra, and Monahan (2012) describes its usage in analyzing data; both of these are described in the paragraph. But the rest do not contain the area under curves and rates of change, rather they focus on the continuity function and historical relation of arithmetic, geometry, and calculus only. Did I miss something here? Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  If I understand the point correctly, the problem is that that the clause about "rates of change and areas under curves" is not supported by the sources. I think the main source for calculus is Kleiner 2012 but you are right that it does not explicitly mention "area under curves and rates of change". I added this clause to clarify to the reader what calculus is. I could add a source for the claim that calculus determines rates of change and areas under curves. If you feel that the expression "in its attempt" is too strong, it could be replaced with "which attempts to" to not imply too much in regard to the role of arithmetic. Or do you have other ideas? Phlsph7 (talk) 09:39, 28 February 2024 (UTC)

  I'm not actually a native speaker of English here. When I read the phrase "in its attempt", my mind thought that those principles have more impact, and more application, on calculating the area under curves and rates of changes, rather than using concepts in calculus such as Riemann integral, definite integral, and whatever those concepts. But meh, I don't mind; hopefully, there are better vocabularies in this case. By the way, the "these principles" before the last sentence should be the principles of arithmetic concepts, correct? Dedhert.Jr (talk) 14:32, 28 February 2024 (UTC)

  Yes, I kept the expression intentionally vague in order not to focus on just one single aspect of arithmetic. Phlsph7 (talk) 11:30, 1 March 2024 (UTC)

• Numbers/Kinds: I see that some of the mathematical formats here use TeX, while others use HTML. Can we simply be consistent with one of these formats? Also, is it important to denote the set of whole numbers? They can be denoted as $\mathbb {N} _{0}$ or others variously such as $\mathbb {N} ^{0}$, or by using the notation of set theory: $\mathbb {N} *\cup \{0\}$ (see the article Natural number#Notation). One of the citations, Khhatar (2001), uses $W$ as the set of whole numbers. By the way, here's the link for Hindry (2011): [3] and https://link.springer.com/book/10.1007/978-1-4471-2131-2, just in case. Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  I formated them all to TeX, I hope I didn't miss any. I added footnotes for the alternative symbols. Phlsph7 (talk) 10:09, 28 February 2024 (UTC)

  @Phlsph7 Okay. But should you cite the notation for natural numbers? Dedhert.Jr (talk) 12:49, 6 March 2024 (UTC)

  This seems to be a minor point but I managed to come up with a few sources. They don't cover $\mathbb {N} ^{*}$ so we would have to do more searching or remove that one if you feel strongly about this. Phlsph7 (talk) 14:16, 6 March 2024 (UTC)

  Never mind, I found a source for $\mathbb {N} ^{*}$. Phlsph7 (talk) 17:18, 6 March 2024 (UTC)

• Numbers/Kinds: Maybe you could summarize the meaning of "repeating decimal" briefly in the rational number paragraph, or give an example of it. Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  Done. Phlsph7 (talk) 12:41, 28 February 2024 (UTC)

• Numbers/Kinds: A somewhat cringeworthy idea from me. I wonder if you can modify a little bit about the irrational numbers here. What I meant here is should you explain in more detail how are these numbers obtained? Maybe the first sentence explains the definition of irrational numbers, and what are those properties, followed by the example of numbers and how are they obtained; for example, The $\left(1,1,{\sqrt {2}}\right)$ right triangle in the illustration is an example of how the number ${\sqrt {2}}$ is obtained. The same thing for the $\pi$ and $e$. Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  Good idea, I added the corresponding explanation. I removed $e$ since the explanation is not that straightforward and two examples should be enough. Phlsph7 (talk) 13:00, 28 February 2024 (UTC)

  Umm... okay. But something's off. The mathematical constant $\pi$ should be sufficiently be defined as $C/d$ the ratio of circumference's circle $C$ and diameter $d$, and the size of $d$ is not neccessarily matter. This is already explained in our FA, Pi; see more specifically in the definition. Dedhert.Jr (talk) 14:41, 28 February 2024 (UTC)

