For example, when I write "n∈ N", I don’t mean that n is an element of the actual infinite set of natural numbers. Rather, I mean that, it is a natural number according to the SB tree (details omitted). — keystone

Good luck with that. Probably of more interest to CS people.

The function x(n) — keystone

A sequence of rationals I assume. If you had two functions on Q then a suitable metric would be the supremum.

An actual curve is an indivisible, one-dimensional object with length but no width or depth. It extends continuously between two actual points but excludes the endpoints. — keystone

How do you define "continuous"? Are you sure it is indivisible?

Sorry, but your list of definitions is mind-numbing. Your top down is becoming way more complicated that bottom up, IMO. And the irrational numbers have yet to appear.

Your best bet would be to find a mathematician willing to deal with your arguments and pay him/her a fee to do so.

Good luck with that. Probably of more interest to CS people. — jgill

Yeah, my view leans heavily on algorithms.

A sequence of rationals I assume. — jgill

Yes. I have since edited the post to clarify this.

If you had two functions on Q then a suitable metric would be the supremum. — jgill

Suitable for what?

How do you define "continuous"? Are you sure it is indivisible? — jgill

Good point. I have since edited the post to clarify this. When defining an actual curve I was providing an informal intuitive explanation where I carelessly used 'continuous'. Ultimately an actual curve is simply an object having an actual interval.

Sorry, but your list of definitions is mind-numbing. — jgill

Isn't anything communicated with absolute precision a bit mind-numbing? Not that I achieved that level of precision, but it was trying to be more precise. I find logic much more mind-number, but that's just me...

Your top down is becoming way more complicated that bottom up, IMO. — jgill

Ultimately, it all reduces to the same calculus used by applied mathematicians today. However, building a foundation on constructive philosophy is likely to introduce more complexity—at least that's how it plays out in logic. Actual infinity is certainly simpler to work with, but is it truly sound? Newtonian mechanics is simpler than relativity, which is simpler than quantum mechanics. So, what should be the foundational choice for physics - the simplest? There's an elegance to QM and I believe the same can be said about the top down view of mathematics.

And the irrational numbers have yet to appear. — jgill

I’ve already outlined the framework for irrational numbers. Both potential coordinates and potential intervals are reinterpretations of real numbers, including irrational ones. If we get past the list of definitions then the next step is to present an example that demonstrates how irrational numbers come into play.

Your best bet would be to find a mathematician willing to deal with your arguments and pay him/her a fee to do so. — jgill

I've tried in the past, but nowhere else has been as beneficial as here. That said, I’m open to recommendations. It’s challenging for an amateur mathematician to find someone with the right skills and interests. I primarily used Upwork.com.

You could start with continuity described as a path in the Euclidean plane or complex plane taken by a moving particle. Or something similar.

There's an elegance to QM and I believe the same can be said about the top down view of mathematics — keystone

Careful. I would not compare if I were you.

If you had two functions on Q then a suitable metric would be the supremum. — jgill

Suitable for what? — keystone

For defining "distance" between functions. When I dabble in the complex plane contours become points in the metric space and the distance between them is the Sup|f(t)-g(t)| over 0<t<1 for instance.

If I were younger I might have more time to try to unravel your presentation. You have wandered from metric spaces to topology and now graph theory, with that dreadful SB-table trailing along. Then you have all these definitions which a mathematician is unlikely to find of interest.

Your best bet would be to find a mathematician willing to deal with your arguments and pay him/her a fee to do so. — jgill

I've tried in the past, but nowhere else has been as beneficial as here — keystone

Try a nearby university where a grad student might want a little extra cash.

Careful. I would not compare if I were you. — jgill

Good point. I've needed to learn this lesson too many times.

For defining "distance" between functions. — jgill

Since the functions I'm working with all converge, I don't believe the supremum is necessary for distance, but it might be necessary for other purposes.

Try a nearby university where a grad student might want a little extra cash. — jgill

I'll look into this. Thanks for the suggestion.

