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    What Is The Geometry Of Spacetime? — What Is Space? — Inner-Product Spaces
    By David Halliday | May 12th 2012 12:00 AM | 102 comments | Print | E-mail | Track Comments
    Since "spacetime" is simply a term for a space that has a component we call "time", we need only concentrate on spaces in a somewhat general sense.  Now, as Derek Potter, on this site, pointed out to me, a little while ago, "To us, a space is somewhere to put a box :)"  So, let's take a short "detour" to address what we will mean by a space, in this series.

    Spaces

    Manifolds and Tangent Spaces

    Einstein did something very different with his General Theory of Relativity.  Something that had not been done in science anytime before that.  He actually separated the space "where we put a box"—the space of "places" or locations—from the space of "directions" or vectors (velocity, acceleration, momentum, electric and magnetic fields, etc.).1

    The space of locations is called a manifold.  This is probably an unfamiliar term for many (except that those familiar with automobile engines know about intake and exhaust manifolds).  You can just think about a manifold as being the set of all possible locations, possibly on some high dimensional curved "surface".2The surface of the Earth as a two dimensional manifold

    When viewed very close to a single location—a single point—a smooth manifold will look like a flat space—a vector space.  The tangent space is the vector space that is tangent to the smooth manifold at a single point—the one that looks the most like the smooth manifold at that point, when viewed very closely.

    Tangent space at a point on a sphereWe are all familiar with a curved manifold, and the tangent spaces to that manifold:  The surface of the Earth (whether considered to be a smooth sphere, or as a sufficiently smoothed version of the actual location complete with hills, mountains, and valleys) is a two dimensional manifold.  The tangent space to that manifold, at any given point on that manifold, is a mathematical (Euclidean) plane, which we can imagine—or, in some cases, partiallyA tangent vector (in the tangent space) along a curve through a point on a manifold construct—at any given point on the Earth's surface.  Some ways in which we may partially construct or realize a tangent space at a point on the surface of the Earth is when we level a portion of the ground for the foundation of a building, or lay a board or sheet of glass on the ground.

    So, from here on, we shall be only considering the tangent space—the vector space.  So we will no longer have to consider the collection of locations that is the manifold, even if it is flat.3

    Vector Spaces

    OK.  Now that we are focusing on the tangent space—a vector space—what is a vector space?

    Mathematicians define a vector space in the following manner (underlined words/phrases are terms that are being defined):

    Let F be a (mathematical) field (like the Real or Complex numbers).  A vector space over F (or F-vector space) consists of an abelian group4 (meaning the elements of the group all commute) V under addition together with an operation of scalar multiplication of each element of V by each element of F on the left, such that for all a,b ∈ F (read a,b in F, or a,b elements of F) and α,βV the following conditions are satisfied:Vector Addition and Scaling
    V1)  aαV.
    V2)  a(bα) = (ab)α.
    V3)  (a+b)α = (aα) + (bα).
    V4)  a(α+β) = (aα) + (aβ).
    V5)  1α = α.

    The elements of V are called vectors, and the elements of F are called scalars.

    Well, some of you have probably seen a definition like this before.  Some of you may not recall ever having seen a definition like this before, but are OK with this definition.  Others are scratching their heads and wondering whether they are going to be able to follow this series at all, if we're going to be having stuff like this!

    Dome C distance sign poleThe important takeaway message is that a vector space contains things we call vectors, that have some "amount" in some "direction" (hence the reason we tend to depict them using arrows of various lengths pointing off in different directions).  These vectors can be added together, without having to worry about what order we add them (just like with ordinary numbers), to produce new vectors.  We can multiply these vectors by scalars—numbers like Real or Complex numbers—to also produce new vectors.  And the operations of adding vectors and multiplying by scalars can be combined in what would seem to be the "usual way" from our experiences with plain old ordinary numbers (Counting Numbers, Integers, Real and Complex numbers), via the operations shown in V2 through V5.  Other than that, the definitions simply make sure everyone is working with these things in exactly the same way, to avoid misunderstandings.

    Remind me, why are we concerned about "vector spaces"?  Because, like the picture to the right, we use directions and "amounts" of things (like distance, speed, acceleration, electric and magnetic field, etc.) both in everyday life, and especially in physics.  Such seem to be reasonably natural, quite expressive, and useful.

    Now that we have vectors, and vector spaces, we've got everything we need, right?  No.  Not quite.  Some readers may have notices that I kept referring to nebulous things like "amount" and direction when talking about vectors.  Shouldn't I have been talking about magnitude and direction?  After all, that's how most have heard of such things when talking about vectors.

    The problem is that, at this point, we don't have any real way to determine magnitude, and even direction—especially of one vector relative to another—is not well defined.

    Not well defined?!?  I hear some shouting.  What's not well defined about such things?  You simply...

    Inner or Dot Products

    OK.  Let's take care of this magnitude and direction issue...

    By a show of hands, who has heard of an inner or dot product of vectors?  I suspect that practically everyone here that has heard of vectors, before this article, has also learned about the dot or inner product of vectors, and how this relates to magnitudes and directions of vectors.

    However, how many of you know that just because you have a vector space does not guarantee that you have a dot or inner product?  You see, just because you have vectors, in a vector space, does not, necessarily, mean that you also have the ability to take dot/inner products of such vectors!  The ability is an additional feature in addition to the structure of a vector space.

    All right, so what's so mysterious about having a dot or inner product?  Well, the real question is what is the inner or dot product?

    Remember, the vector space has vectors, and these numbers called scalars.  In the case of the "real world" the only numbers we truly have to deal with are (drum roll, please ...) the Real numbers.5  ;)  So, from here on, we shall focus only on vector spaces over the Real numbers, or Real vector spaces.6  Therefore, the scalars we will be dealing with will be Real numbers.Geometric interpretation of angle between vectors using an inner product

    The thing to recognize is that the dot or inner product is another binary operation (· or <,> or <|>), or function, g, that takes two vectors, from the vector space, and yields a scalar from the number field associated with the vector space (the Real numbers, in our case).  So for all α,β,γV, α·β = <α,β> = <α|β> = g(α,β) ∈ F (which is the Real numbers, in our case), and satisfies the following (we will simply use the dot notation, α·β):
    D1α·β = β·α (conjugate symmetry becomes simple symmetry for Real numbers)
    D2)  (aαβ = a(α·β) and (α+βγ = α·γ+β·γ (linearity in the first argument)
    D3α·α ≥ 0 with equality only for α = 0 (positive-definiteness)

    This can all be succinctly summarized (for those that are comfortable with the terms) by stating that the dot/inner product is a positive-definite Symmetric bilinear (Hermitian) form.

    So, an inner product space is a vector space with the addition of this inner product operation or function.  If we don't have this additional operation or function, we cannot determine the magnitudes of vectors, nor the relative directions, or angles between vectors.  Similarly, if we are able to determine magnitudes (lengths, etc.) of vectors, and angles (relative directions) between vectors, in a consistent manner, then we have, or can construct this very operation or function.

    The truth is, we now have, with this inner or dot product, the METRIC (hear sound of this shout, in a deep bass voice, reverberating off the mountainsides...).  We simply need to take a closer look, expanding the vector space in terms of a basis (as we always can do for a vector space), and, ultimately, relaxing condition D3, the positive-definiteness condition.

    That's what we will be addressing next time.


    1  Bernhard Riemann, a mathematician, created the geometry that caries his name in the nineteenth century in order to handle the geometry of curved surfaces without having to consider how such surfaces may be embedded in some higher dimensional space.

    2  The inner or outer surfaces of intake and exhaust manifolds can be considered as examples of two dimensional manifolds, in this sense.

    3  A flat manifold can be identified with its tangent spaces:  All the tangent spaces can be identified with one-another (made "the same"), and the flat manifold can then be considered to be "identical" to this tangent space.  This is what "allows" the usual physics where vectors, such as velocity, etc., can "appear" on the same space as displacement vectors (straight line movement from one location/point to another), and actual locations/points (and collections of such, like boxes and other objects).

    4  An abelian group (over addition) is a group where all elements commute:  So, for a,b ∈ A (read a,b in A, or a,b elements of A), a+b = b+a.  Mathematicians define a group in the following manner:

    A group <G, *> is a set G, together with a binary operation * on G, such that the following axioms are satisfied:
    G1)  The binary operation is associative:  (a*b)*c = a*(b*c).
    G2)  There is an element e in G such that e*x = x*e = x for all x ∈ G.  (This element e is an identity element for * on G.)
    G3)  For each a in G, there is an element a' in G with the property that a'*a = a*a' = e.  (The element a' is an inverse of a with respect to *.)

    Some will note that this list of axioms is missing "closure".  Very good!  This is because the text that I have used defines a "binary operation" in the following manner:

    A binary operation * on a set is a rule that assigns to each ordered pair of elements of the set some element of the set.  So, in this case, the definition of a binary operation * on a set already includes the concept that the set is closed under the operation *.

    5  I know that Quantum Mechanics "needs" the Complex numbers.  However, even then, the only numbers we humans ever see from a Quantum Mechanical experiment are, again, always Real numbers.  Besides, we are dealing with classical level "things" (like velocities, accelerations, electric and magnetic fields, etc.) for the sake of this series.

    6  It's actually not difficult to generalize an inner product space to being over the Complex numbers.  It's just that it will help (I believe) if we are a little bit more focused, here.


    Articles in this series:
    (previous article)
    What is the Geometry of Spacetime? — What is Space? — Inner-Product Spaces (this article)
    (next article)

    Comments

    vongehr
    Ha ha - David, you are some lovely character, really. I went back and re-read the introduction to your introduction just to make sure I did not remember wrongly some sort of aim about explaining relativity in a better way ("usually taught or presented often seems to make it appear to be ever so complex, far too abstract and opaque"). And now this, in the very next post, killing off about 100% of the audience that does not already know everything they actually do not need to know ("Let F be ..." - LOL).

    Nobody, me included and I have no problem admitting so, grasps that tangent spaces are not just arrows inside embedding spaces. Only crazy mathematicians think dot products are anything else but what tells you the true lengths of projections. I am not entirely sure where you are headed, but it looks already like, well, how to say this nicely? You know when you browse books in the store and most of them you put back onto the shelf, some even after sticking your thump just into one or two of the first few pages? Now if I were to look for an introduction to relativity ...
    mathematical_investigations
    He has only said that something called a tangent vector space -comprised of arrows- exists at any point on a surface and obeys a given algebra, and that distance doesn't depend on these notions alone. Of course, whole books can be written on manifolds, tangent vectors, and the linear algebra presented here, but if he explains SR using only those aspects vector spaces, this could be very interesting. 
    vongehr
    David writes "from here on, we shall be only considering the tangent space—the vector space.  So we will no longer have to consider the collection of locations that is the manifold". Such is meaningless to who needs a yet better access to the core of relativity because "space is somewhere to put a box". The title of this piece is "What is Space?" His answer is: "Let's forget space and do some abstract maths!"

    Yes, what David does may be very interesting, even revolutionary, who knows, but special relativity emerges in graphene for example and one can understand that without formulas. Abstract mathematics is the most powerful language if given certain tasks, but it is not the core reason, the machine under the hood that he claimed to present.
    mathematical_investigations
    In the previous post he wrote,
    Then, I get into the class on General Relativity: Everything changed! The "shackles" fell away, and I was free, at last! Now, what made the difference? True, there are a lot of very new concepts in General Relativity: Curved spaces, Manifolds, general coordinates and general coordinate transformations, Tangent spaces and their dual spaces, Differential Geometry, etc. etc. etc. So what changed? What did it?The difference was I finally learned a very general and fundamental concept pertaining to any space where one can "measure" lengths and angles: The "metric", AKA the inner/dot product...
    So I fully expected him to begin by presenting the metric. I understand that feeling of everything-coming-into-place that occurs after learning more advanced mathematics in a later course, when you think "why didn't I know that all along?... why didn't they teach that earlier?" The problem is that the mathematics at that level is more advanced so there is some awkwardness in presenting it at the beginning, but a stripped down version of the math could be enough for a beginner to get a glimpse of that deeper understanding that usually comes later. Someone with more familiarity with the material will have a deeper understanding of it. I believe that is what he is attempting to achieve. I didn't get the impression that he was presenting "SR for beginners" or "SR without the math." Two great books along those lines are 1. Relativity&Common Sense by Hermann Bondi and 2.The ABCs of Relativity by Bertrand Russell. But, we've all seen elementary derivations of the Lorentz transformations using light beams and algebra; what Halliday describes in the first post is much more interesting.
    vongehr
    We are talking past each other (that David uses the metric as central was of course already clear the last time - see discussion there).
    mathematical_investigations
    The content of his post didn't surprise me or seem inconsistent with his objectives, despite what he said about other approaches being "too abstract." That is all.
    Thor Russell
    Can you explain more in an article perhaps about what exactly is going on in graphene in a way that the majority of the audience would grasp because I certainly would find it interesting.
    Thor Russell
    Halliday
    Sascha:

    I most certainly would not make any claim to any of this being "revolutionary".  After all, as I stated in my first article, I'm merely trying to reproduce something like unto what I heard that professor talk about (and something like unto what I think would have helped me).

    Now, as for things like "special relativity emerges in graphene for example and one can understand that without formulas."

    First off, the fact that a 2D surface, that exists through time, embedded in a 3+1 Minkowski spacetime inherits SO(1,2) (or, equivalently, SO(2,1)) symmetry is not at all remarkable.

    Secondly, while there are a number of very interesting (dare I say "inspiring")* things that "emerge" in solid state systems (especially insulators and semiconductors)—such as particles and "anti-particles"; particles (charge carriers) with different masses/inertias; finite heat capacities vs. the infinite heat capacities for (hypothetical) continuous matter; etc.—any good theoretician will only take such models as "inspiration" or intellectual "stimulation" in creating theories of fundamental physics.  Such models can do little else (besides being a model "test bed" for the ideas so engendered), since the real test is not such model systems, but fundamental reality itself (as embodied in our experimental observations of the universe around us).

