Talk:Geometry/Archive 2

Archive 1

Archive 2

New lead

As it seems that there is a consensus that the new lead is better than the oler one, and as it is now the new lead that is discussed, I have split this long thread. D.Lazard (talk) 09:57, 10 September 2020 (UTC)

I have written a new version of the lead, and implemented it boldly. I am misplaced for judging it, but it is certainly much better than the previous version, and much closer to the prescription of MOS:LEAD. Also, it answers to the issue mentioned in the heading of this thread. In any case, it is easier to edit for improvements. Be free to improve it, and, if there are possibly controversial issues, to discuss them in a new section. D.Lazard (talk) 15:29, 9 September 2020 (UTC)

I find "omits continuity" a bizarre description of discrete geometry, which is not generally formulated around disregarding properties of the spaces its objects live in. I think discrete fits better into the topics grouped by underlying methods. Maybe you were thinking of finite geometry? Finite and discrete are two different things here, although (like many of these subcategorizations) there is occasional overlap between them.—David Eppstein (talk) 00:00, 10 September 2020 (UTC)

The new lead is better, but still has some issues.

The sentence It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. as written promises too much, and conflicts with the cited reference. How about changing it to Classical Geometry.

The broadening of the field didn't start with the 19th Century; Projective Geometry was a couple of centuries earlier, with antecedents long before the 17th century.

While Affine Geometry can certainly be considered a Euclidean Geometry without, e.g., angles, lengths, much of modern Geometry consists of new structures rather than just ignoring some properties of old structures, e.g., Symplectic geometry. Shmuel (Seymour J.) Metz Username:Chatul (talk) 01:22, 10 September 2020 (UTC)

It seems difficult to write a lead which will be easily understandable to the average reader while also faithfully representing the (ultra)modern viewpoint that "geometry is anything you can study using geometric ideas or methods." I think the first paragraph reasonably represents what geometry is (in the eyes of the non-specialist) as it is, and the rest of the lead does a fairly good job of representing this more modern idea (if not in name) by describing how classical geometry relates to its more modern form.

I make no claim of having a precise reference for these remarks, but at least as I tend to view it, one way of summarising the modern viewpoint might be to say that geometry is concerned with objects, usually sets (but sometimes rings, algebras, and so on) that have been rigidified by imposing some extra structure on them. Just what it means to "rigidify" is not clear (and precisely the discussion at hand) but, for example, just a topology is probably too weak to be called geometry, and a metric structure is definitely geometry, and things like smooth structures, projective structures, affine structures, algebraic/complex structures and so on all sit somewhere inbetween. Here I use "geometry" in the sense of "geometry and not topology" rather than the broader use of "geometry as in geometry and topology". This is neither here nor there as we can just broaden what we mean by rigid to include, for example, a topology. I think the page is aiming somewhere inbetween these two interpretations of the word "geometry"; for example whilst topology is mentioned, it is only given a small section of the page (despite being a relatively large part of "geometry and topology" as a combined field!). This might be worth discussing at some point.

Classically the objects are sets of points, and the extra structures are things like specifying lengths, angles, incidences, and so on, but one could be more modern extra structure such as just volumes (like in symplectic geometry), more discrete relationships of relative position between points/shapes, or coarser rigidifying structures such as holomorphic or algebraic structures (such as in complex or algebraic geometry), and also includes examples where the underlying object is not a set of points, but is an algebra (like C-star algebras in noncommutative geometry). It might make sense to clear up the remarks later in the lead along these lines (I agree that it reads a bit as though geometry is about taking euclidean geometry and removing some of the properties, but some effort has been made to steer away from that viewpoint), but I'm not sure how to make the first paragraph reflect this ultramodern viewpoint, or even if that should be the goal of the first few sentences (after all, the lead should be readily understandable to anyone).Tazerenix (talk) 02:01, 10 September 2020 (UTC)

To editor David Eppstein: I agree with your concern. However, I am not sure of the best way for fixing it. It is easy to replace discrete geometry by finite geometry before "omits continuity", but it is not clear for me whether finite geometry is sufficiently important to appear here. On the other hand, discrete geometry is clearly sufficiently important to appear here, but its place is unclear. Discrete geometry is also known as combinatorial geometry, and this alternative name seems to refer to the underlying methods. However the alternative name is much less common, and it does not refer to the methods but on the nature of the problems that are studied. So, it belongs to a third category of extensions of geometry, the study of specific classes of problems occurring in Euclidean geometry. Fractal theory could appear also in this category. Thus, it seems that this (already too long) sentence deserves to be split for being expanded. Further discussion on this point is needed. D.Lazard (talk) 10:21, 10 September 2020 (UTC)

Note that a degenerate finite pseudometric space is finite but not discrete. It is not, of course, derived from an affine or projective space.

