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. — keystone

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.

I sense you can tell I'm enthusiastic about this viewpoint, but it seems you aren't interested in delving into or critiquing it. — keystone

I'm pretty sure I haven't heard a clear statement of your idea. I could not repeat your idea back to you even to disagree with it. Perhaps I missed it. I'd be happy to critique your idea if you stated it clearly. [If you did state it clearly and I missed it, my apologies]. I could argue your thesis (if I knew what it was) all day, without ever having much interest in the subject. Well maybe not, but I'd kick it back and forth a little.

Maybe you can state your thesis so I know what revolution will ensue if everyone older than you would only die already.

Perhaps after considerable reflection, you've already formed your opinion on these issues and don't find additional discussion worthwhile. — keystone

I have no idea what point you are trying to make. You started this thread with a paradox of probability, which has a solution that's mathematically correct but not intuitively satisfying, namely that there's no uniform probability on the naturals.

Then you changed the subject to encompass many other ideas I consider irrelevant to the OP (quantum, etc) and I lost the ability to follow your thinking entirely. So yes, I don't have much interest in talking about Zeno's paradoxes of motion; but more to the point, I have no idea what is your grand new thesis that the old folks just don't get.

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?

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. — keystone

Apology completely accepted. I'm a little hypersensitive in general. No worries as they say.

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. — keystone

I may have misunderstood a lot because I was focussed on the probability aspect. I'm not reading most of the posts in the Staircase thread and it's all over the map at this point.

Nevertheless, I haven't presented it as a complete picture. — keystone

Ok, that's fair. Would be happy to chat about your idea if you present it.

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? — keystone

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

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?

Even if you believe that the foundations of mathematics and our understanding of continua is rock solid, — keystone

I have never expressed, nor do I presently hold either of those beliefs.

you must acknowledge that it confounds many people. — keystone

I cannot take responsibility for the execrable state of math education, or frankly education in general these days.

Take, for instance, the difficulty in convincing a child that 0.999... equals 1, or the prominance of Cantor cranks. [/quotet]

A byproduct of bad education. Not something I can personally remedy.
— keystone

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. — keystone

Lotta fluff so far. "Where's the beef?" (*)

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. — keystone

More marketing fluff. I'm regretting this already.

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. — keystone

I fear that you're going to wave your hands and present a lot of mathematically naive ideas, and I'm going to find myself back in the .999... wars and all the rest of it. And I'm frustrated that you wrote so many words here without saying anything at all.

Mathematical terminology often comes with preconceived notions; for instance, mentioning a continuum might lead you to assume I am discussing real numbers. — keystone

If not, then what?

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.). — keystone

You're going to rework the whole of mathematics? I'm getting a sinking feeling.

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. — keystone

I regret encouraging this conversation.

Thoughts? — keystone

I apologize for the negativity, but you didn't say a thing yet. And you apparently have some kind of grand unified theory of math that you're going to wave your hands at while I attempt to be open-minded.

I am open-minded, and want to hear your ideas, but you don't seem to be interested in presenting them. I was hoping for a paragraph, but you gave me what looks to be the introduction to a very long and very frustrating exposition.

If you have a paragraph or two that I can sink my teeth into, by all means present it.

If what you've got is as vague as this post, then I humbly apologize for encouraging you to aim this at me. On the one hand I'm curious as to what you are talking about, and on the other, well ... you just haven't said anything but you're promising to say way too much.

Can you boil down what you want to say in a couple of clear paragraphs? Without the marketing about how it's revolutionary and will be understandable to children?

I do welcome your thoughts, but I encourage you to get to the point and try not to turn my curiosity into frustration. Clearly this post sent me over that line.

Here is my response to this post in a nutshell.

(*)
https://www.youtube.com/watch?v=Ug75diEyiA0

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...

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... — keystone

I await your next missive with both curiosity and trepidation. Is it wrong for me to encourage you on the one hand, then give you a hard time the next? I'm conflicted.

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.

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. — keystone

Standing by for something specific. And if it's not too much to ask, can you keep it short? I myself tend to write long-assed posts. I should take my own advice.

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.

Pass the popcorn, please. I am sitting in the bleachers watching with interest. :chin:

Should I abbreviate my explanation, you might resort to conventional thinking to bridge the gap, which could lead to misunderstanding. — keystone

Should you abbreviate it, I might have a chance at reading it.

My first impression -- again, forgive me, but I'm finding some virtue in just telling you the truth of my own experience of seeing this -- my first impression is the overwhelming passion that you have toward this subject; passion that is admirable in you, but makes me reluctant to even try to engage. Pictures, and rulers, and pencils ... it's a little ... off putting. That I would be engaging with someone too obsessed for their own good. I would feel that I need to tread cautiously.

