I'm done with our conversation.

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!

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?

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!

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.

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!

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 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.

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.

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.

EDIT: LOOKING BACK THIS POST HAS ERRORS. IF YOU CARE TO RESPOND PLEASE LET ME KNOW AND I'LL FIX. THANKS.

: While working on my response, I realized it made the most sense to start from the beginning, using clearer and more descriptive terms and definitions. Looking back, I believe this post aligns with the kind of response that @TonesInDeepFreeze and @fishfry were looking for in this thread and in our previous thread, respectively. I hope the length is balanced by enough clarity to make for a fast read. I believe this sets up the foundation for a calculus free of any connection to “actual” infinities. I propose that continuous calculus is not the study of continuous actual structures but rather the study of continuous potential structures.

Definition: Ideal Point
In 1D, a ideal point is -∞ or ∞, such that -∞ is less than any rational number and ∞ is greater than any rational number.

Definition: Actual Point
In 1D, an actual point is a rational number.

Definition: Actual Curve
In 1D, a actual curve is doubleton set {a,b}, where a and b are either actual or ideal points.

Definition: Simple Functions on Actual Curves in 1D
Lower bound function, L: Actual Curve {a,b}→min(a,b). The lower bound of actual curve {a,b} is min(a,b).
Upper bound function, U: Actual Curve {a,b}→max(a,b). The upper bound of actual curve {a,b} is max(a,b).
Length function, d: Actual Curve {a,b}→|b-a|. The length of actual curve {a,b} is |b-a|.

Definition: 1D Actual Structure
A 1D actual structure is a finite, undirected graph in which each vertex represents an actual point, ideal point, or actual curve. Ideal point ∞ and ideal point -∞ must be included. Edges connect these vertices to indicate adjacency between the objects.

Definition: Continuity of 1D Actual Structures
A 1D actual structure is continuous if it satisfies the following continuity requirements:
1. Connections Involving Actual Points:
Each vertex representing an actual point q must be linked with exactly one vertex representing an actual curve for which q is the lower bound and one vertex representing a actual curve for which q is the upper bound.
2. Connections Involving Actual Curves: Each vertex representing a actual curve {a,b} must be linked with exactly one vertex representing actual/ideal point a and one vertex representing actual/ideal point b.
3. Connected: There exists a path between any two vertices.

Definition: Convergence
Convergence of a function: The function x(n): Natural number →Actual Point converges if a constructive proof demonstrates that for any ε>0, natural number N can always be found such that for any n>N, |x(n)-x(N)|<ε.
Convergence of a function to actual point a: The function x(n): Natural number →Actual Point converges to actual point a if a constructive proof demonstrates that for any ε>0, natural number N can always be found such that for any n>N, |x(n)-a|<ε.
Convergence of a function to rational number a: The function x(n): Natural number →Rational Number converges to rational number a if a constructive proof demonstrates that for any ε>0, natural number N can always be found such that for any n>N, |x(n)-a|<ε.
Convergence of a function to another function y(n): The function x(n): Natural number →Actual Point converges to y(n): Natural number →Actual Point if a constructive proof demonstrates that for any ε>0, natural number N can always be found such that for any n>N, |x(n)-y(n)|<ε.

Definition: Potential point (reinterpretation of a real number)
In 1D, a potential point is a function p(n): Natural number→Actual Point such that p(n) converges.

Definition: Potential curve (alternate reinterpretation of a real number)
In 1D, a potential curve is a function c(n):Natural number → actual curve such that L(c(n)) and U(c(n)) converge, and d(c(n)) converges to rational number 0.

Definition: 1D Potential Structure
A 1D potential structure S(n), where n is a natural number, is a finite, undirected graph whose vertices represent:

• Ideal Points: Pseudo point ∞ and ideal point -∞ must be included.
  Actual Objects: Actual points and actual curves.
• Potential Objects: At least one potential point or curve (all of which depend on n), such as a potential point p(n) or a potential curve c(n).

Edges connect these vertices to indicate adjacency between the objects.

Definition: Continuity of 1D Potential Continuum
A 1D potential structure is continuous if it satisfies the following continuity requirements:
1. Connections Involving Actual Points: Each vertex representing an actual point q must be connected to two vertices: One for which q is the lower bound, either:

• A vertex representing a actual curve {a,b}, where L{a,b}=q.
• A vertex representing a potential curve c(n), where L(c(n)) converges to q.

And one for which q is the upper bound, either:

• A vertex representing a actual curve {a,b}, where U{a,b}=q.
• A vertex representing a potential curve c(n), where U(c(n)) converges to q.

