Talk:Exterior algebra

On edits by Timo Omburo

It seems to me that this user rolled back all the changes that were a result of a careful discussion late last year among many editors.

The introduction to this article is, again, horrendously long. We have already discussed this. I do not want to have to rehash the same discussion every time - if they want to do extremely large edits, let them argue their case out in this talk page.

Otherwise, I am rolling back their changes in the lede, because of all the problems already listed below which they did not engage with, in a week or so.

MeowMathematics (talk) 06:54, 26 March 2024 (UTC)

Personally I think the lead in Special:PermanentLink/1178847481 (which I worked on some after D.Lazard) was a better general approach than MeowMathematics's replacement which ultimately settled at Special:PermanentLink/1195382941, or the current version Special:PermanentLink/1215275338, though I'm sure it would be possible to do better than any of these. I don't think MeowMathematics's version can really be characterized as resulting from consensus, and I intended to (someday) get around to reworking it again, but didn't have the energy to wade into. YMMV. –jacobolus (t) 09:20, 26 March 2024 (UTC)

I'm fine with that version of the lede. The MeowMathematics version is, to me, unacceptable. I've gone ahead and put that version in. Tito Omburo (talk) 12:47, 26 March 2024 (UTC)

I think this puts WP:BRD the wrong way around. There was no consensus for the "MeowMathematics" edits. The evidence of this lack of consensus is my rollback to earlier versions of things.

However, on substance, the "MeowMathematics" version was clearly inferior to what had been there before. Firstly, the lede should provide an accessible overview of the article, which the MeowMathematics version did not. Secondly, the purpose of a motivation section is to motivate the definition, so it makes sense to have it be first, before a formal definition. Thirdly, the definition in terms of formal symbols was not technically correct, and also lacked a reference. Various other issues with this article were as follows. Sources had been removed from various places, which I restored. Plucker embeddings and differential forms are discussed much later in the article, and they are out of scope for a section on motivating examples. A lot of the linear algebra section was referenced to a self-published work, and seemed out of scope for this article.

I do not see any consensus on the discussion page for MeowMathematics's edits. In fact, mostly people seemed to be at best neutral to these edits (advising them, for instance, to work on a draft before working here, or else to be bold). My changes to the lede were reverting it to an earlier consensus version based on years of discussions. My changes to the article itself mostly restored the old consensus ordering of the sections, and various consolidations of content to other parts (e.g., differential forms and Plucker embeddings to much later).

Incidentally, my version of the article is about 10% shorter than your version, and your primary complaint seems to be that the article was too long. If you wish to change back to your version from the prior consensus version, please discuss why you think yours is better. Tito Omburo (talk) 12:31, 26 March 2024 (UTC)

Inconsistent nomenclature

A recent edit had the summary Motivating examples: article later uses "anticommutative" for "skew-symmetric", changed for consistency. The article still uses both terms. I believe that both terms should be mentioned as equivalent at first use and that all further uses stick to one or the other. Which should be preferred? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:32, 18 June 2025 (UTC)

Length of lead

The lead is much too long. The second and third paragraphs, dealing with blades, should be in a subsequent section. I'm not sure about the last two paragraphs, dealing with universality and generality. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:43, 18 June 2025 (UTC)

Unfortunately the various shorter leads which have been proposed have been terrible. Tito Omburo (talk) 17:46, 18 June 2025 (UTC)

I think the lead is actually in an okay place at the moment. Some of the historical versions of the lead have been some combination of inscrutable, incomplete, or very wordy. Compare a few selected versions in reverse chrono order: 1192296245, 1179645309, 1164784535, 1036892821, 906722806 –jacobolus (t) 22:21, 18 June 2025 (UTC)

I actually prefer the 906722806 and 1036892821 leads. Tito Omburo (talk) 12:16, 19 June 2025 (UTC)

Just keep the first two paragraphs of the lede?

I rewrote the first paragraph of the lede. I think maybe the other four (!) paragraphs could be maybe moved to a separate place, early in the article? (If only Wikipedia had a "lede" + "abstract", aimed at different audiences!) Tito Omburo (talk) 20:33, 5 September 2025 (UTC)

The current state doesn't seem great. Maybe revert to a stable version and workshop this on the talk page or at talk:Exterior algebra/Lead sandbox or something? –jacobolus (t) 20:38, 5 September 2025 (UTC)

Fair enough. Hopefully you see what I am aiming at though? A two-paragraph lede that conveys the main ideas without undue technical burden. Obviously this was a poor attempt. Tito Omburo (talk) 20:46, 5 September 2025 (UTC)

I just think adding a new 2-paragraph version and then following it by the existing several-more paragraph version, with significant overlap between versions, leaves readers quite confused. This isn't so urgent a change that it has to be initially tested as a Frankenstein mashup. I'm not feeling up to thinking hard about the intro and organization of this article today, but I'll try to come back to it if I remember in a few days. –jacobolus (t) 21:47, 5 September 2025 (UTC)

Your link is now blue. Anyone is obviously free to edit! Tito Omburo (talk) 20:50, 5 September 2025 (UTC)