Now I do want to try to give this a fair reading. I have a few other mentions to get to tonight and I'll put this aside for later. But I must say one thing. It is impossible to prove anything mathematically using physical constructions. There's no way you and I are even having the same conversation, if you think your opening salvo should involve pencil and paper and scissors and ... is that a screwdriver? I don't have to come down to the basement to see this, do I? Why are you closing that door behind me ...? Aiiiiiyyyyy.

Pass the popcorn, please. I am sitting in the bleachers watching with interest. :chin: — jgill

I wish I was up there with you.

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.

You wish to speak and reason in the realm of actual infinities when you cannot do such a thing. Reasoning fails there. So your tool of reasoning is the wrong tool. Well done.
— Chet Hawkins

I don't think you understand my position. I'm playing in the "paradise" which Cantor created (involving infinite sets) not because I believe in it but because I want to convince others that it's a mirage (at least in my view). — keystone

But it is no mirage. The same difficulty arises in math with limits and with the repeating value of constructs like Pi. These are NOT mirages. They are actual and demonstrable within reality. So much of reality answers to the limit functions that their utility and probable inclusion as meaningful and dependable is a great practice. If you wish to dismiss them, I must report that you'd need some fairly compelling, next to miraculous new ways of looking at the entire universe in order to approach success.

The sides should be of the same length.
— Chet Hawkins

Isn't the concept of an infinite-sided die that could fit in your hand intriguing? It’s impossible to construct a die with finite volume if you insist that all sides must be of equal length. — keystone

So YES, it is intriguing and also impossible. As for your second sentence, no, not at all. Unless you misstated what you were trying to say, all regular shapes of equal sides are easily of finite volume at any n where n = length of a side. {picks up D&D dice to prove it. Yup, finite volume. }

And since infinity extends in both directions, or all directions, and not just one direction your arbitrary single bound of natural numbers is yet another nonsensical limit that does not help in any way.
— Chet Hawkins

Do you not believe in natural numbers being bounded by 0 (or 1) on one end? And regarding time, isn't it widely believed that time had a beginning (meaning one boundary of time is t=0)? — keystone

Belief in such a concept has no relationship to stating that the concept is not useful in the example given. In REALITY the infinity goes, BOTH, MULTIPLE, OR ALL; ways at the same time. So, offering examples that do not match reality is ... a mirage ... which I thought was what you were trying to avoid or point out.

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. — keystone

Wow. Man. I read your post. I have no idea what you're talking about. A metric space has a metric. Some topological spaces are metric spaces. But a topological space without a metric can not have a sensible notion of distance. Topological spaces by definition are stretchable. But you can't measure distance consistently in them.

You have totally lost me. I don't know what point you are making.

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. — keystone

You lost me totally. I have no idea what your point is other than that you're stretching a topological space and noting that there's no sensible notion of distance.

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?

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. — keystone

I'm still concerned about that screwdriver ...

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

1) Topological spaces have no sensible notion of distance. — keystone

Perfectly standard.

2) Topological metric spaces have a sensible notion of distance. — keystone

By the definition of a metric space, right? Also perfectly standard.

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). — keystone

Yes ok.

4) If you lived inside a topological metric space, you'd perceive it as a metric space,[/quoute]

Little unclear. 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

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. — keystone

Yes that's true.

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'. — keystone

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

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? — keystone

Is it a topological space? Is there a metric? Then yes.

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. — keystone

Emergent objects become actualized? Bit vague for me.

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.

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. — keystone

Ok.

Is there disagreement or confusion on any of these points? — keystone

Not much disagreement, only confusion about where this is all going. It's perfectly clear that some topological spaces are metrizable and others aren't.

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 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. — keystone

I just don't see where you're going with all this. You're pointing out that some topological spaces aren't metrizable. Right?

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

Oh but there is one. For any universe or set, define a metric as follows: d(x,y) = 1 if x and y are different, and 0 if x and y are the same. This is known as the discrete metric.

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

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

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. — keystone

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

https://plato.stanford.edu/entries/holes/

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

I don't know what's common. Does it matter?

If you return to my photographs, — keystone

I did not understand the photos.

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. — keystone

Why is "what's more fundamental" important? Do you think I hold one view versus the other? What difference does it make?

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. — keystone

Ok I'll keep going as long as I can, but I feel like I'm going down in warm maple syrup.

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. Did I argue the contrary at some point?

From the bleachers: Hasn't MU spoken of the existence of continua prior to points? What of Bergson's notion of time being fundamentally duration? Perhaps not elastic. Is this where the discussion is going? Peter Lynds has written several papers on the non-existence of points in time, as well.

Back to the popcorn.

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).

No, I'm only talking about topological metric spaces. — keystone

A metric space is typically just called a metric space. There aren't "nontopological" metric spaces. Any metric space can be made into a topological space by defining the open sets in terms of the metric.

