The following is not a formal, finalized definition, but I hope sets the stage for the discussion — keystone

Is what you wrote supposed to be informal unfinalized mathematics, or informal unfinalized mathematics infused with philosophy, or informal unfinalized philosophy, or informal unfinalized philosophy infused with mathematics, or other?

Stage for what discussion? Discussion of your stuff. Meanwhile, the stage for the overall discussion has included some actual mathematics.

If you're interested in more than just your own stuff, I suggest you read the mathematics I gave. Then you could compare your own stuff with the product of mathematicians deeply dedicated to the subject. That's a great place to start. Then, if someone such as you wants to advance on the beachhead of an alternative, then at least we can compare that alternative to where we stand on ground won by dint of the combination of profound mathematical imagination and heroic intellectual discipline.

Finite object: Finite in the sense that its complete set of attributes can be fully described without invoking infinite processes. — keystone

Define:

"complete set of attributes"

"can be fully described"

"invoking infinite processes"

Continuous object: In 1D, the proposed fundamental objects are of two types: (1) open-ended curves, which are inherently continuous, and (2) points. A composite 1D object is the union of these fundamental objects and is continuous if, when duplicates are removed, the following conditions are met:
Points are connected to 0–2 curves (but not to other points).
Curves are connected to 0–2 points (but not to other curves).
No objects are disconnected from the composite structure. — keystone

"open-ended curves" means "open curves"?

Define:

"inherently continuous"

"duplicates removed"

"connected to"

"0-2 curves"

"0-2 points"

Potential for arbitrarily fine partitioning: The continuum can be subdivided into an increasingly refined composite object made up of arbitrarily many fundamental elements, maintaining its continuity. — keystone

Define:

"refined composite object"

"increasingly refined"

"fundamental elements"

"maintaining continuity"

Characterized by the cardinality 2^aleph_0: The partitioning process can be described algorithmically, such that no algorithm can be devised allowing for further division. Although this algorithm would not halt if executed, the structure of the algorithm itself reveals that the potential for infinite subdivision aligns with the cardinality 2^aleph_0. — keystone

Define:

"partitioning process"

"algorithm devised allowing further division"

"algorithm reveals"

"aligns with"

I have refrained from providing examples or illustrations for the sake of brevity, though they could help clarify my position. — keystone

To clarify your stuff, the best thing to do would be to define your terms. If your stuff is supposed to be mathematical, then proposing to "define" by adducing yet more undefined, impenetrable verbiage that never resolves to primitives is a mug's game.

Do really think there are people other than yourself who can make sense of your verbiage? Who in the world is supposed to know what you mean by things like "refined composite object", "fundamental elements", "align with" in context of this subject?

If I've asked for definitions that exist already in the literature as clear and rigorous mathematics, then I would like to know where to look them up. Can you direct me to writings in which such verbiage is defined in the sense you use it? Can you tell me what the prerequisite readings are? Or is your stuff couched ultimately and merely as you wish to personally use words, with no way for people to inform themselves of the usage to follow along with you, either to accept your stuff or to show errors in it?

As I told you of books that would be a starting point for mathematics, what books would you direct me to that are starting points for keystonematics? If there are no books to recommend, then your best bet would be to tell me your primitives, your axioms and your train of definitions and proofs.

The axiomatic method is the opposite of exclusionary. Anyone may look at the primitives and axioms, which are stated right up front, and opt in at least to see what they lead to, or, on the other hand to reject them from the start. And when one does follow along to see where they lead, the process of seeing whether a purported proof is indeed a proof is objective; no one, not even the author, may rule by decree in that regard. Checking the purported proof is open to anyone. On the other hand, your style is quite exclusionary. You are the only one who knows (or thinks he knows) what in the world you mean by all that verbiage. You can fashion any arguments you want out of it and no one can verify that you're correct or demonstrate that you're incorrect, because the meanings are ultimately free floating in your own mind, unmoored to a clear (let alone rigorous) system of definitions.

