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. — keystone

Please start any time. I simply have no idea what your overall point is, nor have I understood any of your examples. Start from the top. "I wish to reform the entire corpus of modern mathematics." Then tell me what are numbers, sets, functions, relations, etc.

I just can't figure out what you are doing.

Explain to me as you would if I were standing in front of you, what point you are making with the map.

• 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)

Although I didn't plan to start with directions and maps, I'm glad we ended up here. It's an excellent starting point. — keystone

I don't even get a mention now? That's the only way I know when someone's talking to me.

I don't know what you are talking about. I swear to God, I do not understand what you are doing, what you're talking about, why you're doing this. I am totally lost. I went back over the thread, i simply don't understand what you are talking about. And you flat out refuse to tell me. At one point you were talking about Achilles, so is this something to do with one of Zeno's paradoxes?

Why can't you just give me the top-line summary of what you are doing? A while back we were talking about metric spaces and topological spaces, that at least made some sense. The rest of this, the map, the grid, I just don't know what you are doing. I don't know what is the overall point being made, what I'm supposed to be getting from this. It's very frustrating.

I'll stipulate that you can traverse a grid. Or a line. Your coordinates have two components yet appear on a straight line. That's a little odd. What is your point?

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

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?

It's very frustrating. — fishfry

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

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.

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
— keystone

I find this incredibly annoying. Can't you get to the point?

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

I don't see the point.

I'm developing a framework that applies topological metric spaces to describe continua with arbitrarily fine precision. [ /quote[

Oh. Ok. I understand that. I appreciate this clear, simple statement of what you are doing.

Question: Don't the standard real numbers already do a fine job of exactly that?
— keystone

This might seem esoteric, — keystone

It's not esoteric, it's basic high school math. The real number line.

but achieving this involves turning everything upside down—without dismissing any past mathematical progress. This approach offers a powerful new perspective on mathematics. — keystone

You aren't making a case for that.

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). — keystone

I"ll stipulate to the real number line.

I'm still confused by your describing points on the real line with two coordinates.

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

— keystone

I get that you start at 0, land at .5, and end up at 1. Is that sufficient for your purposes?

I still don't see why you use two coordinates to describe a point on the real line.

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

It's an interval?? What? You are labeling locations on the real line as intervals? That makes little sense. Google maps doesn't do that.

No, that's not right. I referred back to your picture. You described the origin on the line as (0,0). What is the meaning of that?

You described the point commonly notated as .5 as (.5, .5). What am I supposed to take from that?

In fact the notation (.5, .5) is a degenerate open interval. It denotes the empty set. If I take (.5, .5) as interval notation, there are no points at all in it. Do you see that?

So we have two specific questions on the table.

1) What does (.5, .5) represent? In standard mathematical notation, it's the empty set. At best, [.5, .5] would simply be the point .5. But why do that?

2) Don't the standard real numbers already "describe continua with arbitrarily fine precision?" [

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?

Step two entails describing these intervals within the framework of a topological metric space. — keystone

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

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

I always engage directly with anything I understand. I don't dismiss the rest, I clearly say I don't understand. This has been true all along.

What does (.5, .5) represent?
— fishfry
Yes, it represents the point we would conventionally label 0.5. — keystone

And why do you do that, he asked ...

Step one involves defining the journey through the use of intervals. — keystone

Journey through the use of intervals, I don't understand. Not a dismissal. Direct statement that you said something I can't understand. Can't relate it to anything in my experience or knowledge.

Step two entails describing these intervals within the framework of a topological metric space. — keystone

It's just a metric space. It's like saying that I petted my cat mammal.

Secondly, "describing these intervals within" etc? First, they're not intervals. The notation (0,0) denotes an empty interval. So you are not communicating.

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. — keystone

You haven't defined any intervals, you have (0,0) and (.5, .5) and say these are intervals. But as intervals, they are both empty. They denote the empty set. You haven't got any intervals.

Secondly, to define a metric, you need a distance function that satisfies some properties. You haven't done that here. And there's already a perfectly good metric on the real numbers, the usual one.