    So, I, at least, take such interesting "emergence" as just that—interesting.  Little more (at this point).

    David

    *  I have taken my own "inspiration" from such.  That's part of why I know what I am talking about.  (Solid state physics was also one of my subjects of study in obtaining my degree.)

    vongehr

    First off, the fact that a 2D surface, that exists through time, embedded in a 3+1 Minkowski spacetime inherits SO(1,2)

    No, it does not inherit that symmetry. In every system, not just graphene and certain superfluids etc. in our universe is there the inherited relativity. The additionally emergent relativity in those solid state systems is not the same one. You are correct however that it is not surprising that SO(1,d) arises (which is my point).

    any good theoretician will only take such models as "inspiration" or intellectual "stimulation" in creating theories of fundamental physics.  Such models can do little else (besides being a model "test bed" for the ideas so engendered), since the real test is not such model systems, but fundamental reality itself

    "any good theoretician" knows that we still do not know whether what you call "fundamental" in some sort of religious sense is not emergent in the same sense, for example on a stringy membrane in a higher dimensional bulk space. You need to stop thinking fundamental physics is somehow close to god while what we do in the lab is artificial. They are all physical systems we observe.

    * "in the same sense" means here as opposed to that it is in terms of a relational resolution of course emergent anyway - but we talk about a different kind of emergence here

    Actually he didn't say that. He strongly implied that "location" was not going to be used, in which case the arrow has no particular place to go or even to be.
    And even the question of locating the vector is moot. Engineer-level physicists are used to dealing with "forces" that act "through" a point (or along a line), they also act though other points but then you have a turning moment to consider - hence the force vector is not merely "magnitude and direction", it is actually located.
    David, however, has launched straight in by saying that the place to put places, namely the manifold, is put to one side, we only want somewhere to put little arrows. That sounds suspiciously like the arrows are not located. And yet suddenly they are located  after all - on an underlying "surface" (what's a suface?). So are we to understand that the vector "at" a given location is located there - standing in a diiferent relationship to different points, in the same way that a force creates a different turning moment at different points depending on how offset they are?  Or are are we supposed to grudgingly acknowledge the underlying "locational" manifold whch has a vector associated with each location even though the vector itself is purely magnitude and direction, no mention of location in the vector itself?
    Then there is the problem of the underlying manifold. David and every definition I have seen, insists that it is a Euclidean space. Now, a little knowledge is a dangerous thing, but we all know that a lot  ofgeometries are self-consistent but do not obey Euclid. What's so special about Euclid for manifolds? Are we going to build a theory of vector spaces which presuppose an underlying manifold which is strictly Euclidean? There is a huge amount of circularity in the argument already. I do trust mathematicians to the extent that if they say it can be put on a non-circular basis, I'll take their word for it, but David assumes a pretty comprehensive background knowledge of group theory here. This statement alone
    I suspect that practically everyone here that has heard of vectors, before this article, has also learned about the dot or inner product of vectors, and how this relates to magnitudes and directions of vectors.
    is so far removed from reality that I seriously wonder whether anyone is going to benefit. Certainly the material covered in this second article needs to be be recast. It's no good scattering chummy facetious interjections like "(hear sound of this shout, in a deep bass voice, reverberating off the mountainsides...)" to try to make it more accessible. Readers who are prepared to tackle this sort of thing at all do not need to be patted on the head when three years of undergraduate study are dumped on us as undigested dogma. It's the writer's obligation to break it down into bite-sized portions. If that can't be done, so be i, tell us what to take on faith. But the one thing you can't do with an intelligent audience is gloss over the difficult stuff!
     He has only said that something called a tangent vector space -comprised of arrows- exists at any point on a surface and obeys a given algebra, and that distance doesn't depend on these notions alone.
    Actually, Barry, that is a lot clearer as it stands than David's entire article.  Still totally incomprehensible but clear :).

    Well, there's a lot of comments to plough through, no need for anyone to work through my questions specifically, they'll either get answered or not as we go. I'm resigned to this being a marathon of Googlipedia maths. There again, my starting point is that a space is somewhere to put a box so what hope is there?
    Halliday
    Derek:

    I'm sorry you have so misunderstood what I have said, and have tried to convey.  I only started from this manifold perspective because of what you said, before, though I still think it was a good idea.

    Unfortunately, your first misunderstanding is shown in your first statement:

    Actually he didn't say that. He strongly implied that "location" was not going to be used, in which case the arrow has no particular place to go or even to be.

    Actually, I did try to say precisely what Barry has expressed.  While you are correct that after having introduced the tangent space (at each point on/in the manifold) we will not be using "that 'location' ", at least for a while.  However, the arrow most certainly has a "particular place to go":  It "goes" in the tangent space, and each tangent space has a single point in/on the manifold where it "goes" (exists).

    Does that help clarify this part?

    Continuing, you ask:

    ... So are we to understand that the vector "at" a given location is located there - standing in a diiferent relationship to different points, in the same way that a force creates a different turning moment at different points depending on how offset they are?  Or are are we supposed to grudgingly acknowledge the underlying "locational" manifold whch has a vector associated with each location even though the vector itself is purely magnitude and direction, no mention of location in the vector itself?

    I don't know why you say "grudgingly acknowledge the underlying 'locational' manifold".  Is it because I'm not making the manifold a central feature of what I am trying to focus upon?

    Now, from my explanation, above, I hope you now recognize that the correct answer is that the vector, itself, is, indeed, "purely magnitude and direction, no mention of location in the vector itself".  The only "mention of location" associated with a vector is given by the location of the tangent space to which it belongs.

    However, for the next while we will be able to simply focus upon a single tangent space.

    You then state another misunderstanding of what I have tried to convey:

    Then there is the problem of the underlying manifold. David and every definition I have seen, insists that it is a Euclidean space.

    While I have little doubt that "every definition I[you] have seen, insists that it[the underlying manifold] is a Euclidean space."  However, where did I insist upon this?  Is it simply because the example I used (as would be any example I could use that general readers would have had any experience with) happens to be Euclidean?  (Actually, the manifold is not truly Euclidean, only the tangent space, of course.)  It was but an example, that was intended to help.  I'm sorry it has confused.  :{

    You quote my statement:

    I suspect that practically everyone here that has heard of vectors, before this article, has also learned about the dot or inner product of vectors, and how this relates to magnitudes and directions of vectors.

    Then you state that it "is so far removed from reality that I seriously wonder whether anyone is going to benefit."

    What "is so far removed from reality" about that statement?  Perhaps I made the first part overly broad?  You are probably correct that not everyone that has heard of vectors, even before having read this article, has learned about the dot or inner product of vectors.  Perhaps I should exchange "heard" and "learned"?

    I'm OK with suggestions on how to "recast" this article.  :)

    You quote Barry as saying:

    He has only said that something called a tangent vector space -comprised of arrows- exists at any point on a surface and obeys a given algebra, and that distance doesn't depend on these notions alone.

    Then you state that "Actually, Barry, that is a lot clearer as it stands than David's entire article.  Still totally incomprehensible but clear :)."

    You also continue with:

    ... I'm resigned to this being a marathon of Googlipedia maths. There again, my starting point is that a space is somewhere to put a box so what hope is there?

    Boy, I can see I have my work "cut out for me".  We certainly cannot continue with my having done this poor a job communicating.  :(

    Please help me know how this can be improved.  After all, that's one of the purposes I stated from the beginning.

    David

    blue-green

    My phrasing is off since I've been away from academia since the 1970s, and still, I think it would help if your emphasis was more towards the physical and mathematical challenge of comparing distinct objects. I realize that my attempt below to dot product a dog and an ant is too far out for your comfort, since after all, it is a bit of stretch to have a linear algebra composed of distinct species. Be that as it may, methinks you need to work more with the natural tendency of people to make comparisons … (and for that matter, the very origin of mathematics, weights and measures).

    Somewhere down the road you will arrive at the essential inner product in which one can unambiguously distinguish whether objects have a light-light, space-like or time-like character

    If you want people to focus on the inner product and not notions like location or how things might compare after they are parallel transported next to each other, then just focus on how different types of objects are to be compared.

    My introduction to harmonic analysis is not an off-topic diversion. It puts the focus on the objects and steers the attention away from the underlying surfaces, unlike what you have done with your textbook pictures of tangent spaces kissing surfaces. People gaze at the surfaces and forget that this inner product business is really about the objects in the tangent spaces.

    Eventually you are going to get to a basis that spans your tangent space, a somewhat arbitrary chosen basis, and yet one whose elements are conveniently chosen to be linearly independent of one another.

    Here's a way to do it with pictures that works for any number of dimensions. The object that has a unit copy of each basis element pointing in each direction is drawn like a big asterisk ~ a star-burst whose radii are of equal length. Each radius stands for a unit-vector pointing in one of the linearly independent directions (for example, a specific harmonic). The inner-product of any two different  basis elements is zero regardless of whether we have a positive definite metric or a pseudo-metric. The basis elements are mutually perpendicular and pointing off in different dimensions, even if the star-burst picture is drawn on a flat piece of paper. Each radius is orthogonal to the other radii since it is one of the basis elements which all together can be scaled to span the entire vector space.

    A side-by-side comparison of two general objects is now like putting two irregular star-burst patterns next to each other and looking for similarities and differences. There is no assumption here that the inner-product of a single basis element with itself has to be a positive number. It could be a negative number, and therefore, for many combinations C of basis elements, the inner-product product (C,C) can be a sum of positive and negative numbers, and in certain cases sum to zero. Is this where we are going?

    blue-green

    While our proverbial Good Shepherd is off bringing the one that is lost back to the fold, I'll suffer the remaining 99% with another card trick.

    When quantifying distinct objects A and B, we can sometimes array or list their individual features in the form of a matrix or chart. More generally, the array can be three-dimensional or more … as with bubble graphs and the uses of color. A tangent space can be cross-producted with itself to accommodate n-tuples crossed with m-tuples and more. I mention it to get the emphasis away from the spacial geometry of arrows (which can be mistaken for being in the underlying manifold) and more towards the challenge of comparing a wider range of objects. When A and B are matrices or something more complicated, so long as they live in the same type of tangent "space" and therefore have the same dimensionality or rank, then one can entertain a dot product A-dot-B that measures how far apart A is from B .... or how much their basis elements are similar.

    Okay. One of the big problems is the proliferation of mathematical entities. 

    First we have "location". I think most of us are happy that a thing or event has to be somewhere and we need a certain number of coordinates to specify where. Now I would be inclined to leave the locations as an amorphous mass in a sack, but apparently they form a manifold whatever that may be. I don't know what that implies for distances, or any other geometrical idea.

     However the tangent space - another entity - appears to be much the same thing and this is suddenly "Euclidean". Given that our intuition lets us down in relativity, I am surprised that Euclidean geometry is blithely assumed for the tangent space, but perhaps "Euclidean" is an over-statement. Perhaps it just means it has orthogonal axes so as to keep the locational coordinates independent? And therefore can be mapped to paper and pencil.

    Now, at each point on the tangent space, we hypothesise a vector space.  This is totally confusing, but I would infer that this use of the word "space" is overkill. You don't seem to have done anything beyond introducing the idea that you you can have vector values at every point. Only we haven't really said what a vector is, other than a quantity with direction! But calling it a vector space at each location, makes me ask what the concept of a "space" adds to the argument and, more to the point, whether the spaces belonging to each location merge or are isolated bubbles?
     
    Here's a sack of locations being poured out onto a space which has orthogonal axes but very little else. Except we have said there may be vectors associated with each location. What else is implied by the various jargon terms - and therefore assumed in the model?


     
    Finally the dot product arrives. Now, if we already understand Euclidean geometry we can probably relate it to n-tuples... and faint hope that it will be the key to building a final "realistic" geometry at the top.
     
    p.s. Does anyone else have a tendancy to read "n-tuple" as "n-turtle"?

      
    Halliday
    Derek:

    Not too terribly bad, though there are some issues, and, obviously, some things that I apparently need to clarify.

    You start off with "locations" vs. "manifolds":

    First we have "location". I think most of us are happy that a thing or event has to be somewhere and we need a certain number of coordinates to specify where. Now I would be inclined to leave the locations as an amorphous mass in a sack, but apparently they form a manifold whatever that may be. I don't know what that implies for distances, or any other geometrical idea.

    If we "leave the locations as an amorphous mass in a sack" what does "that impl[y] for distances, or any other geometrical idea"?  If there is no "geometric", or other relationship between the "locations" (being "an amorphous mass in a sack"), is there any geometry?

    You see, it's the placement of these "locations" into a "geometric" relationship to one-another, called a manifold, that provides "for distances, [and/]or any other geometrical idea."

    Unfortunately, I have to go, now, so I'll get back to the rest later (almost certainly today).

    David

    That is precisely what I am driving at. Simply needing n coordinates is pretty well an operational definition and that means we have already gone to physics to discover our manifold. Just defining a set of n-nuples doesn't of itself get anywhere without a load of assumptions which I approximately .01% understand (about neighbourhoods which seem to give you surfaces without defined angles or distances fore some obscure reason). So, yes, "points in a sack" was deliberate! I am trying to keep track of where you sneak assumptions about geometry into the picture :)  What is left of space as we know it if you exclude distancxes and thus direction and vectors? This is where I, for one, amd finding it sticky going. Perhaps you should explain exactly what your are putting in, leaving in, or taking out?
      
