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Merge of Abstract mathematical object

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I did a paste/merge of Abstract mathematical object into this one, replacing the existing text which lacked references and also did not conform well with WP:NPOV. Further improvements very welcome. --Vaughan Pratt (talk) 10:46, 1 October 2008 (UTC)[reply]

Quiet an improvement. I will translate it in Dutch shortly JRB-Europe (talk) 22:25, 10 October 2008 (UTC)[reply]

@Vaughan: What about urelements? Shouldn't they be mentioned here?--Hpstricker (talk) 23:18, 31 January 2010 (UTC)[reply]

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I think some of links on the footer are indifferent with "mathematical object as a philosophical concept". —Preceding unsigned comment added by 220.209.7.44 (talk) 14:46, 14 December 2010 (UTC)[reply]

Needs better explanation

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Perhaps you could explain this in PLAIN ENGLISH? 78.86.145.139 (talk) 18:50, 1 February 2011 (UTC) JustSomeBoy[reply]

Ugly opening

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The opening sentence of this article is ugly and simply an attempt to play the Philosophy game. Can the primary editors of this page please fix? 68.168.179.92 (talk) 21:38, 4 March 2012 (UTC)[reply]

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Suggestion: Categorical approach

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This article needs a lot of help. But most notably, it is heavily reliant on philosophical articles, which is fine, but it needs some grounding in modern mathematical notions. I suggest moving this in the direction of how "objects" are defined in category theory. That way this article has some better grounding and direction. Farkle Griffen (talk) 18:22, 20 August 2024 (UTC)[reply]

Neutrality of "In philosophy of mathematics" section

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Much of this section was written by me and is therefore likely to be biased. I ask that others look over and help make the section more neutral before this tag is removed. Farkle Griffen (talk) 20:19, 28 August 2024 (UTC)[reply]

While your editorial care is appreciated, this isn't usually what explicit neutrality concerns are raised over: if you didn't cherrypick your sources and you stuck to what they said you should be totally fine. As such, I'm removing the banner. Remsense ‥ 论 20:42, 28 August 2024 (UTC)[reply]
How are you so sure I didn't cherry pick? Lol
No, but really, I have my own views on the subject, and it's not only possible but probable that it leaked into my interpretation of the sources. And moreover, I didn't look into the possible bias for many of the sources used. I'm mostly just looking for someone to give it a once-over to make sure I didn't write some glaring misrepresentation. Farkle Griffen (talk) 02:13, 29 August 2024 (UTC)[reply]
I getcha! In any case, the neutrality tag comes off as far more severe than it's likely intended here. I thought your contribution was a pretty cogent addition to the article, anyway. Remsense ‥ 论 02:15, 29 August 2024 (UTC)[reply]

List of objects section suggestions

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Given that there are so many diffrent areas of math, it may be useful to begin governing the order of the branches (and maybe objects within them too). I'd like to sort them alphabetically, but this sounds rather tedious. Is there a way to do this automatically? Also, how do we tell future editors to continue in this order? I'd use a comment, but those don't seem to fully display until clicked on.

Second, a quick one sentence summary of the branches and/or objects couldn't hurt, and could only spark more curriosity in readers to click on the links. But mostly, this would help fill the, currently mostly empty, horisontal space of this section.

Last, would it be okay to add an {{Expand section}} tag to this section? It doesn't necessarily need expansion, but it may be helpful to encourage others to add to the list and make it more comprehensive. As it stands now, one could argue that it covers very little of the vast list of all mathematical objects. Farkle Griffen (talk) 03:43, 29 August 2024 (UTC)[reply]

