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I am open to your your criticism.
Ah, thank you.
I apologize in advance for the length and depth of this post, I had originally meant to be succinct; however, I wanted to make sure that I was clear when explaining both the things I didn't quite understand and the assumptions that I made while reading this paper.
I know this isn't currently the topic at hand in the thread; however, I was reading
Organic Mathematics - The Science of Direct Perception, and I had a few questions.
I guess the first one is, what exactly do we mean when we say "1-dimensional?"
The informal definition with which I am familiar would be something along the lines of "An object is n-dimensional if a minimum on n coordinates is required to describe the location of a point on this object."
Thus a point would be 0-dimensional (how many coordinates do you need to specify one point out of one point?)
A curve (e.g. a line, circle, whatever) would be 1-dimensional because, given a start point, we need only to describe how "far" along this line we need to move to reach our point.
And so on, and so forth.
So, that's all fine and dandy... but what happens when we talk about the set {0,1} consisting of two points on the real number line? Its clearly not 0-dim, (at least by my definition) because if we wish to specify a single point of this set, then we need one coordinate to distinguish which of the two points we mean.
So, if this is 1-dim, then this seems like it would be a counterexample to the claim that "No collection of 0-dim elements may cover a 1-dim element," because the points 0 & 1 cover the subset of the number line consisting of the points {0,1}.
So, I guess my first question boils down to:
If this isn't a 1-dim element, why isn't it? Is it because my definition of an n-dimensional element is incorrect/insufficient? If so, what definition should I use? Is it because the kind of object I described (an object consisting of two discrete points) is not something we can talk about the dimensionality of? If this is the case, for what kind of objects can we talk about dimensionality? Do they need to be simply connected? Or, is there some other reason as to why my reasoning is flawed?
A second question I have, is about the concepts of locality/non-locality. I just want to make sure I understand what you mean.
A point is an example of locality, because a point exists only in one specific location, correct?
A line on the other hand, would have a non-local property, because it exists in more than one specific location at once; e.g. In the XY plane, the line y = 0 exists at the points (1,0) and (2,0) (and at many other points) and therefore does not exist in one specific location... giving it its non-local property? Is that the correct line of reasoning?
Likewise, would the object I described earlier (consisting of the points {0,1} along the number line) be an example of something with a non-local property?
Another thing, is a point the only element with the local-property? If not, what other elements may have that property?
And finally, you say that "no collection of 0-dim elements can completely cover a 1-dim element." Which, if I understand you correctly, would imply such things as "The real number line is not completely covered by points." Unless I am misinterpreting what you mean, of course.
Statements such as "The real number line is not completely covered by points" are directly contrary to my intuition. That is to say, I read the statement "The real number line is not completely covered by points" and I think of it as its contrapositive: "There exists at least one spot on the real number line which is not covered by a point."
The way with which I am familiar with constructing the real numbers is the following.
We begin with our additive identity and our multiplicative identity. {0,1}
From here we obtain the ordinals by inductively adding one. That is to say, we define 1+1=2 then we define 2+1=3 and so on and so forth. We get the natural numbers. {0,1,2,3,...}
Next for each ordinal number, we adjoin an additive inverse, and we obtain the integers. {...,-3,-2,-1,0,1,2,3,...}
From here, we adjoin the multiplicative inverses of each number and we get: {...,-3,-2,-1,-1/2,-1/3,...,0,...,1/3,1/2,1,2,3,...}
Next, we adjoin the products and sums of all of the elements we currently have with each other to obtain the rational numbers.
There are several intermediary steps between here and the entirety of the reals that one could take (the constructable numbers, the algebraic numbers);however, they are immaterial as the next step encompasses them both.
To adjoin the irrationals, we proceed in the manner of Dedekind. I find it helps to take a step back and visualize what we are about to do before we do it, for clarity's sake.
Imagine, if you will, the XY plane. The X-axis shall serve as our real number line. Imagine, if you will, that we draw a straight line - any line- so that it intersects the X-axis somewhere, anywhere. Maybe it intersects at one of these nice rational points that we know about. But maybe it doesn't. For our purposes we will assume that the line we drew, did
not intersect the X-axis at a rational point.
If we look at all of the rational numbers on the X-axis, we find that this line splits them into two groups. There are those rational numbers to the left of this line, we will call them, collectively, A. There are also those rational numbers to the right of this line, we will call them, collectively, B. Because, for the purposes of our imaginings, this line did not intersect the X-axis at a rational point, we must have that all rational numbers are contained in either A or B. (We know all rational numbers must either fall to the left of this point of intersection, to the right of this point of intersection or they must be exactly on this point of intersection, and since we are assuming that no rational numbers are on this point of intersection they must all belong to either A or B). Further, with a little thought it becomes clear that A has no greatest element and B has no least element.
Now, with this physical intuition in mind, we turn our attention back to the problem at hand (constructing the rest of the reals from the rationals).
We divide the Rationals up into two sets, A and B. Where A has the property that it has no greatest element, and further that given any choice of an element a in A, every rational number less than our chosen a is also in A. Likewise we choose B so that it has the property that it has no least element, and further that given any choice of an element b in B, every rational number greater than that chosen b is also in B.
This is a very verbose way of saying that we are doing fundamentally the same thing as when we were discussing lines intersecting the X-axis in the XY plane. The sets A and B we formed from those intersections have the same properties as the ones we described.
Armed with our sets A and B, we look and we see that the elements of the two sets A and B get arbitrarily close together, but they never actually meet. So we take the point on the real line, that lies between all of the members of A and all of the members of B, and we adjoin it to the real numbers. And so, we have our first irrational number.
For every possible choice of our sets A and B, we adjoin the numbers between all of the members of those sets as well; This is how we construct the irrational numbers.
This is the method I am familiar with, and it seems to me as if that gives us a real number line which is entirely covered by points.
Going back to what I said earlier, this would not be true if "There exists at least one spot on the real number line which is not covered by a point."
However, if we posit that such a spot exists, we can clearly dismiss the possibility that it is a rational number, because we know those to be on the real line because of how they were constructed.
Since this uncovered spot on the real line is not a rational, we must have that it splits the rationals into two groups A, consisting of those rationals to the left of this uncovered spot and B, consisting of those rationals to the right of this uncovered spot. However, if this uncovered spot partitions the rationals in the described manner, then this spot fits the criteria set forth when we were constructing the irrationals, and therefore must be one of the irrationals that we adjoined.
Therefore, such an "uncovered spot" must not exist.
That argument seems to me as if it suffices to describe a 1-dimensional object which is completely covered by 0-dimensional objects. Although perhaps you may find a flaw with the method of construction, or you may be able to demonstrate the existence of a point that is not covered.
I do have some further questions about the specifics of locality/non-locality linkage; however, it seems to me that I should wait until I am sure I understand locality and non-locality before I decide that I do not understand their linkage.
Once again, I apologize for the tremendous length of this post. And thanks in advance for clearing these issues up for me.