Anyway, like I said, you would serve yourself well by understanding the topological meaning of 'is a continuum'. Then at least you could see how your stuff, from the left field of your ruminating, relates or not to mathematics.

Here's a definition:

A continuum is a decomposition of a hyperspace module with sidewise-partitionable step-wise radii

decomposition: limitless regarding of tangents in non-Euclidean operatives

hyperspace module: arbitrary [1 3] transcendental intersections modulo mono-component recursions

sidewise-partitionable: bounded reaching of proximal antitheses in the quadrant independent ordinal numbers

step-wise radii: graph-based ultra-empty full-product numeric manifestations concentrically regarded

There, now I've really set the stage.

A continuum is a decomposition of a hyperspace module with sidewise-partitionable step-wise radii — TonesInDeepFreeze

And you say you are not a mathematician! :cool:

Continuous object: In 1D, the proposed fundamental objects are of two types: (1) open-ended curves, which are inherently continuous — keystone

So we begin by defining such curves as "inherently continuous". That seems to solve the problem. Why proceed? Why dabble with sets of points that may fill up a curve - or not?
Just chop up such curves and there we are. Bend them a bit to go to two or more dimensions.

My Claim: You asked me to define continua, which I assume was prompted by my earlier claim that "THE continua" is not actually used by applied mathematicians when they describe "physical continua." By "stage," I am referring to the basis on which we can discuss this claim.

Formal Logical System: I fully accept the recommendations from you and others in this forum to study mathematical logic (and intuitionism). I'm in the early stages of actively pursuing this in my spare time, but I haven’t yet reached the point of formalizing my ideas into a logical system.

Established Mathematics: I agree that the more established math I learn, the better. However, my primary concern is the continua used in basic calculus, so I wonder if delving into topology might be more of a distraction. I also question whether your definition of a continuum is unnecessarily complicated. For reference, even ChatGPT struggled to make sense of it and had a humorous take. I do appreciate that the mathematics presented earlier in this thread seems more relevant to this thread's elementary discussion on continua, but that math is rooted in actual infinities. I just don’t see how applied mathematics has any need for or use of actual infinities. I see an alternative.

In any case, I’ll provide the requested definitions in my next post. I hope you continue that discussion at least a little more but if I’m heading down a path you'd prefer not to explore, no worries.

the real numbers are defined as the continuum. They can be proven to exist within set theory, but that has no bearing on what's true in the real world. — fishfry

Some things obviously apply to the world. It is often said that there are no perfect shapes in the world. But we can mentally draw a perfect shape WITHIN any object although there it is surrounded by OTHER matter. The shape does exist as a part of another thing

Some things obviously apply to the world. It is often said that there are no perfect shapes in the world. But we can mentally draw a perfect shape WITHIN any object although there it is surrounded by OTHER matter. The shape does exist as a part of another thing — Gregory

If I adjusted my statement to say that the mathematical real numbers nave no necessary bearing on the world, would that be better?

We don't know whether there's a real number-like continuum in nature. Is that better?

I can tell you've taken care to read my words closely. Even though we sometimes clash, I truly appreciate your effort.

• complete set of attributes constituents: All the fundamental objects that make up the continuum
• can be fully described constructed: Can be built or generated from its constituents
• invoking infinite processes: Using a finite series of steps or operations.
• open-ended curve: A curve that does not have endpoints.
• inherently continuous: By its very nature is continuous.
• duplicates removed: In a mathematical graph representation, multiple vertices representing the same node are removed.
• connected to: As defined in the mathematical graph context.
• 0-2 curves: 0, 1, or 2 curves. (What I'm doing is perhaps in the spirit of Euler's polyhedral formula but for mathematical graphs representing continua)
• 0-2 points: 0, 1, or 2 points.
• refined composite object: broken down into finer and finer elements
• increasingly refined: Progressively more subdivided
• fundamental elements: The basic building blocks or primary components that make up a larger object or structure. In this 1D context, they are open-ended curves and points.
• maintaining continuity: Does not change the continuity (as defined).
• partitioning process: The act of partitioning (as described).
• algorithm devised allowing further division: Such an algorithm describes the partitioning of the object such that further partitions are possible.
• algorithm reveals / aligns with: The cardinality of an algorithm is determined by comparing it to other algorithms whose cardinalities are already known (or assumed), in this case, the reference algorithm is that for indefinitely partitioning a continua one partition at a time.