Sounds reasonable. How do you suggest handling ordering, given differences in notation?
Conspicuously[a] missing are the objects of algebraic geometry, algebraic topology, Functional analysis, homological algebra, measure theory and probability theory. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:05, 29 August 2024 (UTC)[reply]
In alphabetical order by the title of their linked article seems simple enough (though, there will probably be exceptions, like articles starting with "mathematical")
The point is mostly for navigation. If someone wants to add an object or branch, they should know generally where to place it.
It may also be helpful to put subfields in the same section as the main branch. For instance, making algebraic topology as a subsection in the Topology section. But I can how this might cause problems. For instance, whether algebraic topology belongs in the topology section or the abstract algebra section. Or whether addition belongs in number theory or elementary algebra.
I think for now, it would be best to just place it in either, and deal with disputes as they're brought up.
(Unless someone else has a better idea) Farkle Griffen (talk) 15:05, 29 August 2024 (UTC)[reply]
I would say that algebraic topology should be under topology but that homological algebra should be under algebra. But what of homology theory and cohomology theory? They pop up in both Analysis[b] and topology. What of hybrid disciplines, e.g., Lie groups, Banach algebras? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:54, 29 August 2024 (UTC)[reply]
Again, I think just choose a section. It's not that big of a deal, the point of that section is basically just to list a bunch of objects. It's not really about trying to accurately categorize all branches of math.
If someone disagrees with a placement of a topic, we can deal with it then. Farkle Griffen (talk) 17:24, 29 August 2024 (UTC)[reply]

This list has several issues

  • WP:NOTDATABASE applies here
  • It is too long to be useful to anybody (it is unbelievable that somebody come here to know whether something is a mathematical object).
  • "Mathematical object" is a colloquial term that is not mathematically defined. The list suggests the contrary.
  • The list contains many entries whose qualification as mathematical objects is controversial or depends on context. For example, arithmetic operations are generally not considered to be mathematical object when there are used, but are clearly mathematical objects if considered as bivariate functions or as ternary relations.

So, I suggest to remove this list and to replace it with a few well-chosed examples. D.Lazard (talk) 09:11, 31 August 2024 (UTC)[reply]

If someone wants to keep the list, it could be made into a standalone List of mathematical objects, but I agree it does not belong here. –jacobolus (t) 16:04, 31 August 2024 (UTC)[reply]
I've opted to go ahead and create the article since there does not appear to be much opinion against it Farkle Griffen (talk) 23:40, 3 October 2024 (UTC)[reply]
I agree with the first point. However, I do have one note. The list is supported by the lead; all items in the list have been "formally defined," and for all, one may use for "deductive reasoning and mathematical proofs."
And for the arithmetic operations: in introductory number theory, for instance, one often derives properties of addition from the Peano axioms. In these situations one is certainly using addition as an object. Though, I agree, in most situations, arithmetic operations aren't thought of as objects.
I think for the latter two points to be considered, the lead would need to be rewritten to support the change.
I also have one question about replacing the list with "a few, well-chosen examples," By what criteria do we chose which objects can be listed and which shouldn't? Do you have any suggestions?
I do have one suggestion in the direction of this change: I think only terms which refer to specific objects should be listed; for example, in Geometry, "Square" should be included, but "Shapes" should not. Farkle Griffen (talk) 21:24, 2 September 2024 (UTC)[reply]

Notes

  1. ^ A different editor might give a disjoint list of missing areas. ;-)
  2. ^ E.g., de Rham cohomology.

Lead change suggestions

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As talked about in the previous discussion, the lead may need some clarification. I think it would be best if this change were left up to consensus here as it is very likely to be a contentious issue. I believe the current lead is more or less fine, but needs a clairification sentence that explains how what is considered an "object" depends on context.

However, as a more aggressive change, I suggest the following

"A mathematical object is an abstract concept used in mathematics to represent and reason about various structures, patterns, and relationships. These objects include numbers, shapes, sets, vectors and more abstract entities like spaces, categories, and transformations. Unlike physical objects, mathematical objects may not necessarily be tangible or have any physical properties whatsoever. They are typically defined through construction from more fundamental objects or taken as primitives, and described by their properties and relationships to other objects using axioms. For example, the number '3' is a mathematical object that represents a specific quantity; it is not tied to any specific physical representation but rather defined by its arithmetic properties and relationship to other numbers (e.g., '3 + 2 = 5')."

"What exactly constitutes an “object” depends on the context. In mathematical logic and proof theory, concepts like formulas and mathematical proofs are considered objects, whereas outside these branches, they are seldom referred to as such. Bertrand Russell once even suggested that natural numbers themselves are not objects but rather variables and the Natural numbers represents an arbitrary set of objects that satisfies the Peano axioms. However, generally, a mathematical object can be thought of as anything that has been (or could be) formally defined and with which one intends to reason about or derive properties."

