Talk:Geometry

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Note 3 full citation is Greek and Vedic Geometry Frits Staal Journal of Indian Philosophy 27 (1/2):105-127 (1999)

Geometrist

A person who studies geometry is commonly called a 'geometrist' worldwide beyond the USA. Should this not be added to the end of tte first paragraph in the Lead? Billsmith60 (talk) 00:16, 18 June 2022 (UTC)

Really? I thought the word was "geometer". Google ngrams agrees, with "geometrist" far lower in word frequency. Do you have any evidence of "geometrist" being more popular anywhere? —David Eppstein (talk) 00:36, 18 June 2022 (UTC)

Hello, it's not that "geometrist" is more popular just that it's common enough here in the UK:

https://www.collinsdictionary.com/dictionary/english/geometrist Regards Billsmith60 (talk) 10:48, 18 June 2022 (UTC)

How common is "common enough"? nGrams shows it far behind. This is an encyclopedia, not a thesaurus. —David Eppstein (talk) 16:41, 18 June 2022 (UTC)

"Geometric space" listed at Redirects for discussion

The redirect Geometric space has been listed at redirects for discussion to determine whether its use and function meets the redirect guidelines. Readers of this page are welcome to comment on this redirect at Wikipedia:Redirects for discussion/Log/2023 March 12 § Geometric space until a consensus is reached. fgnievinski (talk) 03:04, 12 March 2023 (UTC)

A space is not always geometric

@D.Lazard: Revision https://en.wikipedia.org/w/index.php?title=Geometry&oldid=1144056819 added the text This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space, or simply a space is a mathematical structure on which some geometry is defined. However, the word space can refer to mathematical structures that are not geometric, e.g., vector spaces over arbitrary fields. I'm not sure how it should be worded, since the term Geometry is itself murky. Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:04, 12 March 2023 (UTC)

This depends of your definition of “geometric”. Currently, nobody pretends that algebraic geometry and finite geometry are not geometry, and vector spaces over a finite field belong to both areas. There is nothing murky in geometry. Simply, this is a scientific area and not a mathematical term, and, as such, it is not subject to a mathematical definition. D.Lazard (talk) 19:51, 12 March 2023 (UTC)

How is Geometry not a mathematical discipline? --Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:47, 13 March 2023 (UTC)

Geometry just refers (except in very limited cases in NCG) to any set whose elements we can describe as "points" because in addition the set has some information about how its elements have a "position" relative to each other. "Space" is just a catch all term used to describe such structures, so I think its sort of tautological to say Geometry is the study of Spaces.

There's a more limited definition of geometry in the context of topology which refers to spaces with some particular kind of rigidifying geometric structure on them such as a metric, Riemannian metric, volume form, algebraic structure, etc. But I don't think that really applies to "Geometry" in the large. Tazerenix (talk) 23:09, 12 March 2023 (UTC)

I've never seen an Algebra text refer to the elements of, e.g., a vector space, a Fréchet space , as a point. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:47, 13 March 2023 (UTC)

The requirement is not that a textbook refers to them as "points" but that there is a relation between elements which provides information about their relative position. In the case of a vector space, the relation is linear (you can specify when two elements lie along the same line). In particular there is an affine structure (and more, as there is a distinguished point at the "center", another positional relationship). Of course an algebra book will not think of vector spaces as spaces if its goal is to do algebra, but they certainly don't refer to them as "vector sets". Tazerenix (talk) 23:07, 13 March 2023 (UTC)

In Topology there is no concept of relative position. Does that mean that a topological space is not a space.? --Shmuel (Seymour J.) Metz Username:Chatul (talk) 08:58, 14 March 2023 (UTC)

Closeness is the basis of topology, and is a sort of relative position. However, although although Tazerenix's definition of points and spaces is ingenious, I am not sure that I completely agree with it, and it is WP:OR. So, it is better to say that space, point, geometry, geometric method, geometric space, etc. are what is so called by the community of mathematicians. These terms do not require to be formally defined as they are only used to provide an intuitive support to reasonnings, which otherwise would be more difficult to understand. For example, learning the axioms of vector spaces is easy, but understanding the richness of the concept cannot be done without considering the geometrical aspects of the concept. D.Lazard (talk) 10:31, 14 March 2023 (UTC)

See for example Kuratowski closure axioms in which topology is defined entirely using the concept of a point being "close" to a set. This is an example of information about the relative positions of points: If a point x is close to a set A and a point y is not, then x is closer to A than y! Tazerenix (talk) 22:58, 14 March 2023 (UTC)

Not so. None of the axioms refer to closeness. There is a derived concept of a point being close to a set, but none of the axioms use it. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 00:23, 15 March 2023 (UTC)

If you define the relation "$x$ is close to $A$" as "$x$ is contained in $\mathrm {cl} (A)$" then the axioms of a topology can be specified as

1. No point is close to the empty set
2. Every point of $A$ is close to $A$
3. The points of $X$ which are close to $A\cup B$ are the points close to $A$ or to $B$
4. If a point $x$ is closeto the set of points close to $A$, then $x$ is close to $A$

A set $X$ with a relation between points and sets of "closeness" is equivalent to specifying a topology (precisely, define the closure operator $\mathrm {cl} :{\mathcal {P}}(X)\to {\mathcal {P}}(X)$ by $\mathrm {cl} (A):=\{x\in X\mid x{\text{ is close to }}A\}$). Tazerenix (talk) 02:23, 15 March 2023 (UTC)

Speaking as a topologist, I don't believe that every topological space ought to be described as geometric, however one might reasonably define the term. While there is, of course, a close connection between topology and geometry, I don't think topology is best described as a subset of geometry. Paul August ☎ 16:50, 13 March 2023 (UTC)

I would probably classify Topology as part of Geometry, although topologies not satisfying the separation axioms might be counter-intuitive. I could probably make an argument for considering it to be a part of Analysis, albeit a weak one. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:19, 13 March 2023 (UTC)

Geometric Algebra

The Contemporary Geometry section describes ten different fields of Geometry. Shouldn't Geometric Algebra be in this list?