Talk:Mathematical object

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)

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

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)

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)

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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)