I'm not suggesting this be the final version, it is just my attempt to clarify the topic, and I would like input and other suggestions.

(Edit: added a bit of elaboration and links.) Farkle Griffen (talk) 16:35, 3 September 2024 (UTC)[reply]

IMO, the current version is much better. Nevertheless, the current lead has many issues. I have prepared a project for a new lead at
User:D.Lazard/Lead for Mathematical object.
Before implementing it, I must write sevral sections for expanding the paragraphs of the new lead (Normally, a lead must be a summary of the content of the article).
Also third party opinions are needed in view of a consensus. D.Lazard (talk) 18:20, 3 September 2024 (UTC)[reply]
Could you elaborate on what exactly you dislike about what I have above? Again, it was just an attempt at clarification, and wasn't intended to be a final version. Though your opinion would be helpful in determining what changes need to be made to bring this closer to a final version. Farkle Griffen (talk) 18:33, 3 September 2024 (UTC)[reply]
My main objection is that you introduce a confusion between object (philosophy) and object (mathematics). This is an article on mathematics, not on philosophy of mathematics. Also, almost every sentence is misleading or wrong in the context of mathematics of the 21th century. D.Lazard (talk) 19:50, 3 September 2024 (UTC)[reply]
The first point is a fair objection, that I will respond to in a moment, however, the second point "almost every sentence is misleading or wrong in the context of mathematics of the 21th century," I'm looking over this and, apart from the second-to-last sentence on Russell, I fail to see how any other sentence could be considered "wrong". Farkle Griffen (talk) 01:24, 5 September 2024 (UTC)[reply]
@D.Lazard, To the first point, there is reasonable overlap between these two. Any attempt to completely remove the philosophy notion from this article would be disingenuous to the topic. For instance, the article on Abstract Objects - Stanford Encyclopedia of Philosophy talks almost entirely about mathematical objects. In fact, there are thousands of papers in philosophy about mathematical objects. This article has just as much reason to be about the term's use in philosophy as it does its use in mathematics. Farkle Griffen (talk) 19:02, 9 October 2024 (UTC)[reply]
There is a third type of mathematical object, although it is anethema to some camps; an object whose existence is not asserted by an axiom or shown by a construction, but has been proven by nonconstructive proofs. Some of these involve the axiom of choice. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:15, 17 September 2024 (UTC)[reply]
The version of the lead to which you refer is unclear, as well as the two types to which refers "third". For clarification and for ease of the discussion, I copy here my proposal for a new lead (User:D.Lazard/Lead for Mathematical object).
"Mathematical object", or simply "object" is a colloquial term used in mathematics for anything that has been (or could be) defined in mathematical terms, and whose properties may be deduced with mathematical proofs. Typically, a mathematical object can be assigned to a variable, can be quantified, and therefore can be involved in formulas. Common mathematical objects include numbers, sets, mathematical structures such as field and spaces, functions, expressions, geometric objects, and transformations. Mathematical objects can be very complex; for example, in mathematical logic, theorems, proofs, and even theories are considered as mathematical objects.
Being a colloquial term, there is no formal definition of the concept, and it may depend on the author and the context whether a mathematical entity is considered as a mathematical object. For example, arithmetic operations are not generally considered as mathematical objects when used for computing, but are when studied as bivariate functions or ternary relations.
The term "mathematical object" was introduced in the 20th century, with the generalization of the use of set theory and the axiomatic method, which led to assign to variables and to manipulate new "objects" such as infinite sets, algebraic structures and spaces of various nature.
The objects of a category are mathematical objects, but many mathematical objects, such as numbers, are not objects of any category.
Mathematical objects are weakly related with abstract objects of philosophy: mathematical objects are abstract objects if one accept mathematical Platonism, but the concept of abstract object is much wider than that of mathematical object.
Comments and improvements are welcome. D.Lazard (talk) 21:03, 17 September 2024 (UTC)[reply]
I am no kind of expert in the philosophy of mathematics, but this seems nice to me. –jacobolus (t) 22:56, 17 September 2024 (UTC)[reply]
I'm referring to They are typically defined through construction from more fundamental objects or taken as primitives, and described by their properties and relationships to other objects using axioms. in the version proposed by Farkle Griffen on 3 September 2024; that text does not appear in your version. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:31, 18 September 2024 (UTC)[reply]

Replacement of first image in lead

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There are suggestions by multiple users that the current image of a tesseract is not a good example for the first image, and are attempting to delete it. This post should serve as a discussion of what to replace it with.

