I thought we were making progress on at least having the same conversation when we were traversing the unit interval. — fishfry

I think we're making progress. 2 steps forward, 1 step back.

You're taking that as fundamental? — fishfry

I'm sure there are other ways to define the ordering of rational numbers, that's just my favorite.

and then you declined to respond when I asked you twice how you get from (0, .5) to .5 without invoking a limiting process. Then you changed the subject. — fishfry

I thought I twice answered your question. Let me try again. What you don't seem to appreciate is that with the top-down view we begin with the journey already complete so halving the journey is no problem. If we already got to 1, then getting to 0.5 is no problem. You can't seem to get your mind out of the bottom-up view where we construct the journey from points, which indeed requires limits.

Even taken at face value, I fail to understand how posting stills from the movie relates to anything we're discussing. — fishfry

I take it you're not a fan of analogies.

And like I said, humans make lousy batteries. So the premise of the film is wrong. — fishfry

People once mocked movie scenes where detectives would enhance blurry security camera footage with a simple "refine" button, magically clarifying a suspect's face. Now, AI technology has turned that fiction into reality. Indeed, suspension of disbelief has its virtues.

Great flick though the plot gets a little muddle in the second half. Classic Cronenberg. — fishfry

Okay, I'll watch it.

• Use Metric Space, not Topological Metric Space - Got it.
• Burden on me to be clear - I agree, though I believe I haven't been unclear. Nevertheless, I'll keep my posts shorter to make sure I keep your attention.

If you deny the real numbers then I have no idea what 0 and .5 are, since they are real numbers. What do those symbols mean? — fishfry

0 and 0.5 have distinct positions on the Stern-Brocot tree. If we cut (0,0.5) we'll introduce a new point whose value will lie in the yellow zone. We don't need anything more than the structure of the Stern-Brocot tree to give points and intervals their expected meaning. From the bottom-up view I agree that 0 and 0.5 are real numbers, but from the top-down view 0 and 0.5 are rational numbers.

How do you get to .5 fom (0, .5)? Don't you have to take a limit? This is an important question. You seem to be implicitly willing to take limits, while denying the real numbers. I see that as a problem. — fishfry

Let me try again to explain the top-down view.
Consider the path: 0 U (0,1) U 1

This path is traversed in 3 (not infinite) steps. You start at 0, then you step to (0,1), then you step to 1. No limits needed, just 3 simples steps. You don't have to step through all the points within (0,1) because no points exist on (0,1). If you want a point between 0 and 1 then you have to cut (0,1) which will introduce a point having a value between 0 and 1.

Let me try another analogy. Hopefully you like football more than the Matrix.
Imagine a field with no hash lines, only yard lines every 5 yards.


Let's say the on the play the running back catches the ball between the 10 and 15 yard lines and advances it to the 20th yard line. As a commentator we want to be as precise as possible without giving any false information. He would say "He catches the ball between the 10 and 15 yard lines, now he's at the 15 year line, now he's between the 15 and 20 yard lines, oh and he gets tackled on the 20 yard line."

His continuous journey is described as (10,15) U 15 U (15,20) U 20.

Attempting to provide more precision than what the yard markings allow would be incorrect. Furthermore, commentating an exact play-by-play using real numbers is impossible, as real numbers do not have adjacent values and cannot be listed, which complicates precise location reporting in this context.

I'm just getting around to responding to a few of your earlier comments:

As it happens I hate that stupid movie. It's a kung-fu flick with silly pretensions to pseudo-intellectuality. Also someone did the calculation and it turns out that humans make lousy batteries. Very inefficient. Where is the line between your indulging yourself, and your trying to communicate a clear idea to me? — fishfry

Wow, it's one of my favourite films. To each their own, I suppose. It seems we view things quite differently in several respects. That's exactly why I find this conversation so valuable.

Like a triangular section of the plane? Why? — fishfry

JGill noted that using x and y for my upper/lower bounds was confusing. I think that's why you were confused with my earlier post. Hopefully using a and b is less misleading.

I'm very glad I can help. What is the digital rain? Do you remember the Church of the Cathode Ray from the movie Videodrome? — fishfry

I was suggesting that our discussion around topological metric spaces has warmed me up to the idea of sets being fundamental. I now believe that, if there is merit to a top-down view of mathematics, that is will be described using sets. I certainly didn't hold that view at the beginning of our conversation. I didn't watch Videodrome, it was a little before my time.

Does [stuff with pi] exist in your system? Or are you assuming rational numbers only? — jgill

Pi is just as important in the top-down view as it is in the bottom-up view. However, as with many other things, it just needs a little reinterpretation to fit into the top-down picture. As a number, pi, is inseparably tied to actual infinity, so it will need to be elevated to a higher status in the top-down view to break this connection. I hope the conversation continues long enough where we'll be ready to elaborate on this, but for now let's just say that the upper/lower bound of intervals must be (rational) numbers. No doubt, such a restriction has consequences but I hope we will eventually agree that these consequences are features, not flaws. Anyway, why do you ask about pi?

Is [stuff with epsilon] for small epsilon [allowed in my systems]? — jgill

The upper and lower bounds of intervals need to be (rational) numbers. It seems you're employing epsilon in its traditional role as an infinitesimal, which does not qualify as a (rational) number. Considering epsilon's role in calculus, let me just say that with some reinterpretation, calculus can be elegantly integrated into the top-down perspective without the need for infinitesimals. This is another topic I hoe we will explore more deeply once we've addressed some of the initial considerations. But again, why do you ask about epsilon?

? — jgill

You're right, I meant to say point 1. Thanks for the catch. I've now fixed that post.

Please stop calling it a topological metric space just as I don't call my cat my cat mammal. — fishfry

What should I call it?

But why denote the point 0 as [0,0]? Isn't that obfuscatory and confusing? — fishfry

I did that to facilitate the straightforward definition of the metric. If you permit me to work within a metric space without necessitating an explicit definition of the metric, then I will designate it as point 0.

If you mean what's mathematically called a path, I'm fine with that. — fishfry

I'm talking about a top-down analogue to (bottom-up) paths. By this I mean that (bottom-up) paths are defined using points (real numbers) whereas I'm defining the (top-down) 'path' using continua. I would like to use the term 'path' if you permit me to use it without implying the existence of R.

How do you get from (0,0.5) to [0.5,0.5]? …Mathematically, you take a limit. — fishfry

Let me describe both the bottom-up view and the top-down view.

Bottom-up view
The journey from point 0 to point 0.5 can be constructed as follows:

• Step 1: 1/4 = 0.25
• Step 2: 1/4 + 1/8 = 0.375
• Step 3: 1/4 + 1/8 + 1/16 = 0.4375
• Step 4: 1/4 + 1/8 + 1/16 + 1/32 = 0.4688
• Step 5: 1/4 + 1/8 + 1/16 + 1/32 + 1/64 = 0.4844
• Step 6: 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 = 0.4922

…

Along this journey there is no finite step where we arrive at precisely 0.5. This approach requires something like a 'step omega' and to get to 0.5 requires a limit to 'jump' the gap.

