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.

The length of that union is zero, if the intervals are restricted to rationals. Do you agree with that point? — fishfry

If an interval corresponds to a set of points (and nothing else) then I agree that an interval containing only rationals has no length.

No points. So they're all the empty set? I'm not supposed to push back on this? — 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?

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.

I very much believe in the continuum, which is pretty well modeled by the standard real numbers. — 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 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

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)...

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).

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

Likewise.

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

But you are the one saying that you only have rationals.

Our problem is that you are only allowing points in your sets. — keystone

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.

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

I don't know anything about set theory with urlements.

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

Only adds to my annoyance level. But that's a low bar so no worries.

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

Too deep for me. I don't even know what that means.

My preference is that you accept non-points into sets, — keystone

So set theory with urelements? I don't know much about that subject past the definition.

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

You only believe in rationals. Where are you getting these things?

(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

If you have a line and you have the rationals, you will get the real numbers by Cauchy-completing the line.

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

Your idea is not coherent. If you start with a line (a thing you have declined to define) and say it's populated by the standard rational numbers by cuts, then you can construct the standard real numbers.

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

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?

What are these lines of yours, anyway?

...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

You can't get anywhere as far as I can see.

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

You haven't given a coherent definition of these objects. All along you've been saying they are intervals of rationals. That's at least coherent, even if such intervals lack all properties of being continua.

But now only the endpoints are rational, leaving me baffled as to what those objects are.

ps -- 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.

Perhaps you are getting at some idea like that. Here's one link, you can Google around to find others if this interests you.

https://en.wikipedia.org/wiki/Charles_Sanders_Peirce

His ideas on continuity:

https://plato.stanford.edu/entries/peirce/#syn

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).

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

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.

I don't know anything about set theory with urlements. — fishfry

Urelements are indivisible 'atoms'. The lines that I'm working with are divisible.

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). 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 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.

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, 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:

1) Start at the bottom
2) Build up to the top
3) 'Start' at the top
4) Approach the 'bottom' from the top

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?

But now only the endpoints are rational, leaving me baffled as to what those objects are. — 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.

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.

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

I'm afraid I don't know what a line is, absent the real numbers, unless you mean the original line of Euclid, "A line is breadthless length." I'm not a scholar of Euclid so I really can't say.

I mentioned Peirce to you because it seems to me that you are interested in the "top down" definition of a continuum. I'm deeply unqualified to discuss the matter. I can only give you the standard mathematical interpretation, which is unsatisfying to both of us. I don't know enough about the philosophy of the continuum to comment.

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

Sigh. Your pictures don't help. What is a line? What does the notation [0, 0.5] mean?

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 said that your sets contains things other than sets. You just keep making up your own terminology. I don't think we are making any progress, and I no longer know what we are discussing.

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

That directly contradicts what you said earlier. And I don't know what your notation means.

I see this line as a single object (a line). — keystone

What is a line?

It is not populated by rational points. It is not populated by any points for that matter. — keystone

It's empty? We're going in circles.

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

Ok.

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

Ok. We're going in circles. I have no idea what you're talking about.

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

I'm not qualified to discuss the philosophy of the continuum with you, except as it relates to the standard mathematical real numbers.

Currently, I don't believe I can persuade you that a continuum can exist without points. — keystone

I'm perfectly willing to believe it, I just don't know anything about it.

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

Was this supposed to be helpful?

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

Jeez man ...

But now only the endpoints are rational, leaving me baffled as to what those objects are.
— fishfry
Yes, the endpoints are rational, — keystone

Two seconds ago you denied this.

and the object between any pair of endpoints is simply a line. — keystone

What is 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. — keystone

I did not say that, and there are other characteristics that a line must have. I am perfectly willing to adopt your ontology, if only you will state it clearly.

What is a line?

However, I view the line as a primitive object. — keystone

Ok. Euclid again?

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

You could move this forward by telling me what a line is. But I don't think I'm helping anything by sniping at your ideas in frustration.

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

Just giving a reference to what you seem to be getting at. A continuum that can't be divided into points.

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.

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.

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.

---------------------

Please give the following figure a chance as it captures a lot of what I'm trying to say:



1) In the first row, we have a line with two endpoints, totaling three objects.
2) I can represent this path as a graph composed of three connected vertices. Notice that the lines and points are all represented as vertices in the graph.
3) I want to put the information from all vertices into a set. That's 3 objects in one set. Not just the end points.
4) When the runner travels from 0 to 1, they don't run a path composed of infinite points. They walk the graph, which in this case is the journey from vertex 1 to vertex 2 to vertex 3.
5) If you cut the line, you'll end up with five objects: the two endpoints, a middle point, and the two line segments in between. This is what we have in the second row.
6) The length of the points is zero. In fact, no matter how many times we cut the line, the total length of the points will always be zero.
7) The total length of the line segments is one. In fact, no matter how many times we cut the line, the total length of the segments will always be one.
8) Notice that in the second row, the interval (0,1) is not present because it has been cut.
9) We can continue cutting the line indefinitely, and one particular sequence of cuts is depicted across the subsequent rows.
10) Notice the pattern in the columns labeled 'length of lines' and 'path length'. As we progress downward, we approach the familiar geometric series.
11) Unlike the bottom-up approach, which requires a limit to make the summation total one, the top-down approach results in a summation that totals one at every stage.

I'm not qualified to discuss the philosophy of the continuum with you, except as it relates to the standard mathematical real numbers. — 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.

That directly contradicts what you said earlier. — 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.

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

Didn't I ask you about this several posts ago? Ok, Euclid's line.

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

Ok so you are doing classical Euclidean geometry (not modern Euclidean geometry, please note).

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

Euclid would not recognize that notation; and at this point in our conversation, neither do I. You have variously stated that (0,1) contains only rationals, or that it may even be empty.

In view of my new understanding that by line, you mean Euclid's line, what does the notation (0,1) mean? Euclid did not have numbers as we know them.

Please give the following figure a chance as it captures a lot of what I'm trying to say: — keystone

Utterly baffled. Utterly. Baffled. No idea what it means. 0, 0 + 0, 0 + 0 + 0, no idea what I am supposed to glean from that. And by the way, what is this "+" symbol? Have you defined it? Is this the standard + of the rational numbers?

I feel terrible ignoring these diagrams that you put so much work into, and that hold so much meaning for you. I wish I could be more helpful. I don't mean to just continue to snipe at you. It pains me. I just don't know what you are saying and have no idea how to respond.

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 never claimed to be able to discuss the philosophy of the continuum. On the contrary, I've admitted that I can't. Except, that I know a bit about the real numbers, and they are the standard mathematical model of the continuum. And that counts for something.

I'm 100% certain you have the capacity to understand, discuss, and criticize the top-down philosophy. — keystone

Possibly, but not the inclination. If I could dispatch a clone, I'd have him read Peirce. I'm not a philosopher of the continuum. I'm not a philosopher at all. I only come to this forum to clarify people's mathematical misunderstandings. And it's a full time job :-)

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

Ok. So far, your line is Euclid's original line. Leaving undefined, your notation (0,1), which you have repeatedly pointed out is NOT the open unit interval of real numbers.

ps -- Ok I took another look at your picture. 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.