    This *is* getting somewhere but rhetorical questions like
    If we "leave the locations as an amorphous mass in a sack" what does "that impl[y] for distances, or any other geometrical idea"? If there is no "geometric", or other relationship between the "locations" (being "an amorphous mass in a sack"), is there any geometry?
    You see, it's the placement of these "locations" into a "geometric" relationship to one-another, called a manifold, that provides "for distances, [and/]or any other geometrical idea."
    remind me of the sales pitches of cold-call telephone salesmen. Stating the obvious as if it was the answer to a question when it was in fact the very question I am asking :)
     
    No need to hurry back as far as I'm concerned, I am battling with a flu-like bug and I can't think for more than about 3 minutes at a time. 
    Halliday
    Derek:

    I'm sorry to hear that you are feeling "under the weather", so to speak.  I hope you get feeling better soon.

    I certainly didn't mean to sound like a "cold-call salesman."  I wasn't so certain whether the answers to my "rhetorical questions" were all that "obvious" to you or not.  (In the case where the answers were not so "obvious", they were intended to be genuine questions.  To the extent that they were "in fact the very question [you] [were] asking", they were intended to reformulate the question in such a way that you, and others, may have a better ability to answer the question for your [their] self.)

    OK.  In your earlier post, you said "I think most of us are happy that a thing or event has to be somewhere and we need a certain number of coordinates to specify where."

    This is a very important point.  You see, for the most usual manifolds (based upon "a certain number of" Real numbers, as "coordinates"), this is precisely what makes the "amorphous mass [of locations] in a sack" into something more than simply "an amorphous mass in a sack", but into a manifold.*

    So, a manifold is nothing more than your usual concept "that a thing or event has to be somewhere and we need a certain number of coordinates to specify where."  It simply gives that concept a more specific name, along with more specific definitions (that I haven't felt were necessary to go into).

    David

    *  As I am trying to explain to Hans and Frank, there are other, more general, concepts of manifold that do not require "a certain number of" Real numbers, as "coordinates".  However, for usual physics, and "everyday life" these usual manifolds suffice.

    You see, for the most usual manifolds (based upon "a certain number of" Real numbers, as "coordinates"), this is precisely what makes the "amorphous mass [of locations] in a sack" into something more than simply "an amorphous mass in a sack", but into a manifold.*
    That means nothing to me whatsoever.



    Halliday
    Derek:

    Is your illness causing your mind to simply go too fuzzy to understand that statement?

    Let's try another tack:  What do you mean by stating that "we need a certain number of coordinates to specify where" "a thing or event has to be"?

    Surely such implies more than simply having "the locations as an amorphous mass in a sack", or "a bag full of n-tuples" (as you mentioned in your message below).  (What does "a bag full of n-tuples" mean, anyway?  [That's not a rhetorical question, by the way.])

    David

    Halliday
    Derek:

    Go ahead and take your time recuperating.  I'm simply covering the bases in your message.  It's as much for others as for you.  You can just take your time with all this.

    OK.  You continue with the tangent space:

     However the tangent space - another entity - appears to be much the same thing and this is suddenly "Euclidean". Given that our intuition lets us down in relativity, I am surprised that Euclidean geometry is blithely assumed for the tangent space, but perhaps "Euclidean" is an over-statement. Perhaps it just means it has orthogonal axes so as to keep the locational coordinates independent? And therefore can be mapped to paper and pencil.

    Now, as I asked earlier, "where did I insist upon this [that the tangent space is 'Euclidean']?"

    Unfortunately, especially on Wikipedia (a source I certainly never consider authoritative), manifolds, and their associated tangent spaces (when such exist), are often considered to be locally "Euclidean", or, more correctly, "having a neighborhood that is homeomorphic to a Euclidean space".

    There are a couple of things that should be pointed out to anyone that looks at such things, especially on Wikipedia:

    1. The various branches of Mathematics that deal with geometric-like concepts all evolved using Euclidean geometry as the prototype.  So, is it any wonder that comparisons are often couched in terms of a comparison to Euclidean geometry?
    2. The terms "homeomorphic" or "homeaomorphism" are the topological form of isomorphic or isomorphism.  In other words, they have to do with the "mappings" between topological spaces that preserve all the topological properties.  Such have noting, whatsoever, to do with inner/dot products, angles, orthogonality, norms, or distance measures (except in the more nebulous topological sense of "neighborhood").
    3. Since all the usual manifolds are locally "homeomorphic" to some space with "a certain number of [Real number] coordinates" (we call that Rn, for n dimensions, where R represents the Real numbers), including Euclidean space; and since "homeomorphisms", as with any isomorphism, are invertible (so we can use the "mapping" both ways); all such usual manifolds are also "homeomorphic" to an appropriate dimension of Euclidean space, even if they have very different concepts of "inner/dot products, angles, orthogonality, norms, or distance measures", since such are of no concern to "homeomorphisms".  (So they "therefore can be mapped to paper and pencil.")

    The confusions that can be engendered by the tendency of people to associate all the extra "baggage" (namely the nature of the inner/dot product, and its "concept" of "orthogonality",  "angle", and "distance"/"length") when confronted with the term "Euclidean space", is why I prefer to simply compare the usual manifolds (and their associated tangent spaces, when such exist) to Rn.  While some may yet think of this as "Euclidean", it actually need have no measure of "angle", "inner/dot product", or even "distance" (except in the most primitive "neighborhood" sense actually used in topology).

    So, at this point, the manifold, and its associated tangent spaces, are not actually Euclidean.  In fact, they need not have any ability to "measure" "angles", "orthogonality", "inner/dot products", or "distances" (except in the most primitive "neighborhood" sense actually used in topology:  a concept "inherited" from the n "copies" of the Real numbers that are used as "coordinates" for the manifold, and as scalars for the associated tangent spaces).

    David

    Good grief. I understood that, and it's what I was groping towards saying. But now that you have raised the subject of neighbourhoods, PLEASE explain what of "space" remains in the idea of "manifold"!  I see no reason why a bag full of n-tuples should be homeomorphic to "a surface" unless a) the term "surface" is meaningless or b) you're sneaking in some minimalist ideas of geometry, probably via neighborhoods. But *what* you're sneaking in, and *how*, is not obvious. 
     
    Halliday
    Derek:

    If you truly understood all that, before my having explained it, then what is the trouble?

    However, your question:

    ... But now that you have raised the subject of neighbourhoods, PLEASE explain what of "space" remains in the idea of "manifold"!  I see no reason why a bag full of n-tuples should be homeomorphic to "a surface" unless a) the term "surface" is meaningless or b) you're sneaking in some minimalist ideas of geometry, probably via neighborhoods. But *what* you're sneaking in, and *how*, is not obvious.

    suggests you don't fully understand all that, even after my having explained it.

    Now, are you simply getting hung up on the use of the term "surface"?

    OK.  To help make these things more concrete, let's restrict ourselves to two dimensional cases (so we may have 2-tuples, if you need them), for now.

    What do you mean by "a bag full of n-tuples" (2-tuples, in this case)?

    Is there some relationship between that and "need[ing] a certain number of coordinates [two, in this case] to specify where" "a thing or event has to be"?

    Unfortunately, I think your difficulty is that the concept of a manifold is, perhaps, too simple!  I think you are trying to make it too complicated.  (Just a thought.)

    I'm most certainly not "sneaking in some[thing]".  In a sense, I am probably "deconstructing" what you already think you know about "geometry":  I am most certainly separating the concepts of the "space" of "locations", from the "space" of "vectors" (vector spaces), from the concept of "distances"/"lengths"/"magnitudes", "norms", "angles", "orthogonality", or "inner/dot product".

    David

    No-o-ooo... when I said "I understood that" I was trying to make an encouraging comment that you were making yourself understood, I didn't expect to be cross-examined on precisely what I understand and what I don't. 
     
    I have put it all into a picture to make it all nice and simple :)

    By "bag of n-tuples" I mean a collection of n-tuples - ordered number pairs, triplets etc as in diagram A. I don't attach any significance to the "bag" - my own preference is for them to be heaped up on the back of a turtle as in B. Since neither the bag nor turtle play any active role in the theory, A and B are homeomorphic. As indeed is C with them both where the collection of locations is dumped in a pencil and paper space with no particular significance to the position on the diagram.
     
    At this point I get distracted because a the same n-tuples could represent a physical location or, just as easily a physical vector. Since we have hopefully stripped them of physical/geometrical baggage, a collection of vectors with no particular positions, D, is just as good a representation as  C, the collection of points. Of course already there are some conundrums. There is nothing to stop us taking the difference between two n-tuples. The result is an n-tuple and we may reasonably assume it is a member of the turtle's back-set... well we can define the set so it must be. But we certainly can't assume that it will be a vector with any of the usual connoptations as in E.

    I'm also intrigued by the parallel case F, where a web of vectors seems to suggest a surface just as a collection of points does - and with the same limitation: that until we define distance it's just a collection...

    On the other hand the elements of each n-tuple are numbers and numbers are ordered. So we can arrange the "locations" in order - according to the first element say. This gives us a row of points like G. In fact each point along the line is actually a collection  of all n-tuples with that value in the first element. So it's more like H. We may as well then order these points as well, making a network of criss-crossing rows, I. They are of course actually continuous lines like J. I assume this baby-talk way of putting things captures what you mean by manifold without, as far as I can tell, sneaking in any undesirable geometrical assumptions.


    I am most certainly separating the concepts of the "space" of "locations", from the "space" of "vectors" (vector spaces), from the concept of "distances"/"lengths"/"magnitudes", "norms", "angles", "orthogonality", or "inner/dot product".
    Well, that was the difficulty. You tried to do so but introduced an entity which definitely suggests "geometry" to us mortals. I know you didn't say much of the above but it seems to me that you can arrive at the manifold without making it out to be a surface which isn't a surface. Did you notice that as soon as you extended it to a tangent space, you slipped into pure Euclidean geometry of thr sort you learned at school? And immediately, people were asking whether it had to be flat?
     
    Cheers!


     
    Halliday
    {I have done some editing and correcting of this message.  Such edits are set apart by these curly braces.}
    Derek:

    I most certainly didn't mean to have you feel "cross-examined on precisely what [you] understand and what [you] don't."  However, I was noticing some significant difficulty in our ability to "connect".  That strongly suggested that I was not correctly understanding "what [you] understand and what [you] don't."

    I'm very glad you have elaborated upon what you have had and/or presently have in mind.  This should help tremendously.  (Thanks for the illustration, as well.  :)  )

    Let's start with your concept of this "bag" of "n-tuples" and/or "locations":

    By "bag of n-tuples" I mean a collection of n-tuples - ordered number pairs, triplets etc as in diagram A. I don't attach any significance to the "bag" - my own preference is for them to be heaped up on the back of a turtle as in B. Since neither the bag nor turtle play any active role in the theory, A and B are homeomorphic. As indeed is C with them both where the collection of locations is dumped in a pencil and paper space with no particular significance to the position on the diagram.

    Do you really think that these "n-tuples" and/or "locations" are purely discrete, with no relationship among one-another?

    Even if that were the case, one then has a 0-dimensional (discrete) manifold (even if one has [countably] infinite many "n-tuples" and/or "locations").  So, even then, one has a manifold.  It's just that you haven't thought of it that way.

    You are absolutely correct that they need have no relationship, whatever, with anything besides each other.

    Now, this is highly related to your next comment:  "At this point I get distracted because a[ll?] the same n-tuples could represent a physical location or, just as easily a physical vector."

    Ahhhh.  That's likely a great deal of the problem right there.{*}  The geometry of "physical location[s]" is quite different from the geometry of "physical [or other] vector[s]".

    A manifold has the geometry of "physical location[s]", but not (necessarily) the geometry of "physical [or other] vector[s]".  While, simultaneously, a vector space has the geometry of "physical [or other] vector[s]", but {maybe} not (necessarily) the geometry of "physical location[s]".  {However, it is true that all Real vector spaces—vector spaces where the scalars are from the Real number field—are also manifolds, when one ignores the operations of vector addition and scalar multiplication, but only a small subset of manifolds can even be made into vector spaces.}  (The "not necessarily" proviso is due to the fact that certain classes of manifolds—namely "flat" {infinite} manifolds—{can be given} the geometry of a vector space.  It is the fact that all of physics dealt only with such "flat" {infinite} manifolds {that were also given the geometry of a vector space}, before Einstein [borrowing from Mathematicians long before him] introduced the curved manifolds of General Relativity, that, I believe, leads many to conflate the [mental] concepts of manifold and vector space [with no knowledge that they are doing so, since they don't have the vocabulary—the term/words—to help them even recognize that there even is any distinction].)

    You go on with:

    ... Since we have hopefully stripped them of physical/geometrical baggage, a collection of vectors with no particular positions, D, is just as good a representation as  C, the collection of points. Of course already there are some conundrums. There is nothing to stop us taking the difference between two n-tuples. The result is an n-tuple and we may reasonably assume it is a member of the turtle's back-set... well we can define the set so it must be. But we certainly can't assume that it will be a vector with any of the usual connoptations as in E.

    Actually, we have not, entirely, "stripped them [(n-tuples, locations)] of physical/geometrical baggage".  Whether they have the geometry of a manifold, or the geometry of a vector space, they still have some form of "physical/geometrical baggage".  {In fact, one can see, at least for Real vector spaces and Real manifolds, that the Real manifold has the least "physical/geometrical baggage".}

    As you state in your comment, the ability to take "difference[s]" or sums of n-tuples is a particular form of "physical/geometrical baggage".  It is the "physical/geometrical baggage" of a vector space (when you also allow multiplication by scalars:  Re-scaling vectors).