My experience on this forum is that people don't like figures, but let me just include one as maybe it will help.

So we begin by defining such curves as "inherently continuous". That seems to solve the problem. Why proceed? Why dabble with sets of points that may fill up a curve - or not?
Just chop up such curves and there we are. Bend them a bit to go to two or more dimensions. — jgill

I detect some sarcasm here. Isn’t the conventional approach to start with points, which are inherently discrete? How is that any less of a leap than starting with curves, which are inherently continuous? In fact, I would argue that you've encountered far more actual continua in life than individual actual points.

Why proceed? Because an unpartitioned continuum only holds the potential for usefulness. In its unpartitioned state, it’s effectively a void.

Just as a point can’t be manipulated to form a curve, a curve can’t be manipulated to form a surface. An open-ended surface is its own fundamental object...

You asked me to define continua, which I assume was prompted by my earlier claim that "the continua" [...] — keystone

Again, best to keep things straight:

(1) 'the continuum' is noun that names the set of real numbers.

(2) 'is a continuum' is an adjective that we talking about defining.

Also, the discussion had been, for a while, primarily (pretty much solely) mathematical. Of course, one should be allowed to discuss perspectives other than mathematics alone, but I suggest that one should be clear at each point as to what they are talking about - mathematics, physics, philosophy, or some combination that is explained.

I haven’t yet reached the point of formalizing my ideas into a logical system. — keystone

Let alone that you don't offer primitives, axioms, definitions and proofs, I wonder whether you grasp the idea. Making definitions that rely on yet more undefined terms needs to finally arrive at the base set of primitives, which are undefined but inferences mentioning them are governed by axioms. Or, even if not that formal, at least to state starting notions that are so basic that they at least they can be comprehended intuitively at a basic level. Such things as "increasingly refined composite object made up of arbitrarily many fundamental elements" are so detached from a clear meaning that they are not at all comprehended at a basic intuitive level.

So, even if not formal mathematics, but rather as informal exposition of intuitions, ideas or philosophical views, I can't make heads or tails of whatever it is you're trying to say.

You mentioned providing examples. Ostensive indications and understandings not from explicit definition but rather from gleaning in context are fine and useful, mainly at the stage of basic intuitions. Indeed, I doubt that I could define such terms as 'is', 'in' or 'at' but rather I presuppose that people understand them however they contextually got to that understanding. On the other hand, your notions are not of that basic kind but rather are intricate enough that they need more than ostensive definitions.

However, my primary concern is the continua used in basic calculus — keystone

Then that is mathematics. Or do you mean the study of physical phenomena that calculus is used for?

In any case, as far as I've ever seen, basic calculus doesn't mention the adjective 'is a continuum'. Rather, if at all, basic calculus mentions the continuum = the set of real numbers.

I wonder if delving into topology might be more of a distraction. — keystone

I wanted to provide a mathematical definition of 'is a continuum'. I find it in topology. People have been knocking around the term 'a continuum' in a math context. So what is a mathematical definition? I provided one. That's the opposite of distraction.

And topology is a study that informs analysis and put analysis in a broad context. Understanding the real numbers and the continuum in context of topology is definitely not a distraction. And why would topology be a distraction but your half-baked verbiage not be a distraction?

