keystone

I find myself agreeing with very little of what I've read from MU. Seriously, am I not supposed to mention numbers? Or lines? Or relevant mathematical...

Ok, this was an excellent post! I better understand your criticism. It lies in the fact that I'm using the term 'interval' in an unorthodox manner. I ...

Why is it faulty? That's the thing - such continuous deformations don't affect the metric. They don't affect anything at all. That's why I say those t...

I'm not saying that U(0,.3)UU(.3,.5)U is homeomorphic to anything. Rather, that interval description describes paths which can be transformed into eac...

Continuous deformations cannot alter topology (i.e. convert the path to a different path). Let's illustrate with an example: imagine a class of 100 st...

I concur that rational numbers alone, represented as points, are insufficient for constructing a continuum. That's not the argument I'm making. You ke...

I learned of it and, as I delved deeper, my amazement grew. For instance, from a paper by Niqui, I learned that the Stern-Brocot tree can serve as a b...

You may not realize it but you are asking a loaded question. I believe in 'rational numbers' but not 'the rational numbers'. The difference is that 't...

I'm not suggesting that a single metric space can represent all continuous 1D systems. Rather, for each continuous 1D system we generate using a top-d...

I'm not quite sure what you mean by "believe in the rational numbers." From a top-down perspective, there's no need to assert the existence of either ...

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

I'm sure there are other ways to define the ordering of rational numbers, that's just my favorite. I thought I twice answered your question. Let me tr...

I take it you're not a fan of analogies. People once mocked movie scenes where detectives would enhance blurry security camera footage with a simple "...

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'...

I'm just getting around to responding to a few of your earlier comments: Wow, it's one of my favourite films. To each their own, I suppose. It seems w...

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 reinterpretati...

What should I call it? I did that to facilitate the straightforward definition of the metric. If you permit me to work within a metric space without n...

Thanks. I've made this change to my earlier post and noted that it has been updated. In a previous post I defined the elements of the enclosing set to...

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 poin...

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 ...

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 infor...

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

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

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

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) Sta...

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 fe...

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'...

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 ack...

Indulge me in an analogy. I see the Matrix (pictures): https://i.imgur.com/1obEIHt.png You see 'Digital Rain' (sets): https://i.imgur.com/rQpn7CD.png ...

Point taken. I need to bring this one picture back. https://i.imgur.com/MU11G5S.png Based on this picture, what I want to say is that Achilles can occ...

No, I'm only talking about topological metric spaces. I'm pointing out that their metrics don't extend beyond their boundaries (meaning externally, th...

I understand that as a trained mathematician, you have the ability to articulate complex ideas clearly using descriptive language. I admire that skill...

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 ca...

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

Should I abbreviate my explanation, you might resort to conventional thinking to bridge the gap, which could lead to misunderstanding. My goal is to p...

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 agree, I just wanted one post to set the stage before I get into it... Well, how much beef can one actually put in a paragraph? Have you ever sunk y...

Exactly. Let’s deconstruct the argument: (1) We accept Zeno's premise as valid, asserting that in a presentist world where only a single state exists,...

Good. Then we're on the same page!

Are you suggesting that supertasks cannot be completed? Agreed, but most importantly: (4) apply those intuitions to (the original) experiments. I like...

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, yo...

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, yo...

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 ...

STAIRCASE PARADOX I see your point, and I appreciate your analogy with the interval. However, you need to clarify what happens in the narrative. The p...

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...

I apologize if it seemed like I was implying anything about wishing for your death; that was not my intention at all. I specifically expressed a desir...

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

Max Planck once said "a new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its oppone...

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

STAIRCASE PARADOX This brings to mind Sagan's quote "extraordinary claims require extraordinary evidence." We start with an extraordinary premise—the ...