The term ultramodern seems strange, given that Felix Klein's Erlanger program was in the 19th century. Shmuel (Seymour J.) Metz Username:Chatul (talk) 19:42, 11 September 2020 (UTC)

My point was more that nowadays things that are very far removed even from what the Erlangen program would think of as geometry are treated as geometry, because of a further shift in perspective. Non-commutative spaces (objects that don't actually literally exist), p-adic geometry, and stacks (and even crazy things like Lurie's infinity-stacks) are more what I was thinking of. These are objects that ostensibly don't look almost anything like geometric objects even from last century, but because the tools people use to study them feel very geometric in nature (or are inspired by analogy with quote-unquote "actual geometry") they are viewed as geometric subjects. This seems to be the way that geometry is viewed nowadays at least within some parts of the pure maths community, but as you say perhaps the fundamentals of this viewpoint aren't as modern as I said! I guess the point of my comment was really a query about whether its worth going into this in the lead of the geometry page, or if it is best relegated to a section about non-commutative spaces or stacks in the contemporary geometry section (I think the latter).Tazerenix (talk) 00:46, 13 September 2020 (UTC)

Certainly schemes and such are more modern than Synthetic Geometry and the Erlanger Program, but after half a century they are very much mainstream. I agree that there's no need to discuss them in the lead; my concern is that the current text is wrong. We should either remove or qualify the statement It is concerned with properties of space that are related with distance, shape, size, and relative position of figures., e.g., replace it with Classical Geometry is concerned with properties of space that are related with distance, shape, size, and relative position of figures. or with Many branches are still concerned with properties of space that are related with distance, shape, size, and relative position of figures.. Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:17, 13 September 2020 (UTC)

It is certainly correct to say that Geometry is "concerned with properties of space that are related with distance, shape, size, and relative position of figures", because it is. Saying that doesn't imply that those are the only this with which geometry is concerned. Paul August ☎ 20:41, 13 September 2020 (UTC)

Points at Infinity are not Euclidean

The lead states Until the 19th century, geometry was exclusively devoted to Euclidean geometry, but the work of Desargues in the 17th Century involved points at Infinity, and the resulting Geometry does not satisfy Euclid's axioms; in fact, it is no longer possible to talk of lengths for all segments and there are no longer any parallel lines. — Preceding unsigned comment added by Chatul (talk • contribs)

I think the statement in the lead is a bit disingenuous. For example, mathematicians were working with spherical geometry in ancient greece and even before that (through navigation and geodesy). It is true that mathematicians thought that geometry was synonymous with Euclidean geometry up until the development of hyperbolic geometry and the work of Gauss (evidenced by how shocked everyone was that you could have geometries without the parallel postulate) but in fact they were doing non-Euclidean geometry without realising, for example in Desargues work, and in spherical geometry. Perhaps the line should be changed to something like Until the 19th century, geometry was dominated by the assumption that all geometric constructions were Euclidean. This was challenged by the development of hyperbolic geometry by Lobachevsky and other non-Euclidean geometries by Gauss (etc.etc.) around this time. In fact it was realised that implicitly non-Euclidean geometry had appeared throughout history, including the work of Desargues in the 17th Century, all the way back to the implicit use of spherical geometry to understand the Earth (geodesy) and to navigate the oceans since antiquity.

I agree that the sentence as quoted is wrong, and should be changed. Tazerenix (talk) 15:52, 30 September 2020 (UTC)

I have changed "exclusively" into "almost exclusively", and added Tazerenix's suggestion as a footnote, after having slightly edited it (mainly by adding wikilinks). The reason for a footnote is that, included in the text, this explanation would disrupt the flaw of reading, and would be too WP:TECHNICAL for this second paragraph. D.Lazard (talk) 15:35, 2 October 2020 (UTC)

Escher

change ((Escher)) to ((M. C. Escher|Escher)) 98.239.227.65 (talk) 20:42, 11 November 2020 (UTC)

 Done. Also a similar change for da Vinci. D.Lazard (talk) 21:05, 11 November 2020 (UTC)

shapes cant have no sides

if circles have no sides, how is that possible? if it had no sides it wouldnt exist cuz that not possible. so circles have infinite sides cause the sides are so tiny they dont even exist and youll never see them bc it has infinite sides. am i missing something??? how does it have no sides? :/ 1fractal4 (talk) 17:12, 16 April 2021 (UTC)

