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
The function x(n) — keystone
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
Yeah, my view leans heavily on algorithms.Good luck with that. Probably of more interest to CS people. — jgill
Yes. I have since edited the post to clarify this.A sequence of rationals I assume. — jgill
Suitable for what?If you had two functions on Q then a suitable metric would be the supremum. — 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.How do you define "continuous"? Are you sure it is indivisible? — 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...Sorry, but your list of definitions is mind-numbing. — 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.Your top down is becoming way more complicated that bottom up, IMO. — 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.And the irrational numbers have yet to appear. — 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.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
There's an elegance to QM and I believe the same can be said about the top down view of mathematics — keystone
If you had two functions on Q then a suitable metric would be the supremum. — jgill
Suitable for what? — keystone
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
Good point. I've needed to learn this lesson too many times.Careful. I would not compare if I were you. — 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.For defining "distance" between functions. — jgill
I'll look into this. Thanks for the suggestion.Try a nearby university where a grad student might want a little extra cash. — 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.If I were younger I might have more time to try to unravel your presentation. — 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.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 propose that continuous calculus is not the study of continuous actual structures but rather the study of continuous potential structures. — keystone
A 1D actual structure is a finite, undirected graph in which each vertex represents an actual point, pseudo point, or actual curve — keystone
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
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.A vertex represents an actual curve? — jgill
Imagine how fortunate I (an amateur) feel to have stumbled across it (and Niqui's paper on arithmetic based on it)! :razz:In fact, I had never heard of the S-B tree before it was introduced on this forum. — jgill
Agreed.It is not true that every mathematician will find every math topic interesting. — 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
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
Each indivisible object, whether potential, pseudo, or actual, is represented as a vertex within a structure, regardless of its dimensionality — keystone
(Wiki)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.
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.Hard to imagine a curve is a vertex. — jgill
Yeah, that would be nice, but I really do appreciate you taking the conversation this far. You got me thinking!I wish other mathematicians would chime in on this thread. I am very old and have forgotten what I didn't learn. — jgill
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
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.But here t is a positive real number, which you have not defined yet. — jgill
Incorporating differentiability?incorporating this sort of thing into the definition of vertex assumes what you will probably wish to prove. — 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.I wonder what an "edge" in your graph would be? — 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. — keystone
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.Overall, I think you have started down a path that is far too complicated for the desired result. — jgill
No worries. Thanks for the discussion!Guess it's time for me to quit. — jgill
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
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