Top-down view
We begin with the completed journey from point 0 to point 0.5. Some versions of how the journey can be decomposed are as follows:

• Decomposition version 1: 1/2 = 0.5
• Decomposition version 2: 1/4 + 1/4 = 0.5
• Decomposition version 3: 1/4 + 1/8 + 1/8 = 0.5
• Decomposition version 4: 1/4 + 1/8 + 1/16 + 1/16 = 0.5
• Decomposition version 5: 1/4 + 1/8 + 1/16 + 1/32 + 1/32 = 0.5
• Decomposition version 6: 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/64 = 0.5

…

The various versions correspond to how we might chose to make cuts.
For example, the journey in decomposition version 2 is [0,0] U (0,1/4) U [1/4,1/4] U (1/4,1/2) U [1/2,1/2].
Regardless of how many cuts we make (i.e. regardless of what version we're looking at), the journey is always complete. No limits are required. Limits are only required to make the top-down view equivalent to the bottom-up view (i.e. decomposition version omega = step omega).

The confusion seems to stem from you viewing the interval (0, 0.5) as an infinite collection of points (naturally, since that is a bottom-up perspective of an interval). However, from a top-down perspective, the interval (0, 0.5) represents a single object - a continuum (perhaps I should return to calling it a k-interval to avoid confusion). While this continuum indeed has the potential to be subdivided infinitely (much like an object can potentially have holes), until actual cuts are made, we cannot assert the existence of actually infinite discrete points.

Going back to the set {0 , (0,0.5) , 0.5 , (0.5,1) , 1} , all that exists are 3 points and 2 continua and for a continuous journey we advance through them in this order proceeding from one step to another without taking limits:

Step 1: Start at point 0.
Step 2: Travel the continuum (0,0.5)
Step 3: Arrive at point 0.5.
Step 4: Travel the continuum (0.5,1)
Step 5: Arrive at point 1.

Perhaps use (a1,b1) instead to keep the level of confusion minimized. — jgill

Thanks. I've made this change to my earlier post and noted that it has been updated.

d([2,3],[1,4])=0 ? [2,3] not equal to [1,4] — jgill

In a previous post I defined the elements of the enclosing set to be sets which I called 'continuous sets'. Rewriting what I mean by continuous sets, those are sets whose elements are ordered pairs (a,b) and have the following characteristics:

1) Given any pair of elements (a,b) and (b,c), where a<b<c, there must also exist element (b,b) [I'm trying to say that between any two adjacent continua lies a point]
2) When sorted primarily by the a-coordinate and secondarily by the b-coordinate, the b-coordinate of one element matches the a-coordinate of the subsequent element. [I'm trying to say that there are no gaps and no overlapping intervals]


Returning to your example, (2,3) and (1,4) cannot both be elements of a continuous set so the set you are considering is not included in the enclosing set.

No, because all of your interval notations denote the empty set — fishfry

Okay, I've thought about this further and I think you're right! Do the following 5 intervals make more sense? None of them are empty anymore. For points, let's use closed intervals.

Interval 1: [0,0]
Interval 2: (0,0.5)
Interval 3: [0.5,0.5]
Interval 4: (0.5,1)
Interval 5: [1,1]

First off, I want to reiterate my appreciation for your patience and for sticking with me through this. I won't repeat this in every post, but please know that I'm am always thinking it.

[EDIT: IN A LATER MESSAGE I CONCEDE THAT POINTS SHOULD BE DESCRIBED USING CLOSED INTERVALS. INSTEAD OF HAVING YOU RESPOND TO AN INCORRECT MESSAGE, I'M GOING TO EDIT THIS MESSAGE.]

No, because all of your interval notations denote the empty set and I can't figure out what you are doing. — fishfry

I'm defining the journey from 0 to 1 using the following 5 intervals:

• Interval 1: [0,0]
• Interval 2: (0,0.5)
• Interval 3: [0.5,0.5]
• Interval 4: (0.5,1)
• Interval 5: [1,1]

1,3, and 5 correspond to points.
2 and 4 correspond to continua.

To me, it's obvious that the union of the above 5 intervals completely describes the journey from 0 to 1. Do you agree?

I'm using intervals to describe all 5 parts of the journey because I want to use intervals in my topological metric space. Let me go ahead and do this...

Set M is has following ordered pairs (not intervals) as elements:

• Ordered pair 1: (0,0)
• Ordered pair 2: (0,0.5)
• Ordered pair 3: (0.5,0.5)
• Ordered pair 4: (0.5,1)
• Ordered pair 5: (1,1)

[EDITED FOLLOWING ORDERED PAIRS VARIABLE LETTERS FROM (X,Y) TO (A,B) ACCORDING TO JGILL'S LATER FEEDBACK]

As I mentioned before, the metric between ordered pairs (a1,b1) and (a2,b2) is defined as follows:
d((a1,b1),(a2,b2)) = | (a1+b1)/2 - (a2+b2)/2 |

This metric essentially measures the distance between the midpoints of two intervals. Hopefully this clarifies why I chose to represent points as intervals.

Have you read this?
https://en.wikipedia.org/wiki/Metric_space — fishfry

Partially. To fully claim that one has read and understood the material would mean exploring all the hyperlinks and the nested links within them until everything is perfectly clear. I haven’t done that, but I believe the first few sections sufficiently address our immediate needs.

I can define a continuous "journey," whatever that means, using the identity function on the real numbers f(x) = x. — fishfry

I'll address the real numbers once we've clarified the topics above. It's not feasible for me to provide a satisfactory response if we're not in agreement on these preliminary matters.

You have used this expression frequently. Do you know what you are talking about? Just curious. — jgill

I'm brand new to topological metric spaces, so I might make some mistakes along the way. As I mentioned at the beginning, my idea is still in an informal stage. Nonetheless, I believe there's some value in these initial thoughts. I'm not claiming to have all the answers—I'm here to learn just as much as I am to share my concept. Since you're joining in on the conversation, can you tell me if anything I'm saying makes sense to you?

I do have one specific question. Why do your points on a straight line have two coordinates? What does that denote? — fishfry

I appreciate you asking a specific question about my explanation instead of dismissing it outright. I believe this has helped us move forward.

What does (.5, .5) represent? — fishfry

Yes, it represents the point we would conventionally label 0.5.

But why do that? — fishfry

Step one involves defining the journey through the use of intervals. Step two entails describing these intervals within the framework of a topological metric space. To successfully carry out step two, it's crucial that all elements involved are of the same type. For instance, I assume that defining a metric on a set that includes both points and intervals is not straightforward. As mentioned earlier, rather than defining continua in terms of points, I am defining points in terms of continua, utilizing intervals (at least in the 1D case).

Don't the standard real numbers already "describe continua with arbitrarily fine precision? — fishfry

Before I answer your question, I want to ensure we are on the same page. Do you understand how each of the five steps along the journey from 0 to 1 is represented by intervals, and that the union of these five intervals describes a continuous journey from 0 to 1?

Your coordinates have two components yet appear on a straight line. That's a little odd. What is your point? — fishfry

I'm using interval notation. It's an interval.

It's very frustrating. — fishfry

I'm frustrated too, but I know we can make it past this first hurdle. Thanks for persisting!

I do not understand what you are doing, what you're talking about — fishfry

I'm taking the Google Maps directions/map and making them more 'mathematical'. Let me try iteration 0 and tell me if this is clear:

Iteration 0
1) Start at 6445-6451 Peel Regional Rd 1
2) Travel the road Erin Mills Pkwy/Peel Regional Rd 1 N towards McDonalds
3) Arrive at intermediate destination: McDonalds
4) Travel the road Millcreek Dr towards 6335-6361 Millcreek Dr
5) Arrive at destination: 6335-6361 Millcreek Dr



Do you honestly not see how this relates to the Google Maps screenshot I sent a few posts back?