Until a consensus is reached, deleting the image is not an improvement of the article, so I will be reverting its most recent deletion.

As a response to D.Lazard's last edit summary: "This is not a good choice for the first image, as suggesting a confusion between geometric objects and mathematical objects: a geometric object is a mathematical object, but many mathematical objects have nothing to do with geometry"

This objection goes for any example of a mathematical object. "An object from [area of math] is a mathematical object, but many mathematical objects have nothing to do with [area of math]."

[Edit: I've added emphasis to 'Geometry' in the image description to address the concern temporarily] Farkle Griffen (talk) 21:38, 3 October 2024 (UTC)[reply]

A matrix is a mathematical object made up from other mathematical objects. So the image of a matrix could illustrate the sentence of the lead that asserts that mathematical objects can be complex. D.Lazard (talk) 07:43, 4 October 2024 (UTC)[reply]
My only real issues with this are that matricies are just kinda boring visualy, and for the average reader, wouldn't be very familliar. However, I believe I have a compromise: {{Multiple images}}.
I think the current list of examples in the lead is subject to the same issue as before: its not helpful to the average reader. I think it should be replaced with a short list of the broadest possible classifications of objects that would still be familliar to the average reader, such as "Numbers, expressions, shapes, functions, and sets" (here mimicking the list of subjects in the lead paragraph of the Mathematics article).
Then, we can use a few pictures as examples of the objects, like pictures of a Triangle (for shape), Counting blocks (for number), and the Graph of a function. Of course, these are just a few options. Other suggestions might be a Rubik's Cube, (which seems to be a fairly common image on wikipedia mathematics articles), any image of a set, a matrix as proposed (for function), or re-adding the image of a tesseract. I think multiple images is the only way to get around the original reason for removing the previous image. Farkle Griffen (talk) 16:41, 1 November 2024 (UTC)[reply]

Edit war

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@Farkle Griffen and D.Lazard: There is a string of reverts and re-reverts starting with permalink/1256942694. Contrary to the claim in the summary, the reasons for the disputed edit were not discussed last week. Please discuss here. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:51, 12 November 2024 (UTC)[reply]

There is only two reverts each, the last few are me reverting myself to fix an edit summary. I don't think this is necessary, at least not yet.
My only reason for the first revert was that they reverted the whole first paragraph, rather than just fixing the individual word they seem to take issue with. About the second revert: beyond that individual word, replacing the list of simple objects, was mentioned over a week ago: November 1st, in the discussion just above this one.
At no point did I dispute the reverting of that one individual word.
If we would like to discuss that here, we can, but that had nothing to do with my previous reverts. Farkle Griffen (talk) 14:57, 12 November 2024 (UTC)[reply]
You also removed geometric objects, transformations of other mathematical objects, and spaces, and changed a link, with no explanation.
I would assume that it is those changes, plus the inappropriate symbol, rather than the reordering, that D.Lazard objects to. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:12, 12 November 2024 (UTC)[reply]
To reiterate what I said on November 1st:
"I think the current list of examples in the lead is subject to the same issue as before: its not helpful to the average reader. I think it should be replaced with a short list of the broadest possible classifications of objects that would still be familliar to the average reader, such as "Numbers, expressions, shapes, functions, and sets" (here mimicking the list of subjects in the lead paragraph of the Mathematics article)."
I do not understand the objection here.
The change was mentioned: I said the initial list should be changed to this, and I did. If there are objections to the content, then make those objections. But the objection "It was not discussed before" is not true
Farkle Griffen (talk) 16:41, 12 November 2024 (UTC)[reply]

Object of the article

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The article is very confusing: the lead is only about the mathematical use of "object", while the body is only about the concept of object (philosophy) in philosophy of mathematics.

