Talk:Mathematical object

Merge of Abstract mathematical object

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)

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

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

Links in the footer

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)

Needs better explanation

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

Ugly opening

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)

External links modified (January 2018)

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Suggestion: Categorical approach

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)

Neutrality of "In philosophy of mathematics" section

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)

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)

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)

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)

List of objects section suggestions

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)

Sounds reasonable. How do you suggest handling ordering, given differences in notation?

Conspicuously 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)

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)

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 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)

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)

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)

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)

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)

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)

Notes

Lead change suggestions

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)

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)

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)

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)

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)

@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)

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)

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)

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)

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)

Replacement of first image in lead

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)

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)

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)

Edit war

@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)

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)

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)

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)

Object of the article

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

• Creating an entry "object" in List of mathematical jargon ( Done)
• Creating a WP:redirect page named Object (mathematics) that redirects to this entry ( Done)
• Creating a WP:dab page named Mathematical object (disambiguation) that links to the new redirect and the present article ( Done)
• Moving the present article to Mathematical object (philosophy) ( Done)
• Moving (if I can) or redirecting Mathematical object (disambiguation) to Mathematical object ( Done)
• Updating Mathematical object (philosophy) for focusing only to philosophy (I leave this task to other editors)

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

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)

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)

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)

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)

This discussion has been closed. Please do not modify it.
The following discussion has been closed. Please do not modify it.
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) 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)

Consensus 1: Existence of an exact definition

To preface, this is not the place to propose definitions, nor is this the place to propose descriptions of mathematical objects.

Topic: Does there exist an (accepted or acceptable) exact definition of "Mathematical object"?

Please write a bold Yes. or No. as the first word in your reply to be counted.

After going through the "Further reading" listed in the article, and raiding my university library's section on mathematics over the past few weeks, I've come up somewhat empty-handed on this question. Several books contain discussions similar to what is already in the "In Philosophy of mathematics" section here, however, none have said anything close enough to "Mathematical objects do not have an exact definition", or "A mathematical object is..." to count as a source.

But, different philosophical perspectives on what "Mathematical objects" foundationally are may not be evidence of the non-existence of a meaningful definition; after all, there are different perspectives on what numbers are, but that doesn't mean there is no meaningful definition of "number".

However, I'm not perfect, and I could have easily missed something. So, as to comply with WP:NOR, I am looking for the consensus of other editors before asserting there is no generally accepted/acceptable definition.

To reiterate, this is not the place to propose personal definitions or descriptions for the article. If the consensus decides "No", then I will make a second discussion about proposals for descriptions of the topic. Farkle Griffen (talk) 18:53, 13 November 2024 (UTC)

No. In my opinion, it seems that any exact definition of "Mathematical object" would require having an exact definition of "Math" in general, which is known to be controversial. Farkle Griffen (talk) 18:56, 13 November 2024 (UTC)

(Came here via your notice at WT:WPM.) Meta-comment: with very rare exceptions, discussions on Wikipedia are not votes, and you don't get to prescribe rules about which contributions to the discussion "[are] counted". If normal discussion does not result in the formation of a consensus, you can try an RfC next; however personally I don't think the question you've raised here is well suited to an RfC, at least as currently formulated. --JBL (talk) 21:32, 13 November 2024 (UTC)

Okay, I can delete the post if you believe it is unproductive. (I'm not all that attached to it)

The point was to get general opinion on whether there exists an exact definition, which is a yes or no question. As brought up a few times before, the article currently sounds like "mathematical object" is a precisely defined term, but it doesn't draw a clear line between mathematical objects and other objects.

All the sources I've found have vaguely alluded to the fact that there's no precise definition, but none state it explicitly, and I'm trying to avoid WP:OR by just adding that to the article without a source. What do you suggest I do instead? Farkle Griffen (talk) 21:56, 13 November 2024 (UTC)

As a side note: it wasn't a vote, just a tally. If 4 people said "yes" and 3 said "no", that is clearly not a consensus.