    You see, that's part of the point of the different "physical/geometrical baggage" of a manifold:  In a manifold, the sums or "difference[s]" of these "n-tuples" is not allowed—such has no meaning!  {Note:  It's not that "someone" is "not allowing" such operations as much as the fact that such operations are not a part of what a manifold "is".  Furthermore, it is easy to find manifolds where there is no consistent way to define such vector operations to transform the manifold into a vector space.}

    {This next statement is actually FALSE!}Likewise, the ability to "move" from "location" to "location" is part of the "physical/geometrical baggage" of a manifold, but has no meaning for a vector space ("a collection of vectors with no particular positions", as you put it).

    (Your 'E', incidentally, appears to be the case where we have a "flat" {infinite} manifold, where we are therefore allowed to "identify" the manifold [the space of locations] with its tangent space[s] [the space[s] where vectors exist].  I'm not quite sure what to make of your 'F', unless your taking only a discrete set of vectors, and requiring each vector to only exist at the "head" or "tail" of another vector, or something like that.  Such a case is then called a Directed Graph, or Di-Graph.  That's another form of discrete manifold with additional structure[s].)

    Then you go on with:

    On the other hand the elements of each n-tuple are numbers and numbers are ordered. So we can arrange the "locations" in order - according to the first element say. This gives us a row of points like G. In fact each point along the line is actually a collection  of all n-tuples with that value in the first element. So it's more like H. We may as well then order these points as well, making a network of criss-crossing rows, I. They are of course actually continuous lines like J. I assume this baby-talk way of putting things captures what you mean by manifold without, as far as I can tell, sneaking in any undesirable geometrical assumptions.

    Ahhhh.  Now you're starting to get what a manifold is!

    Like you say, these n-tuples are ordered collections of numbers (usually Real numbers, for the usual concept of manifold), and these numbers (Real numbers, in the usual case) have their own, natural relationship**.

    The thing is, do you actually only have "a network of criss-crossing rows", that "are of course actually continuous lines like J"?

    Just as each "element" of each tuple is continuous (the continuum of the Real numbers), aren't the "continuous lines" actually continuously connected, "side by side" (so to speak), just as the points, of 'G', became "continuous lines" in 'J'?

    For 2-tuples, what does that make/create?  {That's right, a surface!  The important thing is that once we recognize the natural structure the Real numbers, as parts of these n-tuples, impose or provide, we see that we already have a manifold (a possibly higher dimensional "surface"***), without anything else having to be added.}

    David

    {*  I suspect that one of the compounding problems is the way "n-tuples" are "thrown about" as "vectors" (at least as representing vectors, but sometimes as the "real thing"), and as "coordinates", and so many other multi-dimensional "things".  The principle problem is that "n-tuples", in and of themselves, mean very little.  They are a way of representing elements of some product space or set (a Cartesian product of elements from different mathematical collections like numbers, or whole spaces, etc.), but provide no indication of the nature of the product space they may represent.  At best, they represent elements of some "nebulous" or "formless" set.}

    **  Incidentally, it is this very relationship that brings forth the concept of "neighborhood" that you asked about, earlier!

    {***  Why is the word "surface" used in conjunction with things with more than two dimensions?  Because most people have experienced two dimensional surfaces that are not "flat", but curved, in so many interesting ways.  So, while most people have a very difficult time conceiving of a three dimensional manifold being anything besides a Euclidean space—like what they think the space around them is like—they can conceive of curved and convoluted "surfaces".

    Similarly, while all vector spaces are "flat", infinite spaces—like flat plains—manifolds are more often anything but "flat" and may even be finite in extent—just like surfaces, in general.}

    Enough!

    I was just trying to get the idea of the various mathematical entities you have thrust upon us. I was quite pleased to discover a sort of "surface without geometry" emerge from consideration of a set of n-tuples. That was all.

    I still don't understand tangent spaces or vector spaces in these terms at all or why addition of n-tuples is forbidden and who forbids it. 

    I don't understand any of the rest. I thought I did but after reading your reply, it seems I'm all wrong as usual. Please delete my post and my diagram, I'll just shut up and stop confusing everyone.
     
    blue-green
    The diagram is lovely. It be far from the first time that you have played the artist hurling his manuscript into the fire. Keep on pottering.

    ~ it's all in the fibres ~
    Fair comment (I was about to ameliorate what I said but a stray cat wandered into the thread, and I can't edit it now), though I think the previous occasion was due to the same thing. I really do get very pissed when I am struggling to understand something and just get told I'm wrong from start to finish... especially when there appears to be no significant difference between what I said and what the supposedly corrected version is.
    Halliday
    Derek:

    I'm sorry to see your distress.  :(

    Unfortunately, I don't have much time, right now, but I do want to point out a conceptual error that crept in, in my desire to contrast manifolds from vector spaces.  While it is only a single conceptual error, it, unfortunately, manifested itself in multiple statements I made in my message, above.  (Unfortunately, I didn't recognize I had made the error until after I posted the message, and took my wife out for a walk this evening.  It was during the walk that I realized the error I made.)

    While it is most definitely true that "A manifold has the geometry of 'physical location[s]', but not (necessarily) the geometry of 'physical [or other] vector[s]'."  It is much less true that "a vector space has the geometry of 'physical [or other] vector[s]', but not (necessarily) the geometry of 'physical location[s]'."

    This mistake most especially manifests itself when we restrict ourselves to Real vector spaces (vector spaces that have Real numbers as the scalars).  For, you see, all Real vector spaces are also manifolds, even though there are far more (Real, as in Real numbers) manifolds that are most certainly not vector spaces, Real or otherwise.

    I'm sorry for letting this mistake make its way into my message.

    Unless you feel you don't need me to, I'll address all the statements that this mistake affects in my message, above.  Or, if it's OK with you, I could go ahead and edit my message, above, to correct this error.

    Unfortunately, in either case, I'll have to do so at a later time.

    Good night, for now.

    David

    Feel free to do whatever is easiest. I haven't a clue what you're talking about so it won't make any difference to me. It might be an idea to mark any edits though just in case something makes sense one day.
    Please - I am out of this. Just move the subject on in whatever way you think best.


     
    Halliday
    Sascha:

    Are you truly admitting that you don't recognize that tangent spaces do not require an embedding space?  I would be very careful, if I were you, in trying to attribute this same misunderstanding to everyone (through your use of "Nobody").

    Do you not know that the principle reason why Riemann invented the geometry that bears his name was to be able to handle curved surfaces without having to have or deal with an embedding space?

    Are you admitting that you cannot understand General Relativity, and its curved spacetime, without having to embed it within some other space?

    OK.  Now on to your assertion that "Only crazy mathematicians think dot products are anything else but what tells you the true lengths of projections."  Actually, though one may use the dot product to create projections, that functionality actually belongs more properly with the "dual space" (the [vector] space of functions that operate upon vectors, in the tangent space, yielding scalar values).  In fact, one can then actually define projections without even having an inner or dot product.

    However, I would be happy not to have to delve into "dual spaces" with this audience, if you don't mind.  I expect they would be happier without such distraction.

    On the other hand, the ability to determine "true lengths", whether of "projections" or otherwise, along with the ability to have a consistent definition/concept of relative direction (angle, in Euclidean geometry), then one must have an inner product space.  (Contrariwise, if one is able to obtain/measure lengths and relative directions in a consistent manner, one is able to construct the inner/dot product [the metric], thus showing that one actually has an inner product space in which one is working.)

    David

    vongehr

    Are you truly admitting that you don't recognize that tangent spaces do not require an embedding space?

    Where did I ever write anything like that? Please read more carefully.
    I would be happy not to have to delve into "dual spaces" with this audience, if you don't mind.
    David - that is precisely my point. You want to talk to a certain audience, that is what you made clear the last time. That audience you have now blown out of the water with asking them to forget space being about locations and instead embrace so called inner product spaces and tangent vector spaces that cannot be embedded and shouldn't be anyway because they are not really spaces (where you put a box) and all that. I have no idea why you think that dual space is anymore abstract. ;-)
    Halliday
    Sascha:

    You quote me as saying:

    Are you truly admitting that you don't recognize that tangent spaces do not require an embedding space?

    Then you ask:  "Where did I ever write anything like that?"

    Well, you stated:

    Nobody, me included and I have no problem admitting so, grasps that tangent spaces are not just arrows inside embedding spaces. ...

    So, is that not what you meant to say?  Do you wish to rephrase?

    David

    Back then, AN Whitehead wrote a huge rambling treatise on Universal Algebra which gathered from the conventions of Classical mathematics not one inner product but two, which split the time-reversal symmetry. You do have a choice whether to take the Lorentz group in the sense of Poincare or Einstein, which makes the point that things are real to us in more than one sense!

    This now matters for medical research, but you can't do a PhD which challenges the idea of a monolithic scientific "consensus". You can cross the line to Heidegger's anti-science propaganda, but what use is that for medical research?

    Halliday
    Orwin:

    I think I understand the gist of your first paragraph.  Unfortunately, the point of your second paragraph seems to escape me.

    David

    blue-green
    The dot product of a dog and an ant is practically zero. Their rhythms (form-wise, time-wise and chemical-wise) do not match.

    There is an instinctual drive to size things up and compare them. This ability is certainly not limited to humans. All animals do it. Bugs do it. Even at the chemical level, dot products are effectively being performed. It is not just a matter of comparing “size”, it is a matter of measuring how much morphing it takes to project or fit one object onto another.

    To compare apples A with bananas B, we can fit them against a standard coin or candlestick C. We do this at a monetary level on a regular basis. If (A,C) = $7 and (B,C) = $10 then (A,B) = 7/10. That's the currency conversion for a trade or swapping of particles to occur. Yes, I'm killing the notation. Algebraically, I find this to be difficult to spell out without a basis. By (A,B) I actually mean (the value of A) divided by (the value of B). If in general the ratio is near one, then at equilibrium, one can expect there to be a lot of interconversion from A to B and back with no energy demand beyond what is pocketed by the money-changers who eventually devour both A and B.

    With Fourier analysis, each function is a “vector”. A function's harmonic decomposition reveals how much of each harmonic resides in the function. If the dot product of two functions is near zero, then they have very different harmonic structures, albeit, the precise choice of the underlying “structures” or basis elements is up to an arbitrary choice. One can use sine waves, triangular waves or something quite irregular as one's basis elements (your family crest!).

    Spoiler alert. The grand trick in this series to justify the use of a metric or dot product which is not positive definite.

    Properly taught, the Principle of Relativity is utterly natural and the simplest way of reconciling the experimental evidence. And yet, it is difficult to picture how (A,A) = 0 without A = 0. Perhaps, if one first justifies that there are real situations in which (C,C) is a negative number (as in loosing money), then the break even point where a dot product is zero is inevitable and the whole light-cone structure of spacetime comes for free.
    fundamentally
    Space, proper time, coordinate time and spacetime are all very different concepts. Not reckoning this makes it unjustified to call spacetime just another form of space. Spacetime has Minkowski signature, while the space part of it has Euclidean signature. The time part of it is not just another dimension. It is either proper time or it is coordinate time. It depends where the clock is ticking. Does is tick with the observer item? Or does it tick with the observer. However, when the clock ticks, then the model that we are analyzing is progressing. So progression is something that is not equal to proper time. It is not equal to coordinate time and it is not equal to spacetime. And it is certainly independent of space.
    In quantum physics position can be treated as an observable and observables can be considered to be eigenvalues of operators. This does not occur with time. It does neither happen with proper time nor with coordinate time. It also does not happen with progression. Proper time and coordinate time are connected to corresponding items. One is the observed item and the other is the observer. However, progression can be viewed as a global parameter. It is not connected to a particular item. In fact it is connected to universe as a whole. Space also covers the whole universe. In that sense space and progression are two global characteristics of universe. They fit better together than space and time. If I must create a four dimensional space from the ingredients 3D space and a fourth dimension, then I would take progression as the fourth dimension. It will mean that the dynamics of the universe occurs in universe wide steps that go together with a clock tick.
    If you think, think twice
    Bonny Bonobo alias Brat
    This picture of the world and the diagram explaining manifolds is really not making any sense to me. David, can you please explain in more detail exactly what this diagram is supposed to be showing here?



    The wiki article on manifolds also shows this globe diagram and says 'The sphere (surface of a ball) is a two-dimensional manifold since it can be represented by a collection of two-dimensional maps'. 

    Why have they selected the smaller map and shown a triangle with the angles 40, 50 and 90 degrees and then shown that next to the other 3 sided section with its 2 right angles and 50 degrees at the top? Are the two 50 degrees somehow connected or just an irrelevant coincidence?
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    mathematical_investigations
    I've debated whether or not I should answer this because your question is directed to Dr. Halliday. I hope it isn't rude for me to answer. Dr Halliday can provide a better answer, but I believe the diagram is simply showing the following:

    The angles of a right triangle in Euclidean (Flat) Space add up to 180.
    50+40+90=180 

    but the angles of a right triangle in Non-Euclidean (Curved) Space do not. 
    50+90+90=230 

    The picture shows that sufficiently small patches of the Non-Euclidean surface of the earth are Euclidean. On a large scale the earth is a curved surface, but locally it is flat. 

    Manifolds are spaces that have this property of being locally Euclidean, but not necessarily globally Euclidean. 
    Bonny Bonobo alias Brat
    Oh thanks Barry, yes that makes complete sense now, one obstacle to me understanding all this has been cleared, on to the next! 
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    Halliday
    Thanks, Barry.  You did a fine job.
    Halliday
    Helen:

    I'm sorry that that diagram threw you for a loop.  The diagram was intended to help, not to confuse.  :(

    Of course, as Barry has already explained, the angles on the very small triangle (the one where we "zoom in" very close to a single point on the globe) has angles that must always add to 180 degrees, as must always be the case in Euclidean geometry.  However, the large "3 sided section", as you called it, can have any arbitrary angle at the top (from zero through 360 degrees).