There are many ways to define side, and the answer depends on what definition you choose. Depending on the definition, it has 0, 1 or 2 sides. With no reasonable definitition is infinite sides meaningful. Shmuel (Seymour J.) Metz Username:Chatul (talk) 22:37, 16 April 2021 (UTC)

Semi-protected edit request on 7 October 2021

Minor single letter change under Planes

current: A plane is a flat, two-dimensional surface that extends infinitely far.[44] Planes are used in many area of geometry. suggestion: A plane is a flat, two-dimensional surface that extends infinitely far.[44] Planes are used in many areas of geometry. 71.75.132.160 (talk) 18:28, 7 October 2021 (UTC)

 Done —David Eppstein (talk) 19:08, 7 October 2021 (UTC)

Semi-protected edit request on 10 November 2021

Within the ‘points’ subsection, the use of the word ‘moderm’ is incorrect and should be modern. Also within that section, the year range is given for a mathematician’s name as a 5 digit number, it likely is correct as 1919-xxxx. 174.213.161.79 (talk) 17:42, 10 November 2021 (UTC)

 Done ScottishFinnishRadish (talk) 17:56, 10 November 2021 (UTC)

Compass and straightedge

@D.Lazard: The article claims Classically, the only instruments allowed in geometric constructions are the compass and straightedge. That is false; classically, the ancient Greeks discussed constructions using other instruments. The restriction to compass and straightedge is more recent. I changed the text to Classically, the only instruments used in most geometric constructions are the compass and straightedge. but D.Lazard revertd the edit. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:02, 12 January 2022 (UTC)

It is possible that ancients Greeks knew other instruments for geometric constructions, but they did not use them in geometry (they are not mentioned in Euclid's Elements), and they did not allow using them. Otherwise the Duplication of the cube and the quadrature of the circle would not have been important problems for them. So, if you disagree with the current formulation, you must find reliable sources supporting your favorite formulation. D.Lazard (talk) 18:48, 12 January 2022 (UTC)

It is false that the Greeks disallowed other constructions. See neusis, a non-compass-and-straightedge technique common in ancient Greek mathematics. See also Wilbur Knorr's book The Ancient Tradition of Geometric Problems. Compass and straightedge may have been preferred, but other methods were known, used, and allowed. As Chatul says above, making compass-and-straightedge into an absolute requirement is a more modern invention. —David Eppstein (talk) 20:11, 12 January 2022 (UTC)

NPOV: Plane

@D.Lazard: The article gives a definition of plane that is valid in Euclidean and hyperbolic geometry, but not in Elliptic geometry. I corrected the article from A plane is a flat, two-dimensional surface that extends infinitely far. to A plane is a flat, two-dimensional surface that extends infinitely far or indefinitely. and D.Lazard reverted my edit. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:54, 12 January 2022 (UTC)

Notes

None of the formulations is a mathematical definition, as the terms that are used (flat, indefinitely, surface, two-dimensional) are not defined in this context. So, it is somehow pointless to discuss here the case where they apply or not. Moreover, I have not exactly reverted your edit, since I have removed "far" from the original sentence, for taking your objection into account. The mention of non-Euclidean geometries is out of scope here, since this would make the section too WP:TECHNICAL. Also, if one would discuss here non-Euclidean geometries, one should also discuss finite geometries, for which "indefinitely" and "infinitely" are both nonsensical. This does not mean that I agree with the tone of the section, but Chatul's edit is not an improvement, since it adds to confusion. D.Lazard (talk) 18:32, 12 January 2022 (UTC)

It doesn't matter whether they are mathematical formulations; the fact that the statement is incorrect is still relevant. Changing infinitely far to infinitely does not take my objection into account; the statement is still false in general.

You mentioned WP:TECHNICAL. Section Technical content assistance states Making articles more understandable does not necessarily mean that detailed technical content should be removed.; section Avoid overly technical language states (as long as accuracy is not sacrificed). --Shmuel (Seymour J.) Metz Username:Chatul (talk) 22:50, 12 January 2022 (UTC)

The statement is not false in general. It is at worst a bit ambiguous what "plane" is referring to, although a Euclidean geometry plane is a perfectly reasonable assumed referent. Nevertheless, the section "Main concepts" is already strewn with "according to Euclid" and "in Euclidean geometry", so why not add one more. How about: "In Euclidean geometry, a plane is a flat, two-dimensional surface that extends infinitely." This would be unambiguous and it avoids discussing non-Euclidean geometry at an inappropriate time. Danstronger (talk) 14:07, 13 January 2022 (UTC)

New lead changes by Garrett.stephens

@Garrett.stephens has updated the lead from the previous version which was discussed fairly extensively here Talk:Geometry/Archive_2#New_lead recently.