2. Connections Involving actual curves: Each vertex representing a actual curve {a,b} must be connected to two vertices: One for its lower bound, either:

• A vertex representing an actual/ideal point L{a,b}.
• A vertex representing a potential point p(n), where p(n) converges to L{a,b}.

And one for its upper bound, either:

• A vertex representing an actual/ideal point U{a,b}.
• A vertex representing a potential point p(n), where p(n) converges to U{a,b}.

3. Connections Involving Potential Points: Each vertex representing a potential point p(n) must be connected to two vertices: One for which p(n) is the lower bound, either:

• A vertex representing a actual curve {a,b}, where p(n) converges to L{a,b}.
• A vertex representing a potential curve c(n), where p(n) converges to L(c(n)).

And one for which p(n) is the upper bound, either:

• A vertex representing a actual curve {a,b}, where p(n) converges to U{a,b}.
• A vertex representing a potential curve c(n), where p(n) converges to U(c(n)).

4. Connections Involving Potential Curves: Each vertex representing a potential curve c(n) must be connected to two vertices: One which bounds L(c(n), either:

• A vertex representing an actual point a, where L(c(n)) converges to a.
• A vertex representing a potential point p(n), where L(c(n)) converges to p(n).

And one which bounds U(c(n)), either:

• A vertex representing an actual point a, where U(c(n)) converges to a.
• A vertex representing a potential point p(n), where U(c(n)) converges to p(n) .

5. Connected: There exists a path between any two vertices in the graph.

I was speaking of ordinal numbers beyond the naturals. Our definitions of "actual" infinities differ. No big deal. — jgill

Natural number arithmetic does not involve infinities, yet natural numbers are inseparably tied to ${\aleph_0}$. In a similar vein, I argue that real calculus is inseparably tied to $2^{\aleph_0}$. My interpretation of the orthodox philosophy is that both ${\aleph_0}$ and $2^{\aleph_0}$ represent "actual" infinities because they are used to describe complete objects, such as sets. It is in this sense that I refer to orthodox calculus as being tied to "actual" infinities.

It might appear that you are moving in the direction of Discrete calculus. — jgill

The ideas I'm proposing are fundamentally centered on continuous calculus. Concepts like continuity, real numbers, and limits are crucial to my perspective—I simply interpret them through a different lens.

But go ahead. I am curious. — jgill

Great! I'll continue in my next post, though it might not be today as I'm starting to feel tired.

Elementary calculus does not require "actual" infinities. It gets along quite well with unboundedness, or what you might call potential infinity. — jgill

I agree that calculus can work quite well with the concepts of unboundedness and potential infinity, but 'actual' infinities are implicitly assumed throughout the standard treatment. The standard treatment is built on $\mathbb{R}$, the complete set of real numbers—implying an actual infinite amount of numbers and points. As a result, when interpreting the notion of a tangent, one is inevitably led to paradoxical ideas like instantaneous rate of change. When interpreting the notion of area, one is inevitably led to the paradoxical idea that events with zero probability can still occur (dartboard paradox).

Ultimately, calculus is currently treated as the study of objects at the limit, rather than the unbounded process of "approaching" the limit (I use approaching in quotes because that word suggests that there's a destination which I do not believe in). I aim to establish a foundation focused on the journey rather than the destination (i.e. the algorithms themselves rather than their output).

As I have said before, I have written many papers and notes without ever becoming transfinite. — jgill

Have you written calculus papers/notes that are not (implicitly or explicitly) built upon infinite sets like $\mathbb{R}$?

By "actual infinity" I suppose you mean a kind of number that can be manipulated by arithmetic processes. — jgill

I view transfinite cardinal and ordinal numbers as crucial for understanding the nature of infinity, and, as you know, they can be manipulated through (special) arithmetic processes. However, I take issue with using transfinite numbers to describe actual abstract objects rather than potential abstract objects. For instance, (assume we live in an infinite world*) and consider a computer program designed to input any natural number, n, and output the set of the first n natural numbers. (In an infinite world*) the program has the potential to output a set larger than any natural number so the potential output has a cardinality of ${\aleph_0}$. But that program (even in an infinite world*) cannot actually output a set with a cardinality of ${\aleph_0}$. Potential is important and I feel like it's been forgotten in our Platonist world.

*I don’t actually believe in an infinite world, but I’m suggesting that mathematics allows us to speak in general terms without assuming any specific limits.

This is either very deep or shallow gobblygook. — jgill

Deep gobblygook is not an option? :razz: If you're following what I'm saying, a discussion on calculus is not that far off. I just need one more post to provide a formal definition of a real number and then we can advance to 2D. I’d be very interested to hear your thoughts on whether my view contains an implicit actual infinity or if it might be insufficient as a foundation for basic calculus. I'm certainly benefiting from this discussion but I understand that one should not entertain gobblygook for too long.