    Unfortunately, I had no control over what angles the creator of that image used.  Besides, unfortunately, it didn't even occur to me that the particular choice of angles would cause anyone any trouble.  :/  My bad, I suppose.  :{

    By the way, did you notice that the images do link to related Wikipedia articles (the same ones that use the images)?  Did that help, or hurt?

    Did my paragraph on the manifold of the Earth, and tangent space to it (at a point), help, or was it still too confusing?

    I do want suggestions for improvement in the presentation.

    David

    Bonny Bonobo alias Brat
    By the way, did you notice that the images do link to related Wikipedia articles (the same ones that use the images)?  Did that help, or hurt?

    Actually I didn't notice that the diagrams had links incorporated within them, I just came across the same diagrams when I searched Wiki for the relevant articles. I did however repeatedly utilise your links to the numbered explanations at the bottom of the blog and back again, which I thought were quite brilliant. How did you do that?

    Did my paragraph on the manifold of the Earth, and tangent space to it (at a point), help, or was it still too confusing? I do want suggestions for improvement in the presentation.

    Yes that helped a lot, not at all confusing!

    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    Halliday
    Helen:

    I'm sorry you didn't notice that the pictures were also links.  As I've said, I placed them there to help, and I thought that providing links to the related articles from which I borrowed the images would provide additional help, for those that want more.

    As for how I did the footnote links:  I used the second "idiom" on the "Footnotes" page of the HTML 5 specification.  Unfortunately, since this site defines a <base> (set to "http://www.science20.com", rather than the actual article) I had to do additional work (the links had to be explicit to the actual URL of the article, once it is published).

    I'm glad they worked for you.

    David

    Bonny Bonobo alias Brat
    David, thanks for explaining how you linked to the footnotes and back, sounds very clever.
    With regard to the diagrams from Wikipedia, I must say that these have caused me the most confusion here and I find it difficult to understand why you decided to use different symbols to the Wiki article in your blog, ie alpha and beta instead of v and w etc. I don't even know how to type the letter alpha here which makes it harder to even discuss its properties doesn't it? Also, it means I have to compare the axioms here on your article with the axioms in the Wiki article trying to find matches with the different symbols that ultimately mean the same thing. 

    This brings me to my next question, why are there 8 axioms listed under vector space definitions in the Wikipedia article but you seem to be only interested in the 5 axioms or 'conditions satisfied' in V1 to V5 here? 
    V1)  aα ∈ V.
    V2)  a(bα) = (ab)α.
    V3)  (a+b)α = (aα) + (bα).
    V4)  a(α+β) = (aα) + (aβ).
    V5)  1α = α
    Wiki says :-

    To qualify as a vector space, the set V and the operations of addition and multiplication have to adhere to a number of requirements called axioms. In the list below, let uv and w be arbitrary vectors in V, and a and b scalars in F.
     
    AxiomSignification
    Associativity of additionu + (v + w) = (u + v) + w
    Commutativity of additionu + v = v + u
    Identity element of additionThere exists an element 0 ∈ V, called the zero vector, such that v + 0 = v for all v ∈ V.
    Inverse elements of additionFor every v ∈ V, there exists an element −v ∈ V, called the additive inverse of v, such that v + (−v) = 0
    Distributivity of scalar multiplication with respect to vector addition  a(u + v) = au + av
    Distributivity of scalar multiplication with respect to field addition(a + b)v = av + bv
    Compatibility of scalar multiplication with field multiplicationa(bv) = (ab)v [nb 2]
    Identity element of scalar multiplication1v = v, where 1 denotes the multiplicative identity in F.

    Sorry if these are stupid questions, I'm happy to continue asking you about all this on my corkboard at any time, just say if you want me to, I don't want to confuse other people here with my crazy way of looking at things. 
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    Halliday
    Helen:

    By the way, no need to ask these questions in private, or anywhere else.  If you are asking such questions, then there are likely others wondering about similar things, but are just too afraid to ask.

    I didn't bother with making sure my notation matched that used in the Wikipedia articles because my article doesn't, truly, depend upon those articles.  (Besides, I linked to more than one article, and those articles may not use the same notation among themselves.)

    As for why the Wikipedia article has more axioms:

    1. The Associativity axiom is accounted for in the requirement that the vector space is a group under addition.  (See G1), in footnote 4.)
    2. The Commutativity axiom is accounted for in the requirement that the vector space is an abelian group (meaning the elements of the group all commute) under addition.
    3. The (additive) Identity element axiom is also accounted for in the requirement that the vector space is a group under addition.  (See G2), in footnote 4.)
    4. The (additive) Inverse element axiom is similarly accounted for in the requirement that the vector space is a group under addition.  (See G3), in footnote 4.)

    I notice you didn't list the axioms of "closure" with respect to either addition or scalar multiplication.

    My approach, in presentation, is more hierarchical.  It has the utility of not having to repeat notions defined at some lower level, but it has the disadvantage that it's not all laid out in a single list.  It also has the advantage that we have the definition of these lower level concepts already defined, in case we may ever need to refer to them.

    David

    Bonny Bonobo alias Brat
    Thanks David, yes this all seems to make sense to me now. I feel confident that I have understood most of what you have written here and I am looking forward to the next installment :)
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    Bonny Bonobo alias Brat
    I think I now have to retract this statement, I no longer understand most of what is written here but I'm still looking forward to the next installment!
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    Halliday
    Helen:

    What caused the change in your assessment of your level of understanding?

    Does it have anything to do with the exchange between Derek and myself?

    Is it something you read elsewhere?

    I'm concerned.

    David

    Bonny Bonobo alias Brat
    What caused the change in your assessment of your level of understanding? Does it have anything to do with the exchange between Derek and myself? Is it something you read elsewhere?
    David, when I reread your blog I feel as though I understand everything written there but then when I reread some of the many comments here, I start to wonder if I really do. My maths is very rusty but I know from my school days, that as long as I am given diagrams and real examples, I can normally understand most abstract mathematical concepts quite easily. BTW, I find Derek's comments probably the least confusing and the easiest to understand of all the comments here and I love his diagrams but naturally his confusion at times was contagious.

    You said somewhere in this comments section, to Barry I think :-
    Yes, I would favor an approach more like "just present[ing] vectors and how they work and then pointed out that this is all the 'Vector Space Conditions' really say." I tried to do that in the prose of the article, but I agree that examples will help many.

    Yes please, I need more examples and diagrams with the associated mathematical formulae and geometry and less abstract concepts and discussions without corresponding diagrams and examples, if that's possible? Sorry if that sounds childish, especially as it reminds even me of how kids always prefer bedtime stories with pictures. I hope that answers your questions?

    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    Halliday
    Thanks, Helen, that does help.  I shall endeavor to create some graphical examples as soon as I can.

    Since the question of "manifolds" and how they relate to "tangent spaces" is not the focus of this series, and since the only reason I introduced such things was to provide some context for moving the reader on to the vector spaces that are these "tangent spaces", would you be satisfied with graphical examples of the characteristics of vector spaces?

    Or are you, with Derek (as far as I can see), too "hung up" on these other "things" to be able to focus on vector spaces, at this time?

    David

    Bonny Bonobo alias Brat
    No David, please keep it as simple as possible for me. I'm not hung up at all and would love to focus just on vector spaces, so yes graphical examples of the characteristics of vector spaces would be great. 
    I also would be really grateful if every symbol, in every future diagram, is explained and not assumed to be understood, even though most people other than me probably would understand that symbol without an explanation. Believe it or not I had never even heard of the expression 'turtles all the way down' before reading the comments section of Derek's 'the Last Turtle' blog, I just thought that Derek liked turtles :)
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    Halliday
    Helen:

    I shall endeavor to do my best in providing graphical examples of the characteristics of vector spaces.

    I will also strive to keep in mind the potential for confusion with symbols in images vs. text.

    David

    Halliday
    The vector space illustration is below.
    Don't bring me into it :)  I just want you to get on with the next instalment.
     

    Halliday
    Derek:

    I'm glad to see that you seem to be in a better mood.  ;)

    I'll take the time to address Helen's desire for some illustrations of vector space characteristics, then it will be on to the next installment.  (Unfortunately, work and home are keeping me very busy.)

    David

    I'm glad to see that you seem to be in a better mood. ;)
    Pragmatism wins over curiosity in the end :/
    I'll take the time to address Helen's desire for some illustrations of vector space characteristics, then it will be on to the next installment. (Unfortunately, work and home are keeping me very busy.)
    Absolutely! First things first, every time. Certainly no need to hurry on my account - I'm writing a "Quantum Crackpot Randi Challenge Lite" triggered off by the exorcism of Joy Chrisan from Oxford academia plus planning the second instalment of "Time Travel Paradoxes" in my "spare" time. That would be handy wouldn't it? A spare set of 24 hours for each day.
     



     
    Halliday
    Derek:

    It sure would be handy to have "a spare set of 24 hours for each day."  ;)

    I actually had an application from a company, recently, that assumed 15,000 hours per year!  They're close!  ;)

    David

    rholley
    Hello Helen,

    Here is something I did for my History of Maths course a couple of years ago.  I’m putting it here as much for artistic effect as contributing to the discussion (the image is much bigger if you download it).

    Menelaus of Alexandria (roughly 70 – 130 AD)  produced a work called the Sphaerica, which got translated into Arabic and from there to German by a mathematician who died on the Russian front in 1943 or -4 (sad loss).  Our library loaned the book from Germany, and here is it is:

    On the top left is a theorem of Euclid (1:16) which states that the exterior angles of any triangle are greater than either of the interior angles.  Menelaus showed that on the sphere, this only works for triangles up to a certain size (top right).  If the triangle’s area is half the lune (shape of an orange segment), then the exterior and the opposite interior angles are equal (bottom left).  If the triangle is bigger, then the interior angle is bigger (bottom right).

    I don’t know whether it is Menelaus himself, or his Arabic or German translators, but the Sphaerica comes across to me as a much more cheerful work than the Euclid.


    Robert H. Olley / Quondam Physics Department / University of Reading / England
    Bonny Bonobo alias Brat
    Yes, very nice Robert, puts a whole new perspective on peeling and eating an orange doesn't it?
    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    Bonny Bonobo alias Brat
    The truth is, we now have, with this inner or dot product, the METRIC (hear sound of this shout, in a deep bass voice, reverberating off the mountainsides...).  We simply need to take a closer look, expanding the vector space in terms of a basis (as we always can do for a vector space), and, ultimately, relaxing condition D3, the positive-definiteness condition.
    That's what we will be addressing next time.

    Well done David, I at least think that I now understand most of what you have explained here so far, though I think it was a bit confusing to introduce a diagram with a second vector 'W' with no explanation as to what 'W' represents. Wiki's vector space article explains that 'W' is just another vector which is probably pretty obvious to most people here but I found it confusing. I just like everything in equations to be defined clearly beforehand without any prior assumptions :)

    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    Halliday
    Helen:

    I was a little concerned, when I added that image to my article, that the use of v and w, in the image, might cause some issues, since I was using α and β in my definition.  In fact, since you bring up an issue with 'W', but not with 'V', this suggests that you may have actually thought the v (or 'V') in the image was actually related to V, in the definition.  If so, this is an even worse state of confusion than I imagined!  :{

    Perhaps I should have created my own version of this image, more consistent with the text.

    I'm sorry.

    David

    mathematical_investigations
    It might be helpful to provide a simple example of a vector space (like the set of ordered n-tuples with addition defined in the usual manner) and show that it satisfies a couple of the vector space rules. It would also give you something to illustrate the inner product with later.
    Halliday
    Barry:

    Yes, n-tuples are simple, and are especially familiar to computer programmers.  However, the trouble I have with n-tuples is that it gets way too easy for people (especially computer programmers) to confuse tuples with the vectors.  However, I agree that a simple example of a vector space is certainly called for.  The vector space of real numbers is almost certainly too simple, but the vector space R2, of points in a two dimensional plane, should be good.  (It especially helps for the visually oriented members of the audience.)

    Additionally, this provides a good "jumping off point" for talking about different bases for a vector space (which is the more correct "dwelling place" for tuples).

    The reason I didn't do such as a part of this installment was that it was getting rather large already.

    Do you think that we could fulfill the need, reasonably, within this comment section?

    David

    mathematical_investigations
    Do you think that we could fulfill the need, reasonably, within this comment section?

    I believe I can give a quick example of a 2 dimensional vector space (or 2-tuple space) defined over the reals and illustrate how it satisfies the Vector Space Conditions. 

    The Definition of a Vector Space
    Let F be a (mathematical) field (like the Real or Complex numbers). A vector space over F (or F-vector space) consists of an abelian group4 (meaning the elements of the group all commute) V under addition together with an operation of scalar multiplication of each element of V by each element of F on the left, such that for all a,b ∈ F (read a,b in F, or a,b elements of F) and α,β ∈ V the following conditions are satisfied:

    The Vector Space Conditions:

    V1) aα ∈ V.
    V2) a(bα) = (ab)α.
    V3) (a+b)α = (aα) + (bα).
    V4) a(α+β) = (aα) + (aβ).
    V5) 1α = α.

    An Example of a Vector Space: 2 Dimensional Vectors defined over the Reals

    Let F be the real numbers. 

    An example of a Vector Space over the real numbers is the set of 2 dimensional vectors (or 2-tuples) of the form (x,y) where the first and second entries are real numbers.  

    Example (2,3)  and (0,1) are such vectors. 

    {(2,3) can be visualized as an arrow that begins at the origin of a 2 dimensional plane and terminates at the point (2,3) but the mathematics is independent of that particular interpretation.}

    Vector Addition is defined as the addition of corresponding entries.

     Example (2,3) + (0,1) = (2+0,3+1) = (2,4) 

    Multiplication of a real number times a vector is defined by the multiplication of that number by both entries.

     Example 5*(2,3)=(10,15).