Why can't you just give me the top-line summary of what you are doing? — fishfry

I'm developing a framework that applies topological metric spaces to describe continua with arbitrarily fine precision. This might seem esoteric, but achieving this involves turning everything upside down—without dismissing any past mathematical progress. This approach offers a powerful new perspective on mathematics.

It begins with this map example because I want to (1) describe the continuous journey using intervals and (2) show how those intervals can be described by a topological metric space. However, you're not even letting me do step (1).
-------

Please tell me which iteration you are tripping up on: 0, 1, 2, 3, or 4?

• Although I didn't plan to start with directions and maps, I'm glad we ended up here. It's an excellent starting point.
• To move forward, I'll need a few responses from you along the lines of "I understand what you're saying, but I'm not sure where you're heading." If you say "I don't understand anything you're saying," then it will be challenging to proceed.
• When I attempt to outline my argument, you tell me to "show the beef," which is reasonable. Please allow me to directly demonstrate my first point using the map example (as opposed to defining objects as you requested). It didn't work when I started with terminology.

I'm going to go through a few iterations of modifications to the Google Maps directions I screen captured. Please tell me where I lose you. I will make some simplifications along the way, but the essence of the screenshot remains.

Iteration 1
1) Start at point A
2) Travel the road between point A and McDonalds
3) Arrive at intermediate destination: McDonalds
4) Travel the road between McDonalds and point B
5) Arrive at destination: Point B



Iteration 2
1) Start at point 0
2) Travel the interval between point 0 and point 0.5
3) Arrive at intermediate destination: point 0.5
4) Travel the interval between point 0.5 and point 1
5) Arrive at destination: Point 1


Iteration 3
1) Start at (0,0)
2) Travel the interval (0,0.5)
3) Arrive at intermediate destination: (0.5,0.5)
4) Travel the interval (0.5,1)
5) Arrive at destination: (1,1)


Iteration 4
(0,0) --> (0,0.5) --> (0.5,0.5) --> (0.5,1) --> (1,1)

Once again you leave me utterly baffled as to why you posted this. — fishfry

I've been trying to build towards a more important point but I feel like I have to keep going simpler and simpler to find a common ground with you. I'm hoping interpreting a map is the common ground where we can start from. If you acknowledge that you understand how directions and maps work then I will advance with my point.

Wasted on me, hope you got something from it...No idea, eyes glazed long ago. — fishfry

I'll address your other comments later, but for now, let's concentrate on one particular issue. It seems that you're either unable or unwilling to acknowledge even the most basic points I've raised. I apologize if this appears to diverge from your interests, but focusing on the image below, can you see how the instructions on the left relate to the image on the right? (This is not a trick question)

Sets are fundamental, not points. — fishfry

Indulge me in an analogy.

I see the Matrix (pictures):


You see 'Digital Rain' (sets):


Both perspectives accurately correspond to the simulation. So I agree that sets are fundamental, and I could even be convinced that digital rain is more fundamental than the Matrix. But Let's not go there. I'm specifically talking about the (continuous version of the) Matrix where I believe continua are more fundamental than points. But I don't even want to debate this further, I'd rather show you what could be done with a Top-down approach and let you decide.

I bring up the Matrix because, I want you to know that I recognize the unique purity and precision of the digital rain, but there are times, especially in discussions on geometry, when it's more effective to visually interpret the geometry from within the Matrix. Please allow yourself to enter the Matrix, try to understand my visuals, just for a little while. End of Matrix analogy.

Lost me again. In a metric space the distance between two points is 0 if and only if they are the same point. — fishfry

Okay, I lost you because I made a mistake. Let me try again:

Set: { (0,0) , (0,0.5) , (0.5,0.5) , (0.5,1) , (1,1) } where x1 and y1 in element (x1,y1) is a rational number

Metric: d((x1,y1),(x2,y2)) = | (x1+y1)/2 - (x2+y2)/2 |

And as I said, you will have trouble rigorously defining what you mean by outside of your metric space, unless you first say what the enclosing set is. So please do. — fishfry

Upon further consideration, I've decided to significantly restrict my focus to a smaller enclosing set. I am now interested only in what I want to call 'continuous sets' which are those sets where, when sorted primarily by the x-coordinate and secondarily by the y-coordinate, the y-coordinate of one element matches the x-coordinate of the subsequent element. For example, we'd have something like:

If my metric is |x-y| I want to say that there is no Universal Set (within my sandbox) for which my metric yields 0 across the board. — keystone

You're right, |x-y| doesn't qualify as a metric. Let me try again. Forget about Universal Set. Instead, I aim to define a Continuous Exact Set. A set is defined as an exact set if all elements satisfy |x-y|=0. I propose that within my enclosing set, the only Exact Set is the trivial set, containing just one element. Once again, this isn't a groundbreaking revelation; I am simply emphasizing that rational numbers by themselves are insufficient for modeling a continuum.

Sorry what? We're doing Zeno now? I must pass on that. — fishfry

Zeno greatly inspires me, yet from my viewpoint, his paradoxes serve merely as an aside. I assure you, the core thesis I'm proposing is much more significant than his paradoxes. But to save me from creating a new picture, please allow me to reuse the Achilles image below as I try again to explain the visuals.


The story: Achilles travels on a continuous and direct path from 0 to 1.
The bottom-up view: During Achilles' journey he travels through infinite points, each point corresponding to a real number within the interval [0,1].
The top-down view: In this case, where there's only markings on the ground at 0, 0.5, and 1, I have to make some compromises. I'll pick the set defined above and describe his journey as follows:

(0,0) -> (0,0.5) -> (0.5,0.5) -> (0.5,1) -> (1,1)

In words what I'm saying is that he starts at 0, then he occupies the space between 0 and 0.5 for some time, then he is at 0.5, then he occupies the space between 0.5 and 1 for some time, and finally he arrives at 1.

You are trying to invent something more powerful than contemporary math? — fishfry

Inconsistent systems allow for proving any statement, granting them infinite power. While debating the consistency of ZFC is beyond my current scope and ability, my goal is to develop a form of mathematics that not only achieves maximal power but also maintains consistency. Furthermore, I aim to show that this mathematical framework is entirely adequate for satisfying all our practical and theoretical needs.

Sometimes a "point" in a function space can be a function. Sometimes a point is just a tuple of coordinates in Euclidean space. Points aren't fundamental. Perhaps you're thinking of Euclid's original formulation of geometry. — fishfry

I haven't studied his original work, so I can't say with certainty, but I don't believe I'm referring to Euclid's formulation.

For example we can define the real numbers internally, by building them up from the empty set to get the naturals, integers, rationals, and finally reals. — fishfry

I'm familiar with these methods. I believe there is a bottom-up and a top-down interpretation of them. I'm not satisfied with the orthodox bottom-up interpretation of them.

You seem to want to make points out of cuts in a line, but I don't see where you're going with that. — fishfry

I'm getting there, and your feedback has been instrumental in enhancing my understanding of this 'digital rain'. Up until now, my approach has primarily been visual.

Aside: Please note that I will have a house guest for several days, which may cause my responses to be slower than usual.

A metric space is typically just called a metric space. — fishfry

Point taken.

It makes no sense to talk about "outside" the space till we say what set that is — fishfry

I need to bring this one picture back.


