If an interval corresponds to a set of points (and nothing else) then I agree that an interval containing only rationals has no length.The length of that union is zero, if the intervals are restricted to rationals. Do you agree with that point? — fishfry
Our problem is that you are only allowing points in your sets. Suppose I introduce a new concept called 'k-interval' to define the set of ANY objects located between an upper and lower boundary. Would you then consider allowing objects other than points into the set?No points. So they're all the empty set? I'm not supposed to push back on this? — fishfry
I wanted to show you that even if I cut my unit line to contain all rational points between 0 and 1 that there would still be stuff in between the points -- continua. Perhaps I used the wrong tactic by talking about an idea which I don't support. I did say at the start of the paragraph that it was impossible but maybe I could have been clearer.Why on earth do you troll me into arguing with your points, then admitting that you agree with me in the first place? — fishfry
Yes, you believe in continua, but not as 'objects in and of themselves'. You believe that continua can't exist in the absence of points. Please confirm.I very much believe in the continuum, which is pretty well modeled by the standard real numbers. — fishfry
My preference is that you accept non-points into sets, however, if you're unwilling to do that then here's an alternate approach. To move this conversation forward, let's say that when I say 'a line', you can think to yourself that I'm referring to 2^aleph_0 points (which somehow assemble to form a line), and I'll think to myself that I'm simply referring to a line (which cannot be constructed from points). In other words, you can go on thinking that points are fundamental and I'll go on thinking that lines are fundamental. How does that sound to you? All I need from you really is to allow me to restrict my intervals to those whose bounds are rational (or +/- infinity). Could you let that fly? ...Just to see how far my position can go in the absence of the explicit use real numbers (I'm fine if in your eyes their use is implied but I just won't ever mention them)...I am at an utter loss as to what you have been getting at all this time. Can you get to the bottom line on all this? So far I get that your "continua" are either empty or have length 0. Or that they somehow have length 1, despite being composed of only rationals. — fishfry
You’re suggesting that my line, which already extends in space, requires additional points, which themselves individually have no length, to actually have length. I wish you could appreciate the irony in your position. — keystone
If an interval corresponds to a set of points (and nothing else) then I agree that an interval containing only rationals has no length. — keystone
Our problem is that you are only allowing points in your sets. — keystone
Suppose I introduce a new concept called 'k-interval' to define the set of ANY objects located between an upper and lower boundary. Would you then consider allowing objects other than points into the set? — keystone
Why on earth do you troll me into arguing with your points, then admitting that you agree with me in the first place?
— fishfry
I wanted to show you that even if I cut my unit line to contain all rational points between 0 and 1 that there would still be stuff in between the points -- continua. Perhaps I used the wrong tactic by talking about an idea which I don't support. I did say at the start of the paragraph that it was impossible but maybe I could have been clearer. — keystone
Yes, you believe in continua, but not as 'objects in and of themselves'. You believe that continua can't exist in the absence of points. Please confirm. — keystone
My preference is that you accept non-points into sets, — keystone
however, if you're unwilling to do that then here's an alternate approach. To move this conversation forward, let's say that when I say 'a line', you can think to yourself that I'm referring to 2^aleph_0 points — keystone
(which somehow assemble to form a line), and I'll think to myself that I'm simply referring to a line (which cannot be constructed from points). — keystone
In other words, you can go on thinking that points are fundamental and I'll go on thinking that lines are fundamental. How does that sound to you? — keystone
All I need from you really is to allow me to restrict my intervals to those whose bounds are rational (or +/- infinity). Could you let that fly? — keystone
...Just to see how far my position can go in the absence of the explicit use real numbers (I'm fine if in your eyes their use is implied but I just won't ever mention them)... — keystone
So for example, can you allow me to say that there are 5 objects depicted below? You can go on thinking that 2 of the objects are composite objects and I'll go on thinking that all 5 objects are fundamental (they're either 0D or 1D continua). — keystone
No, I'm saying that within an open interval, such as (0,0.5), lies a single objects: a line. Absolutely no points exist with that interval. If you say that 0.25 lies in the middle of that interval, I will say no, 0.25 lies between (0,0.25) and (0.25, 0.5). And what this amounts to is cutting (0,0.5) such that it no longer exists anymore. In its place I have (0,0.25) U [0.25] U (0.25,0.5).But you are the one saying that you only have rationals. — fishfry
Let's move away from directly using sets to describe the path. Instead, we'll describe the path using a graph, and then define the graph with a set.In standard set theory, the only thing that sets can contain is other sets. We can call them points but that's only a word used to convey geometric intuition. Actually sets don't contain points, they contain other sets. — fishfry