    Verifying that 2 dimensional vectors defined over the reals satisfy the Vector Space Conditions V1-V5:

    V1) is satisfied because multiplying a vector (x,y) by a real number, a, gives another vector (ax,ay) whose entries are real numbers.
     
    Example 2*(6,7)=(12,14)

    V2) a(b[x,y])=(ab)[x,y] where a and b are real numbers because a(b[x,y])=[abx,aby]
    and (ab)[x,y]= [abx,aby]

    Example 2*(3*[10,4])=2*[30,12]=[60,24]
                   (2*3)*[10,4]=6*[10,4]=[60,24] 

    If these last two examples are understood, then it should be obvious that the above definition of 2 dimensional vectors defined over the reals satisfies the other Vector Space Conditions.
    Halliday
    Barry:

    It will take me a little longer to produce the visual version.  ;)

    David

    mathematical_investigations
    lol. I'm glad you're joking, because if I thought you were going to draw it, I would have picked better numbers. Looking back at it now, I think the way I presented it would intimidate people who haven't seen arguments like that before. I should have just presented vectors and how they work and then pointed out that this is all the "Vector Space Conditions" really say in this case.     

    ...i also should not have used parenthesis in the usual sense as indicating multiplication and also as the brackets (x,y) indicating a 2-component vector. That is a potential source of confusion.
    Halliday
    Barry:

    Yes, I would favor an approach more like "just present[ing] vectors and how they work and then pointed out that this is all the 'Vector Space Conditions' really say."

    I tried to do that in the prose of the article, but I agree that examples will help many.

    David

    Are you two talking in Klingon-speak or is it Orcish? 
     
    Seriously, David, it is the middle of the night here but if you could wait a bit, I'd like to send you a sketch of what I *think* you are saying.

    No-o-o !!!!

    For goodness sake!



    blue-green
    Is someone having a nightmare?¿?

    Relax, the Good Shepherd will be back.
    mathematical_investigations
    I only suggested it because someone asked. I wouldn't have mentioned it otherwise.  It it isn't as bad as it looks. There are even simpler examples of vector spaces too.
    David,
    I don't understand. You started out wanting to talk about spacetime, but then limit us to discussing the vector space _at_a_single_point_ . If every point has a separate tangent space, it is unclear to me how this could ever tell us about space itself. The metric as you have presented it can only tell us the magnitude of a vector in the tangent space at a single point. It can't tell us the length along a path of points in the spacetime, which appears to be a completely different "vector" according to your presentation here, and that path-length seems to be the real magnitude that we'd want to measure out what kind of spacetime geometry we have.

    I am enjoying the presentation so far, but am already having trouble following despite being comfortable with the definitions you've given so far. So maybe there are some implied pieces you left out?

    Please clarify.
    Thanks,
    Daniel

    Halliday
    Daniel:

    It's interesting that you state:

    I don't understand. You started out wanting to talk about spacetime, but then limit us to discussing the vector space _at_a_single_point_ . If every point has a separate tangent space, it is unclear to me how this could ever tell us about space itself. The metric as you have presented it can only tell us the magnitude of a vector in the tangent space at a single point. It can't tell us the length along a path of points in the spacetime, which appears to be a completely different "vector" according to your presentation here, and that path-length seems to be the real magnitude that we'd want to measure out what kind of spacetime geometry we have.

    I expect that many people have a very similar "problem" with General Relativity, since it has precisely this same feature.

    I could have simply specified that we should assume we have a flat spacetime (the spacetime of Special Relativity), but that adds an additional assumption, that many already "know" is not valid (having at least heard that "gravity is spacetime curvature", whatever that means to them).  Otherwise, I could take a more completely General Relativistic approach, but that involves a lot of extra concepts and mathematics that most are not ready for, and really doesn't add a lot to the understanding.

    However, as I tried to motivate, in the article, even if we have some curved spacetime, we can "zoom in" close enough (to a single point) that it looks like a flat space—a vector space.  Something that is far simpler than curved spacetime manifolds, but valid even with curved spacetime.

    In fact, the vector space (at a single point) is actually even simpler than the flat spacetime (space) of Special Relativity, or Newtonian mechanics (Galilean Relativity), since all vectors have their origins at the same point.

    Now, the geometry of this vector space (at a single point) is actually what gives rise to all of the geometry of Special Relativity, it (along with only a few additional feature/concepts) is even responsible for the geometry of General Relativity.

    The important feature to recognize is that "the length along a path of points in the spacetime" (the "path-length") is the integral of the vector length (given by the inner product) of the infinitesimal displacement vector (at a point) that does exist within the tangent space, at that point, and points along the path.  See, for example, the last picture just before the "Vector Spaces" heading.  (You can also follow the link provided by that image to read much more on the subject, though I cannot vouch for everything you will find over there.)

    Does any of this help?

    David

    "Does any of this help?"

    Kind of. It appears you are making some assumptions which I am uncomfortable with unless they are made explicit. And I'm uncomfortable because I don't really understand the assumptions (I definitely see the idea, I just feel like we're handwaving stuff away.)

    You have defined a manifold to be a collection of locations. So we just have a set of points. It also sounds like at this step there is no notion of "distance" between points. With no notion of distance, how can you define a neighborhood for the points, or dimension of this collection of points?

    At each point, we have a vector space. How do you choose the dimension of the vector space? Since we can pick a vector space at each point, how do you know the dimension is the same at each point?

    Then, ignoring that, we move on and have a vector space (same one) at every point. Define an inner product in the vector space. Now, to get a path length, we need the length of tangent vectors to somehow be identified with a "distance" between points. But we have no notion of distance between points. Say we have a coordinate chart for a collection of these points. How do you fix the ratio in "tangent vector magnitude" = constant * "coordinate chart endpoint value difference"? Just because you shrink the vector and hence coordinate difference to a limit zero when doing your path length integral, doesn't mean we can ignore that constant. How we choose that constant affects the path length answer. This seems to be some kind of extra "structure" we're implicitly adding.

    All the questions are interesting to me, but it really is the last one that is bothering me. It seems at some step you are sneaking in a hidden assumption about that "constant"; somehow you are sneaking in a relation that would let us locally know the difference between a sphere with surface area 4 m^2 and 400 m^2.

    Please feel free to go into as much math depth as you want (I'm a retired electrical engineer. I'm comfortable with complex math, and I've self taught enough to do some GR problems, but the intro books just gloss over the details. So while I can do quite a few calculations, but I don't understand "why" we are allowed to do it the way we do it.)

    I appreciate any extra detail you can give.
    Thanks,
    Daniel

    Halliday
    Daniel:

    Let's see...

    You have defined a manifold to be a collection of locations. So we just have a set of points. It also sounds like at this step there is no notion of "distance" between points. With no notion of distance, how can you define a neighborhood for the points, or dimension of this collection of points?

    Technically speaking, while a manifold need have no notion of distance (in the norm or "metric" sense [where metric, here, is being used in a sense that differs from the sense physicists use the term, so it is not necessarily related to the inner product]), there is a general sense of neighborhood, in a sense very similar to that used in topology.  (Open sets, and open subsets, and all that.)

    I was not so much "defining" a manifold, as providing an access to the concept for the readers I expect here.  (What I said was literally, "The space of locations is called a manifold."  Where "the space of locations" is borrowing from the general reader's concept that "To us, a space is somewhere to put a box :)")  The deeper definitions are irrelevant to such readers, but I'm using the correct terms so, if a reader wants to, they can delve deeper.

    You then go on with:

    At each point, we have a vector space. How do you choose the dimension of the vector space? Since we can pick a vector space at each point, how do you know the dimension is the same at each point?

    Since the vector space at each point is the tangent space (so, not just any vector space [or fiber space]), it inherits its dimensionality from the dimensionality of the manifold (at that point).  How do we determine the "dimension of this collection of points?"  This is where "charts" for the manifold come in.  Again, something quite beyond what is necessary for the reader, but if one truly wants to pursue the deeper aspects, one may.

    Now, I didn't mention "charts" did I?  No, because they are not necessary for getting the reader where I want them to go.  However, all one would need to do is follow the link to the Wikipedia article on Manifolds that the picture of the globe, as an example of a manifold, provides, or look up the concept of a manifold in many mathematical treatments of the subject.

    OK.  Now you continue with:

    Then, ignoring that, we move on and have a vector space (same one) at every point. Define an inner product in the vector space. Now, to get a path length, we need the length of tangent vectors to somehow be identified with a "distance" between points. But we have no notion of distance between points. Say we have a coordinate chart for a collection of these points. How do you fix the ratio in "tangent vector magnitude" = constant * "coordinate chart endpoint value difference"? Just because you shrink the vector and hence coordinate difference to a limit zero when doing your path length integral, doesn't mean we can ignore that constant. How we choose that constant affects the path length answer. This seems to be some kind of extra "structure" we're implicitly adding.

    Boy, you sure seem to want to put the cart before the horse, don't you.  ;)  One can even define the relationship between the tangent space (remember, it's not just any old vector space [like a fiber space] at each point, it is a particular vector space that is tangent to the manifold at a point:  The Tangent Space) without ever having an inner product.  Of course, this is all about Differential Geometry.  Well beyond the scope of this series.

    However, given a sufficiently smooth manifold, one can define the tangent space quite nicely within Differential Geometry.  Furthermore, if one actually has an inner product in the tangent space, one uses that very measure of length to define path lengths upon the manifold.  This is all due to the intimate relationship between "coordinate charts" on the manifold and "tangent vectors" (not just their magnitudes) in the tangent space.

    Now, it is certainly possible to allow the two to be only proportional to one-another, with some (arbitrary) constant of proportionality.  However, that neither gains nor diminishes a thing.  (Once one imposes metric compatibility for "parallel transport", this becomes fixed, even if one allows "torsion".  However, now we are getting even farther afield!  So, yes, I suppose one could consider "metric compatibility" of "parallel transport" to be an additional "structure".)

    It's interesting that you should say:

    All the questions are interesting to me, but it really is the last one that is bothering me. It seems at some step you are sneaking in a hidden assumption about that "constant"; somehow you are sneaking in a relation that would let us locally know the difference between a sphere with surface area 4 m^2 and 400 m^2.

    Because, after all, if we were dwelling upon a sphere, what is the meaning of any unit of length, like a meter?  We could define any length we wish to be called anything we wish.  Then we would use such a measure to "size up" out spherical universe/world.  Who's to say whether our saying such is 4 m2, or 400 m2, is "right" or "wrong"?  It simply is what it is, relative to our chosen measure.  (It's another aspect of "relativity", I suppose.  We are free to choose our measures, and our coordinates.  The answers may look different, but the answers are transformable into one-another [a rather general form of isomorphism]:  That's one of the aspects of General Relativity.)

    Now, you end with:

    Please feel free to go into as much math depth as you want (I'm a retired electrical engineer. I'm comfortable with complex math, and I've self taught enough to do some GR problems, but the intro books just gloss over the details. So while I can do quite a few calculations, but I don't understand "why" we are allowed to do it the way we do it.)

    I'm very glad you provided this information.  I would love to delve into greater detail on these subjects.  However, this is not the right place or time for delving into such details because they are far from the subject of this series.

    However, there is a small, paperback book I have, and use rather often as a reference book on such matters of Differential Geometry (especially when one does not have a metric), that I would love to point you too:  Geometrical methods of mathematical physics, by Bernard Schutz.  It's only about 250 pages, and about the size of a paperback novel.

    I highly recommend it.

    David

    David Halliday wrote (May 12th 2012 12:00 AM):
    > The space of locations is called a manifold. This is probably an unfamiliar term for many [...] You can just think about a manifold as being the set of all possible locations

    So what about the term "location", or even "possible location"? Are you going to provide definitions of those later on, or do you require them to be "familiar" (and unambiguous) from the outset?

    Also: the notion "manifold" (which, like all of mathematics, may readily be presumed in discussions of physics) is generally described as "a set together with ...".

    > When viewed very close to a single location—a single point—a smooth manifold will look like [...] a vector space.

    Is it therefore fair to say that what you mean by "manifold" is (or is some special case of)
    "a set together with a description of all subsets which are a vector space"? ...

    Halliday
    Frank (familiarity.breeds.contempt):

    I've been waiting for you, over here.  ;)

    Why are you still not logging in, since you now have an account here.  I've even "befriended" you (in your logged in persona), already.

    You ask:

    Is it therefore fair to say that what you mean by "manifold" is (or is some special case of)
    "a set together with a description of all subsets which are a vector space"? ...

    No, that is not correct, so it is most certainly not "fair".  ;)  Besides, I did narrow the notion to "smooth manifold" (or "sufficiently smooth").  (We can be especially pedantic and say a "weakly differentiable manifold", but that's a lot more specific than necessary for this discussion.)

    As for "location" or "possible location", the notions the readers have suffices my needs, here.

    The principle purpose was to move the readers/audience from their concepts of "To us, a space is somewhere to put a box :)", to the more narrow concept of a vector space (the tangent space).

    Are you willing to make this move along with them?

    David

    David Halliday wrote (05/14/12 | 16:33 PM):
    > [Is it therefore fair to say that what you mean by "manifold" is (or is some special case of) ...] No, that is not correct, so it is most certainly not "fair".

    At least this should let you know what I'm missing from your article, in this respect.

    > As for "location" or "possible location", the notions the readers have suffices my needs, here.

    Presupposing such notions may of course be avoided by considering geometric relations between participants (such as readers), immediately.

    > The principle purpose was to move the readers/audience from their concepts of "To us, a space is somewhere to put a box :)"

    With this I can identify, considering that "box" denotes some particular geometric relation of eight (principal) participants (as "corners"; perhaps along with additional participants "on the edges").