I also question whether your definition of a continuum is unnecessarily complicated. — keystone

I provided a quite streamlined definition. If an equivalent definition could be simpler then you're welcome to state it. Less formally:

A topological space C is a continuum if and only if C is compact, connected and Hausdorff.

So I provided definitions of 'compact', 'connected' and 'Hausdorff'. Those definitions depend on the definition of 'a topology' so I provided that definition, which I couched with only these non-logical notions: 'element' (primitive), 'subset', 'power set', 'union', 'pair' and 'binary intersection', which are quite basic notions of mathematics (except for 'element', they also can be defined back to the sole non-logical primitive 'element').

For reference, even ChatGPT struggled to make sense of it — keystone

What folly and how insulting!

(1) ChatGPT is so often stupid and dishonest about mathematics. When you posted a purported "proof" by ChatGPT, I even exactly showed you that ChatGPT doesn't know what it is doing*. And that is only one example and not even the worst instance of ChatGPT's stupidity and dishonesty. I could give you as many examples as requested (I'm not going to though since I waste my time only when I feel like wasting it).

* You didn't acknowledge that.

(2) Of course ChatGPT wouldn't be able to understand purely text-symbol rendering of mathematical symbols (for example, if you don't know that 'E' is to stand for the existential quantifier then 'Exy' would be gibberish to you).

My definitions are very straightforward use of just the basic logical symbols and a few basic symbols of set theory.

And if you don't understand the notation (though I specified most of it explicitly in a post only a few posts prior), then you can ask.

And, by the way, the definition I gave is especially efficient as you don't have to read all the chapters in the topology book that lead up to the definition; but instead my definition is extracted from that context while being standalone.

(3) But wait, out of curiosity, I put my definition to ChatGPT verbatim, and ChatGPT replied:

This text outlines definitions related to topological spaces and various properties in the context of set theory and topology. Here’s a breakdown of the key components:

Topology: The first definition establishes the criteria for a topology T on a set
X. It requires that:

T is a subset of the power set of X (denoted PX).

The whole set X and the empty set must be in T.

Any union of sets in T must also be in T.

Any finite intersection of sets in T must also be in T.

Compactness: The second definition describes compactness in terms of a topology T on a set X. A set C is compact if every cover of X by open sets in T has a finite subcover. This emphasizes the idea that even if you have a potentially infinite collection of open sets that covers X, you can always find a finite number of these sets that still cover X.

Connectedness: The third definition characterizes connectedness. A set C is connected if it cannot be partitioned into two non-empty, disjoint open sets. This property ensures that C is "in one piece" in a topological sense.

Hausdorff Condition: The fourth definition introduces the Hausdorff property, which states that for any two distinct points p and q in X, there exist neighborhoods (open sets) around each point that do not overlap. This separation property is crucial for many results in topology.

Continuum: Finally, the last definition states that a continuum is a set that is compact, connected, and Hausdorff. This is a significant concept in topology, as it combines these important properties into a single structure.

Comment
These definitions are foundational in understanding topological spaces and their properties. They help formalize concepts that are essential in analysis and topology. By establishing clear criteria for these properties, they pave the way for deeper exploration into how different topological spaces behave and interact. If you have a specific area you'd like to explore further or questions about these definitions, feel free to ask!" — ChatGPT

that math is rooted in actual infinities. I just don’t see how applied mathematics has any need for or use of actual infinities. I see an alternative. — keystone

For Pete's sake again! The original post in this thread purported to prove that the continuum does not exist. The argument wasn't that it doesn't exist in nature but that it doesn't exist mathematically. The argument was bunk. Then there was discussion of the mathematics, so it was needed to give a mathematical definition of 'is a continuum'. And the definition itself makes no presupposition that there exists an infinite set. But I would bet that we can prove that if <X T> is a continuum then X and T are infinite. If you can provide an finitistic mathematics in which there are finite continua or in which there are no continua, have at it; but that is a task not completed by posturing with a bunch of undefined terminology.