Based on this picture, what I want to say is that Achilles can occupy any position on the continuous line, but, for this specific example where the ruler only has a few tick marks on it, I'm limited to describing his location using one of five specific intervals:

• (0,0)
• (0,0.5)
• (0.5,0.5)
• (0.5,1)
• (1,1)

I believe what I want to do is define a 2D metric space on set S={(0,0),(0,0.5),(0.5,0.5),(0.5,1),(1,1)} where each element is an ordered pair (x1,x2).

While I will eventually explore higher dimensional spaces, for now, let's say that my sandbox is limited to sets of ordered pairs of rational numbers.

I have no idea what the "universal metric" is. You have not communicated that to me. — fishfry

You're right. Scratch the Universal Metric. If my metric is |x2-x1| I want to say that there is no Universal Set (within my sandbox) for which my metric yields 0 across the board. This is yet another trivial conclusion since we know that rational numbers alone cannot model a continuum.

Elements of sets are sometimes called points, but it's possible to do set theory without elements! — fishfry

Is it sets all the way down or do you eventually get to points? Anyway, you don't have to answer that question. I'm willing to agree that it doesn't matter which is more fundamental. What matters is what approach yields the most powerful math. Let's move on.

I don't get the top-down idea. 'Splain me please. — fishfry

I was hoping to get closure on the open topics first, but if you don't have any problems with this post then I think we're there. By the way, if you ever feel like my time is running out then please let me know and I'll plow through. But at the current pace I'm extracting a lot of value from our conversation.

You're pointing out that some topological spaces aren't metrizable. Right? — fishfry

No, I'm only talking about topological metric spaces. I'm pointing out that their metrics don't extend beyond their boundaries (meaning externally, they act like topological spaces without a metric), and internally, they have entirely geometric characteristics (meaning internally, they are indistinguishable from metric spaces without the topological aspects).

You can put the discrete metric on any space of points whatsoever. — fishfry

Interesting! Let's treat the Discrete Metric as a trivial metric, and by Universal Metric I'm considering only non-trivial metric.

There's a whole SEP article on holes. Deep stuff. — fishfry

Wow, it's a deeper topic than I imagined.

I did not understand the photos. — fishfry

It turns out the photos were more helpful to me than to you. You've helped me realize that what I'm actually discussing are metrics.

So far I've got the idea that you think objects are more fundamental than holes. I just don't see why you're telling me this. — fishfry

There are two primary methods for creating core mathematical artifacts:

Bottom-up Approach:

• Starts with tiny building blocks to assemble (or at least define) more complex mathematical objects.
• Points are considered fundamental in this approach.
• This method is akin to assemblage art, where separate elements are combined to form a whole.

Top-down Approach:

• Begins with a larger, unified block and divides it to produce mathematical objects.
• Continua are fundamental in this approach.
• Similar to sculpting, where material is removed from a larger mass to reveal the desired form.

I've observed that orthodox mathematics predominantly favors the bottom-up approach. However, my informal exploration of the top-down method has revealed a perspective where everything seems to fit together perfectly, without any apparent disadvantages, paradoxes, or unresolved issues compared to the bottom-up view. I'd like to share this perspective with you, so you can either help identify any potential flaws (I don't want to waste my time on a dead end) or guide me further (for example, I've already learned from this discussion that I should be describing them as topological metric spaces rather than elastic rulers).

I'm still concerned about that screwdriver ... — fishfry

I understand that as a trained mathematician, you have the ability to articulate complex ideas clearly using descriptive language. I admire that skill, but as an engineer, my strengths lie more in visual thinking. This is particularly true with mathematics, where I sometimes struggle to express my thoughts precisely in words. Consequently, I tend to rely on illustrations to communicate my ideas. I ask for your patience and flexibility in trying to understand the essense of my message.

Who is the perceiver? How do they perceive they're in a metric space? I suppose by applying the basic definition that there exists a distance function satisfying the usual requirements. In which case an internal perceiver and an external perceiver would use exactly the same method of determining that a space is a metric space. — keystone

Yes, that's right.

Ok, but "universal elastic ruler?" That part I don't get. — fishfry

Instead of saying that there cannot exist a "Unversal Elastic Ruler" what if I say there cannot exist a "Universal Metric"?

Emergent objects become actualized? Bit vague for me. — fishfry

Think of it like this: a hole is an emergent property. To have a hole, you first need an object that can contain a hole. In this sense, the object is more fundamental. We begin with the object, which holds the potential for a hole. Then, once we make a cut, what we have is the same object, but now with an actual hole in it.

I've adopted the 'k-' prefix to denote this distinction, as it's common to encounter the reverse belief - that points are fundamental objects and continua are created by assembling infinite points.
— keystone

Losing me. — fishfry

If you return to my photographs, you will see that I start with a continous object and put cuts in it. I call those cuts points. Just as an object is more fundamental than the hole, with my view a continua is more fundamental than the cuts (i.e. points). I used k-continua and k-points instead of continua and points because I wanted to avoid a debate over what's more fundamental. In my sandbox the continua are more fundamental. If you want to grant me that, then perhaps we can set aside all this 'k-' terminology.

Not much disagreement, only confusion about where this is all going. — fishfry

Okay, this feels like progress. Let's iron out the points discussed above and then I'll give you more details on where this is going.

If it's not obvious, I want you to know that I really appreciate you sticking with me on this.

I have no idea what your point is... — fishfry

At this stage, I'm making such minor points that perhaps you are confused why it took me so many words (and pictures) to express it. If that is the case, my apologies.


I think what I'm trying to say is the following:

1) Topological spaces have no sensible notion of distance.
2) Topological metric spaces have a sensible notion of distance.
3) If you lived outside a topological metric space, you wouldn't be able to use it as a measuring tool on external objects (i.e. the metric qualities of the space are not applicable to objects outside of the topological metric space).
4) If you lived inside a topological metric space, you'd perceive it as a metric space, where the topological qualities aren't obvious in everyday experiences. For instance, if our world were a topological metric space and everything, including the space, ourselves and our measuring tools, suddenly grew twice as big, we wouldn’t detect the change because all our measurements would scale up too.
5) If it is always possible for an object to exist outside of a topological metric space, it's notion of distance cannot be universally applied to all objects. I phrased this as, 'there cannot exist a Universal Elastic Ruler'.
6) I'm constructing a topological metric space from the ground up, rather than examining one that already exists in completion. So, in my example, it's a very crude ruler and there is no mention of real numbers. Does this qualify as a topological metric space?

Aside from the topological discussion, I also made the following point:

7) I'm treating continua as fundamental objects and points as emergent objects which become actualized when I make cuts. I've adopted the 'k-' prefix to denote this distinction, as it's common to encounter the reverse belief - that points are fundamental objects and continua are created by assembling infinite points. Perhaps you wouldn't characterize your viewpoint in these exact terms; you might regard points and continua as simply coexisting without one preceding the other. However, it's undeniable that the conventional approach primarily describes continua in terms of points rather than the reverse.


Is there disagreement or confusion on any of these points?

It is impossible to prove anything mathematically using physical constructions. — fishfry

I know that, and I explicitly stated that the photographs are not perfect representations of the abstract objects which I'm actually talking about.