So far I'm not seeing anything beyond a line segment between two points that converge to one. From a continuum to a point. Why should one care about this? — jgill

For the moment, please treat 1/1 and 1.0repeating as distinct objects. Without bringing in the SB tree, let me just say that the former is a fraction represented by a string with finitely many (three) characters, while the latter is a real number represented by a string with infinitely many implied characters.

In my view, when numbered, k-points must have fractional values without exception. It is meaningless to speak of a k-point with a real number value because such a k-point cannot be defined within this framework. Therefore, it is incorrect to claim that a sequence of k-curves converges to a real-numbered k-point.

In my recent posts, I have been establishing that real numbers instead describe potential k-curves. These can be thought of as k-curves yet to be constructed, which, when constructed, will have the potential to become arbitrarily small (but not zero length).

This shift in perspective moves the focus away from a philosophy centered on the destination—limit objects like irrational points—and instead emphasizes the process itself, described by algorithms. By shifting from the destination to the journey, the need for actual infinity disappears. Our discussion sets the necessary groundwork for establishing a calculus that operates without invoking actual infinities.

Cauchy sequences themselves are infinite sets. — TonesInDeepFreeze

I agree. However, the main point of my post was to clarify that I'm not working with Cauchy sequences themselves, but with the algorithm used to construct any arbitrary term. In my figure, I highlighted the Cauchy sequence and noted, '0.9 repeating is not this.' In the subsequent figure, I highlighted what I believe 0.9 repeating actually represents and following that I expanded on this in bold.

I see a mistake in your last figure, typo probably. And I assume -1/0 (meaningless) designates negative infinity, however you define that — jgill

Apologies for the typo. Also, I initially used -1/0 to represent negative infinity because that’s how it appears in the Stern-Brocot tree, but since we’ve skipped over discussing the SB tree, I’ll switch to the more familiar notation.

I see nothing of interest so far. — jgill

I intentionally kept things uninteresting to maintain a sense of familiarity. Now, I'll begin to diverge from the familiar, which will hopefully make things more interesting. Here's part 3...

In my view, 0.9 repeating does not actually correspond to all the infinite highlighted k-curves in the image below, simply because no k-continuum beyond 3 actually exists, as none have been constructed yet. In the spirit of constructivism, one is not justified to use ellipses to represent the completion of infinite work.



Instead, 0.9 repeating represents the following highlighted object in the generalized diagram.



I cannot call that highlighted object a k-curve because, until n is assigned a specific natural number, the object it describes is not yet a k-curve. The same applies to the other objects and labels in the figure, so I will introduce some new terms (nothing fancy, just adding "potential" in front).



Essentially, I’m proposing that 0.9 repeating corresponds to a potential k-interval, which describes a potential k-curve. What I'm leading towards is framing calculus not as the study of actual objects (such as fully constructed k-continua and its constituents), but as the study of potential objects (such as potential k-continua and its constituents), where some or all of the labels remain in algorithmic form for as long as possible. Of course much more is needed to be said about this.

But first, I've been overlooking the fact that real numbers are typically defined as equivalence classes of Cauchy sequences, not just individual Cauchy sequences. In this context, equivalence classes introduce another actual infinity which needs reinterpretation, but let's save that discussion for a future post.

(Aside: If I had the opportunity to redo some earlier posts, instead of k-objects vs. potential k-objects, I would use actual objects vs. potential objects, getting rid of the k- prefix altogether. But I suppose it's too late to make that change now...)

Move on to 2. — jgill

First, I'd like to point out that this part (Part 2) takes some liberties with actual infinities for explanatory purposes (and to keep my individual posts sufficiently small), but these will be addressed and resolved in Part 3. Let's explore the meaning of the real number 0.9 repeating from my perspective. For now, let's set aside equivalence classes and represent 0.9 repeating as the following Cauchy sequence of k-intervals:



Term n in this sequence is defined according to the following equation:



As depicted below, term 1 describes a k-curve in k-continuum 1, term 2 describes a k-curve in k-continuum 2, term 3 describes a k-curve in k-continuum 3, and so on. Generally speaking, term n describes a k-curve in k-continuum n.



A real number, such as 0.9repeated, doesn’t correspond to a single k-point (as a bottom-up view would have it) but rather 0.9repeated corresponds to an infinite sequence of k-curves, shrinking in size as you progress deeper into the sequence in the spirit of Cauchy. [In Part 3, I’ll adjust this explanation to avoid implying the existence of actually infinite sequences].

