(1) To me the numberline itself is a continuum. But the set of Reals isn't. It is just a set of numbers.
In mathematics, the real line is simply the set R of real numbers. [6]. Mathematicians routinely refer to the set of real numbers as the continuum. [7],[8].
Rogue Physicist wrote:
I believe I read somewhere that the old definitions of a line as a collection of points is actually absurd, as are concepts of area and volume (extension in space). That is, there is no way to get to a line by gathering an infinite number of points together.
I encourage you to abandon popularized versions of science and pursue knowledge. The only way to understand mathematics is to learn mathematics. [9].
Rogue Physicist wrote:
But this may not matter, since you have lept ahead with a handy definition of distance that has the appearance of solving all the dilemmas:
Eugene wrote:
The distance is defined by the absolute value: d(x,y) = |y-x|. We have now arrived at a one-dimensional Euclidean space.
Wow! I feel this actually works, except, I have one remaining problem now:
It also seems obvious to me that |y-x| is something *NEW*. It is not a point, and it is not a Real Number, and it is not a location. It is a scalar quantity, but is an entirely new entity to our previously defined universe. This 'thing' called distance is an abstract thing, which has no direct relation to the numberline or the Real numbers marked on it.
Knowledge is continually being built on earlier conclusions and new definitions. Sometimes reconstruction is necessary. That's the way science works. Of course distance is a real number.
Rogue Physicist wrote:
I just don't feel right glossing over the fact that we have created a new abstract entity that has no physical direct connection to our Universe.
Geometry began as a scientific investigation of real space. Physics began with mathematical modeling the real universe.
Rogue Physicist wrote:
Now once again, the math makes me nervous:
Eugene wrote:
It follows that observer O(x) can note the time T1 on his or her own wristwatch when the point x'_1 on the other ruler passes by and also the time T2 when the point x'_2 passes by. Proper velocity u_xx' is defined by the equation u_xx' = (x'_1 – x'_2) / (T2 – T1). This number is assumed to be constant. No matter which observer on gamma performs this experiment or when, everyone will agree on the value of this constant. The notation u_xx' represents the proper velocity of gamma' as measured by any observer on gamma.
This looks meaty, but needs to be broken down for me to really get it:
As I struggle with it, it occurs to me that the 'Proper Velocity' = delta x/delta t is a ratio. In the past, I always thought of such quantities, be they vectors or scalars, as absolute amounts, (although admittedly set by choice of scales or units). Now here for the first time I am confronted with a 'velocity' defined as a ratio. … I just don't feel like I really understand what we have done here properly.
Speed is defined by a ratio, delta miles/delta hours, for example, yielding miles per hour. [10]. How is the ratio 1/3, understood as a real number, and written as a decimal, not an absolute amount?
Speed is actually a derivative (or more properly, the magnitude of a dervivative). Velocity is the time rate of change of position.
A line segment can be simply defined as consisting of the points on some open interval of the x-axis in the x-y plane.
"This 'thing' called distance is an abstract thing, which has no direct relation to the numberline or the Real numbers marked on it."
This is correct. It's only the metric that matters. The coordinate system used is merely a human construct. Although the coordinates are really the abstract things and distance is real. This is one of the main foundations for GR.
"I believe I read somewhere that the old definitions of a line as a collection of points is actually absurd, as are concepts of area and volume (extension in space). That is, there is no way to get to a line by gathering an infinite number of points together."
All of this is handled rigorously in measure theory. The cocnepts of volume and area are not absurd. It can be done.
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