- #31
JOPearcy
- 19
- 0
K.J.Healey,
Good comments. I like your imaginative example about a point particle in space.
Mathwonk,
I do think you also make some good points, but, I want to focus on one narrow and simply example, to keep things focused and hopefully easier to understand for all of us. Kind of like trying to analyze a system be reducing it to as few of degrees of freedom as possible, preferably down to a single degree of freedom when starting.
Instead of picking a point particle in space, I want to pick a simple lab or experimental real physical setting.
We can observe real physical space around us. We know space exists. We can detect it, quantify it, measure it and analyze it. In the process we can, through experimentation and analysis, figure out that we can measure three dimensions in real physical space such that any three mutually perpendicular directions can function as the three dimensional lengths of real physical space.
To consider how this is done, we can look at beginning with a single dimension of length in physical space. We pick two points and draw a straight line between the two. We can then imagine the line extending infinitely from both ends of that line segment defined by the two points.
But, to define this length, we have to be able to detect it, quantify it and measure it. You unit of quantity can be arbitrary. But it is required. As soon as you place the second required point to define the one dimensional length, you have already created the basis for a defined quantity, the distance between the two defining points. Even if that it the only reference you have, it is still a required quantity. Without being able to define a quantity of length you can not define a dimension of length.
Quantity and dimension are inter-related. You must have one to define the other.
Think about the process of observing, quantifying, measuring and analyzing one dimension of real physical space.
Then consider the following formal definitions:
A “quantity” is the property of a phenomenon, body, or substance, to which a magnitude can be assigned.
“Quantities of the same kind” are quantities that can be placed in order of magnitude relative to one another.
A “system of quantities” is a set of quantities together with a set of non-contradictory equations relating those quantities.
A “base quantity” is a quantity, chosen by convention, used in a system of quantities to define other quantities.
A “derived quantity” is a quantity, in a system of quantities, defined as a function of base quantities.
A “quantity dimension”, equivalently phrased as a “dimension of a quantity”, equivalently simply phrased as a “dimension” is a dependence of a given quantity on the base quantities of a system of quantities, represented by the product of powers of factors corresponding to the base quantities.
Apply these definitions to the process of observing, quantifying, measuring and analyzing one dimension of real physical space.
Does this help you understand what dimension means with respect to one dimension of real physical space?
Good comments. I like your imaginative example about a point particle in space.
Mathwonk,
I do think you also make some good points, but, I want to focus on one narrow and simply example, to keep things focused and hopefully easier to understand for all of us. Kind of like trying to analyze a system be reducing it to as few of degrees of freedom as possible, preferably down to a single degree of freedom when starting.
Instead of picking a point particle in space, I want to pick a simple lab or experimental real physical setting.
We can observe real physical space around us. We know space exists. We can detect it, quantify it, measure it and analyze it. In the process we can, through experimentation and analysis, figure out that we can measure three dimensions in real physical space such that any three mutually perpendicular directions can function as the three dimensional lengths of real physical space.
To consider how this is done, we can look at beginning with a single dimension of length in physical space. We pick two points and draw a straight line between the two. We can then imagine the line extending infinitely from both ends of that line segment defined by the two points.
But, to define this length, we have to be able to detect it, quantify it and measure it. You unit of quantity can be arbitrary. But it is required. As soon as you place the second required point to define the one dimensional length, you have already created the basis for a defined quantity, the distance between the two defining points. Even if that it the only reference you have, it is still a required quantity. Without being able to define a quantity of length you can not define a dimension of length.
Quantity and dimension are inter-related. You must have one to define the other.
Think about the process of observing, quantifying, measuring and analyzing one dimension of real physical space.
Then consider the following formal definitions:
A “quantity” is the property of a phenomenon, body, or substance, to which a magnitude can be assigned.
“Quantities of the same kind” are quantities that can be placed in order of magnitude relative to one another.
A “system of quantities” is a set of quantities together with a set of non-contradictory equations relating those quantities.
A “base quantity” is a quantity, chosen by convention, used in a system of quantities to define other quantities.
A “derived quantity” is a quantity, in a system of quantities, defined as a function of base quantities.
A “quantity dimension”, equivalently phrased as a “dimension of a quantity”, equivalently simply phrased as a “dimension” is a dependence of a given quantity on the base quantities of a system of quantities, represented by the product of powers of factors corresponding to the base quantities.
Apply these definitions to the process of observing, quantifying, measuring and analyzing one dimension of real physical space.
Does this help you understand what dimension means with respect to one dimension of real physical space?