And I've definitely seen several other articles using this "please bold the first word Agree or Oppose to be counted" or something like that. It just helps ease tallying when there is a clear answer.

They're not "not counted" toward the discussion, just toward the tally if it's not clear Farkle Griffen (talk) 22:07, 13 November 2024 (UTC)

Well, the notion of object exists in category theory. So as long as you work with the language of category theory, the answer is yes? The adjective “mathematical” is needed to distinguish from say everyday objects, and in mathematical writing, you usually speak of objects. —- Taku (talk) 06:19, 14 November 2024 (UTC)

Yes, but that provides no guidance on the usage of object outside of Category Theory. Overloading of words in Mathematics is common, e.g., closed, field. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:36, 14 November 2024 (UTC)

I guess I'm saying that concepts like “objects” are part of language to do mathematics. Category theory is one language. In another language, there can be a concept called “object”, which may or may not be the same one in category theory. So it’s a matter of which language to do math (and I guess I don’t believe you can do math without a language like category theory). —- Taku (talk) 04:31, 15 November 2024 (UTC)

This article is not referring to objects in category theory, but to mathematical objects (i.e. concepts defined abstractly in mathematics). There are clear differences; for example, one could take the set of people as objects and one morphism of the identity function, but you are clearly not a mathematical object (unless you are purely an abstract entity... which, technically, could be true). BTF Flotsam (talk) 02:27, 16 November 2024 (UTC)

The notion of infinity feels a bit vague and philosophical but can be given precise definitions in set theory for example. But the point here is that set theory here is a language that gives such sense. Category theory is one way to handle objects in rigorous ways in addition to other vague notions like naturalness. Of course, category theory is not the only way to do math but depending on languages, objects can be handled rigorously. —- Taku (talk) 07:46, 16 November 2024 (UTC)

We're talking about if there is a rigorous definition for "abstract". If you think it is rigorously defined, explain if the number one is a mathematical object. BTF Flotsam (talk) 05:48, 17 November 2024 (UTC)

Wittgenstein would say you can’t define anything anyway; so we shouldn’t be discussing whether there is a rigorous definition or not at all, without references. That is, my point is that it’s language-dependent. Within some language, some abstract stuff have perfectly reasonable and working definitions. —- Taku (talk) 13:06, 17 November 2024 (UTC)

This reads like the kind of question that would be asked by an AI, or maybe by someone hoping to become a philosophy student. Why would a thinking person want to know whether there is an exact definition for the sky, or for laughter, or for mathematical objects? What would such a thing even look like? What is the function of "exact" in this question? If you're not sure how to provide an exact definition of a mathematical object, how can you evaluate any definition for exactness? And if the question is phrased with a preface telling us that we cannot describe any definitions, what is an answer to this question even supposed to look like? An up-and-down vote, yes or no, with no explanations? —David Eppstein (talk) 01:01, 15 November 2024 (UTC)

The point was, I don't think an "exact definition" (whatever that means) exists, and to start clearing up what exactly this article is about, I'd like to note that in the article, but I have no source to cite. So, I figured it would leave it up to consensus: First, a consensus that "No, there is no 'exact definition' (whatever that means)", and second, a consensus over how to more precisely describe the subject of the article, without misleading the reader to believe that such a definition does exist. Of course, I could be wrong and a definition could exist, which is why I'm not rushing to just edit my non-findings into the article.

As for what this should look like, I agree I could have been clearer. You can explain your answer, but you shouldn't propose a definition, e.g. "I believe the definition/description should be...", as this is just a "Yes or no: does a definition already exist?"

As for exactness: I have no exact definition of exact, however, the current source in the lead (Oxford Dictionary) defines it as "Designating or relating to objects apprehended not by sense perception but by thought or abstraction.", which is clearly far too broad to be considered an "exact definition" (whatever that means), but feel free to disagree.