You are a sinkhole. — TonesInDeepFreeze

Actually, I think you're the sinkhole. You seem to enjoy destructive conversations.

You need to define "1D analogue of the established term "planar diagram"" in terms that don't presuppose any mathematics that you have not already defined and dervied finitistically and such that it justifies such verbiage as about "embedding in a circle". — TonesInDeepFreeze

I'm working with standard finite graphs, nothing unorthodox about my use of them. As such, I don't need to produce an original definition of them. If you don't like how the informal definition of 'planar graph' uses the word plane then you can instead use Kuratowski's theorem. Admittedly, I haven't studied Kuratowski's theorem...

I don't need to waste my time and energy on you. — TonesInDeepFreeze

It's ironic that you got cold right after I went back, carefully studied, and addressed your comments on topology. That feels harsh, but I suppose I shouldn’t be surprised. In any case, I appreciate the times when you were helpful. We all have limited time, and it’s important not to spend it on things we don't want to do. Wishing you all the best.

Indeed, with the very first predicate 'is a continua' still not fully defined, you've piled on a big mess of more of undefined terminology and borrowing of infinitistic objects while you claim to eschew infinitistic mathematics. — TonesInDeepFreeze

You raised a single issue with my response, which I immediately clarified-specifically, that by "1D drawable," I simply meant a 1D analogue of the established term "planar diagram". You haven't given me a good reason for you to drop out. If your offer to help was sincere, you wouldn't back out the moment I sneezed.

Since you've been gone, the discussion with jgill has allowed me to clarify my position to the point where (I think) he understands what I mean by k-continua. I am not spouting nonsense or doubletalk. You haven't identified any falsity in my current position. Please, give me a chance.

1. should be interesting. — jgill

Perhaps I'll head in this direction and see what you think...

Intuitionism math perhaps. — jgill

I don’t have much experience with logic yet, but from what I know, my perspective seems to align well with intuitionism. My plan is to begin by learning classical logic as a foundation and eventually explore intuitionism.

You have density, but then continuity is next...I thought you were defining these lines as continuous. Fundamental objects. — jgill

Contrary to what my last post may have suggested, in the 1D context, there is always a k-curve between neighboring k-points (i.e. k-points are not densely packed) and k-curves are indeed continuous. Please allow me to clarify:

• Each k-point is assigned a rational number.
• Each k-curve is assigned a k-interval to denote endpoints to which it continuously connects (endpoints excluded). A k-curve which connects k-points a and b is describe by the k-interval <a b>.

Consider the following 3 example k-continua (please note that I'm using 1/0 to denote infinity):



Every possible 1D k-continua can be described using a combination of rational numbers and k-intervals.

ASIDE: When I label a k-continuum using rational numbers and k-intervals, I'm not merely assigning arbitrary strings of characters, but rather indicating a specific structure/ordering—please forgive me—derived from the Stern-Brocot (SB) tree. In fact, the three examples above correspond to the top three rows of the SB tree. I understand you’d prefer not to delve into the SB tree, and as long as you don't question the meaning behind my rational labels, I think we can steer clear of it.

I suppose I see some sort of a way to move forward by taking a lattice graph over an area and allowing the number of vertices and edges to increase without bound leading to a countable number of points in the area. — jgill

Instead of discussing 2D continua and area, let’s simplify by returning to 1D continua and length. Length is not a property of an infinite collection of k-points, but rather an intrinsic property of a single k-curve. This should become clearer once we introduce rational numbers into the discussion.

But this would be inadequate regarding the reals. But you might be able to push into the irrationals some way. — jgill

Irrational numbers will require special treatment, but I believe a treatment inspired by Cauchy sequences will largely address the challenge.

So far it appears everything you have given is uninteresting from a math perspective. — jgill

By introducing the fundamental k-objects (such as k-points, k-curves, k-surfaces, and so on), I've laid out the fundamental building blocks of the top-down approach. I acknowledge that these ideas so far may seem unremarkable, akin to someone attempting to build bottom-up mathematics by focusing solely on the successor function and not doing anything with it. However, if my latest figures made sense, the mundane part is behind us, and we can now move on to more interesting territory.