The real numbers include points and intervals, and the usual metric is a perfectly good metric, and it's very easy to define. distance(x, y) = |x - y|.

Have you read this?

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

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). — keystone

All your intervals so far are degenerate, denoting the empty set. Do you understand that?

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? — keystone

No, because all of your interval notations denote the empty set and I can't figure out what you are doing. The usual metric on the real numbers seems perfectly satisfactory and you are somehow obfuscating it. You are being unclear. That's not a dismissal. I'm telling you that you are not saying anything clearly that I can figure out.

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

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?

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.

No problem, and thank you for the discussion. I will say that, in my view, the conflux of mathematics and magical thinking was formalized by Georg Cantor and has been nearly universally adopted in modern mathematics. If you believe that infinite sets cannot exist, then I am preaching to the choir. — keystone

Paradoxically, Empirical Science "facts" are believed to be true to the extent that they are reducible to mathematical ratios, or other incorporeal abstractions. According to some interpretations of irrational Infinity though, an infinite-sided die is not impossible, only supernatural, in the sense that you can imagine it, as an ideal concept --- e.g. a perfect multidimensional sphere --- but never reach-out and grasp it, in the real world. In what sense does that set of one "imaginary die" exist? :joke:

Math Magic :
Mathematics has a similar structure to certain conceptions of magic. It requires years of studying something entirely incorporeal, it seems to exist independent of the physical realm, it’s very powerful and has the ability to predict and influence the world around us, and it’s practitioners are BIZARRE.
https://www.quora.com/What-are-some-examples-of-math-being-magical

Imaginary Dice :
A ten-sided die of Fibonacci, imaginary, and irrational numbers used to abolish intellectual property
https://rollthedice.online/en/cdice/imaginary-dice

Since you're joining in on the conversation, can you tell me if anything I'm saying makes sense to you? — keystone

Not much, I fear. But I stand aside and try not to impede the ongoing discussion between you and @fishfry, who is more familiar with modern math topics that I am. I am curious about the role of elasticity you describe regarding one of Zeno's paradoxes. I assume this is somehow basic to where you are headed.

As I mentioned before, the metric between ordered pairs (x1,y1) and (x2,y2) is defined as follows: d((x1,y1),(x2,y2)) = | (x1+y1)/2 - (x2+y2)/2 | — keystone

Is (x1,y1) a point in the Euclidean plane? That's a standard designation. Then (x1+y1)/2 requires some kind of a projection of the y1 down upon the x-axis in order for this expression to represent a midpoint of a line segment. Or, is (x1,y1) an interval on the real line? Perhaps use (a1,b1) instead to keep the level of confusion minimized. Maybe my colleague sees something here I don't. Please carry on.

According to some interpretations of irrational Infinity though, an infinite-sided die is not impossible, only supernatural, in the sense that you can imagine it, as an ideal concept --- e.g. a perfect multidimensional sphere --- but never reach-out and grasp it, in the real world. In what sense does that set of one "imaginary die" exist? :joke: — Gnomon

What is "irrational infinity"? Infinite sided die seems like a sphere in 3D.

Mathematics has a similar structure to certain conceptions of magic. It requires years of studying something entirely incorporeal, it seems to exist independent of the physical realm, it’s very powerful and has the ability to predict and influence the world around us, and it’s practitioners are BIZARRE — Gnomon

Does magic influence the world around us? Wow, what bizarre powers I wield! :cool:

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]

"In a metric, the distance between two distinct points as always positive."


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

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.

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 — keystone

I don't know what you are talking about. You and @fishfry can sort all of this out if he is willing. Good luck.

What is "irrational infinity"? Infinite sided die seems like a sphere in 3D. — jgill

Sorry, I misspoke. According to the opinion below, Infinity is not a natural or real number, hence the rational vs irrational labels do not apply. Does that agree with your understanding?