Urelements are indivisible 'atoms'. The lines that I'm working with are divisible.I don't know anything about set theory with urlements. — fishfry
That is not what I believe. I can define a line using no rationals: (-inf,+inf). I see this line as a single object (a line). It is not populated by rational points. It is not populated by any points for that matter. I've drawn it for you below in between points at -inf and +inf. To walk this path from -inf to +inf you don't need limits, you just walk the corresponding graph from vertex 0 to vertex 1 to vertex 2.You only believe in rationals. Where are you getting these things? — fishfry
Yes, I mean endpoints. I used the term 'bounds' because it is a more general term that applies to higher dimensional analogues. I'm searching for a way to keep this conversation going so it doesn't end prematurely out of frustration. Currently, I don't believe I can persuade you that a continuum can exist without points. However, I've come to realize that convincing you of this isn't necessary. Here’s my new approach:By bounds you mean endpoints? So now you are backing off entirely from your last half dozen points, and saying that your ontology consists of intervals with rational endpoints. But the real numbers are indeed present inside the intervals after all? Is that what you are saying? — fishfry
Yes, the endpoints are rational, and the object between any pair of endpoints is simply a line. It doesn't go deeper than that. I understand you see that line as a composite object consisting of 2^aleph_0 points. However, I view the line as a primitive object. Clearly, our starting points differ. To move the discussion forward, could we agree to a compromise where we refer to a line as a "composite" object? This way, by including composite it's evident that I recognize your perspective, but the quotes indicate that my viewpoint doesn't necessitate this classification.But now only the endpoints are rational, leaving me baffled as to what those objects are. — fishfry
I agree with this point. The issue has been the lack of viable alternatives. I see that Peirce was suggesting the use of infinitesimals, and you're aware of my stance on those—the one from the comment where you thought I was just trolling.A forum member once pointed me to the ideas of Charles Sanders Peirce (correct spelling) who said that the mathematical idea of a continuum could not be right, since a true continuum could not be broken up into individual points as the real numbers can. — fishfry
But you are the one saying that you only have rationals.
— fishfry
No, I'm saying that within an open interval, such as (0,0.5), lies a single objects: a line. Absolutely no points exist with that interval. If you say that 0.25 lies in the middle of that interval, I will say no, 0.25 lies between (0,0.25) and (0.25, 0.5). And what this amounts to is cutting (0,0.5) such that it no longer exists anymore. In its place I have (0,0.25) U [0.25] U (0.25,0.5). — keystone
Let's move away from directly using sets to describe the path. Instead, we'll describe the path using a graph, and then define the graph with a set. — keystone
I don't know anything about set theory with urlements.
— fishfry
Urelements are indivisible 'atoms'. The lines that I'm working with are divisible. — keystone
You only believe in rationals. Where are you getting these things?
— fishfry
That is not what I believe. I can define a line using no rationals: (-inf,+inf). — keystone
I see this line as a single object (a line). — keystone
It is not populated by rational points. It is not populated by any points for that matter. — keystone
I've drawn it for you below in between points at -inf and +inf. To walk this path from -inf to +inf you don't need limits, you just walk the corresponding graph from vertex 0 to vertex 1 to vertex 2. — keystone
You would call this green line the 'real number line'. You see this as 2^aleph_0 points. You believe that to walk any length on this green line you need limits. I understand what you're saying. We're just starting from different starting points. You're starting from the bottom and I'm starting from the top. — keystone
Yes, I mean endpoints. I used the term 'bounds' because it is a more general term that applies to higher dimensional analogues. I'm searching for a way to keep this conversation going so it doesn't end prematurely out of frustration. — keystone
Currently, I don't believe I can persuade you that a continuum can exist without points. — keystone
However, I've come to realize that convincing you of this isn't necessary. Here’s my new approach:
1) Start at the bottom
2) Build up to the top
3) 'Start' at the top
4) Approach the 'bottom' from the top — keystone
I see this equivalent to starting at the bottom of the S-B tree -> working our way to the top of the tree -> then proceeding back down to approach the bottom. I know you won't see it that way, which is fine. But to be clear, I still believe things are very ugly at the bottom filled with pumpkins. Nevertheless I do understand how mathematicians think things work at the bottom and if starting at the bottom is the only way you'll allow me to get to the top then I'll go with it. I understand your criticisms of starting at the top, I just don't accept them. Once you allow me to get to (3) I endeavor to show you that (3) and (4) alone fully satisfy our needs and if I'm careful (e.g. by only allowing for rational endpoints) that (1) and (2) are not only superfluous but undesirable. Is that a fair approach? — keystone
But now only the endpoints are rational, leaving me baffled as to what those objects are.