Since your intervals are entirely made up of rationals, the total length must be 0.

Where is the extra length coming from?

I'm willing to let you say that the length of the interval (0,1) is 1 even though it's only made of rationals. I'll stipulate that for sake of discussion, even though it's hard to understand how it works.

But what does it all mean? I'm lost and dispirited. It's not my role in life to feel bad about myself for endlessly sniping at your heartfelt ideas.

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.

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) but informally it's performed using the standard method we teach kids. The formal and informal results are equivalent.

what does the notation (0,1) mean? — fishfry

It describes the line's potential. 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.
• 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 $2^{\aleph_0}$ 'potential points', which have real number coordinates between 0 and 1.
• 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.

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.

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

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.

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

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

[post removed - I prematurely moved to 2D. I've since refined my view of 2D so and since this post didn't generate any response I decided to remove it.]

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

Ok. I wanted to say that 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.

In any event to save us some time, I'll stipulate to your vision, even if it's a bit contradictory and not totally clear.

So then what?

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

LOL You are committed to that idea, I'll give you that.

but informally it's performed using the standard method we teach kids. The formal and informal results are equivalent. — keystone

I'll stipulate to arithmetic on the rationals, I think we can agree on that.

what does the notation (0,1) mean?
— fishfry
uu
It describes the line's potential. — keystone

Ok. With my earlier stipulation, I could say:

(a) What? That makes no sense; or

(b) I sort of get it. The line contains a frothing sea of tiny little micro-continua that are not points. Is that about right?

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

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.

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

Oh my. You are now allowing the reals? Ok. Maybe that's good.

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

I sort of get your thinking. Not sure where you're going but I'll stipulate to all this, even with the vagueness.

Of course all mathematical entities are fictional, so I can't see what the difference is between and actual and a fictional point. Once you stipulate to fictional points, they become actual by virtue of being used and accepted. Just as negative numbers and imaginary numbers once did.

The life cycle of a mathematical idea is is:

Regarded as impossible ---> Fictional but useful ---> Normal everyday stuff.

Same path taken by non-Euclidean geometry. Impossible, then Riemann's curiosity, then Einstein's platform for general relativity

So once you admit a "fictional" entity you might as well grant it actual status, since you will eventually.

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

But here's the thing. I said earlier that in the standard unit interval, the length is stored in the irrationals.

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.

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

Ok. I'm adopting a less rigorous and more intuitive sense of what you are saying.

But you have also met me more than halfway. You have agreed finally that there ARE irrationals on the line, and that they carry, or store, the length. You just call them fictionals instead of reals. But they are the same thing.

[==== next post ====]

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 that the length is stored in the fictionals, which I'll continue to think of as the reals till you claim otherwise.

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

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.

The line contains a frothing sea of tiny little micro-continua that are not points. Is that about right? — fishfry

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.

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

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).

Of course all mathematical entities are fictional, so I can't see what the difference is between and actual and a fictional point. — fishfry

Just as you don't grant infinity actual status as a natural number, 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.

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

The difference is that you believe individual irrationals can be isolated, whereas I think we can only access irrationals as continuous bundles of $2^{\aleph_0}$ 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 $2^{\aleph_0}$ fictional points). No matter how many times we cut it, we will never reduce a bundle down into individual points. 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?

It is not built from anything. (0,1) is one object - a line — keystone

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.

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

Welcome back. I've transitioned from topology to graph theory, which (in this context) maintains similar concepts but is much simpler. To convert a path graph into a cycle graph, I would use vertex identification. Not sure what you're getting at. And really this is beyond the scope of what I'm covering. Right now, I'm just focused on reinterpreting the Cartesian coordinate system.

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

It's the only way I can make sense of what you're saying.

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

[0,1] is standard mathematical notation for a particular set of real numbers. You can't fault me for bringing my preconceptions about that notation. You should use less suggestive notation.

How about "L". If you say, "I have a line, I call it L," then I can't come back and challenge you about that notation.

You want it both ways. You think you can traverse the line from 0 to 0.5 to 1, freely borrowing our high school intuitions about the real number line. And when I bite on that bait, you say, "Oh no it's not the real line!"

You call your line [0,1], you treat it as if it's the usual unit interval, and then you object when I believe you!

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

Ok back to no real numbers.

Just as you don't grant infinity actual status as a natural number, — keystone

It's not a natural number. It's not 0 and it's not the successor of any number. I can PROVE "infinity", whatever you mean by that, is not a natural number.

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

Last post you started believing in the real numbers, but you called them fictitious. Now you're backing off even that.

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

Whatever that means.

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

Ok. Can't agree, can't disagree.

fictional points). No matter how many times we cut it, we will never reduce a bundle down into individual points. — keystone

That's true of the real numbers as well. You know, I think you are just coming to understand the nature of the standard mathematical real numbers.

If you start with the real line and cut it any number of times, you can never isolate a point that way. Do you agree?

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

No, because the real numbers have the same property. I cut [0,1] in half, I get [0, 1/2) and [1/2, 1]. I'm arbitrarily placing 1/2 in the second segment.

If I cut [0, 1/2) in half, I get [0, 1/4) and [1/4, 1/2), and so forth.

No number of cuts will ever isolate a point.

So I think what is happening here is that you are coming to a better intuition of the standard real numbers. Because you can keep cutting the unit interval in half and you will never isolate a point.

Throughout our conversation, my perspective and how I express it have greatly developed, leading me to believe it's best to reformulate and clarify my position. I'll be on a short holiday for the next few days, and I'd also like to take the necessary time to gather my thoughts before responding. For now, let me make two points:

• The essence of my perspective (top-down) remains the same, although it requires some minor adjustments.
• Having to reformulate my view underscores the significant value I've derived from our conversation—thanks once more!

I'll reach out again in a few days. I look forward to continuing this discussion. Enjoy your weekend!

Throughout our conversation, my perspective and how I express it have greatly developed, leading me to believe it's best to reformulate and clarify my position. I'll be on a short holiday for the next few days, and I'd also like to take the necessary time to gather my thoughts before responding. For now, let me make two points:

The essence of my perspective (top-down) remains the same, although it requires some minor adjustments.
Having to reformulate my view underscores the significant value I've derived from our conversation—thanks once more!

I'll reach out again in a few days. I look forward to continuing this discussion. Enjoy your weekend! — keystone

No prob, I'll be here.

Just remember: Once you stipulate to the rationals, you get the reals for free. I thought at one point that those were your fictional numbers, but then you said they weren't.

Please allow me to refine and restate my position on reals.

Grandi's series has no sum but it should be 1/2.
Analogously, I believe a line is not made of points but it should be made of $2^{\aleph_0}$ points.
Analogously, I believe a line is not modeled by numbers but it should be modeled by the real numbers.