In any case, I’ll provide the requested definitions in my next post. — keystone

Then I would be right to ask you to defined the undefined terminology in those definitions, and again until (1) You finally fail to reach primitives or (2) You end up in a circle or (3) You do reach primitives.

I can tell you've taken care to read my words closely. — keystone

Not close enough. If I had, I'd have more bad things to say about them.

Is your stuff supposed to be mathematics, or mathematics infused with philosophy, or philosophy, or philosophy infused with mathematics, or something else?

"fundamental objects"

"by its very nature"

"multiple vertices representing the same node" Vertices are nodes.

"broken down into finer and finer elements"

"change the continuity"

"acts"

"algorithm describes"

"The cardinality of an algorithm is determined by comparing it to other algorithms whose cardinalities are already known" Circular.

"continua one partition at a time" You were supposed to be defining 'continua'.

"connected" (in your sense)

/

Maybe more simply, why don't you get back to me when you've figured out your definitions from a starting point rather than backwards from undefined to undefined?

Meanwhile though, still my question: Is your stuff supposed to be mathematics, or mathematics infused with philosophy, or philosophy, or philosophy infused with mathematics, or something else?

And when you cite ChapGPT for mathematics, I figure that you are as lacking in credibility as it is.

you've encountered far more actual continua — keystone

As I understand, you reject using infinite sets. But you say that we encounter continua. So continua are finite?

an unpartitioned continuum — keystone

You've not defined 'a continuum' (a bunch of itself undefined verbiage is not a definition). So I don't know what you're talking about in mathematics when you use the word 'continuum'.

And every set has a partition. Many partitions for larger and larger sets.

How is that any less of a leap than starting with curves, which are inherently continuous? — keystone

And then straighten out a (continuous) curve and you have a continuum, which the OP argues does not exist. Perhaps you should start a thread entitled "The Continuum does exist".

I've never been able to see where it is you are going. Maybe it's just me, old and weary of days.

But wait, out of curiosity, I put my definition to ChatGPT verbatim, and ChatGPT replied:

This text outlines definitions related to topological spaces and various properties in the context of set theory and topology. Here’s a breakdown of the key components:

Topology: The first definition establishes the criteria for a topology T on a set
X. It requires that:

T is a subset of the power set of X (denoted PX).

The whole set X and the empty set must be in T.

Any union of sets in T must also be in T.

Any finite intersection of sets in T must also be in T.

Compactness: The second definition describes compactness in terms of a topology T on a set X. A set C is compact if every cover of X by open sets in T has a finite subcover. This emphasizes the idea that even if you have a potentially infinite collection of open sets that covers X, you can always find a finite number of these sets that still cover X.

Connectedness: The third definition characterizes connectedness. A set C is connected if it cannot be partitioned into two non-empty, disjoint open sets. This property ensures that C is "in one piece" in a topological sense.

Hausdorff Condition: The fourth definition introduces the Hausdorff property, which states that for any two distinct points p and q in X, there exist neighborhoods (open sets) around each point that do not overlap. This separation property is crucial for many results in topology.

Continuum: Finally, the last definition states that a continuum is a set that is compact, connected, and Hausdorff. This is a significant concept in topology, as it combines these important properties into a single structure.

Comment
These definitions are foundational in understanding topological spaces and their properties. They help formalize concepts that are essential in analysis and topology. By establishing clear criteria for these properties, they pave the way for deeper exploration into how different topological spaces behave and interact. If you have a specific area you'd like to explore further or questions about these definitions, feel free to ask!" — ChatGPT

So, I don't know why ChatGPT "struggled to make sense of it" for you.

But ChatGPT got these wrong:

"A set C is compact if every cover of X by open sets in T has a finite subcover."

No, not a set C. Rather, it should be (in the manner of ChatGPT):

A set X is compact per a topology T on X, if every cover of X by open sets in T has a finite subcover.

"A set C is connected if it cannot be partitioned into two non-empty, disjoint open sets."