  I tried to simplify it but you are correct that this is not the most general definition so I used your suggestion instead. Phlsph7 (talk) 11:37, 1 March 2024 (UTC)

• Additional comments related to the previous section. I'm puzzled by the name of the previous section; any reason why is the section is not supposed to be renamed as "System of numbers"? Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  I don't think that systems of numbers is generally accepted as a technical term for these differences. There is the danger of mistaking systems of numbers for number systems, which is one label for the difference between numeral systems like binary numerals, decimal numberals, etc. I was also considering the label Types instead of Kinds but I decided against it to avoid mistaking it for Type (model theory). I'm not sure how serious that concern is, though. Phlsph7 (talk) 13:08, 28 February 2024 (UTC)

• Number/Numeral system: Maybe template the main article Numeral system? "A numeral is a symbol representing a number", yes, but the "numeral systems are representational frameworks" phrase somewhat gives the implicit definition. Is it possible to explain more about what the "frameworks" mean in this context? Here is the Ore (1984) URL [4]. Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  I added the template. The explanation is given in the following sentences: a framework comprises a set of symbols together with rules for combining them. I'm not sure if there is a better way to explain it but I hope the general characterization becomes clear to the reader through the concrete examples of the different systems in the next paragraphs. Phlsph7 (talk) 13:34, 28 February 2024 (UTC)

• Arithmetic operations: Missing wikilink inverse elements? Also, the second paragraph mentions the identity and inverse element in addition, but not the multiplication. Is it possible to add them as well? The same reason for the third paragraph. In the case of images, why does each image have the wikilink if you already provide the main-article templates? Also, are these images helpful in understanding all of these operations aside from the numbers and symbols as well? From my perspective, I do think it will really helpful if the images can be drawn with some objects along with the numbers showing the amount of objects while operating another one; for example, three apples and two apples, added them becomes five apples, and the numbers are written under the objects with the operation symbols. I do think these can be helpful in some basic operations, such as addition, subtraction, multiplication, and division. Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  Regarding identity/inverse elements: my idea was to only give one example in the introduction to help the reader grasp the general concepts. All the elements for the different operations are given in the following subsections. This way, we avoid repetition.

  I think the main point of the images is to introduce the reader to the technical vocabulary (addend, subtrahend, exponent) visually without needing to describe what number is meant. I like the idea of showing the amount of objects visually along with the numbers. I'm not sure how to handle the problem of limited space: most of the right side of the screen is filled with images in this section. If more are added (or the images become bigger) then they start pushing each other down, which might lead to the multiplication image being displayed next to the exponentiation text. How they are displayed depends on the reader device, like screen size. Phlsph7 (talk) 13:55, 28 February 2024 (UTC)

  What I meant is replacing those images with the proposed one, instead of adding more images. Ah, I should have said it more specifically. Dedhert.Jr (talk) 14:24, 28 February 2024 (UTC)

  I tried a different solution by adding images to the introductory text before the subsections. This way, we can keep the images below as they are, which I think is helpful for the terminology, and have images representing how arithmetic operations are applied to sets of objects. Does that work for you? Phlsph7 (talk) 10:35, 29 February 2024 (UTC)

  Okay. Dedhert.Jr (talk) 10:54, 29 February 2024 (UTC)

  I removed the image links. Phlsph7 (talk) 13:58, 28 February 2024 (UTC)

• Arithmetic operations/Addition and subtraction: Burgin (2022) defines the summation, but not the counting. As well as the source problem, this section may need another image on which it shows the computation of addition and subtraction with the arrow bouncing on each number in real line, for defining the addition and subtraction visually, right? Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  I added a source for counting. I added an image of the number line method to the section "Integer arithmetic" instead because of the limited space in the section "Addition and subtraction". Phlsph7 (talk) 17:37, 29 February 2024 (UTC)