— fishfry
Yes, the endpoints are rational, — keystone
and the object between any pair of endpoints is simply a line. — keystone
It doesn't go deeper than that. I understand you see that line as a composite object consisting of 2^aleph_0 points. — keystone
However, I view the line as a primitive object. — keystone
Clearly, our starting points differ. To move the discussion forward, could we agree to a compromise where we refer to a line as a "composite" object? This way, by including composite it's evident that I recognize your perspective, but the quotes indicate that my viewpoint doesn't necessitate this classification. — keystone
A forum member once pointed me to the ideas of Charles Sanders Peirce (correct spelling) who said that the mathematical idea of a continuum could not be right, since a true continuum could not be broken up into individual points as the real numbers can.
— fishfry
I agree with this point. The issue has been the lack of viable alternatives. I see that Peirce was suggesting the use of infinitesimals, and you're aware of my stance on those—the one from the comment where you thought I was just trolling. — keystone
Yes!!! I agree with Euclid's definition of lines and points. I appreciate that he provides foundational definitions of both as separate, fundamental entities. Thanks for pointing this out.unless you mean the original line of Euclid, "A line is breadthless length." — fishfry
Euclid also said that "The ends of a line are points." When I describe a path as 0 U (0,1) U 1:What is a line? What does the notation [0, 0.5] mean? — fishfry
I believe that someone even as intelligent and knowledgeable as yourself is not qualified to discuss the bottom-up philosophy of a continuum because it is flawed. I'm 100% certain you have the capacity to understand, discuss, and criticize the top-down philosophy.I'm not qualified to discuss the philosophy of the continuum with you, except as it relates to the standard mathematical real numbers. — fishfry
You're right, I did say that the endpoints were necessarily rational numbers. (-inf, +inf) has no endpoints. While there are scenarios where it is useful to include points at infinity, for this discussion, let's agree that the points at -inf and +inf are not real points. I'm only using infinity as a shorthand. I should have been clearer.That directly contradicts what you said earlier. — fishfry
unless you mean the original line of Euclid, "A line is breadthless length."
— fishfry
Yes!!! I agree with Euclid's definition of lines and points. I appreciate that he provides foundational definitions of both as separate, fundamental entities. Thanks for pointing this out. — keystone
What is a line? What does the notation [0, 0.5] mean?
— fishfry
Euclid also said that "The ends of a line are points." When I describe a path as 0 U (0,1) U 1:
(0,1) corresponds to the object of breadthless length and
0 and 1 correspond to the points at the end. — keystone
It seems that some people intepret Euclid as saying that a line without endpoints extends to infinity. I do not think this is necessarily the case. While (-inf,+inf) is a valid line, I believe (0,1) is also a valid line in and of itself. — keystone
Please give the following figure a chance as it captures a lot of what I'm trying to say: — keystone
I believe that someone even as intelligent and knowledgeable as yourself is not qualified to discuss the bottom-up philosophy of a continuum because it is flawed. — keystone
I'm 100% certain you have the capacity to understand, discuss, and criticize the top-down philosophy. — keystone
You're right, I did say that the endpoints were necessarily rational numbers. (-inf, +inf) has no endpoints. While there are scenarios where it is useful to include points at infinity, for this discussion, let's agree that the points at -inf and +inf are not real points. I'm only using infinity as a shorthand. I should have been clearer. — keystone
Didn't I ask you about this several posts ago? Ok, Euclid's line. — fishfry
And by the way, what is this "+" symbol? Have you defined it? Is this the standard + of the rational numbers? — fishfry
what does the notation (0,1) mean? — fishfry
Since your intervals are entirely made up of rationals, the total length must be 0. Where is the extra length coming from? — fishfry
I'm lost and dispirited. It's not my role in life to feel bad about myself for endlessly sniping at your heartfelt ideas. — fishfry
You correctly note that the sum of the lengths of the points is 0. But then you say that the sum of the lengths is 1, and I'm not sure how that follows. — fishfry
But what does it all mean? I'm lost and dispirited. — fishfry
Didn't I ask you about this several posts ago? Ok, Euclid's line.