That I would be engaging with someone too obsessed for their own good. I would feel that I need to tread cautiously. — fishfry

I'm certainly not too obsessed with this. This topic has sat dormant in the back corner of my mind for years before it resurfaced with my unexpected hanging paradox paradox post a few weeks back. Should this discussion not lead anywhere significant, it'll probably return to its quiet corner. Nevertheless, I'm keen to hear your candid thoughts. Whether they nudge me towards new insights or help me lay these concepts to rest, I am welcoming of both possibilities.

Now I do want to try to give this a fair reading. — fishfry

I would really appreciate that. I don't plan to have many photographs in my subsequent posts. This was just my way of laying the groundwork.

And if it's not too much to ask, can you keep it short? — fishfry

Should I abbreviate my explanation, you might resort to conventional thinking to bridge the gap, which could lead to misunderstanding. My goal is to present my ideas with such simplicity and clarity that you’ll effortlessly grasp them, swiftly perceiving their evident truth. So I hope that quick instead of short is acceptable.

So here goes…

PART 1 - The Elastic Ruler

THE MATERIALS
Everything that follows takes place within my abstract sandbox. I intend to construct a ruler in an unconventional manner, starting with an abstract elastic band (i.e. a k-topological object). The graphics that follow are an imperfect representation of what's truly happening my sandbox. For example, my abstract elastic band is a one-dimensional object with the property of perfect elasticity (i.e. it can stretch infinitely far), unlike its physical counterpart. It's crucial that you not ascribe physical characteristics to elements within my sandbox. So, resist the urge to suggest that my elastic band is made up of indivisible atoms or exists amidst a quantum foam.



THE TERMINOLOGY
I am going to cut the elastic band a couple of times. At this point we can begin to label regions.

• a-The void left of the band
• b-The void corresponding to the first cut of the band
• c-The void corresponding to the second cut of the band
• d-The void right of the band
• (a,b)-The band between a and b
• (b,c)-The band between b and c
• (c,d)-The band between c and d

Notice that both the void and the elastic band segments display a sense of duality in that they are defined in relation to one another. They are both important.

While we must remember that we're talking about cuts to an elastic band, the gap size is unimportant. As such it is much cleaner to make the cuts 0-width in illustrations as depicted below.



Let us use the following terminology:

• The cuts - 'k-points'
• The elastic bands - 'k-lines' or, more generally, 'k-continua'
• The labels of cuts (e.g. b) - 'k-numbers'
• The labels of k-lines (e.g. b,c) - 'k-intervals'

THE RULER
I will take the elastic band and cut it 3 times, adding k-number and k-interval labels as depicted below.

And there we have it, an elastic band ruler!

USING THE RULER
Test one: Measuring a pen in my sandbox. Conclusion, pen is 1 unit long.


Test two: Measuring a pen again in my sandbox with ruler stretched. Conclusion, pen is 0.5 units long.


Test three: Measuring a screwdriver in my sandbox with ruler slightly stretched. Conclusion, screwdriver is 1 unit long.


Test four: Measuring a screwdriver in my sandbox with ruler stretched more. Conclusion, screwdriver is 0.5 units long.


Employing this ruler as demonstrated could yield highly questionable outcomes, like equating 0.5 with 1, suggesting all objects are of the same size, or that an object's size is not consistent with itself, among others. This would appear to render the ruler quite ineffective, wouldn't it? Yet, let's explore the subsequent experiment.

Test five: Achilles and the tortoise live on the elastic band. Achilles is at (0,0.5) and the tortoise is at 0.5.



As the elastic stretches, the positions relative to one another remain constant, thus allowing the ruler to accurately depict their placement.

CONCLUSION
1) Elastic rulers are useful only when applied internally (i.e. locally).
2) Elastic rulers are not useful, and in fact misleading, when applied externally (i.e. globally).
3) IF it can be proved that there will always be abstract objects that exist outside of the ruler (similar to how there is no set of all sets), then it would follow that the existence of a Universal Elastic Ruler (one that can measure everything) is not possible.

Are you with me? I know this seems extremely basic (and perhaps inconsequential), but I'm laying the groundwork for a more consequential idea so I hope you stick with me.

I don't want you to go easy on me. I pride myself in my ability to correct my trajectory in the face of new evidence/feedback.

you didn't say a thing yet. — fishfry

I agree, I just wanted one post to set the stage before I get into it...

If you have a paragraph or two that I can sink my teeth into — fishfry

Well, how much beef can one actually put in a paragraph? Have you ever sunk your teeth into an abstract?

Anyway, I don't want to write another long post. My first real post will come tomorrow...I got consumed by the Staircase post this evening...

I suppose that if Zeno actually accepts his (unreasonable) conclusions, then you get something like just that one state. — noAxioms

Exactly. Let’s deconstruct the argument:

(1) We accept Zeno's premise as valid, asserting that in a presentist world where only a single state exists, motion is impossible.
(2) We assume that the world functions according to presentist beliefs.
(3) Our experiences clearly indicate that motion is possible.

These three assertions cannot all be true simultaneously. It’s unlikely that anyone, including Zeno, would dispute (3). You find (1) to be unreasonable. However, consider the possibility that (2) is incorrect.

Here's an alternative approach:

(1) We accept Zeno's premise as valid, asserting that in a presentist world where only a block universe exists, change of the block is impossible.
(2) We assume that the world functions according to eternalists beliefs.
(3) Our singular, consistent historical experience gives us no reason to believe that the block universe is subject to change.

In this scenario, all three points could indeed hold true, suggesting that an eternalist viewpoint might be more suitable. However, quantum mechanics challenges point (3), necessitating a more nuanced argument. Despite this, an eternalist framework—albeit with some adjustments to incorporate the quantum aspects of our universe—appears to be the most rational choice.

Not sure of the difference. If I cut a string, I don't get points, I get shorter strings. — noAxioms

The cuts themselves are the points (think Dedekind cuts).

You can under some interpretations. — noAxioms

One can observe a superposition directly? Please share a link.

Zeno's arguments are of the form (quoted from the Supertask Wiki page):
"1 Motion is a supertask, because the completion of motion over any set distance involves an infinite number of steps
2 Supertasks are impossible
3 Therefore, motion is impossible"

If motion is discreet, then premise 1 is demonstrably wrong. If it isn't, then premise 2 is demonstrably wrong, unless one just begs the conclusion and adopts the 'photo' interpretation. — noAxioms

What I aim to demonstrate is that there is a scenario where local motion is possible and continuous without involving supertasks. This occurs in a block universe where the block itself remains unchanged (i.e., no global motion), yet the entities within it experience change (i.e., local motion).

Necessary only if the first premise is to be accepted. — noAxioms

If the universe is discrete, then Zeno's paradoxes cannot occur as he described them. What I'm suggesting is that in a continuous universe, the scenarios depicted in Zeno's paradoxes can indeed unfold precisely as he described them, without necessitating the completion of supertasks.

I'm asserting that an infinite process is necessarily never completed - by definition. — Relativist

Good. Then we're on the same page!

the process of counting steps is not completable — Relativist

Are you suggesting that supertasks cannot be completed?

The process is:
1) Have fuzzy intuitions;
2) Study some math;
3) Develop far better intuitions. — fishfry

Agreed, but most importantly: (4) apply those intuitions to (the original) experiments.