I don't think you will get a reaction from anyone but me until you produce a plan moving forward from your images of edges, vertices and surfaces. What is your goal and how do you plan to proceed? — jgill

My discussions here rarely go as planned, so please take this plan with a grain of salt:

1. Rational Numbers – Describing any arbitrary 1D k-continua entirely using rational numbers.
2. Real Numbers Part 1 – Describing potentially infinite sequences of 1D k-continua using rational and irrational numbers.
3. Real Numbers Part 2 – Shifting focus to the algorithm for constructing sequences rather than the impossible task of constructing a complete sequence.
4. Real Numbers Part 3 – General definition of a real number
5. Cardinal Numbers – Applying transfinite cardinal numbers to describe potentially infinite processes, avoiding the need for actually infinite sets.
6. 2D Part 1 – Extending the 1D concepts to their 2D analogues.
7. 2D PT 1 - Derivative and Reinterpreting Motion
8. 2D PT 2 - Integral and Reinterpreting Length
9. Ordinal Numbers – Offering a reinterpretation of ordinal numbers in the context of potential infinity.

GOALS:
1. To provide a top-down foundational framework for basic calculus that avoids reliance on actual infinities.
2. To argue that the philosophical issues in quantum mechanics arise from bottom-up mathematical intuitions. Physics at a foundational level is inherently top-down, and by developing new intuitions grounded in top-down mathematics, these philosophical issues in QM can be resolved.

I don't think you will get a reaction from anyone but me — jgill

I'm eager to move forward with this plan if you're open to it. There's no commitment to a lengthy discussion—we can take it one step at a time, and you're free to end the conversation at any point along the way. Of course, if you'd prefer to wait for someone else to potentially lead the discussion, I fully respect that decision as well.

You have done your imagery very well. I will wait and see what comes next. — jgill

I understand that you prefer not to lead the conversation, but I want to sincerely thank you for asking thoughtful questions that have helped me better articulate my perspective. I hope it's now in a form that TonesInDeepFreeze will be willing to engage with.

@TonesInDeepFreeze, would you consider taking a look at my recent message to jgill? The graph I described there represents a k-continuum, partly because it is a planar graph. For instance, if there were an edge connecting vertex 1 to vertex 8, it would no longer be planar and, therefore, wouldn't describe a k-continuum.

Your second figure is bewildering. Maybe go back to 1D and explain the real numbers as you see them. Expressions like k-vertex instead of point are confusing. — jgill

Thank you for taking the time to try to understand the figure and for sharing what didn’t make sense. I now realize that I skipped too many steps and made the figures more complex than necessary. Please allow me to make another attempt, and I hope this version is clearer. Also, I’ve decided to stop using the terms 'k-vertex' and 'k-edge.' The graphs I’m working with follow standard properties, so I’ll simply refer to them using the usual terms: vertex and edge.

• k-continua are composite structures formed by unifying fundamental objects, as illustrated in the exploded view below.

• I want to emphasize that when these fundamental objects are unified, there is no overlap between them. This is because k-curves exclude their endpoints, and k-surfaces exclude their boundaries.

• The purple arrows between pairs of objects indicate that, once the exploded view is collapsed, that pair of objects is connected.

• Edges can be cut (i.e., removed) to partition the graph representing k-continua, but nothing else can be cut—not vertices, nor k-objects.

How all this simplifies normal calculus is questionable. — jgill

I was initially planning to respond with a high-level overview of how this relates to calculus, but I realize that discussing it now would likely cause more confusion until the figures are clear. On further reflection, I think it was really helpful that you asked for the 2D explanation, as the 1D version is too simplistic and can easily lead to confusion between edges/vertices and k-curves/k-points. Introducing k-surfaces forces me to better distinguish the fundamental objects from the graph objects.

Edit: I wrote a response here but I want to reflect further on it so removing...

Why resort to graph theory and call a simple line an edge? — jgill

The edges do not represent objects like lines; rather, they signify connections. When I say that edges can be cut, I mean that these connections can be severed. The edges themselves do not contribute to the 'extension' of k-continua. In the 1D case, the only fundamental objects that have extension are k-curves and they cannot be partitioned. Within graphs they are represented by k-vertices.

Is this an effort to enhance an almost trivial concept of line and point? — jgill

This approach aims to elevate lines to the status of fundamental, indivisible objects, thereby eliminating the need for actual infinity. As long as points remain the foundational geometric elements, an actually infinite number of them will always be required to describe continua and paradoxes such as the Dartboard Paradox will remain.

Again, why not go to 2D? Maybe your ideas will make more sense in that context. — jgill

Well, I think it's important to first establish the meaning of real numbers within a 1D context with my view. But you've asked for 2D multiple times so let me give it to you with no numbers or equations.

The image below depicts a 2D k-continuum, composed of 31 fundamental objects: 10 k-points, 15 k-curves, and 6 k-surfaces. (If it's not clear what each of the 31 fundamental objects are, let me know and I can create an 'exploded view' image.) In the case of k-curve 7, it continuously extends from k-point 14 and k-point 2 without including the endpoints. While greater mathematical value could be extracted by associating each of these objects with specific equations, we are not at that stage yet. For now, the value lies in its particular structure.



