I feel like I am arguing with a bunch of high school kids that can not grasp the meaning of dimension. At this point I am coming to the conclusion that further argument over it is fruitless and that you are not able to grasp the concept of the meaning of dimension.
If the definition of dimension is beyond your grasp to understand then there is little point to arguing about it further.
This kind of BS isn't helpful. We
do understand, at least some of us, so don't be a jerk! If we have misunderstood things you've said, that's mostly
your fault, because you have certainly made it seem that you don't understand that there are several very different definitions of the word "dimension".
The three definitions that I've encountered lots of times are these:
1. Vector spaces. The dimension of a vector space V is defined as the maximum number of linearly independent vectors in V.
2. Manifolds. A class of open subsets that cover the manifold are mapped into R
n by functions called coordinate systems. The number n is called the dimension of the manifold.
3. Your definition.
When you measure the quantity of something of the same nature and characteristics, there is a dimension to that measurement, be it length or be it apples.
If you go to measure something and measure out any arbitrary unit of that something, and then measure out another arbitrary unit of the same something, the units can be added together, even if you have to do some conversion, but both arbitrary units will have the same dimension.
Well said. No complaints from me.
For example, to use that which people are very familiar with, take the distance between two points. We can measure that distance. We can select an arbitrary unit smaller than the distance we want to measure and then count out the quantity of the units that fit along the length. The dimension is what the units consist of. You can arbitrarily define all manner of different units, but if you are measuring the distance between two points all these units consist of the same dimension.
You're asking for trouble by using a direction in space as an example, but I believe you got it right.
In a similar manner, let’s say you have 1,000 apples. You want to measure how many apples you have. You define a unit of measure for the apples as 12 apples = a dozen apples. You can then count out how many apples you have in that unit of measure and say 83 and 1/3 dozen apples. You can also define as a unit of measure of 120 apples = a small gross of apples and count the apples by those units. But the real dimension of what you are counting in is not based on the units and thus the dimension in this case is apples.
This is fine too. I think you've explained it very well actually.
Now, the question is, do you understand that with your definition of a dimension, space has an
infinite number of dimensions, and not just three? (Each direction is a dimension).
If you understand that, you must also understand that theories that say that space has more than three dimensions are talking about a completely different kind of dimensions. They are using definition 2, not 3.