Does that clear things up a bit? Farkle Griffen (talk) 01:53, 15 November 2024 (UTC)

You can't answer yes or no to a meaningless question. —David Eppstein (talk) 02:43, 15 November 2024 (UTC)

That answer works too Farkle Griffen (talk) 04:20, 15 November 2024 (UTC)

I think claiming that such a definition doesn’t exist can be misleading. Nominalists might argue that mathematical objects like numbers or functions are exactly what mind can conceive and we don’t and shouldn’t reject them. (In Wikipedia, we don’t necessarily believe in platonism.) It’s like it may be true there is no God but we can’t categorically say so in Wikipedia. —- Taku (talk) 05:03, 15 November 2024 (UTC)

The problem we are faced to is not to find a definition of "mathematical object". It is to decide whether there is one or several concepts with this name, and to know what is called an object in mathematical contexts. I know three usages of this phrase:

1. Objects of a category. There is no problem here: we have a specific article, and when talking of this sort of object, the category must always be explicited.
2. "Object" is a term of mathematical jargon that is used for elements of unspecified sets or for something that can be assigned to a variable (this is essentially the same). For example: "in the relation ⁠$x\in X$⁠, the right-hand side ⁠$X$⁠ must be a set and the left-hand side ⁠$x$⁠ can be any object." There is no philosophy at all in this usage.
3. "Mathematical object" may also refer to the abstract objects considered in mathematics. In this sense, it is a concept of philosophy of mathematics rather than a concept of mathematics.

Until the first Farkle Griffen's edit on 27 August 2024, only the meaning 2 was considered in the article. Presently, the whole article is about the meaning 3, except for the first paragraph of the lead. Having a single aticle for two different concepts is usual for dictionaries, but not for an encyclopedia (see WP:NOTDICT).

This is a reason for having two different articles for meanings 2 and 3. This why I tried boldy to transform Mathematical object into a disambiguantion page. After being reverted, I consider that this thread is the right place for discussing further my proposal consisting of renaming the present article as Mathematical object (philosophy), and transforming Mathematical object into a disambiguantion page linking to the 3 different meanings, namely Object (category theory), Object (mathematical jargon) and Mathematical object (philosophy) (or Mathematical object (philosophy of mathematics). D.Lazard (talk) 09:43, 17 November 2024 (UTC)

A minor quibble with this otherwise-helpful classification of mathematical usages of "object": alongside the specifical technical meaning of objects in a category in category theory (#1), one should probably also include the inhabitants of a type in type theory. This also starts to shade over towards Object (computer science). —David Eppstein (talk) 23:28, 19 November 2024 (UTC)

Wikipedia does not publish original thought. All material in Wikipedia must be attributable to a reliable, published source. Do you have a source for these claims? Farkle Griffen (talk) 08:06, 18 November 2024 (UTC)

Not all material: see WP:the sky is blue. Wikipedia must mention colloquial usage, and colloquial usage can rarely be sourced. This is the case here. Note that Glossary of mathematical jargon has many entries that use "object". These refer sometimes to Object (category theory), often to Object (mathematical jargon), but never to "mathematical object" in the sense of philosophy of mathematics. As I am not the author of these entries, this shows that the use of "object" in mathematical jargon is not original research. D.Lazard (talk) 11:32, 18 November 2024 (UTC)

What I wrote was an exact quote from WP:NOR. The keyword here is "attributable", not "attributed". We shouldn't have to source "The sky is blue" because it is WP:Common knowledge, it is assumed that there exists a source, and is unlikely to be WP:CHALLENGED. Nevertheless, WP:The sky is blue is an essay, not a policy nor guideline.