    Of course other, perhaps more important geometric relations between participants may be (thought-experimentally) defined and (depending on the given observational data to be evaluated) found, for instance:

    one participant having been the "middle between" two others,
    two participants having been "resting" to each other,
    three participants having been "straight" to each other,
    four participants having been "plane" to each other,
    five participants having been "flat" to each other,
    three participants having been "equidistant" from each other, etc.

    > to the more narrow concept of a vector space (the tangent space). Are you willing to make this move along with them?

    I'm certainly not going to renounce the relativistic point of departure:
    participants observing each other, each judging order or coincidence of their observations.

    As mathematical notions, terms like "vector space" or "tangent space (to a given manifold)" sure are well defined and undisputed.
    They may even turn out to be useful for summarizing or interpolating certain measurements. But they're not going to be a starting point, or an end in themselves, in my blog.

    Halliday
    Frank (if.it.ain~t.broke--don~t.fix.it):

    Do you think?  Then doesn't that mean that, therefore, there is at least a one dimensional manifold in which you exist?

    If there is another participant, or if you are able to observe an extent to your own body with you being able to manipulate it at will (at least within limits), doesn't that significantly enlarge the minimum size and dimensionality of the manifold in which you exist?

    So, the manifold had to exist first, or, at least, had to come into existence concurrent with you.

    So, we need not do things backward.  We already know we have a manifold.  Other elementary observations of our surroundings—ones that all human beings have done, though some more consciously than others—already indicate a dimensionality, and the ability to observe and manipulate directions (vectors, in a primitive sense), lengths, and angles between directions.  So we do observe the existence of not only a vector space (tangent spaces), but one with a dot or inner product (and inner product space).

    To do otherwise—to pretend such don't exist—is to ignore that which already impinges upon your senses.

    Now, the remaining issue is to find ways to characterize this inner product space (and, ultimately, the manifold to which it is "attached" or associated).  One approach is to start blindly doing experiments to see what one can observe and determine about such.  (Historically, a great deal of this has already been done.)

    Another approach is to start from the knowledge that one has an inner product space at one's disposal, and determine what possible forms such can take, in order to focus one's investigation of its nature and characteristics.

    This latter approach is what I have undertaken in this series.

    You are not required to take the same journey, but if our conclusions (concerning the nature and characteristics of the inner product space and the manifold to which it is "attached" or associated) should differ, then there will need to be additional work performed.  However, unless you develop a superior framework for the comparison and evaluation of such conclusions, then we shall use the framework of an inner product space, and, ultimately, the manifold to which it is "attached" or associated, for such comparison and evaluation, since such already embodies so much of humankind's observational and experimental knowledge.

    Other than this, I will not engage in any more of your approach unless and until you address where we left off in another thread.

    David

    David Halliday wrote (05/17/12 | 23:09 PM):
    > [...] therefore, there is at least a one dimensional manifold in which you exist?

    Considering all observations I've collected so far, and even some expectations about observations I might yet obtain, and considering everybodies ability to judge order or coincidence of their observations, at least in principle, I'd call this existence of mine an "ordered set". (And therefore each existence of anyone else, too.)

    Now, do you suppose that any ordered set, for instance the ordered set of all rational numbers, should be called
    "(one dimensional) manifold" ? (I don't.)

    > If there is another participant [...] doesn't that significantly enlarge the minimum size and dimensionality [...] ?

    As explained above (and surely indicated elsewhere already):
    distinguishable participants may well be represented as distinct ordered sets.
    These are of course idealizations, especially considering participants without any (distinguishable, recognizable) parts.

    > or if you are able to observe an extent to your own body with you being able to manipulate it at will (at least within limits)

    Well, one related presumably unambiguous judgement of observation is that distinct participants may "have met" (either "in passing", or "over a stretch").
    Of course, this may be an idealization, too.

    > Other elementary observations of our surroundings—ones that all human beings have done, though some more consciously than others—already indicate a dimensionality

    ... using thumbs and fingers of their two hands, considering only trials in which thumb tip and finger tip of either hand kept meeting throughout, I'd suggest ...

    > and the ability to observe and manipulate directions (vectors, in a primitive sense)

    ... at least: being able to distinguish and recognize various pairs of participants ...

    > lengths, and angles

    Certainly happily pretending to. I'd draw the line so:
    judgement of "same or distinct" (trial, participant, indication) may be presumed from the outset;
    judgement of "equality, or inequality, or differences, or real number ratio" must be derived, i.e. (as operator) constructed from suitable judgements of "same or distinct".

    > So, the manifold had to exist first, or, at least, had to come into existence concurrent with you.

    Apart from the quibble with "manifold" vs. "ordered set(s)", this touches on the relation between (given or hypothetical) observational data, operators to be applied to observational data for deriving (real, unambiguous, mutually agreeable) values, and any one such value obtained from a sufficient set of observational data ("a trial's worth"; either actually collected, or simulated).
    Generally I'd say that the value comes last.

    p.s.

    > [...] unless and until you address where we left off in another thread.

    In my last comment there (04/23/12 | 19:49 PM) I've tried to address your questions quite comprehensively. In your reply (04/23/12 | 23:55 PM), which also seems your last comment there, I'm unable to recognize any follow-up quesion(s); but, as I mentioned already, a fitting closing statement.

    p.p.s.
    Your "&mdash;" appeared in my editor as (octal code) "\227", which in turn was rendered as a dash in my browser.

    Halliday
    Frank (pre.method.blogging):

    You ask (and answer):

    Now, do you suppose that any ordered set, for instance the ordered set of all rational numbers, should be called
    "(one dimensional) manifold" ? (I don't.)

    Do you not recognize that the ordered set of the real numbers is a one dimensional manifold?  It most certainly is.  So is the ordered set of the rational numbers (since they, too, form open sets).

    In fact, even a countable ordered set, like the integers, can be considered to form a topological manifold, by considering (defining) each distinct element to be "an open set".

    Additionally, a completely ordered set can have no higher dimensionality than one.  (If your observations form a [completely] ordered set, then you must have no simultaneous observations.  How interesting, or dull, depending on how one considers such.  ;)  )

    So, now "how do you plead?"

    David

    P.S.  I'm sorry I left some unconverted &mdash elements lying around.  That (with a semicolon ending) is the HTML code for a dash that's about as long as the width of a capital M.  It's supposed to look like '—'.

    fundamentally
    Please re-read the notions of open and closed sets for example in wiki. The real numbers are NOT countable, but the rational numbers are. Both sets are closed for arithmetic operations, except division through zero. The reals are closed for the limit operation (of converging series). The rationals are not. The cardinatity of the rationals and the integers is the same. The cardinality of continuums like the real numbers, a line, a surface, the complex numbers, the quaternions and the octonions are all the same.
    An infinite dimensional separable Hilbert space is countable but it is closed under its norm. The rational numbers are countable but not closed. The reals are not countable but they are closed.A Gelfand triple (falsely called rigged Hilbert space) is not countable in its dimensions. It is closed with respect to is norm.
    If you think, think twice
    Halliday
    Hans:

    You have completely misunderstood the sense of "closed" vs. "open" sets and subsets used when talking about manifolds.  It is not at all talking about closure under some set of operations.

    Furthermore, when talking about dimensionality and completely ordered sets, this has little if anything to do with cardinality.

    You do recognize that while the cardinality of a line and a surface are the same*, the dimensionality are not.  Right?

    You do recognize that while the cardinality of a line and a surface are the same*, the line forms a completely ordered set, while the surface does not.  It is completely analogous to the way the complex numbers are not an ordered set.  Right?

    David

    They only have the same cardinality in a certain sense, since there are multiple, not completely consistent ways to determine/define cardinality of infinite sets (especially once one goes beyond countably infinite sets).  So, technically speaking, one would need to state what cardinality measure one is using.

    fundamentally
    Wiki (http://en.wikipedia.org/wiki/Manifold#Other_curves )says: Manifolds are never countable.
    If you think, think twice
    Halliday
    Hans:

    As I always say, I do not consider Wikipedia to be an authoritative source for anything!

    However, that being said, you should reread the reference you provided, for it most certainly does not say that "Manifolds are never countable."  What it does say is "they are never countable, unless the dimension of the manifold is 0."

    That last part is eminently important!

    Do you think a manifold with "dimension 0" is only the trivial manifold of a single point?  It most certainly is not!  Reread that same reference you provided.  Read about how manifolds need not be connected!

    Now consider a manifold of an infinite (countably infinite) number of (not "connected") points.  (I've been considering such manifolds off and on for the past quarter century!)

    Now, if one can have a manifold of an infinite number of such (not "connected") points, and one can have a manifold of any number of copies of the Real numbers (n-tuples, or Cartesian product[s] of n Real number lines [or segments thereof]), then what is to preclude "manifolds" of Cartesian products of other ordered* (mathematical) number Fields, such as the Rational numbers?

    You see, there is no reason (besides pure pedantry) for not considering manifolds formed from Cartesian products of other ordered (mathematical) number Fields anymore than there is any reason to not allow vector spaces to be based upon such number Fields.  Yet, vector spaces are most certainly allowed to be based upon such number Fields!

    David

    *  Manifolds do seem to have some "troubles" being based upon non-completely ordered (mathematical) number Fields, such as the Complex numbers.  However, unless one is simply going to "tack" vector spaces based upon the Complex number Field onto a manifold (as some form of "fiber space"), rather than the more natural Tangent space, then one needs to consider Complex manifolds as well.  (This is something else I have considered off and on for the past quarter century.)

    David Halliday wrote (05/18/12 | 22:54 PM):
    > [...] a countable ordered set, like the integers, can be considered to form a topological manifold, by considering (defining) each distinct element to be "an open set".

    This definition of "open" (sub-)sets (among others) makes the given ordered set a topologigical space. But a topological manifold?? (I don't think so.)

    Given your suggested "open sets" (separately, or collectively) does there exist a homeomorphism to the open interval "(0, 1)" of real numbers with its standard topology?
    (Vgl. for instance http://en.wikipedia.org/wiki/Real_number_line#As_a_topological_space, etc.)

    > Do you not recognize that the ordered set of the real numbers is a one dimensional manifold? It most certainly is.

    Sure it is. My point was to provide some example of an ordered set which is not a one dimensional manifold (despite having "a lot of elements").

    > [...] the ordered set of the rational numbers (since they, too, form open sets).

    For each ordered set there exist its ("natural", "obvious", "appropriate") "order topology", which makes it a topological space. But this is not sufficient for being a topological manifold, i.e. for being topologically equivalent to the topology of the "real line". (The "ordered set of all rational numbers" lacks "(topological) local compactness".)

    > (If your observations form a [completely] ordered set, then [...])

    The elements of the ordered set pertaining to any one particular participant can be called "indications", each of which contains all observations collected in coincidence by the participant under consideration. (The terminology should be familiar to you from our previous correspondence.)

    p.s.
    David Halliday wrote (05/18/12 | 23:09 PM):
    > I was challenging you. [...]

    I'm trying to focus my efforts accordingly, starting with
    http://www.science20.com/ping_parlor/relativistic_pregeometry
    and before long covering the Einstein-Synge distance definition and its implications, too.

    Halliday
    Frank:

    You focused, first, upon my mentioning that even countable, or even finite sets can be formed into a generalization of a topological manifold.

    Now, actually, this generalization of a topological manifold is certainly not a topological space, in the usual sense, since it is not continuous.  (Even the rational numbers are not, strictly speaking, since they are not continuous, since the continuum is defined as the real numbers (even though the rational numbers are dense within the real numbers)!)

    Furthermore, since the usual definition of a topological (or otherwise) manifold is likewise defined in terms of the continuum (direct products of the real numbers), it doesn't fall into strictly into this category, either.  (That's why I called it a generalization, above.)

    It's like unto the way the usual definition is defined as being locally Euclidean, or the way mathematicians distinguish between Riemennian Geometry (being locally Euclidean) and what they refer to as pseudo-Riemannian Geometry (that is not locally Euclidean).

    So, yes, one can be pedantic and distinguish between such things, but such distinctions are seldom useful in any general sense.  (I should have known that you would be overly pedantic, if for no other reason than to be argumentative.)

    David

    P.S.  In general, for countable or finite (open) sets to be formed into a generalization of a manifold (whether topological or [pseudo-]metric) there are additional relationships that must be defined.  On the other hand, such is not required for ordered countable or finite (open) sets.

    David Halliday wrote (05/19/12 | 18:01 PM):
    > [...] that even countable, or even finite sets can be formed into a generalization of a topological manifold.

    It is only now (based especially on on your comment (05/26/12 | 21:04 PM), in correspondence with Hans van Leunen) that I even come to consider "manifolds of dimension 0" at all. In some of my earlier statements concerning "manifolds (in general)" (above) substitutions to "manifolds of dimension at least 1" might therefore be in order.

    However, as recently as in your comment (05/18/12 | 21:57 PM) (or (05/17/12 | 23:09 PM)?) and my reply (05/18/12 | 16:28 PM) our topic had apparently been "manifolds of dimension at least 1"; such as your suggestion "at least a one dimensional manifold in which you exist".

    Now, not to get bogged down in too many side tracks (concerning only terminology):
    do you agree that a (given) ordered set is not necessarily a (given) one dimensional manifold ?

    Also, in trying to recall how you used the notion "manifold" in the blog article above, and in how far I object to related outright assumptions:

    for the sake of discussion we may well assume that "my existence" could indeed be given or described as a one dimensional manifold; and likewise the individual "existences" of all other participants.

    Which consequences do you suppose this might have for (the description of) the (geometric) relations between "us" participants?

    Halliday
    Frank (Metric.as.Foundation.of.All---Box.13.1-Metric.distilled.from.Distances*):

    As I had already pointed out, so long as you are a being of greater than zero dimensional spacial extent, the manifold already has to be greater than a single dimension.