Just as Grandi's series only sums to 1/2 in a very particular light, my view amounts to the belief that there is great mathematical value in irrationals, but that they only make sense in a very particular light - when considered collectively as bundles, rather than individual, isolated points. This is the essence of the top-down view where we start with such a bundle of 2^aleph_0 points - a line in this case - and then we make cuts to selectively isolate segments of this line. I refer to any point nested within such a bundle, as opposed to being isolated, as a potential point.

Revisiting the analogy above, when I utilize an interval to describe a range, I am referring to the underlying and singular continuous line between the endpoints, which should correspond to the set of real numbered points contained within these endpoints.

Let's look at two examples:

0D cut example - a cut of line (0,2) at 1.5:
$(0, 2) = (0, 1.5) \cup [1.5, 1.5] \cup (1.5, 2)$

• Potential points in (0,1.5) = $2^{\aleph_0}$
• Actual points in [1.5,1.5] = 1
• Potential points in (1.5,2) = $2^{\aleph_0}$

Length of continua (a,b) = b-a

• Length of (0,1.5) = 1.5
• Length of [1.5,1.5] = 0
• Length of (1.5,2) = 0.5

1D cut example - a cut of line (0,2) around $\varphi$:
$(0,2) = (0, \varphi - \epsilon_1) \cup [\varphi - \epsilon_1, \varphi + \epsilon_2] \cup (\varphi + \epsilon_2, 2)$

Where

• $\epsilon_1$ and $\epsilon_2$ are arbitrarily small positive real numbers such that $\varphi - \epsilon_1$ and $\varphi + \epsilon_2$ are rational.
• $\varphi$ is the golden ratio number

Potential points in $(0, \varphi - \epsilon_1) = 2^{\aleph_0}$
Potential points in $[\varphi - \epsilon_1, \varphi + \epsilon_2] = 2^{\aleph_0}$
Potential points in $(\varphi + \epsilon_2, 2) = 2^{\aleph_0}$

Length of $(0, \varphi - \epsilon_1) = \varphi - \epsilon_1$
Length of $[\varphi - \epsilon_1, \varphi + \epsilon_2] = \epsilon_1 + \epsilon_2$
Length of $(\varphi + \epsilon_2, 2) = 2 - \varphi - \epsilon_2$

I believe performing an arbitrarily small 1D cut around $\varphi$ instead of a 0D cut is more true to the Cauchy definition of $\varphi$. With this approach, $\varphi$ is never isolated as an actual point.

What do you think?

Please allow me to refine and restate my position on reals.

Grandi's series has no sum but it should be 1/2. — keystone

Why on earth would you think that? It clearly has no sum, since the sequence of partial sums has no limit.

Analogously, I believe a line is not made of points but it should be made of 2ℵ0
2
ℵ
0
points. — keystone

It's not made of points but it's made of points? How am I supposed to understand that?

Analogously, I believe a line is not modeled by numbers but it should be modeled by the real numbers. — keystone

It's not but it is?

Just as Grandi's series only sums to 1/2 in a very particular light, — keystone

I can't imagine what that light is. The Wiki page is misleading on that point.

my view amounts to the belief that there is great mathematical value in irrationals, but that they only make sense in a very particular light - when considered collectively as bundles, rather than individual, isolated points. — keystone

Needs explanation.

This is the essence of the top-down view where we start with such a bundle of 2^aleph_0 points - a line in this case - and then we make cuts to selectively isolate segments of this line. I refer to any point nested within such a bundle, as opposed to being isolated, as a potential point. — keystone

Cuts as in Dedekind cuts? If you already have continuum-many points, why do you need cuts?

Revisiting the analogy above, when I utilize an interval to describe a range, I am referring to the underlying and singular continuous line between the endpoints, which should correspond to the set of real numbered points contained within these endpoints. — keystone

But that is exactly the standard view.

I believe performing an arbitrarily small 1D cut around φ — keystone

I don't know what an "arbitrarily small cut] means. It conflicts with your previous use of cut.

What do you think? — keystone

There are no arbitrarily small real numbers.

The golden ratio?

This isn't working for me.

Why on earth would you think that? It clearly has no sum, since the sequence of partial sums has no limit. — fishfry

I agree that it lacks a sum, but do you think that terms like Cesàro summation and Ramanujan summation are completely misnomers? Do you truly think that there's no meaningful way to assign a value of 1/2 to that divergent series? I'm taken aback by this, though perhaps debating Grandi's series is merely a distraction.

Needs explanation. — fishfry

I think there's a bit of confusion around what I mean by "bundle." Let me explain using an analogy. GULP. Consider a fitness membership that includes access to cardio equipment, swimming pools, sauna rooms, group classes, and more. When you join the club, you pay a single price for this all-inclusive membership bundle. This means one price covers numerous amenities. There isn’t a separate charge for the sauna or the swimming pools. However, there should ideally be underlying individual prices, right? Like, when setting the bundle price, the gym owner should have calculated costs for each component. But what should have been done doesn't necessarily reflect what is—a single price for the entire bundle.

Similarly, in my scenario, the bundle of interest (a line) is represented simply as (0,2). Just as there's no itemized pricing for each gym amenity, there's no infinite set detailing every coordinate on the line.

Cuts as in Dedekind cuts? If you already have continuum-many points, why do you need cuts? — fishfry

Dedekind cuts have perfect precision. I claim that the best we can do is plan to cut an arbitrarily narrow line surrounding an irrational number. Cuts are used to decompose the bundle. Initially, the bundle price for the membership is established, and it's only afterwards that we attempt to deconstruct it into an itemized price list. Itemizing a membership can become an endless endeavor, breaking the price down into increasingly smaller fragments—from the cost of each toilet to each square of toilet paper, and even down to the cost of each atom in that toilet paper. Attempting to detail a gym membership to such minute components is a fool's errand. The same goes for breaking a line into individual points.

I don't know what an "arbitrarily small cut] means. It conflicts with your previous use of cut. — fishfry

The process of making cuts involves two distinct phases: (1) planning the cut and (2) executing the cut.

(1) We can devise a perfect plan. During the planning phase, we don’t commit to specific values for epsilon; we only recognize that it can be arbitrarily small. This stage is the realm of mathematicians.

(2) Conversely, executing the cut requires selecting specific values for epsilon, which inevitably introduces some imprecision. Applied mathematicians handle the execution, often employing approximate values for irrationals like pi, such as 3.14. While this approach might seem dirty, it's also quick, and this has allowed applied mathematicians to significantly improve the world.

You’re correct that previously, I was focused on the execution, but I've realized that the planning phase is indeed more critical for this discussion.

I agree that it lacks a sum, but do you think that terms like Cesàro summation and Ramanujan summation are completely misnomers? — keystone

I think they are chainsaws, not to be trifled with by the untrained masses.