I am not challenging that the word "object" is used in math, I am challenging that (1) your 2nd definition has any WP:RS, and (2) that the term "mathematical object" in Philosophy of mathematics and the term "mathematical object" in mathematics are not the same has any RS. This is, quite simply, a very bold claim, and goes against my experience of this subject. I am challenging it as WP:OR. Per WP:CHALLENGE, you must provide a reliable source. So I'll ask again: do you have a source for these claims? Farkle Griffen (talk) 19:28, 19 November 2024 (UTC)

You have no source either that "mathematical object" in philosophy of mathematics is the same as "mathematical object" in mathematical jargon. On the opposite, Jacobolus provided below a highly reliable source (the Princeton Companion to Mathematics) that uses "object" in a mathematical context that is away from any philosophy, with exactly the meaning given in my definition 2. So, this is you, who are trying to impose your original research, by pretending without any source that the two concepts are the same. D.Lazard (talk) 19:49, 19 November 2024 (UTC)

"this is you, who are trying to impose your original research, by pretending without any source that the two concepts are the same"

Please see Wikipedia:Assume good faith. If you believe I am imposing WP:OR, you can simply ask if I have a source. For a beginners introduction to the philosophical aspects of the subject, see Thinking about Mathematics: Philosophy of Mathematics, Section 2.1: Object, by Stewart Shapiro, which has been linked on this article well before my first edit. A primary source in this case, but a source nonetheless. In the first paragraph, he explains the topic as follows:

"One global issue concerns the subject-matter of mathematics. Mathematical discourse has the marks of reference to special kinds of objects, such as numbers, points, functions, and sets. Consider the ancient theorem that for every natural number m>n. It follows that there is no largest prime number, and so there are infinitely many primes. At least on the surface, this theorem seems to concern numbers. What are these things? Are we to take the language of mathematics at face value and conclude that numbers, points, functions, and sets exist? If they do exist, are they independent of the mathematician, her mind, language, and so on? Define realism in ontology to be the view that at least some mathematical objects exist objectively, independent of the mathematician."

He does not introduce any other notion of "object" before or after this section. Any reasonable reading of this would suggest that he is talking about the same "numbers, points, functions, and sets" as any mathematician. Unless you're also accusing Shapiro of being misleading.

"Jacobolus provided below a highly reliable source (the Princeton Companion to Mathematics) that uses "object" in a mathematical context that is away from any philosophy, with exactly the meaning given in my definition 2."

Please provide the quotation where they explicitly state your definition.

And I believe you're misundertanding what I'm saying. I'm certainly not saying "mathematical object" is a philosophical term. If anything I would be more willing to assert that "mathematical object" is a common language term. "object" with the common meaning, and "mathematical" meaning relating to mathematics. This seems reasonable since no book on Philosophy of math that I've ever read has said "We define a 'Mathematical object' as:...", since they expect you to understand what the term means. Similarly, no book on mathematics I've ever read has either (Category theory being somewhat of an exception).

You, on the other hand, seem to be asserting that there are three distinct, explicit definitions apart from common language. So I'll ask you again: do you have a source for these claims? Farkle Griffen (talk) 01:06, 20 November 2024 (UTC)

I wouldn't make the claim that these are different types of "objects" being talked about, but rather that the concerns, interests, and methods of discourse of philosphers are largely ignored by and often considered irrelevant to mathematicians in their practice. So focusing on topics of interest to philosophers may give a misleading view to any reader who is looking to know what mathematicians mean when they say "object". Mathematicians don't generally concern themselves (at least in their practical work) with, e.g., what it means for the rational number ⁠${\tfrac {1}{2}}$⁠, the sine function, the regular dodecahedron, the symmetric group of 3 symbols, or the differential operator del to "exist". Instead they ask questions such as "how can I combine these objects?" or "how can I define this object in terms of previously defined objects?" or "do these two characterizations result in the same properties?" or "do these relations between different objects have essentially the same structure or not?" –jacobolus (t) 19:50, 19 November 2024 (UTC)

I agree with this. Currently the article has an WP:NPOV issue only covering the philosophical aspects of the subject. I've been putting off adding the sections about the mathematical uses of the topic, trying to find an explicit definition, which was the purpose of this discussion topic.