— fishfry
Sorry, I didn't appreciate the point when you first mentioned it. Yes, I'm starting from classical Euclidean geometry. — keystone
And by the way, what is this "+" symbol? Have you defined it? Is this the standard + of the rational numbers?
— fishfry
Yes. Formally the arithmetic is performed as described here (https://www.sciencedirect.com/science/article/pii/S1570866706000311) — keystone
but informally it's performed using the standard method we teach kids. The formal and informal results are equivalent. — keystone
what does the notation (0,1) mean?
— fishfry
uu
It describes the line's potential. — keystone
I'm going to provide a shorthand answer involving real numbers that I don't want you to take literally. If this explanation lands, great, otherwise forget it.
No points exist on lines, including the unit line (0,1). To put it another way, there are no 'actual points' present on that segment. (Actual vs. potential is discussed below).
Cutting line (0,1) in two will introduce an 'actual point' between the two resulting line segments. That point will have a rational coordinate between 0 and 1. — keystone
In my last post, I noted that -inf and +inf are not 'actual points' but rather are used as helpful shorthand. I should have called them 'potential points'.
With a similar shorthand, we can say that on line (0,1) exist 2ℵ0
2
ℵ
0
'potential points', which have real number coordinates between 0 and 1. — keystone
The rational 'potential points' can become 'actual points' through cuts.
The irrational 'potential points' are permanently confined to their 'potential point' status.
I want to reiterate that 'potential points' don't actually exist. They're just a fiction that may help us comprehend the potential in continua. If you don't think potential points are a useful concept we can just drop.
The interval "(0,1)" describes the potential of the corresponding unit line. — keystone
Since your intervals are entirely made up of rationals, the total length must be 0. Where is the extra length coming from?
— fishfry
The length of a line comes from its potential. — keystone
Sometimes it’s a bit frustrating when my explanations don’t connect, but this conversation is exactly what I need right now, so please don’t feel bad. I'm very appreciative that you've stuck around. — keystone
Path Length = Length of Lines + Length of Points
Path Length = Length of Lines + 0
Path Length = Length of Lines
So referring to row 3 of that figure...
Path Length = Length of Lines
1 = 1/2 + 1/4 + 1/4 — keystone
I'll stipulate to your non-rigorous conception of a continuum of being made of tiny little continua "all the way down," with no need for actual points, if that's your idea. I think this is what Peirce is getting at. — fishfry
The line contains a frothing sea of tiny little micro-continua that are not points. Is that about right? — fishfry
Well here you are in trouble. If you allow "cuts" then à la Dedekind we have the real numbers. But you don't want to go there so ok. There are cuts but not so many as to allow the reals. — fishfry
Of course all mathematical entities are fictional, so I can't see what the difference is between and actual and a fictional point. — fishfry
You are saying the exact same thing, but changing the name of irrationals to "fictionals." I don't see how that changes anything. You just changed their name but they're the same irrationals. — fishfry
How do you propose to pass from a finite line to a circle, say? If you are considering topological transformations, how can you express them? Sorry for butting in, but I remain curious. — jgill
I also think that's what Peirce was getting but that's definitely not what I'm getting at. Remember when I "trolled" you by introducing a scenario involving infinitesmals? I believe that approach aligns with Peirce's thinking and I believe it's wrong. — keystone
You keep trying to concieve of my line as something built from smaller more fundamental elements (before points, now infinitesimals). It is not built from anything. (0,1) is one object - a line. The smaller elements emerge from the line, not the other way around. — keystone
I'm not allowing a single real number. We can partition the S-B tree at a rational node (e.g. 1/2), but we cannot partition it at a real node (because real nodes don't exist). — keystone
Just as you don't grant infinity actual status as a natural number, — keystone
I don't grant irrational points actual status as points. After all, infinity and irrational points are inseparably linked in the S-B tree, since irrational points become actual points at row infinity. If there is no actual row infinity, there are no actual irrational points. — keystone
The difference is that you believe individual irrationals can be isolated, whereas I think we can only access irrationals as continuous bundles of 2ℵ0 — keystone
fictional points. A mathematical 'quanta' if you will. In a 1D context, I refer to this continuous bundle as a line. And if we cut a line, we have two lines (i.e. two bundles of 2ℵ0 — keystone
fictional points). No matter how many times we cut it, we will never reduce a bundle down into individual points. — keystone
Since we can only ever interact with these bundles, it is meaningless to discuss individual irrationals - they are fictions. The bundles are not. Do you see the distinction? — keystone
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