The image below is a mathematical graph representation of the k-continuum depicted above capturing its structure (i.e. connections). The edges connecting the vertices indicate the adjacencies between fundamental objects. For instance, since k-surface 6 is adjacent to k-curve 7, there is an edge linking them. Importantly, these edges represent only the connections between objects, not the objects themselves.

Can a vertex be partitioned? Like saying a point can be partitioned. — jgill

No, k-vertices cannot be partitioned. The only objects in these graphs that can be 'cut' are the edges.

Sorry, it looks like you are taking a line segment and dividing it into two smaller segments. Then comparing. If you think there is something significant here you had better present a philosophical argument supporting it. — jgill

I am doing quite the opposite. K-continua can only be partitioned at the edges connecting the vertices. As such, k-continuum 1 cannot be transformed into k-continuum 2. Instead, k-continuum 2 can be manipulated (entailing unions) to reflect the structure of k-continuum 1. This distinction is philosophically significant because it enables us to work with and construct continua without the need to invoke actual infinities.

There is virtually no mathematics so far. — jgill

I understand this is a fairly basic concept, but it sets the foundation for a constructive approach to mathematics. Instead of working with objects that have 'actual' infinite divisibility, we will focus on constructing k-continua (which are by definition finite in every way). With this view, the essence of calculus lies not in the properties of any single k-continuum, but in the properties of the processes designed to construct particular endless sequences of k-continua. You might notice a resemblance here to Cauchy sequences.

...the first step is to accept that k-curves are indivisible. k-vertices in these graphs cannot be partitioned. I was presenting my position in a very different way back in our previous thread with fishfry, but this was the main sticking point.

What structure? A line segment has structure? One line segment has the same "structure" as another? You must see something there that eludes me. But I am old and a lot gets past me. — jgill

k-continuum 1
k-continuum 1 is represented by the 3-vertex graph: a - <a b> - b, where:

• vertices a and b represent k-points.
• vertex <a b> represents a k-curve.
• - indicates an edge connecting adjacent vertices.

Properties:
1. k-point a identifies location a.
2. k-point b identifies location b.
3. k-curve <a b> continuously connects a and b without including the endpoints a and b.

In this model, k-points and k-curves are indivisible objects, and k-continuum 1 has no additional properties beyond these.

k-continuum 2
k-continuum 2 is represented by the 5-vertex graph: a - <a c> - c - <c b> - b where:

• vertices a, b, and c represent k-points.
• vertices <a c> and <c b> represent k-curves.
• - indicates an edge connecting adjacent vertices.

Properties:
1. k-point a identifies location a.
2. k-point b identifies location b.
3. k-point c identifies location c.
4. k-curve <a c> continuously connects a and c without including the endpoints a and c.
5. k-curve <c d> continuously connects c and d without including the endpoints c and d.

Again, k-points and k-curves are indivisible objects, and k-continuum 2 has no additional properties beyond these.

Relationship between k-continuum 1 and k-continuum 2
k-continuum 2 can represent all the structure of k-continuum 1.

Here's how:

1: Both k-continua identify location a in the same way.
2: Both k-continua identify location b in the same way.
3: The k-curve <a b> in k-continuum 1 can be represented as the union of 3 vertices in k-continuum 2, namely: <a b> = <a c> U c U <c b>

This means that the combination of the k-curves <a c> and <c b>, along with the k-point c, continuously connects a and b without including the endpoints a and b, just like the k-curve <a b> does in k-continuum 1.

Therefore, k-continuum 1 can be seen as a unification or simplified version of k-continuum 2, where the additional structure in k-continuum 2 can be collapsed into the simpler form of k-continuum 1.

True. I hope there is something of interest coming from this discussion. But we've been through metric spaces and topology and now are venturing into graph theory with some sort of hope of connecting that with calculus. I have my doubts, but am trying to keep an open mind. — jgill

I share your sentiment on all accounts - from the hopes to the doubts. Thank you.

You seem not to understand how the mathematical method of handwaving works. It's not ZF or PA or one of those; it's the theory BS. — TonesInDeepFreeze

There's an important distinction between handwaving and BS. Handwaving involves vagueness or imprecision, where the core idea might be sound but lacks detail or rigor in its current form. BS, on the other hand, is fundamentally incorrect—an argument that doesn't hold up under scrutiny and lacks substance from the start. A good idea often begins with some handwaving as it's being formed, but through refinement and rigorous thought, it can mature into a precise and well-supported explanation. In contrast, a bad idea rooted in BS remains flawed because it has no solid foundation to build on. While both terms carry subtleties, I hope this captures the essence of the difference.