• Arithmetic operations/Multiplication and division: The note in the early paragraph, in the early sentence has somewhat terrible sources. There are no pages of the source Cavanagh (2017) included here, and for the Weisstein, the MathWorld, I do remember that they are not reliable sources, so maybe you could find a better one? Also, should you link Product (mathematics)? Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  I added a page number for Cavanagh 2017. It is published by CRC Press, which belongs to Taylor & Francis and should be reliable. From what I gather, Eric W. Weisstein is a respected and well-published author in the field. Are you aware of any past discussion about Wolfram Mathworld? It is not listed at Wikipedia:Reliable sources/Perennial sources as an unreliable source. To me, the claim for which these sources are used does not sound particularly controversial. I added the wikilink to Product (mathematics). Phlsph7 (talk) 11:54, 1 March 2024 (UTC)

  I have asked in WT:WPM, and there are some discussion about Mathworld whether it is dubious, reliable, or whatever it is. Someone is already replaced the source; see the talk. Dedhert.Jr (talk) 12:50, 6 March 2024 (UTC)

  Thanks for the link, I was not aware of the discussion. Phlsph7 (talk) 13:42, 6 March 2024 (UTC)

• Arithmetic operations/Exponentiation and logarithm: Should you also explain what if the exponent is 0 for any base, except for 0, and what about if the both base and exponent are also 0? Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  Good idea. I added it in a footnote to not put too much emphasis on this detail. Phlsph7 (talk) 12:03, 1 March 2024 (UTC)

• Types of arithmetic/Integer arithmetic: the "multiplication and repeated addition" contains the debate about whether educators should teach those operations in education; is it actually helpful to readers to understand? I mean, what's the point of adding that link, and does this link relate to the topic? I would expect that the article may be linked in the section in which the operation, multiplication, is defined as the repeated addition. Also, in the same section again, there are HTML and {{math}} being used in a mixed manner. But we already have TeX in the previous section; again, can you consistently use one of them? Also, again, the reference Prata (2002) does not mention the inaccuracies when rounding them. Methods to calculate logarithms include the Taylor series and continued fractions: Did you mean the logarithm in general, right? But the sources show the expression of sum and integrals in a natural logarithm, an $e$-base logarithm. Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  I removed the link, the name of that article is a little misleading. I defaulted to tex, it's probably best to use that everywhere. The inaccuracy of rounding is implicit in Prata 2002 but I added an additional source to make this more explicit. Regarding logarithms, I added an additional source that does not explicitly mention this restriction. I'm not sure that this is necessary since it's possible to use the natural logarithm to calculate the logarithm for another base as well. Phlsph7 (talk) 12:51, 1 March 2024 (UTC)

• Types of arithmetic/Number theory: The Yan (2013) reference gives the wrong page containing the definition of the set of systems of numbers. I don't think that analysis and calculus may need to be included in defining the elementary number theory. Also, maybe you can explain briefly what are the other fields of arithmetic in the last sentence, especially in the applied one. Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  It seems I relied on two different versions of Yan's book, which is why the page numbers didn't match. I defaulted to the 2002 version, which has the list of subfields of number theory on page 12. I removed the mention of analysis and calculus and expanded the explanation of the additional subfields. Phlsph7 (talk) 10:08, 2 March 2024 (UTC)

• Axiomatic foundations: As far as I know, the axioms start with 1, which is a natural number, but I am surprised that 0 is included in these primitive axioms. Mind-blowing!. Speaking of citations, I do think that Taylor 2012 should put in the footnote nearby, and replace it with other sources: [5]; the URL page Ongley & Carey 2013 have a slight problem. Axiomatic foundations of arithmetic try to provide a small set of laws, so-called axioms Should the phrase "so-called" be avoided under WP:WTW? Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

  This part about the axioms was also confusing to me. It would have been helpful if the different versions had different names. I moved Taylor 2012, added the source suggested by you, and fixed Ongley & Carey 2013. I rephrased the so-called to be on the safe side. Phlsph7 (talk) 10:28, 2 March 2024 (UTC)