Does anyone know what the phrase "spatial (static) patterns" means. I am a professional research geometer and have absolutely no idea what this means, and it certainly doesn't seem to capture most of the geometry I've ever seen, which is among other things about objects, not patterns and is can be highly dynamic. The previous opening sentences "Geometry is concerned with properties of space that are related with distance, shape, size, and relative position of figures" seem to me to be more accurate and more understandable to a layperson, so I don't understand why they've been moved to the second paragraph of the lead and replaced by something obscure and non-standard. Tazerenix (talk) 10:29, 11 April 2022 (UTC)

I agree that Garrett.stephens's version is not an improvement. As it introduces new concepts, it should have been discussed here first. According to WP:BRD, I'll revert it. D.Lazard (talk) 11:00, 11 April 2022 (UTC)

First, I want to apologize for the etiquette error of updating it before making a new section in the talk page. In math, it seems the answers are self apparent, and the 'previous v. current' look at that proposed change (I thought) seemed to display that clarity. Perhaps I have been found to be wrong.

My 1st question in light of the response to my error is: What is an object? In object-oriented logic, for example, one would be talking about variables (X,Y,Z, etc). In that sense, Algebra is more appropriately the study of "objects"... If I could I'd like to prompt clarification on that distinction.... In terms of Physics as well, mass (object) v. energy has also caused a lot of fuss in the field...

2nd question is where you say "not about patterns". I guess I'd just like clarification on why geometry is not a study of spatial patterns. Take topography for instance. If we declare geometry is on object [instead] of pattern, I feel the way is not prepared for topography, spatial analysis, tensor geometry, fluid dynamics, quantum dynamics, pattern recognition programming in computing, etc., for their fair shake of "Geometry" if that makes sense. These are all fields that deserve a fair path to consideration of their people being geometers per their having evolved from the ancient geometry of Euclid, who began geometry with allowable spatial movements and exercises prompting readers to reach QED.

Garrett.stephens (talk) 17:16, 11 April 2022 (UTC)

Perhaps an instance of this discussion's importance is in the works of Mathematicians Ralph Abraham and Robert Shaw "Dynamics--the Geometry of Behavior" https://g.co/kgs/gbq8ce Garrett.stephens (talk) 17:37, 11 April 2022 (UTC)

Add me to the list of people baffled by the attempted new phrasing "spatial (static) patterns such as [list of things that are planar not spatial and are shapes not patterns]". I don't think the addition was an improvement. —David Eppstein (talk) 17:50, 11 April 2022 (UTC)

Ok, I yield...

To me, I hear: Geometry is similar to Arithmetic in being a study the ancients did. When you do Geometry, you will work with terms like [term 1], [term 2], [term 3], [term 4] ...etc

Geometry is the study of spatial (static) patterns.

Why (static)? Well, because that paves the way for what a shape is. It's a pattern in space that is static enough to yield itself to analysis. (Can be said to be in "stasis"). Garrett.stephens (talk) 18:25, 11 April 2022 (UTC)

Both versions of the lead have issues, but the previous lead strikes a better balance among, e.g., accuracy, brevity, clarity. I too have a problem with, e.g., dynamic, pattern, static. Is a timelike curve in a Lorentzian manifold static or dynamic? --Shmuel (Seymour J.) Metz Username:Chatul (talk) 04:33, 13 April 2022 (UTC)

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)

Geometric Algebra

The Contemporary Geometry section describes ten different fields of Geometry. Shouldn't Geometric Algebra be in this list? — Preceding unsigned comment added by 50.206.176.154 (talk) 05:17, 26 April 2023 (UTC)

From the geometry point of view, geometric algebra is only a tool used in Euclidean geometry. So, it must not be listed among the main parts of modern geometry.

