We need a qualifier in there to makes sure we’re not just comparing the number of points, objects, etc. The dimensionality comes from the additonal structure. Like I said any overarching definiton of dimension is necessarily imprecise (and therefore) vague) and open to exceptions so I don’t actually mean anything exactly by continuous. Of course if you formalize it you’d get something along the lines of the defintion of the dimension of a manifold.
Of course if we stick to physics just about every single space we care to use will be locally metrizable.
Pretty sure you can define continuity without a metric so long as you have a topology (i.e. something like, for any open set on the manifold, there exists an open set in coordinate space that maps into it) but yeah.
All spaces used in physics may be metrizable, but there’s not always a natural choice of metric for them. I expect, though, that any non-absurd choice of metric in those cases will still lead to the same topology.
This is why the third dimension is still the best. It’s the last easily explained dimension. 
This is a very nice thread. Thanks for starting it, OP.
It’s one of those things that’s “not even wrong.”
He starts explaining dimension 5 by saying that there are all these other places that the timeline could have gotten you, and that those “places” are separated from us along the fifth dimension. Well since a dimension is anything used to define a mathematical space, I suppose it is trivially true that you could define that range of possible outcomes from a starting point as varying across a fifth dimension. That makes it a fifth dimension, but not the fifth dimension, and it is NOT what physicists and mathematicians typically mean.
Then he gets even weirder by assuming that these other timelines a) actually exist and b)can be travelled to by folding through an even higher dimension. Then he gets even weirder and starts talking about alternate universes and assuming that they a) exist and b) exhibit some sort of quasi-spatial relationship analogous to ordinary spatial location.
Sure, there’s the “Many Worlds” interpretation of QM, but that’s pure speculation unconnected AFAIU to the higher dimensions used in String Theory. If we lived in the sci-fi world that video described, then it might (or might not) make sense to describe it that way, using those dimensions. But there is no evidence that that’s the world we live in and plenty of evidence that the guy making the video made all that stuff up after watching too many episodes of Star Trek.
And again, nothing that real physicists trying to describe our actual world rather than Star Trek has anything to do with alternate timelines or multiple universes. That’s simply not where the idea of eleven (or however many) dimensions in String Theory comes from.
Imagine a baseball in mid-air. I can tell you where it is using three coordinates (say, latitude, longitude, and altitude). Thus, the position of the baseball is represented by a 3-dimensional space. “Space” here is an abstract concept, even though in this case the three dimensions can be trivially connected to the concept of physical space.
However, if you were a baseball player hoping to catch the ball, I haven’t given you anywhere near enough information. In particular, you need to know what direction the ball is traveling and how fast is is moving. There are a few ways I can specify this. One might be: “It’s currently dropping in altitude at 5 ft per second, and it’s headed east at 2 ft per second, and it’s heading north at 1.5 ft/s.” I could also choose to specify the direction of its motion (one angle relative to vertical and another relative to north) along with a speed (5.56 ft/s). But regardless of how I choose to specify this information, it takes three quantities to do it.
Thus, if I am to tell a blindfolded outfielder what he needs to know to catch this ball, I need to specify six numbers in total: three for the current position and three for the current velocity. We say that the trajectory of the baseball is represented by a 6-dimensional “space”. Just as the position of the ball is a point in a 3D space, the trajectory of the ball is a point in 6D space.
In principle, an outfielder might also want to know the spin of the baseball, since the future path of the ball depends on the spin (as with a curve ball). Or, he might want to know the current air density, which will affect drag. These would add dimensions to the space required to specify the situation. The more aspects of the situation (or, more jargon-y, the more aspect of the “system”) you need to specify, the higher the number of dimensions you need.
Only in a limited range of particular applications are higher numbers of dimensions (above 3, say) intended to represent new physical spatial dimensions.
Indeed, and I guess you are giving the abridged description of orbits, though orbits are usually given by 6 different numbers, since (x,y,z) and velocities don’t give the most intuitive feel or mathematically satisfying description for an orbit.
Indeed, abridged. I didn’t imagine that a different basis for this space would be all that helpful to the OP.
Yes certainly,continuity is defined for any function between any topological spaces.
Speaking specifically about manifolds, their defining characteristic is that for every point there exists a neighbourhood (of that point) for which a bicontinuous function (i.e.a homeomorphism) exists between that neighbourhood and R[sup]n[/sup].
Though to sound a note of caution, that doesn’t mean that all open sets on a manifold are homeomorphic to R[sup]n[/sup]
Yes. There’s a formulation of physics in which the dynamics of a system is expressed as the motion of a point in a high-dimensional space - Phase space - Wikipedia
Quoth Pasta:
Actually, though, it is a good and relevant point that a space can be expressed in terms of many different bases, and that some are more convenient than others for various purposes. No matter what basis you use, though, you always need the same number of coordinates, and that number is the dimensionality.
Agreed - that’s why it’s jarring to hear “width” described as one of three dimensions; you could just as easily say the three dimensions are radial distance, azimuth and elevation as say they were length, breadth and width (and in relativity, your time dimension is not necessarily the same as mine).
Infact in relativity it’s possible to have a null basis where each basis vector is null (i.e. “lightlike”). People ‘like’ bases such as Caretsian and Minkwoski bases as they are constant and orthonormal (thoguh obviously they only work for certain sapces).
Okay, I’ll bite. What’s the dimension of a set with cardinality c under the cofinite topology? Damned if I know.
I don’t know, but I’d bet that it’s either 1, aleph-0, or c.
It probably depends on what notion of dimension you want to use. For covering dimension, I believe it’s infinite dimensional. (Simply meaning that it’s not finite dimensional, no notion of type of infinity enters.)
Yes, if you had a bunch of dimensions, then you could call them coordinates. The problem is we’re not defining “coordinate” nor are we defining “space” nor are we defining "dimensions. We’re solely defining “dimension”.
A dimension is not a set of coordinates, but rather a set of all possibilities for a single coordiate. It’s not {3", 60 C, yellow}, but rather {1", 2", 3", 4",…}
So why would my definition need to specify that coordinates are independent when we’re not defining coordinates, independence, or spaces?
But, again, a dimension isn’t like 43N,63W. That’s a coordinate, not a dimension. A dimension is the set of all possible “norths and souths”. Another dimension is the set of all possible “easts and wests”. How do you know that 43N and 24N are in the same dimesion? Because in the space we’re (likely) talking about, a thing can only have either but not both.
Do any of you remember a guy who came here around eight years ago to propagandize his grand unified theory that used four spatial dimensions and no time dimension?
Couldn’t have been too many real physicists in the thread because he built a solid wall of woo that deflected all criticism until I finally penetrated it by pointing out that orbits were not stable in four spatial dimensions.
That shook him but he volleyed back with the greatest single bit of pseudoscience I’ve ever read.
He was going to investigate the situation by seeing what happened with a fourth dimension that wasn’t perpendicular to the others.
There. He ran rings around us logically.
re. the OP…
A dimension is a measurement in a co-ordinate system.
This many inches per side of a box.
That many rotations per minute for a motor.
You could define a system with temperature, electromagnetic charge, mass, chromodynamic color, spin, etc.