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Sorry if this is a really basic question but I've searched and googled everywhere and haven't found any relevant answers so I have to ask my question here. Please excuse me for any misuse or ignorance of basic concepts and terminology.

In my understanding, in general, the position of an object should be absolute. What if you have relative positions of objects that disagree with each other? For example, imagine a 1 dimensional world (a line) composed of 3 objects, X, Y and Z:

For X, Y is 5 units to its right, and Z is 10 units to its right.

For Y, X does not exist, and Z is 10 units to its right.

For Z, X is 5 units to its right, and Y does not exist.

Now obviously these 3 statements are inconsistent with each other, you cannot draw a map of the 3 objects X, Y and Z. But you CAN draw 3 maps, one for each object. These 3 maps do not agree with each other, but the whole idea is that this concept of relative space encompasses "absolute" space, where the 3 statements agree with each other and are consistent.

So, to me, the concept of relative space, which INCLUDES sets of inconsistent statements such as the ones I made above, is MORE GENERAL than the concept of absolute space where these sets of statements must be in agreement with each other i.e describe the same map.

Now what I'm asking here is, is there an EVEN MORE GENERAL concept than the idea of relative space that I described above? A more general concept that removes more restrictions, that is what I'm after.

What I really hope for is if the idea of coordinates is generalized away altogether to be replaced by something more general.

Thanks for reading my question, I hope it made sense.

user2108462
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    In your framework, is there some sort of structural rule that retains some form of consistency between the pictures? For example, in our world, if we both look at two events (an "event" is a 4-D coordinate in spacetime), if we're angled and moving differently, then we disagree on the $\Delta x$, $\Delta y$, $\Delta z$, and $\Delta t$ between the two events, but we will always agree on the $\sqrt{\Delta (c t)^2 - \Delta x^2 - \Delta y^2 - \Delta z^2}$, i.e. there's a metric that makes our coordinate spaces consistent. Do you have a rule like that, or anything goes? – Bridgeburners Sep 23 '14 at 20:45
  • Anything goes, since I want it to be as general as possible. If we start without any restrictions, we can always apply restrictions later to obtain the particular systems, right? – user2108462 Sep 23 '14 at 20:54
  • "Draw 3 maps, one for each object" strongly brings to mind the concept of a "manifold", where local maps (called "charts" so you don't confuse it with the "function" meaning of the word "map") combine together to give information about the whole space. – Mark S. Sep 24 '14 at 00:21
  • "In my understanding, in general, the position of an object should be absolute." -- why ?? What does it even mean? There are no coordinates in "real life". – osa Sep 24 '14 at 01:17
  • So you're not requiring the triangle inequality? – Nishant Sep 24 '14 at 04:17

3 Answers3

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It's possible to encompass your notion, at least in the sense I read it, in that of a generalized quasimetric space. This kind of space has a weak distance function $d$ which takes pairs of points to nonnegative real numbers or $\infty$. We don't require $d(x,y)=d(y,x)$, but we do require that if $x\neq y$ then $d(x,y)$ and $d(y,x)$ are greater than $0$ and that $d(x,y)+d(y,z)\geq d(x,z)$. So your "line" could be described by a quasimetric with $d(x,y)=5,d(x,z)=10,d(y,x)=\infty,d(y,z)=10,d(z,x)=5,d(z,y)=\infty$. I do think this will satisfy the triangle inequality just stated-for a space that doesn't, you could just ignore the triangle inequality, so that you're using a premetric, which Wikipedia says is a somewhat nonstandard term. I will point out that this model of your problem loses the 1-dimensionality that you wanted. One way to get it back would be simply to put an ordering on the underlying set, which would be completely independent of the quasimetric.

And it is indeed possible to generalize even further. One well-known generalization is the topological space, which can be described as a set $X$ together with a collection of symmetric relations $r,s,t,...$ where $r(x,y)$ can be interpreted as "$x$ is $r$-close to $y$". These relations satisfy some axioms to make them reasonably space-like, but can encompass much stranger phenomena even than the one you describe.

Topological spaces aren't the only such possible generalization-it's a question of finding a balance between great generality and retaining something geometric. Others that come to mind mostly require notions from category theory, which might not fit your needs well. Anyway, as you already admit, we have to add more structure back in to get much of anything interesting-the diversity of topological spaces is far beyond comprehensibility. But you might want to pick up an introduction to topology, if this intrigues you.

Kevin Carlson
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What can you do with your idea of "relative space"? Just ask questions "How far is X from Y?"

Then the answer is straightforward: the most general notion of relative space given a collection of objects is nothing more than a list of numbers, one for each ordered pair of distinct objects, and the answer to "how far is X from Y?" is simply the number attached to $(x,y)$.

  • Is this kind of list more general than topological spaces? – user2108462 Sep 23 '14 at 21:43
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    Neither of these is more general than the other. Topological spaces have many different answers to "how far is $x$ from $y$," but each answer for $(x,y)$ is the same as $(y,x)$, so that an asymmetry has to be encoded in many relations. Hurkyl's spaces have asymmetry and don't follow any axioms but only one kind of distance at a time. There is an obvious common generalization of the two: a list of lists of numbers, answering "how far is $x$ from $y$ according to distance $i$?". – Kevin Carlson Sep 23 '14 at 22:07
  • Would this common generalization be, to have a list of distances to other objects for each object? Then we will have n lists for n objects, each list containing n-1 items. – user2108462 Sep 23 '14 at 22:09
  • And for each question we want to ask, we add a list of n-1 items containing answers to that question to each object? – user2108462 Sep 23 '14 at 22:10
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To answer your question definitively, from a conceptual point of view: no.

It's really altogether contingent on the fact of relativity. You see, relativity is (naturally) not of the same conceptual ilk as the concept of absoluteness. When you essentially ask, 'is the there an absolutely relative relativisation of mathematical entity x', it's a loaded question, because the second you abandon absoluteness, you thereby necessarily forego anything which resembles a truly coherent answer. In mathematics this is acceptable, as the relativisations are themselves being posed and operated upon in some likewise relative framework, but if you're asking the question in a casual, non-mathematical manner (where 'mathematical' specifically refers to the admittance of relativisation), then you are not going to receive an answer which can be presented in the same common or 'casual' manner.

There are progressively relative, well, relativisations, of the concept of (Euclidean) space, such as the ones described be other commenters, but you're enquiring as to the definitive (absolute) logical conclusion of this relativity, which, again, is itself a contradiction.

So, relatively speaking, the answer to your question is sure, there are progressively relative conceptualisations of space which can be possibly conceived of formally. On the other hand, absolutely speaking, the answer to your question is a definitive no, there is not a concept which is more general than the relativisation of space itself.

ricœur
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