Do you truly think that there's no meaningful way to assign a value of 1/2 to that divergent series? — keystone

I do. But I assume it's possible that there's some clever Ramanujan insight to get some deep mathematical context that makes the equation come out. And it might even mean something. But as a factoid to be dropped into casual mathematical conversation, no. Not any more than the execrable 1 + 2 + 3 + ... = -1/12. That's also "False as stated, and true only in rarified mathematical contexts of no relevance to a general audience."

I'm taken aback by this, — keystone

That I think there's no sensible way to assign a value to the series, regardless of whether there's some deep mathematical context? Why?

though perhaps debating Grandi's series is merely a distraction. — keystone

Yes.

I think there's a bit of confusion around what I mean by "bundle." — keystone

There are fiber bundles in math. A hairbrush with bristles sticking out is a fiber bundle. Off topic but reminded me of the name.

Let me explain using an analogy. GULP. Consider a fitness membership that includes access to cardio equipment, swimming pools, sauna rooms, group classes, and more. When you join the club, you pay a single price for this all-inclusive membership bundle. This means one price covers numerous amenities. There isn’t a separate charge for the sauna or the swimming pools. However, there should ideally be underlying individual prices, right? Like, when setting the bundle price, the gym owner should have calculated costs for each component. But what should have been done doesn't necessarily reflect what is—a single price for the entire bundle. — keystone

Ok, it's an aggregate price where the components haven't necessarily been priced. So you have aggregate lengths, but no individual ones. Something like that?

Similarly, in my scenario, the bundle of interest (a line) is represented simply as (0,2). Just as there's no itemized pricing for each gym amenity, there's no infinite set detailing every coordinate on the line. — keystone

You know, you can recover the real line just from its open intervals. Is that what you mean? In fact the rational intervals are good enough, the cover all the reals anyway.

I honestly think that what you are doing is coming to understand, in your own way, the nature of the real numbers. Which in my opinion is one of the most worthy endeavors a person can do. Clarifies so much formerly bad thinking about infinite processes.

Dedekind cuts have perfect precision. I claim that the best we can do is plan to cut an arbitrarily narrow line surrounding an irrational number. — keystone

Well sure, every irrational can be identified with a descending sequence of open intervals. I can locate pi in the sequence (3, 4), (3.1, 3.2), (3.14, 3.15), (3.141, 3.142), ...

Does that idea resonate with you?

What I mean "identified with" is this. I don't mean that my descending sequence of nested intervals "traps" the number pi or closes down on it.

No. I mean that the sequence itself IS the number pi. If pi didn't already exist, we'd just define it to be this sequence; or more precisely, the equivalence class of all such sequences. That's how mathematicians think. Pi doesn't have to be any particular thing. If you can define it as something that behaves the way pi is supposed to behave, then you might as well just consider it to be pi.

Does this make sense to you?

Which is why I say that once you grant me the rationals, you get the reals for free. If you believe in the rationals there's no point in trying to make the irrationals second class citizens in number country. Once you have the rationals the reals are already right there on equal footing. They're first class numbers.

You just have an ... ahem ... irrational prejudice against irrational numbers.

Cuts are used to decompose the bundle. Initially, the bundle price for the membership is established, and it's only afterwards that we attempt to deconstruct it into an itemized price list. Itemizing a membership can become an endless endeavor, breaking the price down into increasingly smaller fragments—from the cost of each toilet to each square of toilet paper, and even down to the cost of each atom in that toilet paper. Attempting to detail a gym membership to such minute components is a fool's errand. The same goes for breaking a line into individual points. — keystone

You do appear, to my humble and untrained eye, to be recapitulating the notion of descending down to points on the line by means of a sequence of downward nested open intervals; which is exactly the method of defining a real number as an equivalence class of Cauchy sequences. That's an alternative construction to Dedekind cuts but it gives you the same set of real numbers, since there is only one set of real numbers up to isomorphism. That's a handy thing to know. No matter how you conceive them, there is essentially only one set of real numbers.

I don't know what an "arbitrarily small cut] means. It conflicts with your previous use of cut.
— fishfry
The process of making cuts involves two distinct phases: (1) planning the cut and (2) executing the cut. — keystone

This bit about planning and execution is a little off the mark. I'm with you descending down to points via sequences of open intervals. In math when we conceive a thing it's automatically done. Would that the rest of the world were so simple!

(1) We can devise a perfect plan. During the planning phase, we don’t commit to specific values for epsilon; we only recognize that it can be arbitrarily small. This stage is the realm of mathematicians.

(2) Conversely, executing the cut requires selecting specific values for epsilon, which inevitably introduces some imprecision. Applied mathematicians handle the execution, often employing approximate values for irrationals like pi, such as 3.14. — keystone

I see this as making an infinite sequence of descending intervals as above.

pi = {(3, 4), (3.1, 3.2), (3.14, 3.15), (3.141, 3.142), ...}

Do you relate to this at all?

While this approach might seem dirty, it's also quick, and this has allowed applied mathematicians to significantly improve the world. — keystone

Does the world seem improved to you? I'm afraid that one of my bad habits is following the news, so I can't share your enthusiasm. And a lot of the trouble I see comes directly from the data mungers trying to eat us all. Hope you're glad you asked my opinion about that :-)

You’re correct that previously, I was focused on the execution, but I've realized that the planning phase is indeed more critical for this discussion. — keystone

In math you can do it all at once. You can decree that the number pi be located via a decreasing sequence of nested open sets converging on pi. And if you don't believe pi is really there, then no problem. You just define pi as the sequence of nested open intervals and you've got an object that, if it's not the "real" pi, is just as good. That's how they build the reals from the rationals.

I think they are chainsaws, not to be trifled with by the untrained masses. — fishfry

This sounds a lot like gatekeeping. Cesàro summations are very simple. Nevertheless, let's set aside Grandi's series for now. It doesn't have any relevance to my position...until further on.

There are fiber bundles in math. A hairbrush with bristles sticking out is a fiber bundle. Off topic but reminded me of the name. — fishfry

Wow, I feel like a generation alpha kid trying to come up with an email address - all the good names are already taken! Anyway, I'd actually rather call the bundles 'quanta', but to avoid QM-washing this discussion I'll keep calling them bundles.

Ok, it's an aggregate price where the components haven't necessarily been priced. So you have aggregate lengths, but no individual ones. Something like that? — fishfry

Further. bundles have interval descriptions but individual points (within a bundle) do not. In other words, the bundle is the fundamental unit. Sure, we can perform a cut actualize a 0D bundle, such as [0.5,0.5], but that point is emergent.

I honestly think that what you are doing is coming to understand, in your own way, the nature of the real numbers. — fishfry

I don’t believe that’s the case. It seems there are three factors involved here. (1) I'm refining my ideas—thank you for your assistance with this. (2) I'm improving how I communicate my ideas—again, thanks for your help. (3) You are starting to understand that my perspective doesn’t undermine any established mathematics; it mainly reinterprets it (making bundles fundamental).