Blade terminology

I understand there has been some discussion on the introduction of the article which I am not up to speed with. However, I think the use of non standard terminology of blades for the wedge products of vectors (which I think most mathematicians would tend to call a k-vector or multivector, or just k-form in the more specific context of the exterior algebra of the dual of some vector space) is at best misleading in such a prominent position. Is changing it to more standard terminology a reasonable proposal? Arnaumas (talk) 23:55, 3 April 2026 (UTC)

Google scholar turns up thousands of papers using the name 'blade' in this context. The name was apparently invented because "multivector" and "k-vector" both typically mean something different. Specifically, a "k-blade" is an object that can be expressed as the wedge product of k vectors, a "k-vector" is, more generally, an arbitrary sum of k-blades, and a "multivector" is an arbitrary sum of k-vectors with arbitrary (i.e. possibly different) values of k. Conflating multiple separate types of objects by using the same name for them seems more confusing than consistently distinguishing them, as long as the definitions are clear within an article. Some sources seem to use the name "simple k-vector" instead of "blade". –jacobolus (t) 01:16, 4 April 2026 (UTC)

I do not doubt it is used but after looking at the search results you link, but it is just not terminology one finds in the standard textbook accounts like Lee, Tu, or Aluffi, just to name a few. If I am not mistaken, the aim of Wikipedia tends to be to reflect the broad consensus, rather than the specific views of the editors. My point is that using this terminology in such a prominent position is simply not reflective of the standard accounts of exterior algebras and of multilinear algebras more generally.

Additionally, going into specifics in the lede is, in my view, not particularly helpful. It distracts from the fundamental properties of the exterior algebra, namely multilinearity and exterior algebra. Arnaumas (talk) 18:07, 4 April 2026 (UTC)

What counts as a "standard textbook" depends substantially on where the reader is coming from. I assume the books you are talking about are Lee's Introduction to Smooth Manifolds, Tu's Differential Geometry, and Aluffi's Algebra: Chapter 0? All three of these are books aimed at mathematics grad students assumed to have done years of difficult background preparation. Two are about differential geometry, and one is an algebra book with a category theory slant. None of these books is going to be an appropriate introduction for, say, an engineer or scientist who wants to use the exterior product in their concrete calculations. The mathematicians have an extremely high-level and abstract focus, and the required background makes their often terse and jargon-filled explanations basically impossible for non-specialists to make sense of without spending literally years of effort to catch up. Limiting the audience to people who can follow the most general and abstract treatment is a really unpleasant kind of gate-keeping, which we should try to avoid wherever possible. It also often obscures the basic ideas, even for specialists. –jacobolus (t) 20:41, 15 April 2026 (UTC)

An example resource well known in computer graphics is Keenan Crane's Discrete Differential Geometry, whose sections about the exterior product are much more concrete and accessible than the books you mention, intended for practitioners with something like an undergraduate engineering background. –jacobolus (t) 20:59, 15 April 2026 (UTC)

As a different example, this American Mathematical Monthly article, Khosravi & Taylor (2008) "The Wedge Product and Analytic Geometry", JSTOR 27642558 has a quite accessible treatment which is readable by anyone who finished high school geometry and introductory undergraduate linear algebra. We should try to reflect the spirit of examples like these from within the first few sections of this article, and defer our discussion of tensor products, multilinear maps, and universal properties, etc. –jacobolus (t) 21:09, 15 April 2026 (UTC)

Or if you want a book chapter, from a quick skim Mikusiński & Taylor (2002). "K-Vectors and Wedge Products" from An Introduction to Multivariable Analysis from Vector to Manifold doi:10.1007/978-1-4612-0073-4_6 seems pretty accessible. ("Apart from mathematicians, we expect that physicists, mechanical engi­neers, electrical engineers, systems engineers, mathematical biologists, mathemati­cal economists, and statisticians engaged in multivariate analysis will find this book extremely useful.") –jacobolus (t) 21:12, 15 April 2026 (UTC)

Lede needs revision

I think the lede of this article needs serious revision. I made an edit which seems to be too terse, but surely a happy medium can be reached. A lot of the material that is currently there can be safely moved to a section on examples or even to applications. I understand that wikipedia's audience is broad, but I don't think the current state of the introduction is particularly helpful to anybody. Looking at other mathematics articles here, it does not seem to be the spirit to provide a gentle introduction, but rather to record the current consensus on the topic, which I think the current introduction fails to do. The emphasis on oriented areas is simply too narrow for such a fundamental topic like the exterior algebra. Arnaumas (talk) 18:14, 15 April 2026 (UTC)

Also, the section titled Alternating tensor algebra could probably be subsumed inside a of a section on the Cosntruction of the exterior algebra, which exhibits it both as a quotient of the tensor algebra and as a subalgebra of it, with the different normalisation conventions, which I think would be a useful thing to record since it is a common matter of confusion amongst students. Arnaumas (talk) 18:17, 15 April 2026 (UTC)

