I'm going to respond in two posts. This post covers topics not directly related to my ideas and my next post will cover topics more directly related to my ideas. I hope this covers the important points we've been discussing but let me know if I skipped over something important.

Topology: I now have a basic understanding of the definition of a continuum as compact, connected, and Hausdorff. Connectedness seems straightforward, and I believe it’s a necessary part of any definition. Compactness is more intriguing because "the continuum" itself isn't compact, which is surprising since it means "the continuum" isn't a (topological) continuum. While I understand that compactness is important for results like the Extreme Value Theorem, I don’t yet fully grasp why it's a crucial property of continua. Given that I want define the sort of object suitable for calculus, I'm inclined to not include this property in my definition of continua.

The Hausdorff property, on the other hand, is a more subtle aspect I hadn't considered, likely because most (if not all) spaces I’ve encountered are already Hausdorff. I think it's important and makes sense, but I am unsure how it fits in with my view. This is something I’ll need to explore further in the future. You're right that topology is relevant to this discussion.

Philosophy vs. mathematics: Based on your criticisms, I need to rethink my position. I believe that basic mathematics operates on two parallel tracks: (1) the objects and operations themselves, and (2) the descriptions of those objects and operations. To use a Matrix analogy, this would be like (1) the lived experience within the matrix, and (2) the underlying code or digital rain that describe/define it. Both tracks are fundamental. In basic mathematics, the objects are what we depict, such as points, continua, etc., while the descriptions are the formal logical statements we write down.

Though I’m still learning the basics of logic, I find intuitionism most appealing—especially since you've made it clear that ZFC leaves no room for flexibility when it comes to the ontology of infinite sets (and my focus is on transitioning from actual infinities to potential infinities). To be frank, I'm not willing to accept infinitely many axioms or statements holding a truth value so it's not just a matter of avoiding the Axiom of Infinity. So you're right, I do have an issue with ZFC.

However, my primary focus isn’t the description itself (or the supporting philosophy), but rather the objects and operations they describe. Specifically, I’m interested in the philosophy that justifies the existence of mathematical objects and our operations on them. The orthodox view is a whole-from-parts (point-based) philosophy, which aligns well with ZFC. I, on the other hand, propose the opposite: a parts-from-whole (continuum-based) philosophy. Perhaps this view might align better with CZF, I don't know.

Infinity plays a central role in both (1) mathematical objects and operations, and (2) logical descriptions. I realize now that I’ve been unclear in distinguishing between the two. For instance, my concerns with infinite sets and Cauchy sequences likely fall under (2), the realm of logical descriptions. To address these issues properly, I recognize that I need to deepen my understanding of logic and intuitionism. However, my primary focus is on the objects themselves, such as the Cartesian coordinate system. I believe this system needs a parts-from-whole, continuum-based reinterpretation, as the current understanding relies heavily on the notion of actual infinity.

In short, I would like to see the replacement of the actual infinity based whole-from-parts + ZFC mathematics with the potential infinity based parts-from-whole + ?CZF? mathematics. However, I'm in no position to talk about ZFC vs. CZF.

For reference, here are the axioms of Z set theory — TonesInDeepFreeze

The idea behind these axioms seem sensible yet I'm wary of schemas which imply infinitely many instances of those axioms.

the answer is that a calculus book is about setting forth the most basic mathematics that is put to use in different fields of study; it's not about very much understanding the foundations of that mathematics or its broader mathematical context. — TonesInDeepFreeze

Understood.

However, my primary focus is on the objects themselves, such as the Cartesian coordinate system. I believe this system needs a parts-from-whole, continuum-based reinterpretation, as the current understanding relies heavily on the notion of actual infinity. — keystone

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.

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

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.

I wonder if Calculus on Finite Weighted Graphs is the direction you are headed? This is a topic even less popular than mine, with a scant 8 views per day on Wikipedia.

The article mentions several applications connected to data processing and CS. But calculus approached this way is obscure and unlikely to replace elementary calculus as it it is currently taught. Just my opinion. You are probably not pursuing this line of thought.

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

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.

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

I seem to lack your insight in this example. It appears you simply take a real line and divide it into several line segments by inserting "k-vertices". You are assuming the existence of these points on the line. Indeed, the line segments are continua. In the example A-B-C what if instead you used the square root of two as the dividing k-vertex? You seem to be assuming the common notion of the real line. Maybe if you extend your ideas into 2D they will seem to be more than trivia? As a constructivist, what are you constructing other than a few line segments?

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.

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.

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.

All graphs are 1D drawable (in that each can be embedded in a circle without any of its edges crossing) — keystone

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. But you say you're going to do this without infinite sets. You don't know what you're doing and you don't know what you're talking about. And if you were to say something like "a circle that's only a potential infinity" or whatever, then that would not cut it, since you haven't given a mathematical definition of such a thing. Just as I mentioned, you start to give mathematical definitions, but they finally end up relying on even more complex notions that are themselves presumptuously undefined and only gestured at with half-baked handwaving. You're a vortex.

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

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.

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. You need to familiarize yourself with its advanced techniques as exemplified by @keystone.

You seem not to understand how the mathematical method of handwaving works — TonesInDeepFreeze

True enough. But I keep hoping there is something profound I am missing in all this. :roll:

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.

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.

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.

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. — keystone

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.

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.

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.

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. There is virtually no mathematics so far. Except for interpreting vertices and edges from graph theory, which only complicates a vacuous scenario.

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.

..the first step is to accept that k-curves are indivisible. k-vertices in these graphs cannot be partitioned — keystone

OK, you have a line that is indivisible. But it has k-vertices that "cannot be partitioned". Can a vertex be partitioned? Like saying a point can be partitioned. Concise language is very important in math, not so much so in philosophy.

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.

The only objects in these graphs that can be 'cut' are the edges — keystone

Why resort to graph theory and call a simple line an edge? Is this an effort to enhance an almost trivial concept of line and point? Again, why not go to 2D? Maybe your ideas will make more sense in that context.

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.

I appreciate the graphs you have drawn. You have 2D surfaces that are defined by interiors of edge figures. The surfaces, edges and vertices seem to constitute fundamental objects. Time for a few axioms.

How all this simplifies normal calculus is questionable.

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

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.

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.

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