It's only a finite number of steps back, even from infinity. — fishfry

I like where you're going with this. To navigate between the staircase and omega (and back), one must leap over infinite steps. This concept becomes more palatable if we consider that the steps become progressively smaller towards the bottom. However, let me try to rephrase your perspective: Icarus requires a finite number of strides to reach the bottom and a finite number to return to the top, thus avoiding any supertask. When Icarus adds 1/2, then 1/4, then 1/8, he gets bored and chooses to make a final leap. On his final leap, instead of adding an infinite series of smaller terms, he simply adds another 1/8 and reaches omega, where his calculator displays exactly 1. In this case, the infinity in the paradox describes the steps which he potentially could have traversed (and seen), not what he actually did (and saw). Since he never actually observed all steps, he is in no position to confirm that there were actually infinite steps...but there could have been...potentially. Paradox solved?

This thread's fine. The Staircase thread's hopeless, way too many side issues. It's nice and peaceful in here. — fishfry

Agreed. Okay, let's begin!

It's elementary
Even if you believe that the foundations of mathematics and our understanding of continua is rock solid, you must acknowledge that it confounds many people. Take, for instance, the difficulty in convincing a child that 0.999... equals 1, or the prominance of Cantor cranks. By contrast, I believe children would grasp my concept more easily because it is fundamentally simple, albeit it requires adopting a different viewpoint towards the foundations of math. To use an analogy, my perspective is less like a target that's difficult to hit and more like one that's difficult to spot.

Why I believe it's important
The validity of my ideas is still up for evaluation, but if they prove to be correct, deep truths often end up having practical relevance, even if their complete implications are not immediately apparent. Nevertheless, I am convinced that my theories could enhance mathematics education, resolve many paradoxes, and shape our understanding of reality, particularly in the context of physics. Ironically, coming from an engineer, I don't anticipate any significant impact on applied mathematics, as practitioners in such fields typically do not focus on the foundational aspects of math. I also want to clarify that my work is not meant to suggest that previous efforts by mathematicians were wasted.

How I'm going to share my ideas
I understand that for an idea to gain acceptance in the mathematical community, it needs to be formalized. I'm just not there. I don't have a formal paper to share with you, but instead, I plan to share my ideas gradually, in a manner akin to our ongoing discussions. Just as we can introduce children to the basic concepts of Cartesian coordinate systems without heavy formalities, I hope you can allow me the same flexibility in explaining my ideas with a similar level of informality.

Mathematical terminology often comes with preconceived notions; for instance, mentioning a continuum might lead you to assume I am discussing real numbers. To avoid these assumptions and start with a clean slate, I'll be using a 'k-' prefix in front of familiar terms (like k-points, k-curves, k-continua, etc.). By the end of our discussions, I hope you'll not only find my approach more appealing but also recognize that it aligns with the mathematics that applied mathematicians have been practicing all along. At that point, it may be justified to remove the 'k-' prefix.

Thoughts?

This thread's fine. The Staircase thread's hopeless, way too many side issues. It's nice and peaceful in here. — fishfry

Agreed. Okay, let's begin!

It's elementary
Even if you believe that the foundations of mathematics and our understanding of continua is rock solid, you must acknowledge that it confounds many people. Take, for instance, the difficulty in convincing a child that 0.999... equals 1, or the prominance of Cantor cranks. By contrast, I believe children would grasp my concept more easily because it is fundamentally simple, albeit it requires adopting a different viewpoint towards the foundations of math. To use an analogy, my perspective is less like a target that's difficult to hit and more like one that's difficult to spot.

Why I believe it's important
The validity of my ideas is still up for evaluation, but if they prove to be correct, deep truths often end up having practical relevance, even if their complete implications are not immediately apparent. Nevertheless, I am convinced that my theories could enhance mathematics education, resolve many paradoxes, and shape our understanding of reality, particularly in the context of physics. Ironically, coming from an engineer, I don't anticipate any significant impact on applied mathematics, as practitioners in such fields typically do not focus on the foundational aspects of math. I also want to clarify that my work is not meant to suggest that previous efforts by mathematicians were wasted.

How I'm going to share my ideas
I understand that for an idea to gain acceptance in the mathematical community, it needs to be formalized. I'm just not there. I don't have a formal paper to share with you, but instead, I plan to share my ideas gradually, in a manner akin to our ongoing discussions. Just as we can introduce children to the basic concepts of Cartesian coordinate systems without heavy formalities, I hope you can allow me the same flexibility in explaining my ideas with a similar level of informality.

Mathematical terminology often comes with preconceived notions; for instance, mentioning a continuum might lead you to assume I am discussing real numbers. To avoid these assumptions and start with a clean slate, I'll be using a 'k-' prefix in front of familiar terms (like k-points, k-curves, k-continua, etc.). By the end of our discussions, I hope you'll not only find my approach more appealing but also recognize that it aligns with the mathematics that applied mathematicians have been practicing all along. At that point, it may be justified to remove the 'k-' prefix.

Thoughts?

So your point was that if everyone older than you dies, you'd win the argument?

Your use of Planck's quote makes not a lick of sense. He was talking about older scientists not being able to get on board with radical new ideas accepted by younger ones. But there's no radically new theory of Zeno that old scientists are rejecting, except for your own personal theory, which as far as I can tell you have not clearly articulated. So it's a failed analogy. — fishfry

Fine. What matters is that you're being very generous with your time to me and I offended you. I don't want to waste the time I have with you arguing over this. Again I'm sorry and I grant that you're entirely right on this. I hope we can put to rest this specific topic.

I'd be happy to critique your idea if you stated it clearly. — fishfry

I've been sharing aspects of my perspective here (but I feel like you never read it, perhaps because it seemed tangential), and other details have emerged in the Staircase thread. Nevertheless, I haven't presented it as a complete picture. Should we continue such a discussion in this thread, which has become like our private chat room, or would you like me to start a new thread?

STAIRCASE PARADOX

Can you see that? It's actually the exact same example as 1, 2, 3, 4, ... ω
. Any step back takes you to a number that is only finitely many steps from the beginning. You don't need infinitely long legs. In fact your legs can be arbitrarily small. Any step backward (or up the stairs) necessarily jumps over all but finitely elements of the sequence. — fishfry

I see your point, and I appreciate your analogy with the [0,1] interval. However, you need to clarify what happens in the narrative. The purpose of this narrative is to ensure that one cannot simply retreat behind formalisms. This mathematical observation doesn't change the reality that Icarus would need to jump over infinite steps. If you're suggesting he doesn’t have infinitely long legs, then perhaps he possesses infinitely powerful legs that enable him to leap over infinite steps. This might explain how he returns to the top, but it essentially sweeps the infinite staircase under the rug.

I've bethought myself and realized that the step numbers will only align if the number of steps is odd. If it is even, they won't be such a point. — Ludwig V

This brings us to another paradox - Thomson's Lamp - in that the last step can neither be even nor odd.

So the staircase down defines the staircase up. — Ludwig V

Now explain how your algorithm works for infinite stairs.

So why don't you just link me to the reading materials that would lead me to believe that the supertask you described in your op is possible to complete? That specific supertask, not supertasks in general. Let's not beat around the bush, let's get right to it. — flannel jesus

Instead, please present any supertask you consider viable, and I will demonstrate its connection to Icarus descending the staircase. For instance, do you agree that the sum of the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ... equals exactly 1?

Once you decide to make this supertask accomplishable by *a human mind*, then you run into brand new problems that don't exist in a purely mathematical context. — flannel jesus

I'm unclear on whether you're disputing the existence of supertasks or merely the ability of humans to perform them. Do you believe it's conceivable for anyone physical or abstract, perhaps even a divine being like God, to accomplish a supertask?