  The same reason for the "real arithmetic" about the phrase "so-called". Dedhert.Jr (talk) 12:52, 6 March 2024 (UTC)

  Done. Phlsph7 (talk) 13:45, 6 March 2024 (UTC)

I will stop here for a moment, and for some reason, I will respond to your reply, if it's sufficiently required. More comments are possibly coming in the next few days. Dedhert.Jr (talk) 06:57, 28 February 2024 (UTC)

@Dedhert.Jr: Thanks for the detailed comments. I tried to respond to them and I hope I didn't miss any. Phlsph7 (talk) 10:30, 2 March 2024 (UTC)

More comments:

• History: Umm... do you have to keep the main article History of arithmetic despite that it is not sufficient enough to describe the history of arithmetic entirely, and I'm pretty sure this section is solely a summary? Dedhert.Jr (talk) 12:41, 6 March 2024 (UTC)

  I'm not sure what you mean. Are talking about the template that links to the article or the article itself? Phlsph7 (talk) 17:19, 6 March 2024 (UTC)

  The article History of arithmetic I refer to, which do not entirely explain the history of arithmetic. Rather, it remains incomplete to translate the content from the FA Russian article ru:История арифметики. To put it plain, this article is seemingly short and does not helpful the audience. Dedhert.Jr (talk) 02:20, 7 March 2024 (UTC)

  I get your idea, thanks for pointing this out. I turned the article into a redirect and explained the main reasons at Talk:History_of_arithmetic#Changed_to_redirect. Phlsph7 (talk) 09:23, 7 March 2024 (UTC)

• History: "Some historians suggest that the Lebombo bone (dated about 43,000 years ago) and the Ishango bone (dated about 22,000 to 30,000 years ago) are the oldest arithmetic artifacts but this interpretation is disputed" — Two sources I could access Burgin 2022 and Thiam & Rochon 2019 described the range of Lebombo bone around 44,000 and 43,000 years ago; should this be included entirely? Also, should you explain what makes those interpretations become disputed? Dedhert.Jr (talk) 12:41, 6 March 2024 (UTC)

  The article Lebombo bone says between 43,000 and 42,000. Our section says "dated about 43,000 years ago", which implies that it's not a precise date. As far as I'm aware, the dispute is mainly because its rather difficult to say now for what purpose it was used back then and historians can only speculate. To keep the section concise, it might be better to leave it as it is. Phlsph7 (talk) 17:33, 6 March 2024 (UTC)

• History: Neither of the facts in both sources Burgin 2022 and Ang & Lam 2004 about the early arithmetic evolved in 3000 BCE. Did you mean 4,000 BCE, or did I miss something? Dedhert.Jr (talk) 12:41, 6 March 2024 (UTC)

  Burgin says "first authentic data on arithmetic knowledge...third to second millennia BCE" and mentions arithmetic in India in the third millennia BCE. The text also talks about token use by the Sumerians 4000 BCE but as it is discussed there, I don't think it qualifies as "complex and structured approach to arithmetic". Our formulation is relatively vague so I don't think we are committing to too much here. But if you are concerned, we could leave the number 3000 BCE out since Burgin talks explicitly about arithmetic in ancient civilizations. Phlsph7 (talk) 17:49, 6 March 2024 (UTC)

  Oh. I see. Dedhert.Jr (talk) 02:21, 7 March 2024 (UTC)

• History (third paragraph): Brown 2010 does not explicitly about the utilization of arithmetic in the era of ancient Greece. Should you write examples about the early civilizations that primarily used numbers for concrete practical purposes and lacked an abstract concept of numbers? Burgin 2022, pp. 20–21 and Bloch 2011, p. 52 describe the irrational numbers in geometrical features; except for the Burgin 2022, p. 34. Dedhert.Jr (talk) 12:41, 6 March 2024 (UTC)

