[Don_1]

I'll not ask what Miss Scarlet did in BR2 with a candlestick. I fear for my sobriety should I get an answer.

[DoctorBeaver]

I'll not ask what Miss Scarlet did in BR2 with a candlestick. I fear for my sobriety should I get an answer.

She banged in a nail. Couldn't find a hammer.

[Don_1]

You had me worried there when I read 'She banged...' I wasn't sure if I should read any further...

[lightarrow]

So my thought is that dimensions do not exist in a rigid and axiomatic mathematical way and are merely a construct.

A thing is the concept of dimension in mathematics, another is its application in physics. The two things are always different.

[LeeE]

Yes, but if n=0 then there is nowhere for it to exist except as an abstract concept.

Matter, as we know it, is three-dimensional but this on it's own doesn't seem to directly account for it's possession of rest-mass; a spatial size of n cubic metres doesn't equate to a particular quantity of kilograms.  We can use Density, of course, to relate the two, but density is just the measured quantity of rest-mass in a volume, not the reason that the rest-mass exists and has a non-zero size.  Rest-mass then, appears to need three spatial dimensions within which to exist but whatever it is, that is manifesting/expressing itself as rest-mass, doesn't appear to have a particular spatial size.

If we look at light though, we've got something that certainly seems to exist, and once again, it seems to need three dimensions to exist within, but unlike matter it has no rest-mass; whatever it is that manifests/expresses itself as rest-mass in matter seems to be absent in light, so its size in light is zero.

I agree that point objects would only seem to be able to exist as an abstract, but any non-infinitely regressive analysis of the universe will lead to such abstracts.  If we now go back to thinking about light again, it could be described as semi-abstract; it doesn't exist as matter, but it definitely exists as something.

With a zero-sized point-object, it seems to me that we could assign a value to it, representing the size of a particular quality, but if that quality isn't spatial size then the quantity of whatever quality it is will just occupy a geometric point that exists at a particular spatial (or temporal) coordinate.  Only if the value (quantity) of its quality is zero would it not exist at all.

The ultimate abstract would be where all different qualities were reconciled at this level, in which case all you'd need would be the value because there would only be one possible quality.

[lyner]

Dimensions don't have to be 'xyz'. You can describe a position using polar coordinates just the same.
The point of having dimensions is that you can specify the position (and other conditions) of an object using a number of quantities which can be independent of each other. So you can change your x dimension without altering your y dimension by moving along a horizontal line on your ordinary graph paper. The x and y dimensions are independent (the posh word is orthogonal).  If you used polar coordinates you still have independent dimensions  so, for example, you can keep the radius constant and vary the angle  and the points will lie on a circle. Using r and theta, you can specify any position on a flat sheet of paper (a plane). Radius and two (orthogonal) angles can specify any point in simple space.There are other sets of coordinates you can use but you still need three dimensions.
When you calculate hard things like the Universe and all that, you can arrive at equations which involve more than just x, y and z. The other dimension which you get from String Theory, for instance, have to be there in order to explain, in one fell swoop, how objects will interact.

So, on a simple level, xyz are  enough to describe the position of objects. If you want to describe how they move about, you need to introduce time - a fourth dimension. If you need to describe the effects of electromagnetic fields and gravity all together, you can (they claim) do it using  extra dimensions  which, of course, we can't SEE. What they say is that moving along, say, the x dimension, we are, in fact moving through lots more dimensions. Although another object may be distant in the x dimension, its relation to us in its other dimensions may affect the way we move in the x dimension - i.e the other dimensions somehow explain how we experience the force of gravity from the Sun. String Theory replaces the theory which describes things in terms of Fields and masses. (Remember - Fields are not necessarily any more 'there' than anything else; we just use them as a model to predict and explain what goes on)
And, of course, when we talk of something with zero length, it may not have zero values for its other dimensions.
I am not too convinced that the two uses of 'dimension' are really that distinct from each other - except that Maths is a limited set of ideas (axioms). So, perhaps Mathematical Dimensions are just a sub set of Dimensions.
That's never stopped us from using all the rest of Maths (subset) to deal with the physical (complete set) World, though.

[yor_on]

Very nice description SC. It is this Idea that string theory seems to build on, that those 'dimensions' xyz are independent from each other. That makes it mathematically reasonable to see a 'one dimensional' string having qualities as 'vibration' aka 'mass' and 'tension. But that is if spacetimes dimensions really are independent isn't it?

[lyner]

Those quantities could be functions of more than one dimension, of course, so it might be expected they could 'talk to each other'.

[DoctorBeaver]

If a particle has momentum in an extra dimension but none in the 3 spatial dimensions we are used to, then that momentum will look to us like rest mass. Maybe particles that do have rest mass are free to travel in an extra dimension whereas photons aren't. Why that should be, I don't know. Maybe the answer lies buried somewhere in string theory.

[LeeE]

That's an interesting idea Dr. B.

Instead of photons not being able to move in the 'momentum' dimension, they simply may not have any presence in it.

If it's not possible for something to exist in a limited set of dimensions, it must exist in every possible dimension, which brings us back to having to deal with an infinite number of dimensions again.  However, by allowing something to exist in just a limited set of dimensions, out of a possible infinite set, we don't have to worry about infinite dimensions any more; only the ones that are being occupied are relevant.

[DoctorBeaver]

That's an interesting idea Dr. B.

Instead of photons not being able to move in the 'momentum' dimension, they simply may not have any presence in it.

If it's not possible for something to exist in a limited set of dimensions, it must exist in every possible dimension, which brings us back to having to deal with an infinite number of dimensions again.  However, by allowing something to exist in just a limited set of dimensions, out of a possible infinite set, we don't have to worry about infinite dimensions any more; only the ones that are being occupied are relevant.

Precisely. Less infinities=good news

[yor_on]

Ethos discussed if the universe differed between left and right before. I don't think the universe uses our definitions at all. I'm not even sure that our definitions of a left or right handed spin have anything to do with what's really 'there'. We put on descriptions of 'stuff' like colourforce etc, but we can't see what we describe so it is quantitatively different than our macroscopic reality where a bike can be described as well as seen. At a QM level this type of definitions disappear and what is left is a mental guessing game where we find puzzle pieces that we label as being of certain properties. But there they may change shape, colour, and name at a moments notice, as a better theory comes.

[DoctorBeaver]

yor_on:

That's absolutely correct. But without labels we wouldn't know what other people were talking about. Where spin is concerned, left & right are meaningless except for us to be able to differentiate between how the particles behave. We may just as easily have used up & down or backwards & forwards. I'm not sure how chirality entered the picture but I suspect it has to do with the terminology from polarity.

[LeeE]

Yor_on & Dr B: these are examples of the abstracts you end up with as a consequence of any top-down analysis.  We assign them a quality and a value, but we don't really know what it is that we're dealing with, other than in abstract terms.

[DoctorBeaver]

LeeE - I agree. But without labels we'd be buggered.

[LeeE]

LeeE - I agree. But without labels we'd be buggered.

Yup - even if we don't know exactly what it is that we're talking about, at least we know we're talking about the same thing.