Well sure, every irrational can be identified with a descending sequence of open intervals. I can locate pi in the sequence (3, 4), (3.1, 3.2), (3.14, 3.15), (3.141, 3.142), ... I mean that the sequence itself IS the number pi... Does that idea resonate with you? — fishfry

ABSOLUTELY (except for the underlined part). I did note this 2 days ago when I said that 1D cuts around φ are more true to the Cauchy definition of φ than 0D cuts. I also chose the golden ratio in that message because it has a beautiful description using the SB-algorithm. Look at the figure below. Instead of focusing on the yellow tree, consider the blue intervals with the red labels. The golden ratio is not any particular interval, but rather the descending sequence of open intervals captured by the sequence ([], R, RR, RRL, RRLR, RRLRL, ...) simply written as $R\overline{RL}$.

You just have an ... ahem ... irrational prejudice against irrational numbers. — fishfry

I've never denied the significance of irrationals. My view is simply that because irrationals are always encompassed within bundles, or rather are the bundles themselves, they differ distinctly from isolated points/rational numbers.

I'm with you descending down to points via sequences of open intervals. — fishfry

Wait, I'm not proposing that an irrational is a descent down to a point. Rather, I'm proposing that irrationals are infinite descents involving arbitrarily smaller intervals. The interval never has a length of 0 whereby a single irrational point is isolated.

And if you don't believe pi is really there, then no problem. You just define pi as the sequence of nested open intervals and you've got an object that, if it's not the "real" pi, is just as good. — fishfry

Ah, okay, so you don't require a point at pi. Awesome. It seems like we're making progress. Once we're completely aligned, I'd like to explore what I believe are the unseen and surprising consequences of this perspective with you.

This bit about planning and execution is a little off the mark. In math when we conceive a thing it's automatically done. Would that the rest of the world were so simple! — fishfry

Distinguishing between planning and execution is paramount. The inability to differentiate between them is precisely why there are so many infinity cranks. Cranks reject the concept of completing a supertask. On the other hand, mathematicians refuse to reject supertasks (or ideas implicitly associated with them) because they carry profound aesthetic and practical value. I find myself in the middle ground. What I suggest is that mathematicians would find complete satisfaction in merely planning the supertask, without concerning themselves with the imperfections of its incomplete execution.

Does the world seem improved to you? — fishfry

There’s a lot wrong with the world today, but would you really want to live an Amish or Mennonite lifestyle? Personally, I appreciate living in the most interesting of times, despite the uncertainty of our future.

I think they are chainsaws, not to be trifled with by the untrained masses.
— fishfry

This sounds a lot like gatekeeping. — keystone

I won't argue this point with you. I stand by it.

Cesàro summations are very simple. Nevertheless, let's set aside Grandi's series for now. It doesn't have any relevance to my position...until further on. — keystone

I can't see that you have a position. I see you as struggling to understand the nature of the mathematical continuum.

There are fiber bundles in math. A hairbrush with bristles sticking out is a fiber bundle. Off topic but reminded me of the name.
— fishfry

Wow, I feel like a generation alpha kid trying to come up with an email address - all the good names are already taken! — keystone

Funny.

Anyway, I'd actually rather call the bundles 'quanta', but to avoid QM-washing this discussion I'll keep calling them bundles. — keystone

Bundles is is. Should I think of them as tiny little wriggly micro-continua?

Further. bundles have interval descriptions but individual points (within a bundle) do not. In other words, the bundle is the fundamental unit. Sure, we can perform a cut actualize a 0D bundle, such as [0.5,0.5], but that point is emergent. — keystone

Ok, You have all the intervals, but no individual points. Is that right I can live with that. But I already showed that if you have all the intervals, you can derive the points as downward nested sequences of intervals.

I don’t believe that’s the case. It seems there are three factors involved here. (1) I'm refining my ideas—thank you for your assistance with this. (2) I'm improving how I communicate my ideas—again, thanks for your help. (3) You are starting to understand that my perspective doesn’t undermine any established mathematics; it mainly reinterprets it (making bundles fundamental). — keystone

I am not the lord high defender of the mathematical realm. I don't care if you overturn mathematics. I'm not defending mathematics. I'm trying to clarify ideas about mathematics, and trying to frame your ideas in the context of what's already known about mathematics.

Well sure, every irrational can be identified with a descending sequence of open intervals. I can locate pi in the sequence (3, 4), (3.1, 3.2), (3.14, 3.15), (3.141, 3.142), ... I mean that the sequence itself IS the number pi... Does that idea resonate with you?
— fishfry

ABSOLUTELY (except for the underlined part). — keystone

Nothing showed up underlined so I don't know what you are referring to. But if you agree that a descending stack of intervals can be taken as the definition of a point, that's a major agreement between us.

I did note this 2 days ago when I said that 1D cuts around φ are more true to the Cauchy definition of φ than 0D cuts. I also chose the golden ratio in that message because it has a beautiful description using the SB-algorithm. — keystone

It's irrational. I've joined your church. I no longer believe in irraionals.

What, now you believe in irrationals? You know the S-B tree is not the only kind of tree structure that represents the real numbers. I don't know why you are fixated on it.

Look at the figure below. — keystone

I don't relate. Sorry.

I've never denied the significance of irrationals. My view is simply that because irrationals are always encompassed within bundles, or rather are the bundles themselves, they differ distinctly from isolated points/rational numbers. — keystone

Well, irrationals are downward nested stacks of intervals. That's the next best thing. Can we agree on that?

But ... so are the rationals! Right?

Wait, I'm not proposing that an irrational is a descent down to a point. Rather, I'm proposing that irrationals are infinite descents involving arbitrarily smaller intervals. — keystone

That's exactly what I said. There is no point. All there is is the downward nested stack. We identify the stack with the point that it represents, even if the point isn't "there."

The interval never has a length of 0 whereby a single irrational point is isolated. — keystone

No real number is isolated. Any interval that contains it necessarily also contains infinitely many other real numbers on either side of it. But if you mean that a point has length 0, and an interval has a positive length, the unsigned difference of its endpoints, we agree.

Ah, okay, so you don't require a point at pi. Awesome. It seems like we're making progress. — keystone

Yes. I don't require a point at pi. I simply define pi to be the nested stack "above" where it should be. Or more accurately, the equivalence class of all the nested stacks above where it should be.

Once we're completely aligned, I'd like to explore what I believe are the unseen and surprising consequences of this perspective with you. — keystone

Ok! Very much making progress. But do note that what I'm describing is actually the standard mathematical way of thinking about the real numbers. The method of nested stacks is a small variation on the method of Cauchy sequences.

Distinguishing between planning and execution is paramount. The inability to differentiate between them is precisely why there are so many infinity cranks. Cranks reject the concept of completing a supertask. — keystone

Forget supertasks please. There's been enough silly talk about them.

On the other hand, mathematicians refuse to reject supertasks (or ideas implicitly associated with them) because they carry profound aesthetic and practical value. — keystone

Mathematicians in general have no interest in supertasks. They're mainly a curiosity for the computer scientists as I understand it.