Continually halfing the time it takes to perform the subsequent step does not just contradict the physical laws of our world but is a metaphysical impossibility. With these paradoxes we shouldn't be looking for some answer that is consistent with the premises but should accept that they prove that the premises are flawed. — Michael

Reading your posts gives me a sense of calm. :D

ou have provided no propositions or premises whatsoever, to conclude that 60 seconds may actually elapse. — Metaphysician Undercover

I said he "reached the bottom of it in just a minute." Thus, the premises address both the completion of the supertask and the passing of a minute. It seems you are challenging the incorrect premise.

There's nothing contradictory with the EXISTENCE of an actual infinite, but it's not accepted that an infinity can be traversed in a supertask. — Relativist

I would contend that all of the infinity paradoxes clearly illustrate contradictions inherent in the concept of actual infinity. Furthermore, I would argue that every definition of real numbers inherently suggests that supertasks are completable.

So a complete (i.e. well-defined) mapping shouldn't be conflated with a completed PROCESS. — Relativist

We can also map the steps to the elapsed time (1 → 0.5, 2 → 0.75, 3 → 0.875, etc.). If we conclude that a full minute has elapsed, doesn't this imply that he has traversed all the steps?

Analogously, a limit entails an abstract operation applying to a mathematical series and shouldn't be conflated with a consecutive process. — Relativist

Why not?

ZENO'S PARADOX

I don't think what you describe can be validly categorized under the term 'presentism'. — noAxioms

You're correct that presentists don't explicitly hold this belief. However, what Zeno's Paradoxes demonstrate is that if their ideas are taken to their logical conclusion, this belief is implicitly suggested.

There is no 'past, present. future' defined under eternalism. All events share equal ontology. The view differs fundamentally from presentism only in that the latter posits a preferred location in time, relative to which those words have meaning. — noAxioms

Instead of presentism vs. eternalism, let's talk about the photo vs. movie reel. For the photo and every frame of the movie reel the characters believe they're in the present. So if you're saying that the experience of the present has nothing to do with Zeno's Paradox, then I agree with you. But there is a very significant difference between a photo and a movie reel.

Irrelevant, but I prefer the one that doesn't posit the additional thing for which there is zero empirical evidence. This is my rational side making that statement. — noAxioms

Reconciling general relativity with presentism is quite challenging. Therefore, if empirical evidence influences your thinking, eternalism might be a more suitable perspective to adopt. Plus, adopting eternalism helps to render Zeno's Paradoxes largely non-paradoxical.

a attempted demonstration that a nonzero thing cannot be the sum of zeroes, a sort of analysis of discreet vs continuous. — noAxioms

You're approaching this with a whole-from-parts mindset, where you aim to construct everything from smaller components. Thus, you believe the only options are to assemble a continuous line from infinite points or from discrete line segments. Consider reversing this perspective: adopt a parts-from-whole approach. Start with a single continuous line and then, as if it were a string, cut it to create discrete points (which correspond to the gaps). I encourage you to explore this mindset; I'm eager to discuss it more with you.

While my explanation might differ from how Zeno would phrase it, I believe it aligns with his philosophical approach. He is quoted to have said “My writing is an answer to the partisans of the many and it returns their attack with interest, with a view to showing that the hypothesis of the many, if examined sufficiently in detail, leads to even more ridiculous results than the hypothesis of the One.”

But he cannot indicate a time that isn't represented by such a point, so I don't think he's shown this. — noAxioms

However, you're working under the assumption that a timeline consists only of discrete points in time. You cannot directly observe a particle in a superposition state, but this doesn't mean that superposition states are merely fictional. I bring in QM, not to sound fancy, but there is an analogy here between observed states (which are like points) and the unobserved a wavefunction (comparable to a line) that lies between them.

The block universe can still be interpreted as discreet or not, just like the presentist view. — noAxioms

I believe you are discussing whether time is discrete or continuous. In the context of Zeno's Paradoxes, it's necessary to consider space and time as continuous (as you later noted). I'm not sure what you're referring to with time being continuous or discrete from a presentist perspective, especially since Zeno's arguments suggest that time does not progress in a presentist's view of the world.

You do if it is discreet. A physical string is very much discreet — noAxioms

I explicitly wrote abstract string.

Nonsense. It says no such thing. — noAxioms

Perhaps it's not my place to speak for others, but let’s say that adopting an eternalist perspective allows someone to reframe the impossibility of supertasks, turning it's non-existence from having unacceptable consequences to acceptable consequences.

This also seems irrelevant since none of his paradoxes seem to reference observation or comprehension. — noAxioms

Additionally, none of the paradoxes explicitly rule out this as a possible solution.

Surely it would take forever to comprehend the counting from 1 on up. Michael's digital counter runs into this: the positing of something attempting to measure the number of steps at a place where the thing being measured is singular. — noAxioms

If there is a continuous film reel capturing the ticking counter, the limits of observation dictate that there are just some frames that we cannot see. They're blacked out. In fact, I would argue that we can only ever observe countably many frames so in fact, most of the frames remain unobserved (in a superposition of sorts). This allows the story to advance and avoids singularities.

And so conversely, if an infinite task may not be completed in a finite amount of time then we must agree that time is not infinitely divisible. — Michael

This only applies if you adhere to a whole-from-parts construction approach. As I mentioned in my discussion with NoAxioms, a seldom considered alternative is that the universe is constructed parts-from-whole. I really hope you will engage with me on this possibility.

THOMSON'S LAMP

Depends on if the calculator is required to follow the mathematical theory of convergent infinite series. If yes, 1, If no, then it can be anything at all. — fishfry

In this scenario, the calculator isn't equipped to perform calculus; it's a basic model tasked with adding each term of the infinite series. While mathematical theory predicts that at 60 seconds, it will display 1, it's true that the narrative does not specify what should appear at that moment. I am even welcoming of the idea that it turns into a black hole at 60 seconds. Nevertheless, isn't it concerning to you that there's a discrepancy between mathematical theory and your intuition? I completely agree that freshman calculus is invaluable, and I'm not suggesting that infinite series or any aspect of calculus are without merit. I use aspects of it everyday. Instead, I propose a new interpretation of what these infinite series represent. The story of the calculator isn't really about what it displays at 60 seconds; it's about the approach to 60 seconds. Likewise, I suggest that infinite series don't actually sum up to a specific number, but rather they outline a continuous, unbounded process. We don't have to assert that there's a least upper bound to this process.

That's the problem with all these puzzles. — fishfry

Your argument that the paradox is nonphysical is a red herring. This narrative takes place in the abstract realm, and unless you can pinpoint a contradiction within that context, we should consider it as abstract and possible and acknowledge its validity. Perhaps you lean towards theoretical perspectives, but it's important not to undermine the significance of thought experiments. They have arguably been among the most influential types of experiments conducted by humans.

In what sense do you regard Zeno's paradoxes as new ideas? That doesn't make sense. — fishfry

There are no new original records of Zeno's paradoxes so they are not new ideas. However, I think that Zeno's paradoxes remain unsolved, and I have an original perspective that resolves these and many other paradoxes in a way that they no longer seem contradictory. I sense you can tell I'm enthusiastic about this viewpoint, but it seems you aren't interested in delving into or critiquing it. Perhaps after considerable reflection, you've already formed your opinion on these issues and don't find additional discussion worthwhile. That's completely acceptable.