Nevertheless, section § Euclidean geometry should be expanded for linking to Geometric Algebra (book), and possibly Geometric algebra. This should be done along the following lines: Euclidean geometry can be defined through axioms (synthetic geometry) or through coordinates and linear algebra (analytic geometry). The equivalence of the two approaches has been proved by Emil Artin in his book Geometric Algebra. The algebraic approach to Euclidean geometry led to the introduction of various algebraic concepts such as vectors, quaternions, dual spaces, and over all, Geometric algebra. D.Lazard (talk) 11:11, 26 April 2023 (UTC)

The previous commenter is talking about something different from (though not entirely unrelated to) Artin's book. It is also not accurate to say that geometric algebra is "only a tool used in Euclidean geometry". For more context, you may perhaps be interested in Hestenes (2002) "Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics". For use beyond Euclidean geometry, see e.g. Hestenes (1991) "Projective Geometry with Clifford Algebra". –jacobolus (t) 14:13, 26 April 2023 (UTC)

My suggestion for the content of the article shows clearly that I am aware of the two meanings of "Geometric algebra". Also my sketch for the content of § Euclidean geometry does not imply that geometric algebra is not applicable outside Euclidean geometry; simply that it has been developed for the need of Euclidean geometry. Similarly, vectors and dual spaces are widely used outside Euclidean geometry. My opinion is that, for geometry, geometric algebras are not more important than, say, tensors. Both seem to be too technical and too specialized for having more than a single mention in this general article. D.Lazard (talk) 15:30, 26 April 2023 (UTC)

You said “From the geometry point of view, geometric algebra is only a tool used in Euclidean geometry.” I am just pointing out that that is not right. Of the sections listed in this article (which to be honest seem like a kind of arbitrary assortment), geometric algebra is a tool relevant to at least Euclidean geometry, Differential geometry, Non-Euclidean geometry, Algebraic geometry, Complex geometry, Discrete geometry, Computational geometry, Convex geometry.

developed for the need of Euclidean geometry – this doesn’t seem right either. Grassmann's work was pretty general and later mathematicians applied his products to all sorts of contexts. Clifford was very interested in modeling non-Euclidean geometry (though he died young and never got the chance to fully develop his ideas). Hestenes started out explicitly trying to model (both flat and curved) spacetime.

I don't think focusing on Artin's book as in your sketch here is the right approach to a section about Euclidean geometry (per WP:DUE), though IMO the current section ("geometry in its classical sense" etc.) is pretty useless.

Inre (Grassmann/Clifford/Hestenes style) geometric algebra I think it would be better to instead add 'vectors' and 'multivectors' to Geometry § Objects. While we're at it, the current way both projective geometry and geometric transformations are shoved into the "symmetry" subsection also seems like a poor choice. These should probably both be elevated to (separate) sections. –jacobolus (t) 16:15, 26 April 2023 (UTC)

Removal of pleonasm "periodic periods" in Algebraic geometry section

The second sentence in the Algebraic geometry section reads "It underwent periodic periods of growth...". The use of the adjective "periodic" to describe the noun "periods" seems pleonastic and therefore hampers readability. I suggest changing the sentence to open with either:

1. It underwent periods of growth---implying that "Algebraic geometry" underwent "a length of time" of growth (see period).
2. It underwent periodic growth---implying that the growth of "Algebraic geometry" was "happening repeatedly over a period of time" (see periodic).

Given the context and subsequent text in the sentence, the first option seems more appropriate than the second. Kyle F. Hartzenberg (talk) 01:23, 6 September 2023 (UTC)

I agree that the sentence is confusing. It seems that the intended meaning is that, since its origin, algebraic geometry had several distinct period of growth, implicitly separated by periods of relative stability; this is a controversial assertion. Moreover, the provided list (projective geometry, birational geometry, algebraic varieties, and commutative algebra) does not correspond to the beginning of the sentence, and is essentially non sensical as a list, as the first item is not specific to algebraic geometry, the second and the third are subjects of study in algebraic geometry, and the third is the fundamental tool for linking geometry and algebra into algebraic geometry. Moreover, this sentence is too vague for having any encyclopedic value.

So, the whole paragraph must be rewritten. Clarifying only "It underwent periodic periods of growth" cannot be done without introducing a controversial assertion. So, before the needed rewrite, it seems better to remain ambiguous. D.Lazard (talk) 08:45, 6 September 2023 (UTC)

Rewrite done. D.Lazard (talk) 17:51, 6 September 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)

We hear geometry-related words all the time: ‘what’s your angle?’ and ‘everyone should eat three square meals a day!’ and ‘she ran circles around me!’, often with little thought to how fundamental those shapes are to the discipline called geometry. Barrista hex (talk) 10:36, 20 December 2023 (UTC)

wanna learn from you... Barrista hex (talk) 10:32, 20 December 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)