I find myself in the middle ground. What I suggest is that mathematicians would find complete satisfaction in merely planning the supertask, without concerning themselves with the imperfections of its incomplete execution. — keystone

I don't know why you think that supertasks are an interest of mathematicians.

There’s a lot wrong with the world today, but would you really want to live an Amish or Mennonite lifestyle? — keystone

It's a thought. I don't believe I'd take well to getting up at 5am to milk the bull.

Personally, I appreciate living in the most interesting of times, despite the uncertainty of our future. — keystone

I like the modern world, but I don't think that applied mathematicians are universally engaged in creating good.

Bundles is is. Should I think of them as tiny little wriggly micro-continua? ... Ok, You have all the intervals, but no individual points. — fishfry

You persist in searching for an infinite set made up of tiny fundamental building blocks to assemble, like a mosaic. This bottom-up approach colors your perspective, but it's not feasible to represent the interval (0,1) as a union of such micro-continua—it simply doesn't work. I am seeking a construction that, at least theoretically, could be explicitly written down.

In contrast, a top-down approach begins with a singleton set that includes a large fundamental bundle to trim, like a sculpture. Each cut can split it into finitely more, smaller bundles. Although we can continue making cuts indefinitely, there is no necessity to complete a supertask and produce an infinite set of tiny micro-continua. Our strategy only needs to involve a finite number of cuts to produce the necessary elements for the computation at hand. Why do you believe it's necessary to have all the intervals?

I'm trying to clarify ideas about mathematics, and trying to frame your ideas in the context of what's already known about mathematics. — fishfry

And I greatly appreciate this.

Nothing showed up underlined so I don't know what you are referring to. But if you agree that a descending stack of intervals can be taken as the definition of a point, that's a major agreement between us. — fishfry

Oops, I meant to edit the quote as follows with the underlined part being the part I disagree with: "I mean that the sequence itself IS the number pi".

What, now you believe in irrationals? You know the S-B tree is not the only kind of tree structure that represents the real numbers. I don't know why you are fixated on it. — fishfry

In our discussion, I've always acknowledged the value and beauty of irrationals. However, I believe they don't share the same status as rational numbers. Rationals correspond to singleton intervals and represent specific points, whereas irrationals correspond to non-degenerate intervals and represent lines, albeit arbitrarily small ones. You're correct, the S-B tree isn't fundamental to my perspective.

Well, irrationals are downward nested stacks of intervals. That's the next best thing. Can we agree on that? — fishfry

Yes, but let me qualify my position as I think we will disagree on some details. We can execute cuts to isolate rationals within singleton intervals. We can plan to isolate computable irrationals within arbitrarily small intervals. However, even that plan alone is not feasible for non-computable irrationals. The best we could plan for is to isolate non-computable irrationals within a finite length interval. I hold this view because any plan we devise must, at least theoretically, be expressible in a finite number of characters. Besides, why would we even need to isolate non-computables? They're social creatures that like to live in large communities.

But ... so are the rationals [downward nested stacks of intervals]! Right? — fishfry

No, I view rationals as singleton intervals. I can devise a plan to target an irrational whose midpoint is arbitrarily close to a rational, but when I actually execute the cut, I must choose a positive epsilon value, and the resulting distance between the point and the resulting line segment's midpoint will necessarily be non-zero. It is for reasons like this why it is critical to distinguish between the plan and the execution of the plan. With the top-down view, there is an inherent approximation in the act of executing a plan - a principle analogous to the Uncertainty Principle in QM.

But if you mean that a point has length 0, and an interval has a positive length, the unsigned difference of its endpoints, we agree. — fishfry

Excellent. This is a crucial point that I will revisit as we continue our discussion.

Mathematicians in general have no interest in supertasks. They're mainly a curiosity for the computer scientists as I understand it. — fishfry

I acknowledge that most mathematicians are Platonists and therefore see no necessity for supertasks. However, constructivist (and people like me) needs supertasks to arrive at the objects that Platonists consider to exist. And my stance is that supertasks are impossible, so while maintaining my perspective, I can never fully align with yours. Instead, I am interested in understanding why you think my position falls short.

I don't believe I'd take well to getting up at 5am to milk the bull. — fishfry

You know you've made the wrong life choices if you're trying to milk a bull.

I like the modern world, but I don't think that applied mathematicians are universally engaged in creating good. — fishfry

I don't think anyone is universally engaged in anything. What's concerning is that within the next decade, it might only take one rogue individual to cause problems for everyone else.

You persist in searching for an infinite set made up of tiny fundamental building blocks to assemble, like a mosaic. This bottom-up approach colors your perspective, but it's not feasible to represent the interval (0,1) as a union of such micro-continua—it simply doesn't work. I am seeking a construction that, at least theoretically, could be explicitly written down. — keystone

The nested interval construction can be explicitly written down. I perhaps am not sharing your vision here.

In contrast, a top-down approach begins with a singleton set that includes a large fundamental bundle to trim, like a sculpture. Each cut can split it into finitely more, smaller bundles. Although we can continue making cuts indefinitely, there is no necessity to complete a supertask and produce an infinite set of tiny micro-continua. Our strategy only needs to involve a finite number of cuts to produce the necessary elements for the computation at hand. Why do you believe it's necessary to have all the intervals? — keystone

Finitely many cuts won't get you enough of the points. Your continuum will be full of holes. The set of real numbers approximable by finite sequences is only countably infinite.

Oops, I meant to edit the quote as follows with the underlined part being the part I disagree with: "I mean that the sequence itself IS the number pi". — keystone

The sequence is defined as pi. And thereafter, it might as well be taken for pi since, by suitably defining arithmetic on the set of sequence stacks, it will have all the required properties of pi.

In our discussion, I've always acknowledged the value and beauty of irrationals. However, I believe they don't share the same status as rational numbers. — keystone

The reals are logically constructed from the rationals. If you have the rationals you get the reals for free.

Rationals correspond to singleton intervals and represent specific points, whereas irrationals correspond to non-degenerate intervals and represent lines, albeit arbitrarily small ones. You're correct, the S-B tree isn't fundamental to my perspective. — keystone

Glad to hear that!

Yes, but let me qualify my position as I think we will disagree on some details. We can execute cuts to isolate rationals within singleton intervals. We can plan to isolate computable irrationals within arbitrarily small intervals. However, even that plan alone is not feasible for non-computable irrationals. — keystone

I don't see why. You haven't mentioned algorithms, and it's the existence of certain algorithms that distinguishes computable from noncomputable reals.

The best we could plan for is to isolate non-computable irrationals within a finite length interval. I hold this view because any plan we devise must, at least theoretically, be expressible in a finite number of characters. — keystone

Ok. But now you're saying that just because you can't express something, it doesn't exist. Maybe things exist that are beyond our ability to express.