If I were younger I might have more time to try to unravel your presentation. — jgill

I’ve just revised the post to remove unnecessary mention of objects, making it shorter. If you skip the sections on the definitions of continuity, the post is only 444 words. I mention the continuity section because it's wordy but obvious. For instance, we already know that the interval ⟨0 5⟩ linked with coordinate 10 can’t be continuous, as 10≠0 and 10≠5, implying a gap between them. I just explicitly lay out all scenarios to capture the obvious. I hope you might reconsider giving it another look, but I completely understand if you choose not to continue. This discussion has already been incredibly helpful to me.

You have wandered from metric spaces to topology and now graph theory, with that dreadful SB-table trailing along. Then you have all these definitions which a mathematician is unlikely to find of interest. — jgill

I’ve admittedly wandered off track at times, and you've been patient with the many detours along the way. However, I’m a bit surprised that once I introduced a more mathematical approach—like discussing the Stern-Brocot tree and providing proper definitions—you felt the discussion was becoming less interesting to mathematicians. I had expected the opposite.

I propose that continuous calculus is not the study of continuous actual structures but rather the study of continuous potential structures. — keystone

@fishfry: Would you be open to re-engaging with me on this topic (that we discussed months back)? I believe the post quoted here will give you something much more concrete for you to chew on. Plus—no figures this time! I'd really appreciate your advice.

A 1D actual structure is a finite, undirected graph in which each vertex represents an actual point, pseudo point, or actual curve — keystone

A vertex represents an actual curve?

I’m a bit surprised that once I introduced a more mathematical approach—like discussing the Stern-Brocot tree and providing proper definitions—you felt the discussion was becoming less interesting to mathematicians. I had expected the opposite. — keystone

In fact, I had never heard of the S-B tree before it was introduced on this forum. It is not true that every mathematician will find every math topic interesting. (Wiki lists well over 25,000 if I recall). Had I been a number theorist or a CS person I may have known of it. I see it averages about 47 pageviews per day on Wiki, and classed as low priority. But that's not trivial by any means. My own page gets only 15.

A vertex represents an actual curve? — jgill

Each indivisible object, whether potential, pseudo, or actual, is represented as a vertex within a structure, regardless of its dimensionality. This approach underscores the fundamental indivisibility of these objects. The only object that is divisible is a structure.

In fact, I had never heard of the S-B tree before it was introduced on this forum. — jgill

Imagine how fortunate I (an amateur) feel to have stumbled across it (and Niqui's paper on arithmetic based on it)! :razz:

It is not true that every mathematician will find every math topic interesting. — jgill

Agreed.

I see it averages about 47 pageviews per day on Wiki, and classed as low priority. — jgill

Sometimes the significance of a discovery isn't recognized until many years later.

I see it averages about 47 pageviews per day on Wiki, and classed as low priority. — jgill
Sometimes the significance of a discovery isn't recognized until many years later. — keystone

It's had 164 years. We'll see. :cool:

Each indivisible object, whether potential, pseudo, or actual, is represented as a vertex within a structure, regardless of its dimensionality — keystone

From the point of view of graph theory, vertices are treated as featureless and indivisible objects, although they may have additional structure depending on the application from which the graph arises; for instance, a semantic network is a graph in which the vertices represent concepts or classes of objects.

(Wiki)

OK. But it gets a bit anti-intuitive. Hard to imagine a curve is a vertex. But then, I treat curves in the CP as points in a metric space.

I wish other mathematicians would chime in on this thread. I am very old and have forgotten what I didn't learn. :roll:

Hard to imagine a curve is a vertex. — jgill

An actual curve in 1D is unique in that it is fully defined by its endpoints. However, in 2D and higher dimensions, a curve is determined not only by its endpoints but also by an equation. Perhaps incorporating that equation into the vertex might make the concept more digestible.

I wish other mathematicians would chime in on this thread. I am very old and have forgotten what I didn't learn. — jgill

Yeah, that would be nice, but I really do appreciate you taking the conversation this far. You got me thinking!