    Now, once you add " 'us' participants", the dimensionality of the manifold can never decrease.  The dimensionality will either increase without any apparent bound (so, possibly an infinite dimensional manifold), or will come to a point where it no longer increase in dimensionality with any increase in the number and variety of "participants" (a finite dimensional manifold).**

    On the other hand, without the connections provided by a manifold**, the ability to have continued, self connected existence, let alone "relations between 'us' participants" are precluded.

    So, once again, given such relationships, one can conclude the existence of a manifold—at least in some reasonably expanded sense.***

    David

    *  If a metric (in the sense used in my article, and as typically used by physicists, relating to inner/dot products) already exists, one may "distill" its form in multiple ways, including from distances (though this is facilitated by a positive definite metric, since non-positive definite metrics have fewer constraints that can be used to overcome measurement uncertainties).  On the other hand, if one does not already exists, then there is no fundamental regularity from which to obtain any self-consistent metric from distances or any other methods.

    (The assumption of the existence of a metric, when "distilling" such from any method, is something to "watch out for" when evaluating any method or concept that makes any claims to being able to "derive" a "metric" without [the existence of] a "metric".  Such can be immensely subtle.)

    **  Even if the "manifold" were discrete (so, formally zero dimensional), the existence of beings that encompass multiple such points (essentially) "simultaneously" (at least in some finite, "over all" sense, as in the existence we experience), and an ability to have seemingly connected existence (again, as in the existence we experience), indicates a "connection" between these discrete points—both spatially and temporally—thus providing a topology, of sorts (like a Graph, as in Graph theory), that goes beyond the formally zero dimensional manifold, even though it is still far from a continuous manifold (not even as continuous as the rational numbers).

    However, this "wrecks havoc" on the concept of a tangent space in a form like unto what I used in the article.  On the other hand, the connections between the discrete points may be allowed (at least) to provide for a vector space, spanned by these connections (actually, their "directions" from and to each individual point).

    This can provide for a similar framework, from which to build, as what I have used within the article.

    ***  Basically, one could define a "manifold" to be "locations" (spatially and temporally, if you will) with some appropriate relationship among them, whether imposed (naturally) by the (ordered) continuum (like the Real numbers, or even the Rational numbers [with a weaker sense of continuum], or any of the infinite number of mathematical Fields that exist in between these two), or as some form of discrete set with connections between them (like a Graph, as in Graph theory), or some other conceivable (or, maybe, even presently inconceivable) form.

    David Halliday wrote (06/14/12 | 17:53 PM):
    > Basically, one could define a "manifold" to be "locations" (spatially and temporally, if you will) with some appropriate relationship among them

    Of course we have a precise definition of the mathematical notion "manifold": as a set with certain (mathematical) properties.

    However:
    Starting out from the notion of participants (each as ordered set of indications) which is surely familiar and self-evident and (therefore) a basic notion of RT, and even admitting the notion of "event" (as set of "coincident" indications of distinct participants), it is questionable how your notion "location" might be defined at all, and whether such a separate notion is even necessary in order to express (geometric) relationships between given participants.
    (There's my FQXi essay topic already! ;)

    > If a metric (in the sense used in my article, and as typically used by physicists, relating to inner/dot products) already exists, one may "distill" its form in multiple ways, including from distances [...]

    This view of separating claims or assumptions of existence from concrete evaluation again calls to mind Einstein's famous declaration:
    The concept does not exist for the physicist until [...] .

    Therefore: evaluating "distances" (in the sense of MTW; or at least distance ratios) apparently must be possible before and without having any description of (the above mentioned) "metric" available.

    Continuing this argument it is required to evaluate "distances" without having values of "distances" already available.

    > On the other hand, if one does not already exists, then there is no fundamental regularity from which to obtain any self-consistent metric from distances or any other methods.

    But obviously participants have the ability, in principle, to jugde and agree on certain "regularities" in their mutual relations; for instance:
    for each indication stated by participant M, M may judge whether he/she/it observed a corresponding echo indication stated by participant A in coincidence with a corresponding echo indication stated by participant B, or first one and then (possibly) the other.
    Therefore the great importance of the related "pre-geometric" thought-experimental definitions at the foundation of RT.

    > [...] an ability to have seemingly connected existence (again, as in the existence we experience), indicates a "connection" between these discrete points—both spatially and temporally—thus providing a topology, of sorts

    A condition even for talking about "topology" is to admit the notion of "set", and thus of distinct elements (except in the most trivial cases).

    Halliday
    Frank:

    You should also take a look at my response to Hans, above.

    David

    Halliday
    Frank:

    You are correct that "where we left off in another thread" did not end with me asking "any follow-up quesion(s)".

    No, I was not asking a question.  I was challenging you.  I directly challenged your assertion (that you had formulated as a question to me).

    So, I suppose the implied question is:  Are you going to take up the challenge, or quietly admit "defeat"?

    What say ye?

    David

    Halliday
    [Frank Wappler placed a reply to one message within a reply to a different message.  Below is the part of his other message that should be placed here:]

    David Halliday wrote (05/18/12 | 23:09 PM):
    > I was challenging you. [...]

    I'm trying to focus my efforts accordingly, starting with
    http://www.science20.com/ping_parlor/relativistic_pregeometry
    and before long covering the Einstein-Synge distance definition and its implications, too.

    Frank W (@~ R (not verified) | 05/19/12 | 15:01 PM
    Halliday
    Frank (farewell.to.g_mu_nu.dx~mu.dx~nu):

    When you say you're "trying to focus [your] efforts accordingly", does that mean you are actually going to take on my challenge?

    Are you actually going to address the fact "that Einstein (Marconi?, Hertz?, ...) were not by default [or otherwise] considering indications[your term for personal, local only observations] with plain first observations"?

    So, are you actually going to address the fact that they were all considering only one special class of "signal" (namely light!)?

    Are you actually going to acknowledge the fact that whether this one special class of "signal" (namely light!) they were considering would be before or after any other "indications with plain first observations" was completely irrelevant?*

    Unfortunately, from the apparent title of your "efforts" (namely "Relativistic Pregeometry"**), it looks like you are "trying to focus [your] efforts" away from addressing my challenge.  Is that so?

    David

    *  Of course, relative to a "first" anything, anything else can only be concurrent or after, never before.  However, your assertion, to which I was responding, was constructed in such a way as to require answering both a "before" as well as an "after".

    **  If you are doing what it appears you are doing, this has nothing to do with pre-geometry, but is an effort to try and "observationally" (though not with actual observations, but "made up" "observations" based upon [hidden] assumptions) determine a pre-existent or pre-existing geometry with some minimal set of assumptions.  The only problem is that you (obviously taking your ques from Synge [not Einstein]) hide a great many of your assumptions!

    I insist upon far more intellectual honesty!  To be doing anything close to something that can be called "science" demands it!  Even to be doing anything within the realm of mathematics further demands it!

    David Halliday wrote (05/19/12 | 16:36 PM):
    > Are you actually going to address [...]

    Hertz had been looking for observations to be made coincidently with a capacitive (primary) spark discharge (as source of el.-mag. waves). Such happened to be "(secondary) side-sparks" in adjacent coils or subsequently selected wire loops with gap.
    Cmp. http://en.wikipedia.org/wiki/Wireless_telegraphy#Heinrich_Hertz and references therein.

    Marconi and users of wireless telegraphy in general, I imagine, considered already the first occurences of readable output (such as "Are you ready") successful transmissions.

    Einstein, in his definition of simultaneity, wrote of A and B having been hit by lightning strikes, and M (as middle between A and B) deciding whether it perceived these flashes coincidently, or not. I take that to mean Ms first perception of A having been struck, and of B having been struck, respectively.

    The condition of considering the first perceptions, rather than any perhaps more specific reminders, stands out for being immediately and generally comprehensible, not requiring any notions of geometry (such as "speed", "polarization", "isotropy" etc.) and for this very reason well suited for defining geometric relations between participants.

    > The only problem is that you [...] hide a great many of your assumptions!

    It is difficult to dispute this in a comment. That's exactly why I've tried to focus my efforts on presenting a self-contained write-up; just as you've apparently started already. (Still fishing another week for a catchy opening sentence. ;)

    Halliday
    Frank (are.we.there.yet):

    It's interesting that you should choose to address my challenge with examples that actually prove my point, which was (since you chose not to quote it):

    Are you actually going to address the fact "that Einstein (Marconi?, Hertz?, ...) were not by default [or otherwise] considering indications[your term for personal, local only observations] with plain first observations"?

    So, are you actually going to address the fact that they were all considering only one special class of "signal" (namely light!)?

    Are you actually going to acknowledge the fact that whether this one special class of "signal" (namely light!) they were considering would be before or after any other "indications with plain first observations" was completely irrelevant?

    Yes.  Indeed.  They were all strictly considering only one special class of "signal", namely "el.-mag. waves", a.k.a. light.

    The only extent to which there was any consideration of "The condition of considering the first perceptions" was the avoidance of reflections, and other delayed occurrences of this selfsame one special class of "signal", namely "el.-mag. waves", a.k.a. light.

    Thank you for making my point.  In every case, they were most certainly "not by default [or otherwise] considering indications[your term for personal, local only observations] with plain first observations" of just any old kind, but only of one special class of "signal", namely "el.-mag. waves", a.k.a. light.

    David

    rholley
    Folks!  Why don’t y’all have a peek at this, from the MAA Online book review column, a review of “Geometrical Vectors” by Gabriel Weinreich?



    Robert H. Olley / Quondam Physics Department / University of Reading / England
    blue-green
    That is quite a fine and creative use of a melon baller!

    Robert, have you seen the 1970's large paperback book on differential geometry by the professor who loved to draw swans?

    ((I guess David's environmental science colleague and proof-reader is having some time getting to or over the need for quadratic equations .... and yet, only a 2 in the exponent would make any sense in pure physics and math. With data fitting, it's whatever shoe fits.))
    Halliday
    I apologize to everyone for being away for so long.

    I've been "under the gun" at work (out here in the "real world").  I have at least one project that is going to go past its deadline, so I've been having to concentrate far more on such things, lately.

    Fortunately, I have this three day weekend off, so I should be able to get some things caught up here.

    David

    Halliday
    For those that wish to see a presentation of example vectors and how they work:

    Let's take a look at the vector space R2, the two dimensional vector space based on Real numbers.*

    Here is a representation of a portion of R2, near the origin or zero vector, with some representative points/vectors highlighted:

    Points and OriginThe point that looks like a star is the origin, the "zero" vector, the additive identity of the vector space.

    Of course, most of you are probably more used to seeing vectors represented as arrows, so here's the same space with some of the points/vectors represented as arrows:

    Vectors as ArrowsThe principle operation (binary operation) in vector spaces is vector addition.  Here's an example of a vector addition:

    Vector AdditionI'm sure many of you have seen vector addition done in this parallelogram fashion many times.  An important thing to note is that if we consider that adding the red vector to the blue vector is like placing the tail of the red vector on the head of the blue vector, then that is like the bottom two sides of this parallelogram.  While, similarly, adding the blue vector to the red vector is like the top two sides.  So, this parallelogram fashion of showing vector addition is simply making it explicit that it doesn't matter which order you add the two vectors, you always get the same result (the purple vector, in this case).

    Looking back at the first two diagrams, I expect you can see that adding the vector represented by the star, in the first diagram, to any other vector (or even to itself), will never change any vector, it will simply give the other vector (or itself) right back to us.  So we can see that this vector does act like the additive identity, the zero vector of this vector space.

    The next diagram illustrates the additive inverse ("negative") of a vector:

    Additive InverseIf we take the sum of these two vectors we simply get the zero vector (the one marked by the star in the first diagram), the additive identity.  So these two vectors are additive inverses ("negatives") of one another.  I expect you can all see that this can be done with any vector we may choose in R2.

    The second important operation in a vector space is multiplication of a vector by a scalar (a Real number, in this case).  This is a scaling operation:  The resulting vector points in the same or opposite direction, compared to the original vector, depending on whether the scalar is positive or negative, and will be "longer" or "shorter" than the original vector depending on whether the scalar has absolute magnitude greater than or less than one (unity).

    Of course, I'm reasonably certain that everyone can see that multiplying by the unity of the scalars (one [1], in the Real numbers) will leave any vector unchanged (V5, in the article).  However, the following two diagrams depict the scaling and addition of two differently scaled vectors (the first shows the red vector as a positive rescaling of the blue vector [or vice versa], while the second shows the red vector as a negative rescaling of the blue vector [or vice versa]):

    Scaling and AdditionAs you can see, the results are always simply a rescaled vector.  So, the results is always a Real number times the original vector.  This is really all that V1 through V3 (in the article) are saying.

    The only thing left is V4, in the article.  However, I'm reasonable certain that you can see that that would simply be the third diagram already presented, with some suitable rescaling of all the vectors involved in the vector addition.

    Does this provide the desired illustrations?

    David

    *  When one forgets about the vector space operations (vector addition and multiplication by scalars [Real numbers, in this case]) this is also the manifold R2, the infinite two dimensional manifold based on the Real numbers.

    Bonny Bonobo alias Brat
    Does this provide the desired illustrations? 
    Yes David, these diagrams are marvellous and make everything seem clearer to me anyway. Thanks :)


    My article about researchers identifying a potential blue green algae cause & L-Serine treatment for Lou Gehrig's ALS, MND, Parkinsons & Alzheimers is at http://www.science20.com/forums/medicine
    Coincidentally, Doug is really struggling with the concept of vectors, vector-spaces, and spacetime in the discussion here:
    http://www.science20.com/standup_physicist/blog/scalars_vectors_and_quat...

    Can you please help?
    He has a vector space in 4 dimensions, but says the first component is different and just a scalar since he claims it only has magnitude and not direction in this vector space.

    No one seems to be able to reach him but you, and we're all getting to our limit of patience with Doug. Please help clarify the situation for everyone.