And yes, mathematicians don't care about whether these objects exist since existance is not a predicate, that is, the existence of numbers does not change the meaning of the statement "1+1=2," or analogously, the existence of unicorns does not change the meaning of the statement "Unicorns have horns." But philosophers do care, and just because different subjects cover the same topic does not mean they need a seperate article; that would be a WP:REDUNDANTFORK: both articles are of the same type (outlines) and about the same topic (mathematical objects).

Given how this discussion has gone, I think it's reasonable to say (at least until proven otherwise) that mathematicians do not sponsor an explicit definition, which is fine, and that can be worked around. I'll work on adding more sections about the mathematical use, and try to avoid misleading the reader to believe a definition exists. Farkle Griffen (talk) 20:45, 19 November 2024 (UTC)

Another source that might be useful is Davis & Hersh (1981) The Mathematical Experience (and perhaps also Hersh's 1979 essay "Some Proposals for Reviving the Philosophy of Mathematics"). –jacobolus (t) 01:51, 20 November 2024 (UTC)

If you search through a source like the Princeton Companion to Mathematics you'll find many uses of "object" (but no explicit definition), e.g. here are some examples from within the first few pages:

Extended content

"[...] algebraic topology is almost entirely algebraic and geometrical in character, even though the objects it studies, topological spaces, are part of analysis." "One algebraist might be thinking about groups, say, in order to understand a particular rather complicated group of symmetries, while another might be interested in the general theory of groups on the grounds that they are a fundamental class of mathematical objects." "A topologist regards two objects as the same if one can be continuously deformed, or 'morphed,' into the other;" "This means that algebraic geometry is algebraic in the sense that it is 'all about polynomials' but geometric in the sense that the set of solutions of a polynomial in several variables is a geometric object." "[Category theory] differs from set theory in that its focus is less on mathematical objects themselves than on what is done to those objects—in particular, the maps that transform one to another." "[...] it says that two objects, 5 and the square root of 25, are in fact one and the same object," "Broadly speaking, a set is a collection of objects, and in mathematical discourse these objects are mathematical ones such as numbers, points in space, or even other sets." "one of the great advances in mathematics was the use of Cartesian coordinates to translate geometry into algebra and the way this was done was to define geometrical objects as sets of points, where points were themselves defined as pairs or triples of numbers" "in geometry one often wants to consider the entire circle as a single object (rather than as a multiplicity of points, or as a property that points might have), and then set-theoretic language is indispensable." "A second circumstance where it is usually hard to do without sets is when one is defining new mathematical objects. Very often such an object is a set together with a mathematical structure imposed on it, which takes the form of certain relationships among the elements of the set." "Sets are also very useful if one is trying to do meta-mathematics, that is, to prove statements not about mathematical objects but about the process of mathematical reasoning itself." "One of the most basic activities of mathematics is to take a mathematical object and transform it into another one, sometimes of the same kind and sometimes not." "Over and over again, throughout mathematics, it is useful to think of a mathematical phenomenon, which may be complex and very un-thinglike, as a single object." "If ⁠$f$⁠ is a function, then the notation ⁠$f(x)=y$⁠ means that ⁠$f$⁠ turns the object ⁠$x$⁠ into the object ⁠$y$⁠. Once one starts to speak formally about functions, it becomes important to specify exactly which objects are to be subjected to the transformation in question, and what sort of objects they can be transformed into." "To specify a function, therefore, one must be careful to specify two sets as well: the domain, which is the set of objects to be transformed, and the range, which is the set of objects they are allowed to be transformed into." "(Such functions may not be useful as individual objects, but they are needed so that the set of all functions from one set to another has an interesting mathematical structure.)" "For the resulting grammatically correct sentence to make sense, the nouns should refer to numbers (or perhaps to more general objects that can be put in order). A mathematical 'object' that behaves like this is called a relation, though it might be