However, in 2D and higher dimensions, a curve is determined not only by its endpoints but also by an equation. Perhaps incorporating that equation into the vertex might make the concept more digestible — keystone

I explore various properties of contours in the complex plane, defining a metric space whose "points" are contours. $z(t)=u(t)+iv(t),\,\,\,0\le t\le 1$.
"Distance" is defined $d\left( {{z}_{1}},{{z}_{2}} \right)=\underset{t}{\mathop{Sup}}\,\left| {{z}_{1}}(t)-{{z}_{2}}(t) \right|$. But here t is a positive real number, which you have not defined yet. Usually, the u(t) and v(t) are differentiable, giving a smooth curve. So incorporating this sort of thing into the definition of vertex assumes what you will probably wish to prove. I wonder what an "edge" in your graph would be?

But here t is a positive real number, which you have not defined yet. — jgill

A real number corresponds to a specific subgraph within a potential structure. In the 1D case, this is represented by a potential curve and the two potential points that are directly connected to it.

incorporating this sort of thing into the definition of vertex assumes what you will probably wish to prove. — jgill

Incorporating differentiability?

I wonder what an "edge" in your graph would be? — jgill

An edge signifies adjacency between objects. For example, in conventional interval notation, an edge would exist between the curve (0,5) and the point [5,5] due to their direct adjacency. In contrast, the curve (0,5) is not adjacent to (5,10) because a gap exists between them (at point 5), so no edge would connect the vertices representing the two curves.

A real number corresponds to a specific subgraph within a potential structure. In the 1D case, this is represented by a potential curve and the two potential points that are directly connected to it. — keystone

You've lost me. Guess it's time for me to quit. Overall, I think you have started down a path that is far too complicated for the desired result. However, if the result you seek is more philosophical than mathematical you may have something.

Overall, I think you have started down a path that is far too complicated for the desired result. — jgill

Someone could say the same thing about the epsilon-delta formulation of a limit, which was introduced to give calculus a more rigorous foundation. After all, infinitesimals produced the desired results and were simpler to work with.

Guess it's time for me to quit. — jgill

No worries. Thanks for the discussion!

Overall, I think you have started down a path that is far too complicated for the desired result. — jgill

Someone could say the same thing about the epsilon-delta formulation of a limit, which was introduced to give calculus a more rigorous foundation. — keystone

At first I thought this is not true, but that is because I followed a learning curve that incorporated analytic geometry before calculus, and this allows clear illustrations and examples en route to limits. You work in CS, however, and what seems like unintuitive definitions to me probably make more sense to you. Had I taken a course in graph theory what you are proposing might seem less opaque.

Continue, if you like, and I will comment from time to time as I learn more about graph theory.

I suggest this thread be placed in the Lounge since it obviously has limited appeal to the general audience, but has merit in philosophy of mathematics. Just my opinion.

You work in CS — jgill

I work in simulation as an engineer, not a computer scientist.

Had I taken a course in graph theory what you are proposing might seem less opaque. — jgill

I'm new to graph theory and currently only utilizing the basics. Even though the concepts I'm using are fundamental, they require a significant paradigm shift, which can be challenging to adapt to.

Continue, if you like, and I will comment from time to time as I learn more about graph theory. — jgill

I’ll give others some time to respond, which will also allow me to think through my ideas as I work on generalizing them across higher dimensions. But since my big post from 6 days ago was opaque to you I'm quite discouraged from continuing on.

I suggest this thread be placed in the Lounge — jgill

When you mention "the Lounge," are you referring to an actual place, or do you mean taking a pause in the conversation until others join in?

When you mention "the Lounge," are you referring to an actual place, or do you mean taking a pause in the conversation until others join in? — keystone

The "Lounge" is part of TPF.

But since my big post from 6 days ago was opaque to you I'm quite discouraged from continuing on. — keystone

The one trait I share with Feynman is understanding a concept through examples. You started with the imagery in 2D. Maybe more?

The one trait I share with Feynman is understanding a concept through examples. You started with the imagery in 2D. Maybe more? — jgill

I share that trait as well—I need some time to fully generalize my ideas. As I work through this, I’m noticing that some adjustments are necessary, so I’d like to minimize making changes mid-conversation. I really appreciate your continued interest, and I’ll be back soon. Thanks!