The current lede has the advantage that it explains what many of the terms used. For example, what it means that it is an algebra, a graded algebra, associative algebra, alternating. Oriented areas provide a model for what the product "means", useful for an introduction. And the current lede discusses more applications, why it is called the exterior product, the wedge product, the exterior power (which redirect here). Per WP:TECHNICAL, any technical terms used in the lede must be explained. The fact that many mathematics articles do not is, in fact, a complaint that Wikipedia editors regularly raise. (This lede was expanded many years ago in response to a long such debate at WT:WPM). Sławomir Biały (talk) 18:39, 15 April 2026 (UTC)

Writing a good lead is tricky. You want it to be short, define the terms of art that it uses and be comprehensible to the layman. Wikipedia could use some good tech writers, but a volunteer organization doesn't have the luxury of hiring some. I agree that the lead is way to long, but I don't have a realistic idea of how to shorten it. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:59, 15 April 2026 (UTC)

"Looking at other mathematics articles here, it does not seem to be the spirit to provide a gentle introduction" – Mathematics articles here are notorious for their inaccessibility. I routinely see complaints like "I have a physics PhD but I couldn't understand the first sentence of XYZ math article". While that may sometimes be an exaggeration or impossible to improve, we really shouldn't be trying to make those articles which do a decent job at accessibility worse, just to match a prevailing bad example.

"I don't think the current state of the introduction is particularly helpful to anybody." – I can't agree with this. Maybe by "anybody" you mean "specialists in the topic"? I think the current lead seems like quite helpful context for anyone who didn't go through math grad school. –jacobolus (t) 20:24, 15 April 2026 (UTC)

Disagreeing about priorities isn't the same as trying to make them worse. WP:AGF -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:30, 16 April 2026 (UTC)

I didn't mean any specific people were trying to make anything worse. By "we really shouldn't be trying to ...", I mean 'we' as in the collective group of Wikipedians and "trying to" as in "making editorial choices with the effect of". Sorry if that was inelegant. What I meant is: (1) I think being inaccessible is bad, and (2) we shouldn't use some articles' inaccessibility (again: in my opinion this is bad) as a justification for making currently somewhat accessible articles less accessible, which would make them worse. –jacobolus (t) 19:05, 16 April 2026 (UTC)

It's a trade-off between accessibility and accuracy. Accessibility should not be used as an excuse for turning an accurate lead into nonsense. A good tech writer is worth his weight in platinum. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:34, 16 April 2026 (UTC)

The present lead is not nonsense. It covers all points of the proposed revision (and a few things not covered). I do not think there are any inaccuracies, so I do not see a trade-off here. I do not think it should be reduced: it is just long enough to explain everything in a non-technical way, and no longer. [User:Sławomir Biały|Sławomir Biały]] (talk) 19:40, 16 April 2026 (UTC)

A plausible alternative could be to move most of the lead to a new section immediately following, and make the lead itself more of an article summary. But the terms and concepts introduced there definitely seem important to cover close to the start. –jacobolus (t) 20:16, 16 April 2026 (UTC)

Maybe, but in that case the lead would have to be kept pretty uninformative. It would not resemble what is being proposed. Sławomir Biały (talk) 04:41, 17 April 2026 (UTC)

I agree that the lead deserves a serious revision.

The first paragraph is correct, although I have not a clear opinion whether it must remain the first paragraph or it must preceeded by a paragraph about the context (see below).

The second and third paragraphs do not belong to the lead; they must be merged into § Formal definition, which must be renamed "Definition" and must be expanded with some explanations.

The sentence (4th paragraph) about the universal property deserves to be moved just before "The exterior algebra is named after Hermann ...".

A paragraph about the context (why the exterior algebra is useful and important?) is lacking. It could be something like: Tthe exterior algebra provides a basic framework for multilinear algebra. It allows systematic proofs of the numerous formulas relating minors of matrices. It allows defining the rules for manipulating differential forms of higher degree, and explaining their role in the computation of areas and volumes, as well as for changes of variables in multiple integrals.

After a second though, it seems that a good structure could be

• A first paragraph consisting of the above suggested paragaph, followed by an adaptation of the sentence about the name.
• A second paragraph consisting of the current first paragraph, followed by the sentence about the universal property.
• A third paragraph about the generalization to modules and vector fields.

D.Lazard (talk) 12:15, 17 April 2026 (UTC)

That proposed first paragraph is pretty good I think. The third paragraph could be where to put the "graded algebra" view, which I think can be condensed from what is there now. In principle, I agree that the geometric explanation of gradedness/simple products could be moved lower to motivate the main definitions. Sławomir Biały (talk) 12:24, 17 April 2026 (UTC)

I recommend that everything other than the current first paragraph and the proposed preceding paragraph be moved out of the lead. They are certainly important, and should remain in the article, but they are details that don't belong in a summary. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:48, 17 April 2026 (UTC)

That would exclude applications like differential forms (and graded algebras, universal properties). I'm not completely opposed, but it would seem to require more substantial thought about how to cover it early on in the article. A section right below the lede might be acceptable, but what would its exact scope be? (To be clear: I am not saying that this is an impossible task, but perhaps something that needs to be considered first.) Sławomir Biały (talk) 12:51, 17 April 2026 (UTC)

Maybe § Overview as the first section, with links to the sections with the details? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:50, 17 April 2026 (UTC)

Notes