Talk:Complex space

Introductory explanation

I think this page would benefit from a simple introduction to the idea of a complex space having both real and imaginary components. I would describe these components as "dimensions" and for the complex number line I would introduce the Argand diagram. I am from a geometrical background in which a metric (e.g. coordinates) should not be assumed. Are there any other aspects, such as say "dimension", which should similarly not be assumed? Here is how this page used to look before it was re-purposed and moved to complex affine space. — Cheers, Steelpillow (Talk) 17:46, 27 October 2015 (UTC)

I disagree on several counts:

• A disambiguation page never has discussion or introduction, so your suggestion would entail changing it back into an article.
• An article about a complex space of any kind should not introduce, define or construct the concept of a complex number; it should link to Complex number as a primitive. (You may note that none of the linked articles refers to the construction of complex numbers in terms of reals.)
• The use of "dimension" that you mention (of complex number over the reals) is at odds with your apparent preference for a coordinate-free definition. The only use of "dimension" in an article about a type of complex space should be of the space over the complex numbers.
• I doubt whether the general concept of complex spaces as a whole is treated as a notable topic; rather, each of the topics tends to be a concept best suited to its own article.

I feel that this page should stay as it is: a disambiguation page. —Quondum 02:14, 28 October 2015 (UTC)

Thank you for your constructive points. I take on board some of what you say but would still argue the other way on other points and in general I am quite confused about all of this. I would note:

• Wikipedia currently treats the "complex plane" not as C², as you seem to imply, but as the plane of the Argand diagram (visualising C¹). Coxeter and Shephard in their 1952 paper on Regular Complex Polytopes talk of the unitary plane as having two dimensions, meaning what I write here as C², while they reserved "complex plane" for what I write here as C¹. It does not seem sensible to me to suggest that the unitary plane is not a complex plane, but what do I know?
• In string theory, physicists talk of Calabi-Yau spaces having six dimensions. Shing-Tung Yau, in his book The Shape of Inner Space (2010), talks occasionally of "Complex space" (p.77), says that a complex manifold must have an even number of dimensions (p.80) and then apparently contradicts himself by remarking that a + ib is a one-dimensional complex number (p.81). He begins to resolve the dilemma by remarking on p.91 that "in complex dimension two, which is real dimension four,..".
• In his book on Projective Geometry (Revised 2003), Edwards mentions complex numbers only in order to introduce the idea of imaginary points in an otherwise wholly geometric account of certain phenomena.

My problem is that I do not always believe the opinions of my fellow Wikipedia editors, I try to understand what the sources are saying. And that has let me down! For every firm assertion, there seems to be a source to the contrary. I hardly know how to express rationally the dreadfulness of the word salad presented to the interested reader by mathematicians on this topic, and I just cannot help feeling that there is a WP:PRIMARYTOPIC lurking here to be dragged out into the open, and that "dimensions" do come into it. — Cheers, Steelpillow (Talk) 17:34, 28 October 2015 (UTC)

To add to that: all the articles/topics currently listed share some property called "complex" in common. At least there should be an article explaining what that common property is, and it would seem reasonable for that article to be the primary topic. — Cheers, Steelpillow (Talk) 20:50, 28 October 2015 (UTC)

"I do not always believe the opinions of my fellow Wikipedia editors, I try to understand what the sources are saying." – this is not a "problem"; it is highly commendable.

A comment: mathematical terminology is not consistent (as you say, a word salad). The case of "complex plane" seems to be a case in point: the term refers to a real plane used to represent a vector space of one dimension over the complex numbers, the affine counterpart being the complex line. The term "complex plane" is thus an exception (remaining in modern terminology through historical precedence) to the more regular and general terminology, in which it would be taken to mean a space of two dimensions over the complex numbers and is quite different from the "complex plane". "Unitary plane" might have arisen to fill the gap (though it seems to imply a little additional structure: an inner product). Such clashes/inconsistencies of terminology have caused me much confusion and frustration in the past, but they seem to be pervasive in mathematics and must be accepted.

In terms of the use of the term "complex space" in sources, one should try to determine what is meant. I suspect that you'll find that it generally is a shorthand for a particular space dictated by the context (such shorthand expressions are quite typical, though at times explicitly introduced, e.g. "A complex vector space, hereafter a complex space ..."). Filtering these out, one can try to analyze the remainder and see whether they are referring to a more general concept. I would expect the more general concept to be a complex manifold, since the others (with the technical exception of a complex analytic space) are specializations of this. And it is already an article.

This is my own perspective; I hope it helps you to get closer to what you're trying to express. I don't particularly care how it pans out; I'll be happy if I've facilitated any clarity. —Quondum 23:21, 28 October 2015 (UTC)

Thank you again. There are two approaches to creating these pages. One, which I think is yours, is to ask, "How best can we present the mathematics?" The other, which is mine, is to ask, "How best can we help the baffled reader to find information based on the term they are trying to understand?" When the maths uses the terms inconsistently, we have a problem.

Some observations: The motivation for the idea of complex space does universally appear to be the complex numbers (or at least, I have never found any other motivation, and assuming "motivation" is the right word). The article on complex analytic space says that it arises as a generalization of a complex manifold. That on Inner product space says that "Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces". Affine spaces are really just a class of complex space, along with say projective spaces, so I think it best not to disambiguate at this level - unless complex affine space has some special importance that I am not aware of.

So I am wondering if we can at least structure the disambig links to present a hierarchy of ideas, something like this:

A complex space is a mathematical space motivated by complex numbers. "Complex space" can mean:

In algebraic geometry

• Complex analytic space
  • Complex manifold - a smooth analytic space

In coordinate geometry

• Complex coordinate space
  • Complex vector space, including unitary space and complex Hilbert space.

Would this make sense? — Cheers, Steelpillow (Talk) 09:10, 29 October 2015 (UTC)

To me, at least, this does not particularly strike a chord; it is not how I'd classify the topics relative to each other, from either of the perspectives that you mention. I don't think it is up to the editors to so either: it would have to be based on some text. Without a source, I don't see it going anywhere. I am also most certainly not an authority; I am merely expressing my perceptions and opinions. —Quondum 15:34, 30 October 2015 (UTC)

Thank you. I'll try something a bit simpler and see if any of it sticks. — Cheers, Steelpillow (Talk) 17:30, 30 October 2015 (UTC)