I'll assume that your wish for my death did not come out the way you meant it. Way over the line. — fishfry

I apologize if it seemed like I was implying anything about wishing for your death; that was not my intention at all. I specifically expressed a desire for you to have a long life. My main point was about the acceptance of new ideas, highlighting that they often gain traction because a new, possibly more open-minded audience emerges over time. The longevity of those holding old beliefs isn't the crucial factor.

Your argument is that Zeno's paradox is so new and revolutionary that I'm too old to see it? — fishfry

Zeno was significant, but the concepts and solutions I'm advocating are not entirely his ideas.

What perspective do I have and why on earth are you going on about it like this? — fishfry

I believed we agreed to confine the Zeno discussion to the Staircase thread, which is why I was vague here. However, I offered detailed criticisms of your perspective in that other thread. I'm not trying to be cagey.

STAIRCASE PARADOX

The "ground", thus defined, is a point that cannot be reached from the stairs, being infinitely far below it. Similarly, you cannot reach the stairs from that point, as every stair is infinitely far above it. That's why the man on the "ground" can't see any stairs as described in the OP story. They are all too far away above him. By making such a definition, we are essentially dividing our thought-experiment-world into two parts, neither of which can reach the other. — andrewk

You are effectively arguing that supertasks cannot be completed since if he cannot reach the ground, he must still be on the stairs.

No chance. — fishfry

Max Planck once said "a new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it." Certainly, I hope you have a long and fulfilling life, but your response brought this quote to mind.

Didn't do any good, nobody understood a word I said. — fishfry

I think I understand what you said; I just have some issues with your perspective.

STAIRCASE PARADOX

It's always only a finite number of steps from infinity back to zero — fishfry

This brings to mind Sagan's quote "extraordinary claims require extraordinary evidence." We start with an extraordinary premise—the existence of infinite stairs and supertasks—and to resolve it, we resort to an equally extraordinary solution: he has infinitely long legs, enabling him to ascend to the top in just one stride. This doesn't strike me as a satisfactory resolution.

You described it as endless, and yet claim he reached the end... The "paradox" is just you choosing to invent a story with contradictory concepts. — flannel jesus

What you seem to overlook is that I'm beginning with a premise widely accepted within the mathematical community: the existence of actually infinite objects (like these infinite stairs or the set, N) and the completion of actually infinite operations (such as traversing the stairs or calculating the sum of an infinite series). If you do not accept the concepts of infinite sets or supertasks, then this paradox is not aimed at you. If you claim that an old woman is 2 years old, then you're not basing your argument on any widely accepted concepts of age.

But if a staircase down can be created by our, or your, say-so, another one, going up, can be created in the same way. — Ludwig V

If there is a parallel staircase where the steps start at 1 and increase as you go up, then there must be a point where the step numbers on both staircases align. What would that step number be?

But the end is not reached. — Metaphysician Undercover

Then your argument should be that supertasks are impossible, not that 60 seconds cannot elapse.

ZENO'S PARADOX

I suggested that movement was discrete, not that space was discrete — Michael

Consider linear motion. If you plot position against time, are you suggesting that the resulting curve, when examined closely, appears stairstepped rather than smooth? If that's the case, what would be the width of these incremental steps? This presents the same issue, as I could always plot a more accurate curve of motion using even smaller incremental steps.

I wouldn't say that. — Metaphysician Undercover

This response does not adequately address my reinterpretation of Zeno's ideas.

I don't see how Zeno's paradoxes work any differently under presentism than under eternalism. — noAxioms

Zeno contends that change is impossible, leading to stark implications depending on one's philosophical stance on time. Under presentism, this translates to an unchanging, static present—life as nothing more than a photograph. In contrast, the eternalist perspective views this as a static block universe, a continuous timeline that encompasses past, present, and future—akin to a film strip. Which view do you think is more reasonable? Of course, this raises profound questions, such as why we experience time's flow, but that discussion is for another thread.

Let me reframe Zeno's argument in different terms more relatable to a modern audience. Consider whether it is easier to draw a one-dimensional line by assembling zero-dimensional points consecutively or to cut a string (akin to dividing a line into segments). Zeno would argue that the first option is impossible: a timeline cannot be constructed from mere points in time. Instead, modern Zeno would suggest that the entire timeline already exists as a block universe, and our experience is merely about observing different parts of it, similar to making cuts in a string. However, there's a twist: abstract strings, like time, are infinitely divisible. No matter how many cuts we make (one after another), we never actually reduce the string to mere points. Each cut still leaves a segment of string, however minuscule. This introduces new challenges (for which there are answers) but as it relates to the discussion at hand, the eternalist perspective reframes the impossibility of supertasks from an unacceptable notion—that motion itself is impossible—to a more acceptable one—that observing every instant in history is impossible. This essentially echoes Aristotle's proposal, but it is only in the quantum era that such a solution becomes truly acceptable.

For only two of the three following premises can be true of a sequence: i) The length of the sequence is infinite. ii) The sequence is countable iii) The sequence is exhaustible — sime

The issue arises if Achilles toggles Thomson's Lamp with each stride, leading to a contradiction: his feet suggest that the sequence is exhaustible, but his hand indicates it is not.

SINGLE DIGIT COUNTER PARADOX

Assuming that paradoxes are metaphysically impossible then the counter is metaphysically impossible, and that suggests that it's metaphysically impossible for time to be infinitely divisible. — Michael

First, instead of using decimal, let's switch to binary, where the counter can only be 0 or 1. You suggest that quantum mechanics resolves this by introducing indivisible units, perhaps akin to Planck time. Looking to QM for inspiration is a good idea. However, the idea of Planck time doesn't hold up because in the abstract realm, we can always conceptualize a smaller increment. I propose that the correct solution is that at 60 seconds, the counter is in an unobserved state where its status fundamentally remains unknown. It could be either 0 or 1, so let's say it's in a state of (0 or 1). If we wish to steal technical terms from QM, we might refer to this state as being in superposition.

The paradox is that given the premise(s) what happens at the limit is undefined, and yet something must happen at the limit. This is a contradiction, therefore one or more of the premises must be false. — Michael

What if the undefined state is fundamentally unobservable? This raises the question similar to "If a tree falls in a forest and no one is around to hear it, does it make a sound?" The limitations I'm suggesting on observation should not be surprising to a generation that has grown up in the era of quantum mechanics.

It is metaphysically necessary that there is a limit to how fast something can change — Michael

Yet, it's impossible to determine what this limit might be. Would you argue that there is a limit to the slope of a line?

THOMSON'S LAMP

Why on earth must there be a behavior defined at the limit? — fishfry

Suppose that with each flick of the lamp, the lampholder adds another term to a cumulative total: first 1/2, then 1/4, then 1/8, and so forth. What does his calculator show at 60 seconds? Why on earth must we assert that it displays 1? After all, the narrative doesn't specify what his calculator must indicate at 60 seconds. It seems to me that you're contesting the very idea which you support - that infinite series can have definitive sums.

By the law of excluded middle and non-contradiction, after 60 seconds the lamp must be either on or off. — Michael

Yeah, that law needs updated. I propose "for every proposition, either this proposition or its negation can be measured to be true." This introduces the possibility of a third, unmeasured state—when we're not observing, the lamp could either be on or off, placing it in a state of being (on or off).