keystone

I believe the main issue is that new topics are added more often than old ones are removed, leading to bloated posts. I'll not respond to a few of you...

I was trying to convey that the representation itself is not important; what matters is the behavior. If in my mind x+x=2, then x behaves like 1. Simi...

Agreed. But sometimes the person will not see the flaw in their argument unless explicitly identified by someone else. By "digest" I didn't mean to su...

That's what I was trying to convey—I don't have axioms and formal rules, but rather concepts relevant to a typical practitioner of calculus. My approa...

Trying to make my ideas clearer so that your eyes might not glaze over has indeed helped me collect my thoughts. I've also benefitted in other of your...

I think only at that time would we enjoy talking with each other. https://www.youtube.com/watch?v=jreGFfCxXr4

Oh ok, that's great to hear. Yeah, sorry for the large number of line items... You're right, I'm likely a constructivist/intuitionist. I say 'likely' ...

I understand the previous post was lengthy, and I know you don't owe me anything. However, I wonder if this marks the end of our discussion. I'm unsur...

In my post directed to both you and fryfish, I either directly or indirectly addressed a lot of your shared points. Below are my responses to your uns...

In my post directed to both you and TonesInDeepFreeze, I either directly or indirectly addressed a lot of your shared points. Below are my responses t...

I've further refined my thinking on numbers and algorithms so please allow me to start from scratch here without us bringing in any of the out-of-date...

I'm very keen to respond but there's a lot to reflect on and respond to and I'm completely overloaded with work and family responsibilities so I might...

EDIT: In case you already read this post, I should mention that I made significant edits to it since it was originally posted. Yay!! Thanks a tonne :)...

I was trying to go along with your idea of engineering math. If the mention of an engineer and a mathematician working together doesn't help, then fin...

Thank you! I wish you very many wonderful years ahead.

Well I'd say infinite sets are pretty pervasive in modern mathematics. The reason is that sets are very useful and finite sets are very restrictive - ...

Yes and no. Yes - The person tasked to execute the cut is an engineer doing engineering math. He knows he'll never be able to cut the line exactly at ...

At the heart of my view is a simple idea: that infinity is a potential, not an actual. This idea leads to many consequences, one being that points are...

I believe the following: 1) The following two algorithms (written with a finite number of characters as infinite series) correspond to e and pi: e = \...

I'm sure you can appreciate the problem of substituting rational number approximations of irrational numbers too early in a computation. The best appr...

Yes. This is the whole idea behind discrete calculus, right? The analyst discretizes their model such that their machine can manage the computation. T...

Discrete calculus is certainly important to my view but it's not what I'm talking about. There are two steps to a cut: Step 1: Planning the cut with a...

With the top-down view we start with a continuous bundle of real numbers forming a line. In other words, we have one object (an interval), not 2^{\ale...

As I said earlier, I've got the points bundled into a continuous line, but not all of the points can be isolated. So if by 'completeness' you mean a l...

I don't have an alternate construction of the complete set of isolated real numbers. I don't have the intermediate value theorem or the least upper bo...

I'm really tempted to respond to all your latest comments, but you're getting impatient, so I'll hold back and move forward. In later posts, I aim to ...

Without seeing where I'm going with this, amidst all of my non-technical dialogue, it is admirable that he has stuck around for so long. Suffice it to...

LoL. I want to make it clear that a line cannot be constructed from/defined by infinite isolated points (numbers) or micro-lines (intervals). If that'...

As a sickly child, when I felt ill, I would imagine myself as heroically fighting severe illnesses, attributing my survival to extraordinary strength....

Why do you want to know?

Hello Sime, thank you for your comment. It prompted me to explore point-free geometry a bit online. I found this paper that adopts intervals instead o...

Thanks! Mostly better now. Allow me to further clarify my position. I can write (0,1) as the union of arbitrarily many disjoint intervals. However, I ...

I'm currently feeling unwell and will reply shortly. Cheers, and thank you for continuing this dialogue.

Based on your description: n=1 (1,0) n=2 (1/2,1/2) n=3 (1/3,2/3) etc. I don't follow. What exactly are you combining into a union? -------------------...

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

You persist in searching for an infinite set made up of tiny fundamental building blocks to assemble, like a mosaic. This bottom-up approach colors yo...

This sounds a lot like gatekeeping. Cesàro summations are very simple. Nevertheless, let's set aside Grandi's series for now. It doesn't have any rele...

I agree that it lacks a sum, but do you think that terms like Cesàro summation and Ramanujan summation are completely misnomers? Do you truly think th...

Please allow me to refine and restate my position on reals. Grandi's series has no sum but it should be 1/2. Analogously, I believe a line is not made...

Throughout our conversation, my perspective and how I express it have greatly developed, leading me to believe it's best to reformulate and clarify my...

Welcome back. I've transitioned from topology to graph theory, which (in this context) maintains similar concepts but is much simpler. To convert a pa...

I also think that's what Peirce was getting but that's definitely not what I'm getting at. Remember when I "trolled" you by introducing a scenario inv...

Path Length = Length of Lines + Length of Points Path Length = Length of Lines + 0 Path Length = Length of Lines So referring to row 3 of that figure....

Sorry, I didn't appreciate the point when you first mentioned it. Yes, I'm starting from classical Euclidean geometry. Yes. Formally the arithmetic is...

Yes!!! I agree with Euclid's definition of lines and points. I appreciate that he provides foundational definitions of both as separate, fundamental e...

No, I'm saying that within an open interval, such as (0,0.5), lies a single objects: a line. Absolutely no points exist with that interval. If you say...

You’re suggesting that my line, which already extends in space, requires additional points, which themselves individually have no length, to actually ...

So I figured out a better way to talk about this instead of using metric spaces. Instead, it is better to use Graph Theory. Consider the following pat...

I fully understand your criticism. The problem is that you are missing my point (or perhaps I should say you are missing my 'continua'). Let's continu...