more accurate to call it a potential relationship." "As with functions, it is important, when specifying a relation, to be careful about which objects are to be related. Usually a relation comes with a set ⁠$A$⁠ of objects that may or may not be related to each other." "There are many situations in mathematics where one wishes to regard different objects as 'essentially the same,' and to help us make this idea precise there is a very important class of relations known as equivalence relations." "If ⁠$P$⁠ and ⁠$Q$⁠ are statements (note here the mathematical habit of representing not just numbers but any objects whatsoever by single letters) [...]" "More generally, a variable is any letter used to stand for a mathematical object, whether or not one thinks of that object as changing through time." "There are some technicalities to sort out, but even these can often be avoided if one allows not just sets but also numbers as basic objects." "A number system consists of some objects (numbers) together with operations (such as addition and multiplication) that can be performed on those objects." "We can think of symmetries as 'objects' in their own right, and of composition as an algebraic operation, a bit like addition or multiplication for numbers." "The objects that belong to the vector space are usually called vectors, unless we are talking about a specific example and are thinking of them as concrete objects such as polynomials or solutions of a differential equation." "So we define two expressions ⁠${\tfrac {a}{b}}$⁠ and ⁠${\tfrac {c}{d}}$⁠ to be equivalent if ⁠$ad=bc$⁠ and we regard equivalent expressions as denoting the same number. Notice that the expressions can be genuinely different, but we think of them as denoting the same object." "Many other important objects in modern geometry are defined using quotients. It often happens that the object one starts with is extremely big, but that at the same time the equivalence relation is very generous, in the sense that it is easy for one object to be equivalent to another. In that case the number of 'genuinely distinct' objects can be quite small. This is a rather loose way of talking, since it is not really the number of distinct objects that is interesting so much as the complexity of the set of these objects." "What is absolutely not essential is the nature of the objects that have the structure: for example, one group might consist of certain complex numbers, another of integers modulo a prime p, and a third of rotations of a geometrical figure, and they could all turn out to be isomorphic." "At the top is another function, ⁠$T$⁠, but the 'objects' that it transforms are themselves functions: [...]" "Mathematics took a huge leap forward in sophistication with the invention of calculus, and the notion that one can specify a mathematical object indirectly by means of better and better approximations." "The notion of limit applies much more generally than just to real numbers. If you have any collection of mathematical objects and can say what you mean by the distance between any two of those objects, then you can talk of a sequence of those objects having a limit." "A conic section is the intersection of a plane with a cone, and it can be a circle, an ellipse, a parabola, or a hyperbola. From the point of view of projective geometry, these are all the same kind of object." "Informally, a ⁠$d$⁠-dimensional manifold, or ⁠$d$⁠-manifold, is any geometrical object ⁠$M$⁠ with the property that every point ⁠$x$⁠ in ⁠$M$⁠ is surrounded by what feels like a portion of ⁠$d$⁠-dimensional Euclidean space." "As we have seen in earlier articles, mathematics is full of objects and structures (of a mathematical kind), but they do not simply sit there for our contemplation: we also like to do things to them." "Because ⁠$x$⁠ and ⁠$y$⁠ can be very much more general objects than numbers, the notion of solving equations is itself very general, and for that reason it is central to mathematics."

Despite the lack of any explicit definition, these examples do give some ideas for topics that could be covered in a less philosophical article about mathematical objects. For example, it could discuss means of combination of objects, means of definition of various kinds of objects, ways in which different types of objects can have the same structure and be essentially the same, and so on. –jacobolus (t) 19:40, 18 November 2024 (UTC)

Thanks for the quotations. They allow sourcing the entry object of Glossary of mathematical jargon with a footnote such as "For examples of this use of object, see the first pages of the Princeton Companion to Mathematics". This would satisfy the policy of WP:Verifiability.

You suggest implicitely to expand this entry into a separate article. My impression is that there is not enough sourceable material for this. D.Lazard (talk) 18:21, 19 November 2024 (UTC)