Besides, why would we even need to isolate non-computables? They're social creatures that like to live in large communities. — keystone

Well, some of them can be isolated, if by that you mean defined. Most can't.

No, I view rationals as singleton intervals. — keystone

Yes. Agreed. But they can ALSO be taken to be nested stacks. And then there is no difference in status between the rationals and the irrationals.

I can devise a plan to target an irrational whose midpoint is arbitrarily close to a rational, but when I actually execute the cut, I must choose a positive epsilon value, and the resulting distance between the point and the resulting line segment's midpoint will necessarily be non-zero. It is for reasons like this why it is critical to distinguish between the plan and the execution of the plan. With the top-down view, there is an inherent approximation in the act of executing a plan - a principle analogous to the Uncertainty Principle in QM. — keystone

I don't follow all this but maybe it's ok.

But if you mean that a point has length 0, and an interval has a positive length, the unsigned difference of its endpoints, we agree.
— fishfry

Excellent. This is a crucial point that I will revisit as we continue our discussion. — keystone

Ok good.

I acknowledge that most mathematicians are Platonists and therefore see no necessity for supertasks. — keystone

No. Supertasks are simply not part of math. Not because anyone's a Platonist or a realist or a formalist or whatever, but because the subject never comes up. It's like you're learning to drive, and you ask, well, how do I peel the apple? The answer is, peeling apples is not part of learning to drive. And you say, "Oh, is that because most drivers are Platonists?"

Makes no sense. Apples have nothing do do with driving, supertasks have nothing to do with math.

However, constructivist (and people like me) needs supertasks to arrive at the objects that Platonists consider to exist. — keystone

If that's true, then you are saying that supertasks are a formalism or a concept that let you reproduce standard math, while pretending that you reject parts of standard math.

The nested interval construction can be explicitly written down. I perhaps am not sharing your vision here. — fishfry

I can write (0,1) as the union of arbitrarily many intervals. However, I cannot write (0,1) as the union of infinitely many intervals.

For example, consider describing (0,1) as the union of N equal-length non-degenerate intervals (plus a bunch of points).

Length of (0,1) = Length of each interval * number of intervals
Length of (0,1) = (1/N) * N

This equation is valid when N is any positive natural number, but it is not valid when N is infinity. Therefore, it is not sensible to define the interval (0,1) as the union of infinitely many intervals. And what I'm saying is that since we have to pick a finite number, why not just pick N=1?

Finitely many cuts won't get you enough of the points. Your continuum will be full of holes. The set of real numbers approximable by finite sequences is only countably infinite. — fishfry

I don't have to cut (0,1) at all to give it length. All $2^{\aleph_0}$ points that you are looking for are there from the start, albeit bundled together in one single object. Cuts don't create length, all they do is divide length.

The sequence is defined as pi. And thereafter, it might as well be taken for pi since, by suitably defining arithmetic on the set of sequence stacks, it will have all the required properties of pi. — fishfry

• We can sensibly devise a plan for arithmetic on rationals, AND we can completely execute arithmetic on rationals.
• We can sensibly devise a plan for arithmetic on irrationals, but we cannot completely execute arithmetic on irrationals.
• There is a distinction here that gets lost when you give rationals and irrationals the same status.

The reals are logically constructed from the rationals. If you have the rationals you get the reals for free. — fishfry

Nothing is for free, not even the rationals. When I start with path $(-\infty, +\infty)$ I have no numbers. Instead I have $2^{\aleph_0}$ points bundled together in a single object. Again, with the fitness membership bundle, there's not a price for every atom (or rather every point) in the gym. There's just a price for the bundle. We don't get the price per point for free. What would that price even be - seriously? $0/point? A bottom-up pricing model is absolute nonsense.

You haven't mentioned algorithms — fishfry

What is a plan if not a form of algorithm?

But now you're saying that just because you can't express something, it doesn't exist. — fishfry

I'm not suggesting that non-computable points don't exist within the (0,1) bundle; rather, I'm saying it's impossible to isolate such points. Humans have never isolated a non-computable point and never will—it's simply unfeasible to come up with a plan to do so. It seems almost as if you're adopting a stance based on faith. But why? What is the need to isolate non-computable points?

Well, some of them can be isolated, if by that you mean defined. Most can't. — fishfry

I'm not referring to a verbal description, but rather to isolating it using cuts. For instance, you might attempt to pinpoint Chaitin's number within the interval (0.007, 0.008) for a specific Turing machine. However, this interval has a finite length of 0.001. It is not feasible to devise a plan to successively refine these intervals to confine Chaitin's number within an arbitrarily narrow range.

Yes. Agreed. But they can ALSO be taken to be nested stacks. And then there is no difference in status between the rationals and the irrationals. — fishfry

A point can never be perfectly represented using a line, no matter how small that line is.

If that's true, then you are saying that supertasks are a formalism or a concept that let you reproduce standard math, while pretending that you reject parts of standard math. — fishfry

I see (0,1) as a bundle of $2^{\aleph_0}$ points. You see (0,1) as $2^{\aleph_0}$ isolated points. In order for me to get to where you are I need to make $2^{\aleph_0}$ cuts. I need to absolutely mince that line until it's made of individual objects that have no length. This requires a supertask. It's not possible. It's not sensical.

I can write (0,1) as the union of arbitrarily many intervals. However, I cannot write (0,1) as the union of infinitely many intervals. — keystone

(0,1) is the union of (1/n, 1 - 1/n) as n goes to infinity. I just wrote (0,1) as the union of infinitely many open intervals.

For example, consider describing (0,1) as the union of N equal-length non-degenerate intervals (plus a bunch of points).

Length of (0,1) = Length of each interval * number of intervals
Length of (0,1) = (1/N) * N

This equation is valid when N is any positive natural number, but it is not valid when N is infinity. — keystone

Infinity is not a natural number.

Therefore, it is not sensible to define the interval (0,1) as the union of infinitely many intervals. — keystone

I just showed you exactly how to do that.

And what I'm saying is that since we have to pick a finite number, why not just pick N=1? — keystone

Lost me entirely with this line of thought.

I don't have to cut (0,1) at all to give it length. All 2ℵ0
2
ℵ
0
points that you are looking for are there from the start, albeit bundled together in one single object. Cuts don't create length, all they do is divide length. — keystone

Ok ...

We can sensibly devise a plan for arithmetic on rationals, AND we can completely execute arithmetic on rationals.
We can sensibly devise a plan for arithmetic on irrationals, but we cannot completely execute arithmetic on irrationals.
There is a distinction here that gets lost when you give rationals and irrationals the same status. — keystone

I just don't follow your idea of assigning status to various classes of numbers. That can be done historically, but not mathematically. The rationals can be extended to the reals by a logical process. If you believe in the rationals you must believe in the reals, since the reals are what you get when you plug all the holes in the rationals.

