Infinitesimal point coordinates or not?
Do you occasionally enjoy speculating? Even if one has technical training, if he speculates outside his field of expertise, he is opening himself up for possible difficulty. What he speculates, if he voices it, could result in his being labeled someone who doesn’t know what he is talking about.
But I never was bright enough to avoid speculation, though I always acknowledge it for what it is. So humor me here… So if you know something of all this and what I’m writing either doesn’t jive, or you can ‘supercharge’ it, please let me know.
Mathematicians’ Points – Reality?
Generally, a point in space is seen as a dot in space, having infinitesimal point coordinates, that is, no dimensions of height, width, or depth. In Cartesian coordinates, the location may be written (a, b, c), where a, b, and c are specific value consonants.
This locates the point (1.3, -2.1, 4) at
x = 1.3
y = -2.1
z = 4
The volume contained within the point is zero, since, if viewed as a sphere, the point would have radius zero.
In the Real World
In a previously QuirkyScience article, Discrete Mathematics: What is a Point and What a Line, the possibility that in the real world an infinitesimal point might not exist was discussed. But if no point is infinitesimal, how would a point be written using the coordinate system?
This article will suggest a way of describing a ‘real’ point using the standard coordinate notation, and for purposes of discussion, it is assumed the point has equal dimensions in all directions.
Let the dimensions of the point in each direction be written as r. Then the number that represents the extremities of the point could be included. The coordinates become,
a +/‒ r
b +/‒ r
c +/‒ r
The point, written out without specialized notation, would be
(a +/‒ r, b +/‒ r, c +/‒ r)
Infinitesimal Point Coordinates – Really?
But suppose we backtrack, and suggest the three r values are actually different? Perhaps they would be different if they aren’t viewed statically. If one travels through space, could the r-values elongate or shorten, accordingly? If so, let’s re-label the points in terms of r1, r2, and r3. The expression becomes,
(a +/‒ r1, b +/‒ r2, c +/‒ r3)
How would the mathematics play out in matters of moving reference frames, such as space travel?
Note: You might also enjoy Discrete Mathematics: What is a Point and What a Line?
References:
Only from the perspective of one single observer. As long as the movement is going forward in a constant velocity and the point of reference is the observer relative to the event then it can be measured fairly accurately. But it would not be infinite from this point of view. It would be apparently finite. Numbers in and of themselves have no real value but their relationship to one another in an equation have specific values of a finite not infinite value. But infinity is more of an abstraction.