I believe that my core idea is sound, but it needs further refinement and I'm not communicating well. Your frustration is justified. But I could really use your help.

What structure? A line segment has structure? One line segment has the same "structure" as another? You must see something there that eludes me. But I am old and a lot gets past me. — jgill

It's not you, I should have provided a description of the structure I was referring to. I intend to reply to you but I likely will not find time for several days. Please stay tuned.

Thank you. You saved me a lot of time and effort. Because my prediction that you would resort to half-baked handwaving is confirmed, so I am done with trying to help you formulate your stuff into mathematics. A circle is an infinite set. — TonesInDeepFreeze

What I termed '1D drawable' is intended to be the exact 1D analogue to a planar graph, nothing more. I'm presenting my ideas informally, but certainly not BS'ing.

Are you saying that one can't use the concept of planar graph until a plane is defined? Are you saying that such finite planar graphs necessarily require the existence of infinite sets? I think one can capture the essence of planar graphs without explicitly mentioning planes in the definition or asserting the existence of actual infinity. If so, then perhaps I could do the same with '1D drawable'.

I would have hoped that this disagreement didn't justify ending the conversation. I kindly ask that you give me another chance.

But I see trouble ahead anyway with your personal notion of partitioning. I pessimistically predict that ultimately you won't be able to get past handwaving with it. — TonesInDeepFreeze

Let's have another take at this. In this take, I will not mention numbers. I also will not use 'partition' anymore as it suggests that indivisible objects can be decomposed. Rather, I'm going to use unify/unification.



In the above figure, I have 3 independent k-continua:

• A
• BaC
• BaDbE

BaC can be unified into A because BaC can be treated as a whole that captures all the structure of A. Perhaps I can write it as follows:

BaC = (BaC) = A

BaDbE can be unified into BaC because DbE can be treated as a whole that captures all the structure of C. In other words,

BaDbE = Ba(DbE) = BaC

A is a unification of BaC. A is also a unification of BaDbE.
BaC is a unification of BaDbE.

A is not a unification of BaC because its elements cannot be unified to capture the structure of BaC.

I seem to lack your insight in this example. — jgill

I was wrong to mention numbers/intervals without properly establishing the notion of a continua. I'm going to leave that message there, but please allow me to refrain from responding here as I believe a discussion involving numbers will not help. Months back you said:

Define a continuum as an abstract entity and not in terms of the real line. As a matter of fact, use another word for your creation. State the properties of the continuum, again not referencing the real line or numbers. This is a tall order. Metric spaces and topological functions are perhaps inappropriate in this regard. I don't know. You will be going into unexplored territory. — jgill

I'm quoting you here because your recommendation is exactly the path I endeavor to go own.

As a constructivist, what are you constructing other than a few line segments? — jgill

Constructing 1D k-continua from (indivisible) k-points and k-curves.

Why don't you jump right into calculus concepts in 2D instead of dwelling on the trivial, incredibly boring 1D case. Either that or make the 1D case something interesting, to capture the attention of a reader. Just a suggestion. — jgill

I've come to understand that although 2D visuals may seem more impressive than 1D, the core of my perspective can largely be conveyed in 1D. Introducing calculus and 2D concepts too early might disengage the reader, as it would lack the foundation of a fully developed idea.

The link you provided is fascinating, especially since calculus and graphs are central to my perspective. However, I believe the connection stops there. While I appreciate discrete calculus—after all, numerical methods are what I work with professionally—the concept I’m developing here is rooted in continuous calculus.

To clarify, discrete calculus deals with finite objects like graphs, while the approach I'm pursuing involves potentially infinite operations. These operations, however, are applied to finite structures, like graphs.

In this context, a mathematical Platonist wouldn’t need to focus on operations, as mathematical objects are seen as complete entities. But for a constructivist, operations are essential, as constructing these objects is at the core of the philosophy.

The biggest hurdle for an intelligent but amateur mathematician is rediscovering a result established some time ago. Hence, my words of caution. — jgill

These are wise words of caution.

Not sure what you mean by actual infinity. Are you speaking of infinity as a sort of number that can be arithmetically manipulated, or infinity as unboundedness? I have always used the concept of the latter rather than the former. But set theorists use both I think. Please provide an instance of "actual infinity" in the Euclidean plane. A projection onto a sphere is not allowed. — jgill

I will use the Euclidean line instead of the plane, since we're currently focused on 1D.

I'm interested in distinguishing between objects and operations, particularly in relation to infinity.

In the orthodox view, the Euclidean line is thought to be fully formed, with all (an infinite number) of its points existing simultaneously. This kind of infinity, which counts objects, is what I refer to as actual infinity. (Note, I’m not referring to infinity as a coordinate, which I believe is what you're asking about.)