Nothing is for free, not even the rationals. When I start with path (−∞,+∞)
(
−
∞
,
+
∞
)
I have no numbers. Instead I have 2ℵ0
2
ℵ
0
points bundled together in a single object. Again, with the fitness membership bundle, there's not a price for every atom (or rather every point) in the gym. There's just a price for the bundle. We don't get the price per point for free. What would that price even be - seriously? $0/point? A bottom-up pricing model is absolute nonsen — keystone

Lost me again.

What is a plan if not a form of algorithm? — keystone

Plans are far more general than algorithms. Not every plan is computable. Chaitin's Omega is "planned" if you like, but not computable. Chaitin's number illustrates the subtle distinction between definable and computable.

https://en.wikipedia.org/wiki/Chaitin%27s_constant

I'm not suggesting that non-computable points don't exist within the (0,1) bundle; rather, I'm saying it's impossible to isolate such points. — keystone

Chaitin's Omega is one such noncomputable number that can be specifically defined.

You're right that most of the noncomputables have no unique definition and can't be "isolated," but so what?

Humans have never isolated a non-computable point and never will — keystone

Chaitin again. You're wrong.

—it's simply unfeasible to come up with a plan to do so. It seems almost as if you're adopting a stance based on faith. But why? What is the need to isolate non-computable points? — keystone

Please read the Wiki page on Chaitin's Omega. It's a noncomputable number that is perfectly understandable and specific enough, depending only on the chosen encoding of Turing machines.

I'm not referring to a verbal description, but rather to isolating it using cuts. For instance, you might attempt to pinpoint Chaitin's number within the interval (0.007, 0.008) for a specific Turing machine. However, this interval has a finite length of 0.001. It is not feasible to devise a plan to successively refine these intervals to confine Chaitin's number within an arbitrarily narrow range. — keystone

Ok whatever. Not sure where this is going. All real numbers are defined by Dedekind cuts.

A point can never be perfectly represented using a line, no matter how small that line is. — keystone

Did I say the contrary? I don't recall doing that.

I see (0,1) as a bundle of 2ℵ0
2
ℵ
0
points. You see (0,1) as 2ℵ0
2
ℵ
0
isolated points. — keystone

No real numbers are isolated.

cuts. I need to absolutely mince that line until it's made of individual objects that have no length. This requires a supertask. It's not possible. It's not sensical. — keystone

Whatever. This is depressing me a bit. I no longer know what we're talking about.

(0,1) is the union of (1/n, 1 - 1/n) as n goes to infinity. I just wrote (0,1) as the union of infinitely many open intervals. — fishfry

Based on your description:

• n=1 (1,0)
• n=2 (1/2,1/2)
• n=3 (1/3,2/3)
• etc.

I don't follow. What exactly are you combining into a union?

-----------------------------

I've summarized key aspects of my argument in the following table, where I provide two analogous examples (a fitness gym vs. a path). Can you please tell me which cells in the table you disagree with or do not understand? This will help us identify the confusion and hopefully advance the conversation. You don't have to read the column for the fitness gym. I've only included it to ensure that our thinking is grounded in reality.

Plans are far more general than algorithms...Chaitin's Omega is one such noncomputable number that can be specifically defined. — fishfry

Okay, forget about 'plans'. The notion of algorithms much better captures what I'm talking about.

You're right that most of the noncomputables have no unique definition and can't be "isolated," but so what? — fishfry

My argument is that the top-down perspective lays equal claim to the irrational points. You can't claim that there are gaps in my intervals just because the non-computable irrational points in my view do not have corresponding numbers. Both of our views of a line involve the exact same $2^{\aleph_0}$ points. The difference is that my points are bundled together as a single package (thus not needing numbers) whereas your points are independent (thus needing numbers).

A point can never be perfectly represented using a line, no matter how small that line is. — keystone
Did I say the contrary? I don't recall doing that. — fishfry

You mentioned that a rational number, which is a point, can also be considered as nested stacks of intervals, essentially an arbitrarily small line. However, no matter how small that line is, it can never truly be a point.

No real numbers are isolated. — fishfry

Interesting. It sounds like you might agree that the best one can do is isolate a small bundle around the real number. What is the length of such a bundle - positive rational? Zero? Ever shrinking?

Whatever. This is depressing me a bit. I no longer know what we're talking about. — fishfry

We don't need to discuss supertasks. They're not relevant to either of our positions.

The inability for dimensionless points to be reconciled with the continuum is what motivated Whitehead's point-free geometry, a precursor to the field of Pointless Topology, as for instance formalised using Locales whose distributive law characterizes the meaning of a "spot". (It might be useful to test this law in relation to the SB tree, for both the truncated and infinite version).

Thanks for introducing math topics that I was only barely aware of into the discussion. Good to learn something.

(0,1) is the union of (1/n, 1 - 1/n) as n goes to infinity. I just wrote (0,1) as the union of infinitely many open intervals.
— fishfry

Based on your description:

n=1 (1,0)
n=2 (1/2,1/2)
n=3 (1/3,2/3)
etc.

I don't follow. What exactly are you combining into a union? — keystone

Yes you're right, I should say that n goes from 3 to infinity here, forget 1 and 2.

So I'm unioning (1/3, 2/3), (1/4, 3/4), (1/5, 4/5), ...

You can see that if x is in (0,1), then x is in a least one (actually all but finitely many) of the sets (1/n, 1 - 1/n).

I've summarized key aspects of my argument in the following table, where I provide two analogous examples (a fitness gym vs. a path). Can you please tell me which cells in the table you disagree with or do not understand? This will help us identify the confusion and hopefully advance the conversation. You don't have to read the column for the fitness gym. I've only included it to ensure that our thinking is grounded in reality. — keystone

I would have to give this some thought. Would it make progress if I stipulate to your metaphysics? I don't know what to say anymore.

My argument is that the top-down perspective lays equal claim to the irrational points. You can't claim that there are gaps in my intervals just because the non-computable irrational points in my view do not have corresponding numbers. Both of our views of a line involve the exact same 2ℵ0
2
ℵ
0
points. The difference is that my points are bundled together as a single package (thus not needing numbers) whereas your points are independent (thus needing numbers). — keystone

Yes ok, so if you have an alternate way of getting to the same real numbers, what does it matter? The real numbers are categorical. That means that any two models of the second-order real numbers are isomorphic. Second order basically means that you can express the least upper bound property.

You mentioned that a rational number, which is a point, can also be considered as nested stacks of intervals, essentially an arbitrarily small line. However, no matter how small that line is, it can never truly be a point. — keystone

Not a line, a nested collection of lines. The point zero is (-1, 1), (-1/2, 1/2), (-1/3, 1/3), etc.

No real numbers are isolated.
— fishfry

Interesting. It sounds like you might agree that the best one can do is isolate a small bundle around the real number. What is the length of such a bundle - positive rational? Zero? Ever shrinking? — keystone

Every interval containing a given real number, necessarily contains other real numbers. That's the definition of (not) being isolated.

We don't need to discuss supertasks. They're not relevant to either of our positions. — keystone

That was my only point.