1. You explained very well (using apples as an example) what you mean by "dimension". I told you several times that your definition implies that space has infinitely many dimensions (each direction being a dimension). And yet space has three dimensions. How do you explain that? (My explanation is that space has three algebraic/geometric dimensions and infinitely many of "your" dimensions).
To understand why we fail to understand each other, you need to understand the meaning of dimension, which of course you think I do not understand. 3 dimensional space has 3 dimensions. Things can exist within 3 dimensional space which have more than 3 dimensions, but this does not change the fact that 3 dimensional space has 3 dimensions.
My simply trying to explain the definition is failing to bring any understanding. So, to try and reach an understanding, let’s start simplifying the topic, hence the question of what is space.
I picked space because of the accepted view of space, the physical space around us which we exist in and can detect, is generally considered to have 3 dimensions and it is accepted by both of us that these dimensions exist. Thus we can begin on a common ground simpler than discussing all possible dimensions especially as we disagree on what a dimension is.
So let’s begin where we agree.
I pick space rather than 1 dimension of length, because I have had physicist refer to space as a fundamental quantity as opposed to length being a fundamental quantity. I really want to boil it down to a single dimension of length, but when we observe space around us, it does not exist as a 1 dimensional length.
So, to get from what we can observe to a 1 dimensional length, we need to start with what we can observe, space.
2. You asked for an example of a dimension that isn't a dimension of the kind that you like to talk about, and I gave you one: The one-dimensional subspace of the vector space of odd periodic functions with period 2*pi, that consists of functions of the form af, where a is a real number and f is defined by f(x)=sin(5x). Why did you ignore that? Do you think that I failed to give you a correct example? If that's what you think, then what's wrong with my example?
To begin a topic over which we are disagreeing and reach out for an example complicated enough to only increase the confusion of trying to define a simple concept is counterproductive. Are you talking about the 1 dimensional result of “a * sin(5x)” as describe by a vector perpendicular to x as x varies between minus to positive infinity?
Can you measure it? Can you quantify it? Can you assign units of quantity to it? If so, it has a dimension and the dimension is the quantities and units of quantities consist of.
But again, instead of making the issue more complicated, let us try to simplify it.
Let’s begin where we can agree.
How do we know space exists?
3. In one of the other forums you got this very good (except for how he spelled "Minkowski") and very relevant question, and you ignored it:
Is this a question?
I have heard real physical space defined as a fundamental quantity because in our real world observations of real physical space it does not exist in 1 dimension or 2 dimensions but 3 dimensions (by basic models of 3D space).
I have also heard length defined as a fundamental quantity because in our real world observations of real physical space our basic model defines it as having 3 dimensions of length.
I don’t think it really matters that much which one is called fundamental. Each point of view has its arguments, but both points a view agree on the same basic model of 3 dimensional physical space.
Is this something we can agree on?
If so, how do we know space exists?
