keystone

:up:

The following is for your amusement only. It's not worth our time to discuss the validity of this response. https://i.imgur.com/HCUkvxH.png https://i....

Agreed. My focus is on the philosophy underlying the rigorous mathematical foundations that support the informal mathematics used in physics. I have n...

I detect some sarcasm here. Isn’t the conventional approach to start with points, which are inherently discrete? How is that any less of a leap than s...

I can tell you've taken care to read my words closely. Even though we sometimes clash, I truly appreciate your effort. complete set of attributes cons...

My Claim: You asked me to define continua, which I assume was prompted by my earlier claim that "THE continua" is not actually used by applied mathema...

The following is not a formal, finalized definition, but I hope sets the stage for the discussion: A continuum is a finite, continuous object with the...

I keep writing a response to this and then deleting it. I'll be back on this ..

@"TonesInDeepFreeze": Apologies if this isn't the best forum etiquette, but I'm genuinely curious about your background. Would you mind sharing a bit ...

I believe his arguments indirectly captured the spirit of the Zeno's Paradoxes. I believe Zeno and MoK communicated their ideas informally but neverth...

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I hadn't lost interest but I needed time to reflect and read. Actually, I still need a lot more time to do that but this forum is too hard to resist. ...

@"MoK" I know your wording taken literally is in agreement with Tones' view, but were you originally trying to prove that "the continuum" does not exi...

I accept this definition of 'the' continuum. It's a definition after all so there's nothing to question. My issue is with using 'the continuum' to des...

I like this subtle distinction as it draws a clear line between your interests/arguments (related to the continuum) and mine (related to continua in g...

I had debated Enderton vs. Mendelson at the start and had landed on Mendelson because the topics in the TOC looked much more interesting. However, it'...

Yeah, cleaning training data is certainly a challenge that gets harder with time. I wonder if they'll end up giving pre-GPT data more weight. I'm opti...

Thanks for the analysis. That's disappointing. What's unfortunate is that my textbook by Mendelson has many examples yet provides answers to only a sm...

Both you and the other poster's definition of continua were point-based. I acknowledge that that's the standard mathematical treatment of 'mathematica...

Not only are people different, but times are changing. If now is not the moment for AI, the near future will be.

I suppose it depends on what we're talking about. If we're talking about developing original ideas on continua I think more than a basic understanding...

I don't know what chatbot you used and I'm not skilled enough at logic to evaluate the output of OpenAI's latest o1, but I'd be keen to know what you ...

I like your argument but I would say that the conclusion that follows is that D is an inadequate definition of a continuum. A continuum cannot be comp...

Sounds good!

I see no need to mention those topics any time soon. Without sufficient priming, you will almost certainly reject the 2D images I intend to share. I'm...

Our relationship was mostly one-sided, with me being the main beneficiary of our conversations. I'm glad to hear that you found some benefit in meetin...

Indeed, I went in another direction and 1D is boring. This is exactly what I want to return to. Here's what I'm thinking: If we keep going, I'll summa...

Well it's been a couple of months but we're finally at the state you predicted. I know you've been on the bleachers largely but now and again you've i...

You stuck with me for an incredibly long time, and while I wasn't specifically looking for a sounding board, it turns out that's exactly what I needed...

Did you try to make sense of my 'elevator pitch'? I wasn't communicating nonsense. It wasn't even my work I was talking about... Epsilon, eh? Will a f...

No, it's just a paper which perhaps only I believe is important. I'm guessing I'm the only person to have used the term 'Niqui arithmetic'. And it's o...

Niqui arithmetic: Niqui's method allows you to take as input a symbol and a pair of locations in an unlabelled tree and it returns a corresponding loc...

NIQUI ARITHMETIC Niqui arithmetic doesn't start by populating the nodes of a tree with fractions using the mediant operation. Rather, it doesn't assig...

As I was suggesting to fishfry, in the context of Stern-Brocot, what fundamentally exists is the algorithm for Niqui arithmetic. The illusion of the S...

EDIT: After posting this I realized that there might be some confusion about Niqui Arithmetic. I have since posted another message entitled NIQUI ARIT...

Please allow me to respond in the context of the SB-tree. Fractions correspond to nodes. Reals correspond to arbitrarily long paths (well, almost but ...

I'd like to distinguish between a fraction and a real. The fraction description is finite (e.g. \frac{1}{1}), whereas the real description is infinite...

I haven't made a point yet. I just wanted to clarify this as I previously found it a stumbling block in our conversation. I'm a computational fluid dy...

That is a fair understanding of my point but I do want to highlight one thing: it's not always about the computation. If I want to focus on algorithm ...

I'll try to do a better job this time. But first, one other area of confusion has been the distinction between infinite and arbitrary as it relates to...

I think it's fair to say that my top-down view likely fits under the constructivism umbrella, but my view does not represent constructivism as a whole...

From my perspective, we can only discuss objects that can be explicitly constructed. Since the complete output of the function cannot be generated all...

In that thread they state that "any set of sentences can be a set of axioms." I want to distinguish between what is (i.e. actual) and what can be (i.e...

This is my statement you're referring to: "I can write (0,1) as the union of arbitrarily many disjoint intervals. However, I cannot write (0,1) as the...

Bringing your original comment back... What I'm trying to convey is that f(0) U f(1) U f(2) U ... doesn't fit into any computer, just as complete infi...

You clearly have a lot of knowledge, and many of your posts, including your recent ones, are informative and well-intentioned. However, sometimes I fe...

Certain areas of mathematics, like combinatorics, are sufficiently distant from foundational issues and actual infinities. These areas transcend the l...

I'm putting this comment in a separate post because I wanted my main post to be 'smallish' and I'm not expecting a response from this. Here are quotes...