I'm pointing out that their metrics don't extend beyond their boundaries (meaning externally, they act like topological spaces without a metric), — keystone

This is kind of muddled. Typically we start with a set and put some structure on it -- a metric, a topology, whatever. It makes no sense to talk about "outside" the space till we say what set that is. For example, what's outside the real numbers. Well the complex numbers are, but so are all the animals on Old McDonald's farm. The complement of any set is the entire rest of the universe; and if you don't say what universe you're working in, you run in to the "set of all sets" paradox. The unrestricted complement of a set is not a set. So it would be good if you could clarify this point. What's outside your metric space of interest?

and internally, they have entirely geometric characteristics (meaning internally, they are indistinguishable from metric spaces without the topological aspects). — keystone

Metric spaces are indistinguishable from metric spaces, yes. But isn't that a trivial remark?

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. By analogy, if you wish to discuss what's outside the real numbers, you have to say if you're talking about the complex numbers, the quaternions, or everything in the entire mathematical universe, which turns out to not be a set. Because the set of all sets that don't contain themselves is a member of the "outside" of the real numbers. Hope I'm making this clear.

Interesting! Let's treat the Discrete Metric as a trivial metric, and by Universal Metric I'm referring to a non-trivial universal metric. — keystone

As it happens, the trivial topology is already defined as the opposite idea. The discrete metric has the most possible open sets. The trivial metric has the fewest open sets. Only the empty set and the entire space are open.

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

But you can't just eliminate the one metric that falsifies your idea, there could be other weird ones. You have to say exactly what you mean.

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

There's a whole SEP article on holes. Deep stuff.
— fishfry
Wow, it's a deeper topic than I imagined. — keystone

Holes are deep!

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. — keystone

Ok.

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: — keystone

You just ignored my comment and steamrollered over it. Why do I care which is more fundamental? I don't even know what that means. Sets are fundamental, then you add properties. That's how it works.

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. — keystone

Sets are fundamental, not points. Elements of sets are sometimes called points, but it's possible to do set theory without elements! All you actually need is to describe the relationships among sets, without regard for the internal contents of the sets.

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. — keystone

I can't imagine how you would get anything done that way. And you are not getting me to believe you have a coherent idea about it.

I've observed that orthodox mathematics predominantly favors the bottom-up approach. — keystone

Starting from sets, yes. Lot of mindshare the past century and a quarter. There's also type theory, which I imagine you'd see as another bottom up approach. I don't know what a top down approach to mathematical ontology would look like.

However, my informal exploration of the top-down method has revealed — keystone

Not to me. Maybe to you. You have not yet communicated to me what is a top-down development of math. How would you top-down construct or define the real numbers? Unless you mean axiomatically. Is that what you mean?

a perspective where everything seems to fit together perfectly, without any apparent disadvantages, paradoxes, or unresolved issues compared to the bottom-up view. — keystone

Where's the beef? That's handwavy, tells me nothing.

I'd like to share this perspective with you, — keystone

I'd like to hear it. What is a top-down construction of the real numbers? Of the integers? Of the number 6?

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). — keystone

A metric space is a metric space. If you are interested in metric spaces there's a large literature on the subject.

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

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.

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) — keystone

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

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. — keystone

I do not know what you are talking about now.

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. — keystone

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

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. — keystone

It's sets all the way down. In set theory everything is a set.

Points are just elements of a set. 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.

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

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. [/quoote]

I don't understand what you are doing. Seems like random flailing.

— keystone

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. — keystone

I'm fine.

By the way I wanted to mention that there are often two ways of describing a mathematical object, internal and external. 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.

Or, we can define the reals as the unique Dedekind-complete totally ordered set. That characterizes the reals without bothering to construct them. Perhaps you're getting at this.

You also talked about cuts, and perhaps you're interested in Dedekind cuts, which are used to construct the reals out of sets of rationals.

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

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

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.

Indulge me in an analogy.

I see the Matrix (pictures): — keystone

This entire idea was completely lost on me.

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. — keystone

Digital rain is more fundamental than the Matrix. That's very poetic.

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. — keystone

You know, it might be better if you would write a draft then edit it. This seems like stream of consciousness. It has a groovy vibe to it but it doesn't say anything.

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. — keystone

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?

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 | — keystone

No idea what you are trying to do, what you are doing, why you are doing it, and why you are telling me about it.

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: — keystone

Like a triangular section of the plane? Why?

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. — keystone

I just don't know what you're doing. You seem to be having fun, and I don't mind because this like taking a rest after the mortal combat of the staircase thread.

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) — keystone

Wasted on me, hope you got something from it.

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. — keystone

No idea, eyes glazed long ago.

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. — keystone

Quite a tall order.

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. — keystone

Well Euclid considered points fundamental, along with lines and planes. But modern set-theory based math takes sets as fundamental. In fact there is nothing other than sets. You start with the empty set and the rules of set theory and build up everything else.

The word point is only used casually, to mean an element of some set, or a tuple in Euclidean space, or a function in a function space.

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. — keystone

I'm just throwing things out that seem related to what you're saying.

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. — keystone

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?

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

No problem, take your time. I hope you and your guest have a lovely visit.

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)