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.

Now that you've moved into graph theory 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. But this would be inadequate regarding the reals. But you might be able to push into the irrationals some way. Speculation. You need to actually start moving beyond your pictures. I am not familiar with graph theory, but perhaps @fishfry and @Tones are. And some on the forum who are or were CS professionals.

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

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.

1. should be interesting. You have density, but then continuity is next. Intuitionism math perhaps.
I thought you were defining these lines as continuous. Fundamental objects.

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.

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. — keystone

That's BS. BS includes nonsense, doubletalk and falsity. And handwaving is not necessarily just lack of rigor to be supplied later. And you presume that your "core ideas" are "sound".

I said I'd be willing to check you out to the extent that we could turn your ruminations into primitives, definitions and axioms. I predicted that right after the first round you would resort to yet more undefined handwaving and I said that I would drop out when that happened. Indeed, with the very first predicate 'is a k-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. You disrespect my intellectual interest that way, just as occurred several months ago with a different half-baked and self-contradictory proposal of yours. You are a sinkhole of a poster. You need to obtain an understanding of the basic concepts of primitive, definition, axiom, and proof. I'm done with providing you assistance of this kind.

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.

1D analogue of the established term "planar diagram" — keystone

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 derived finitistically and such that it justifies such verbiage as about "embedding in a circle".

But don't bother if it is to re-enlist me. I was willing to take it step by careful step with you. But you can't discipline yourself to do that, as instead you just jump to whole swaths of handwaving. I said that at the very first point you invoked anything not previously justified by you then I'm out. I don't need to waste my time and energy on you. You are BS.

You haven't identified any falsity in my current position.

You haven't even defined enough to get the stage of consideration of truth or falsity.

Please, give me a chance. — keystone

I have! Many times! And previously too. But you abuse my time and effort. I'm done.

If your offer to help was sincere — keystone

How dare you question my sincerity that has been demonstrated over and over in careful attention to details, in my labor to explain things for you, in this thread and in one several months ago? Get a load of your narcissistic self. You are full of yourself and full of BS ... though that is redundant.

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.

It's ironic that you became distant right after I went back, carefully studied, and addressed your comments on topology. — keystone

Apparently, you don't recall the post in which I said that I'm willing to indulge you only up to the point that you go past the process of definitions.

You don't need to concern yourself with my decisions about how I spend my time and energy. Instead, you need to start by at least getting a grasp of the basic ideas of primitive, definition, axiom and proof.

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 didn't ask for a definition of 'planar graph'. You didn't read what I said about this a few posts ago. You are a sinkhole.

You are a sinkhole. — TonesInDeepFreeze

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

Move on to 2.

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].

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

Keep digging your sinkhole deeper.

I see a mistake in your last figure, typo probably. And I assume -1/0 (meaningless) designates negative infinity, however you define that. I see nothing of interest so far.

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...)

I've been overlooking the fact that real numbers are typically defined as equivalence classes of Cauchy sequences, not just individual Cauchy sequences. — keystone

Cauchy sequences themselves are infinite sets.

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.

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?

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.

By moving the focus from the destination to the journey the need for actual infinity vanishes. — keystone

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

In my recent posts, I have been establishing that real numbers instead describe potential k-curves, which can be thought of as yet to be constructed k-curves which when constructed have the potential to be arbitrarily small (but always retain a non-zero length). — keystone

This is either very deep - or shallow gobblygook.

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.

But that program (even in an infinite world*) cannot actually output a set with a cardinality of ℵ0. Potential is important and I feel like it's been forgotten in our Platonist world. — keystone

Elementary calculus does not require "actual" infinities. It gets along quite well with unboundedness, or what you might call potential infinity. As I have said before, I have written many papers and notes without ever becoming transfinite.

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 $\mathbb{R}$?

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. — keystone

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

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 R? — keystone

Of course I have used R, but not a transfinite number. Unless I occasionally use the "point at infinity" in complex analysis. Which I rarely do since it is a projection upon the Riemann sphere. It might appear that you are moving in the direction of Discrete calculus. But go ahead. I am curious.

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.

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.