Since infinity is un-real and non-dimensional, I assume the infinite die would be a spheroid in all non-real non-dimensions. :joke:

Infinity neither rational not irrational :
Infinity can be expressed as any fraction a0 where a is a natural number. Due to the denominator being zero, it is not rational. An irrational number is a real number that is not rational. As infinity does not exist in the real number system, it is not irrational.
https://www.quora.com/Is-infinity-a-rational-or-irrational-number

Does magic influence the world around us? Wow, what bizarre powers I wield! :cool: — jgill

I think the quote was merely making an analogy between Magic & Math --- not to be taken literally. However, perhaps you can apply your bizarre mathematical powers in a "possible world". :nerd:

What is the relationship between mathematics and reality?
Answer: When plugged into a possible world, mathematics gives us the tools to analyze the logically possible outcomes. Therefore, when a possible world that is expressed mathematically sufficiently aligns with reality, mathematics becomes effective at expressing relationships and outcomes.
https://brainly.ph/question/31124265

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. — keystone

Ok, that at least makes sense. But why denote the point 0 as [0,0]? Isn't that obfuscatory and confusing?

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

The union covers the closed unit interval. I don't know what a journey is. If you mean what's mathematically called a path, I'm fine with that. If you mean that you can get from 0 to 1 by first going from 0 to 0, then from 0 to not quite .5, then jumping to .5, then jumping from just above .5 to just before 1 ...

How does all this jumping take place? To get from (0, .5) to 1 involves taking a limit. How do you do that in your journey-mobile?

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... — keystone

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

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) — keystone

How do you accomplish those limit jumps? As a set-theoretic union they're fine, but as a "journey" you have a problem. How do you get from an open interval to its limit?

[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. — keystone

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. — keystone

Just wondering about those jumps to the limit.

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] — keystone

How do you get from (0,0.5) to [0.5,0.5]?

Do you understand that any point in [0.5,0.5] is a nonzero distance from .5? How do you jump that gap?

Mathematically, you take a limit.

But how does your "journey" take a limit?

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.

From the Bleachers: Does

$[\pi ,\pi ]\cup (\pi ,\pi +1)\cup [\pi +1,\pi +1]\cup (\pi +1,5)\cup [5,5]$

exist in your system? Or are you assuming rational numbers only?

Is

$[5,5]\cup (5,5+\varepsilon )\cup [5+\varepsilon ,5+\varepsilon ]\cup (5+\varepsilon ,5+2\varepsilon )\cup [5+2\varepsilon ,5+2\varepsilon ]\approx 5$

for small epsilon?

Step 5: Arrive at point 0. — keystone

?

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.

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.

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? — keystone

A metric space. A metric space is already a topological space, as a cat is already a mammal.

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. — keystone

Calling 0 by the name 0 would be far less confusing. Who was it that said that when discussing transcendental matters, be transcendentally clear. Looked it up, Descartes. Smart guy. You have the burden of being as clear as you can possibly be.

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. — keystone

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?

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. — keystone

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] — keystone

I apologize. All this makes my eyes glaze. It makes no sense to me.

And since you denied believing in the real numbers, I don't know what those symbols mean. Perhaps you can start there.

But look. I asked you this last time.

"Step 2: Travel the continuum (0,0.5)
Step 3: Arrive at point 0.5."

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.

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. — keystone

Even taken at face value, I fail to understand how posting stills from the movie relates to anything we're discussing. And like I said, humans make lousy batteries. So the premise of the film is wrong. I agree with it as a metaphor for media and government treating us all as tax cattle.

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. — keystone

a and b is less confusing than x and y? I better go back and re-read the thread.

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. — keystone

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

As a number, pi, is inseparably tied to actual infinity, — keystone

Pi is a computable real number and only encodes a finite amount of information.

• 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.

0 and 0.5 have distinct positions on the Stern-Brocot tree. — keystone

You're taking that as fundamental?

I like football but these picture posts aren't doing much for me. We were at least having the same conversation about getting from 0 to 1 on the real line. Then you said you don't believe in the real numbers, and then you declined to respond when I asked you twice how you get from (0, .5) to .5 without invoking a limiting process. And now you're changing the subject.

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.