  I assume you mean our sentence "This changed with the ancient Greek mathematicians, who began to explore the abstract nature of numbers rather than studying how they are applied to specific problems". Brown 2010 talks about the notion of number as pure magnitude ... first elaborated by Euclid. I added an example about the earlier practical applications. I removed page 34 from the citation of Burgin 2022. Phlsph7 (talk) 18:07, 6 March 2024 (UTC)

• History (fifth paragraph): "empty/missing positions" avoid the slash here; such a symbol has ambiguous meaning. Lützen 2023 does not explicitly say anything about the usage by Cardano in complex numbers, rather it shows only the impossibility of imaginary numbers in the solution of a quadratic equation. Dedhert.Jr (talk) 12:41, 6 March 2024 (UTC)

  I removed the slash. You are right, Lützen 2023 is only used to support the first part of the sentence. The second part about cubic equations is supported by Burgin 2022. Phlsph7 (talk) 18:14, 6 March 2024 (UTC)

• History (last paragraph): Weil 2009 does verify the fact of the number theory foundation, but Karlsson 2011 rather mentions their theorem in some chapters and other mathematicians were working on zeta function. Dedhert.Jr (talk) 12:41, 6 March 2024 (UTC)

  I removed Karlsson 2011 since it only supports the claim indirectly and the other sources do a better job. Phlsph7 (talk) 18:19, 6 March 2024 (UTC)

@Dedhert.Jr: I wanted to enquire whether the changes so far meet your expectations and whether there are more points that should be addressed. Phlsph7 (talk) 08:05, 14 March 2024 (UTC)

@Phlsph7 Ah, sorry. I haven't finish the review yet. Please spare me more time to complete the review, despite the article had some changes by another user, which I have to take a step on the second reading. Dedhert.Jr (talk) 08:58, 14 March 2024 (UTC)

Second reading

Okay. I'm ready to read for the second time. I postponed because I had to wait for the article to stabilize after some previous edits suddenly appeared during reviewing (GACR5). I haven't reviewed the section "In various fields", so it might take more time to review them, as well as review the whole article again.

• For the section definition, etymology, and related fields, I wonder if you could remove the paragraph about the related fields, as you already wrote them in the "In various fields" section. Dedhert.Jr (talk) 11:56, 20 March 2024 (UTC)

  Done. Phlsph7 (talk) 17:02, 20 March 2024 (UTC)

• Do you even have to link the article 18 (number) and 19 (number) in the section Number theory? Dedhert.Jr (talk) 11:56, 20 March 2024 (UTC)

  I linked them because their properties are discussed, such as factorization and being a prime number. I don't think it's important and I don't feel strongly about this either way. Phlsph7 (talk) 17:06, 20 March 2024 (UTC)

• It is usually introduced in relation to concrete scenarios, like counting beads, dividing the class into groups of children of the same size, and calculating change when buying items. Common tools in early arithmetic education are number lines, addition and multiplication tables, and counting blocks I wonder if you could also add that the students may also count numbers with an abacus. Dedhert.Jr (talk) 11:56, 20 March 2024 (UTC)

  Done. Phlsph7 (talk) 17:08, 20 March 2024 (UTC)

• They introduce the students to different types of numbers Maybe you should explain more specifically the phrase "they" in this case, the second paragraph of the first sentence in the education section. Dedhert.Jr (talk) 11:56, 20 March 2024 (UTC)

  I reformulated this sentence by linking it to the previous sentence. Phlsph7 (talk) 17:10, 20 March 2024 (UTC)

• I am not good in philosophy mathematics, but is the article Quine–Putnam indispensability argument may be related to and somehow fit into the article? Dedhert.Jr (talk) 11:56, 20 March 2024 (UTC)

  It's most relevant to Platonism in the philosophy of mathematics. I added a short footnote. Discussing it in detail may go beyond the scope. Phlsph7 (talk) 17:21, 20 March 2024 (UTC)

Okay. I think it's all done. I hopefully do not miss anything after closing this review. Passing. Dedhert.Jr (talk) 05:49, 21 March 2024 (UTC)