In my view, however, there is no such thing as a complete Euclidean line. Instead, we start with a single k-curve (-∞,+∞) and can partition it as much as we like, without ever exhausting it. The continua we construct in this process are always finite (in the sense that they are represented by graphs with a finite number of k-vertices). It is endless operation of partitioning that is infinite, and this infinity is a potential infinity. I don't believe in supertasks, so I would argue that all infinities related to operations are potential infinities, never fully realized.

I do think infinite cardinals and ordinals are very valuable, but not at describing the size of completed objects (e.g. sets) but rather at describing the potential of endless operations.

It seems that a k-continuum is a certain kind of finite(?), undirected(?), loopless(?) graph whose k-vertices are either k-points or k-curves. And (I surmise) no k-vertex is connected to itself. — TonesInDeepFreeze

• Finite - correct
• Undirected - correct
• Loopless - correct (doesn't this mean that no k-vertex is connected to itself?)

CONNECTED: I've realized that I was loose with my use of the term 'connected'. I see that 'connected' means that there is a path between vertices (which can consist of one or more edges). Moving forward, I'll use 'adjacent' to indicate that two vertices are connected by a single edge.

1D Continua: I just want to highlight that all of the following is in the context of 1D continua.

The following are not k-continua:
-a graph in which occurs a k-point not connected (Not connected)
-a graph in which occurs a k-point adjacent to another k-point (Not connected in the topological sense, even if connected in the graphical sense)
-a graph in which occurs a k-curve adjacent to another k-curve (Not connected in the topological sense, even if connected in the graphical sense)
-a graph in which occur two or more k-curves but at least one of them is not connected to another k-curve (Not connected)
-a graph in which occurs a k-curve adjacent to more than two k-points
-a graph in which occurs a k-curve adjacent to another k-curve adjacent to a k-point (Not connected in the topological sense, even if connected in the graphical sense)

The following are k-continua:
-one k-curve (Not compact, but I'm inclined to include to capture the essense of the real line)
-one k-curve adjacent to one k-point (Not compact, but I'm inclined to include to capture the essense of the real line)
-a graph in which occurs a k-curve adjacent to a k-point adjacent to another k-curve

Is there a natural number n>1 such that there is no k-continuum such that there occurs n number of k-curves connected to one another? (In other words do you disallow that a k-continuum may have arbitrarily finitely many connections of curves from one to another?) — TonesInDeepFreeze

A k-continuum can be composed of arbitrarily finitely many vertices.
If there are n k-curves (where n is a positive natural number), there must be between n-1 and n+1 k-points.
If there are n k-points (where n is a natural number), there must be between max(1,n-1) and n+1 k-curves.

I'm not sure that exhausts all possible configurations. You should figure it out to define 'is a k-continuum'. — TonesInDeepFreeze

Does this cover it?
For k-continua:
1) All graphs are finite, undirected, and loopless
2) All graphs are 1D drawable (in that each can be embedded in a circle without any of its edges crossing) (Since I'm focused on continua used for calculus, perhaps this should be a line)
2) All k-vertices are connected
3) All k-points are adjacent to 1 or 2 k-curves (and no k-points)
4) All k-curves are adjacent to 0, 1, or 2 k-points (and no k-curves)

But I see trouble ahead anyway with your personal notion of partitioning. I pessimistically predict that ultimately you won't be able to get past handwaving with it. — TonesInDeepFreeze

[EDIT: I've realized that in the following response I made a bad move in fast forwarding to numbers and intervals. Please ignore for now. I'd rather discuss continua in a simpler sense, free of numbers for now. However I'm leaving it in this post since JGILL has already responded.]

IGNORE-->
I need to rethink the use of the word 'Partitioning'. That word suggests that k-objects can be divided into smaller, perhaps more fundamental, objects. I want to treat k-objects are indivisible wholes. Let me try to explain my view of partitions by means of an example. I'm going to utilize numbers and intervals, even though we haven't discussed it yet.

I have two continua described by Graph 1 and Graph 2, respectively.

Graph 1: A-B-C where
A is k-curve (-inf,5)
B is k-point 5
C is k-curve (5,+inf)

Graph 2: E-F-G-H-I where
E is k-curve (-inf,5)
F is k-point 5
G is k-curve (5,10)
H is k-point 10
I is k-curve (10,+inf)

These are distinct graphs, but Graph 2 is a partition of Graph 1 because it captures all of it's structure. In other words, Graph 2 can behave exactly like Graph 1 if E acts as A, F acts as B, and G-H-I acts as C.
<--IGNORE