This confusion seems the main cause of the recurrent edit wars about this article. The only way I have found for resolving the problem consists of

D.Lazard (talk) 14:29, 12 November 2024 (UTC)[reply]

This seems like overkill. The most recent "edit war" had nothing to do with "philosophy" versus "mathematics", and it seems like an unnecessary WP:Content Fork and WP:RMUM applies here. It is a very bold claim that the use of "Mathematical object" in philosophy of mathematics has nothing to do with "mathematical object" in mathematics, which you seem to be making baselessly, and moving whole articles without consensus, nor any discussion at all.
I very much disagree with this change, so per WP:BRD, it should be reverted, and for such a large change, resistance should be met with a user report for disruptive editing if you do not discuss before trying to redo this effort. Farkle Griffen (talk) 15:13, 12 November 2024 (UTC)[reply]
The previous situation was: a reader who want information on a term of mathematical jargon is led to an article of philosophy of mathematics, where the difference betwee the mathematical meaning and the philosophical meaning(s) is never explained, and a reader interested in philosophy starts with a mathematical definition that seems rather unrelated with the content of the article.
In any case, when a mathematician talks of an "object" outside any philosophical context, this is never the philosophical meaning that is involved. The only relation between these two concepts is that, if one adopt the philosophical view of Platonism, the objects considered by mathematiciens are philosophical objects. The reverse is false: for example, the concept of integration may be considered as a philosophical object, but it is certainly not an object for mathematicains. So, Object (mathematics) and Mathematical object (philosophy) are two different subjects, and do not belong to the same article.
Also, the use of "object" in mathematics being jargon, no other content is available than the short definition given in List of mathematical jargon. So there is not enough content for a specific article. Talking of WP:Content Fork for a single sentence fragment seems excessive.
Before reverting me or accusing me of disruptive editing, you should better to suggest a solution that is more convenient to most readers. You could also rewrite the lead of Mathematical object (philosophy) (whichever name it has) to be conform to the guideline MOS:LINE that says The lead section should briefly summarize the most important points covered in an article, in such a way that it can stand on its own as a concise version of the article. D.Lazard (talk) 16:11, 12 November 2024 (UTC)[reply]
I am not going to respond to the content of this before the original article is moved back. This discussion should have happened before the article was moved, not now. Such major changes to long-standing articles, especially in a way that now seems to require Admin assistance to undo, without consensus or any discussion whatsoever is extremely disruptive. So, again, I refuse to entertain this discussion before the change is undone. Farkle Griffen (talk) 16:30, 12 November 2024 (UTC)[reply]
It seems to me that the "long-standing" version is special:permalink/1239697033, which consisted of a quick definition + a mediocre list, which has since been moved to List of mathematical objects. (Personally I don't particularly mind replacing this with a philosophy article. Not sure how much there is to say otherwise.) –jacobolus (t) 16:42, 12 November 2024 (UTC)[reply]
I agree with you here. D. Lazard's claims here frankly seem to be academic conflict of interest editing: there are very active conflicts over these issues in the academy, D. Lazard has expressed a definite favor for a particular side without acknowledging others (i.e. not WP:DUE), and his professional mathematical legacy is a competitive interest. My naive judgment from reading other recent COI discussions would suggest this is a legitimate issue to raise here. This is also informed by a reading of the recent Algebra Good Article review, such as the DUE concerns in this thread, Wikipedia:Featured article candidates/Algebra/archive1#c-XOR'easter-20240930160500-Mathwriter2718-20240930005400, where D. Lazard's comments had a similar quality and again his professional interest seemed clearly to create a potential COI. RowanElder (talk) 16:36, 12 November 2024 (UTC)[reply]
This accusation is way out of line, in my opinion. Please see Wikipedia:What Wikipedia is not § Wikipedia is not a battleground, Wikipedia:Harassment § Wikihounding, Wikipedia:No personal attacks, and for general advice Wikipedia:Civility. –jacobolus (t) 16:43, 12 November 2024 (UTC)[reply]
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