No, not a set C. Rather it should be (in the manner of ChatGPT):

A set X is connected per a topology T on X if it cannot be partitioned into two non-empty, disjoint open sets in T.

@keystone

As of a recent amendment to site guidelines, using ChatGPT as an academic source is grounds for a warning. You're not being warned for it now, I just thought I'd bring it to your attention, so that you can avoid doing it again.

Engineering claringly uses math as if it applies to reality. You seem to be saying there is nothing contradictory about continuums or that there would only be such only if they were in the real world. So then there is something about physical matter that in its properties is not entirely mathematical as we understand that. That may be true, although I would like to hear reasons why some day. Where do we draw the line when applying math to matter? How do we know we've gone too far?

String theory vs loop quantum gravity. One has little points that are really strings (1 dimension in 0 dimension?) And the other discrete space. The biggest question in physics (quantum gravity) wants to settle the question of the continuum. They don't want to just throw their hands up

(1) 'the continuum' is noun that names the set of real numbers. (2) 'is a continuum' is an adjective that we talking about defining. — TonesInDeepFreeze

Agreed.

Then that is mathematics. Or do you mean the study of physical phenomena that calculus is used for? — TonesInDeepFreeze

My focus is on the philosophy underlying the rigorous mathematical foundations that support the informal mathematics used in physics. I have no issue with the informal mathematics itself (e.g., the calculus of Newton and Leibniz), but I believe there is a disconnect between these informal methods and the rigorous foundations (e.g., Analysis of the continuum) meant to support them. This disconnect isn’t due to flaws in the rigorous foundations but rather stems from a misinterpretation of their meaning and philosophical implications. Specifically, I argue that the concept of actual infinity, which plays a central role in these interpretations, should not be part of the framework. With that said, I want to stress that the work of Cantor, Cauchy, etc. play a central role in my philosophy.

I can't make heads or tails of whatever it is you're trying to say....your notions are not of that basic kind but rather are intricate enough that they need more than ostensive definitions. — TonesInDeepFreeze

I believe an ostensive definition with figures is exactly what you need to make sense of what I'm saying. I'll provide that in a separate post. I understand that this approach isn't for everyone, so no worries if you choose not to respond to it.

Understanding the real numbers and the continuum in context of topology is definitely not a distraction. And why would topology be a distraction but your half-baked verbiage not be a distraction? — TonesInDeepFreeze

I want to learn topology, but in this thread I think it makes sense to focus on the topics that are absolutely essential for understanding continua (such as real numbbers). I don’t believe topology, at least in the way you’re suggesting, is necessary for this understanding. Of course, my current ideas are half-baked, but I believe the direction I’m taking is key to grasping the true nature of continua.

A topological space C is a continuum if and only if C is compact, connected and Hausdorff. — TonesInDeepFreeze

I'm not going to challenge this definition because (1) I'm not well-versed in topology and (2) I still recognize its value. However, the philosophy behind applying this definition to describe non-trivial continua assumes the existence of infinite sets, which I believe is unnecessary.

When you posted a purported "proof" by ChatGPT, I even exactly showed you that ChatGPT doesn't know what it is doing*. * You didn't acknowledge that. — TonesInDeepFreeze

I didn't? If I didn't, I accept that it gave an invalid proof.

My definitions are very straightforward use of just the basic logical symbols and a few basic symbols of set theory. And if you don't understand the notation...then you can ask. — TonesInDeepFreeze

A few questions wouldn't be enough. I suspect I'd have to read at least one textbook to truly understand what you mean by "A continuum is a decomposition of a hyperspace module with sidewise-partitionable step-wise radii."

The original post in this thread purported to prove that the continuum does not exist. The argument wasn't that it doesn't exist in nature but that it doesn't exist mathematically. — TonesInDeepFreeze

Maybe I should do as jgill recommends and create a new thread. Thoughts?

Then I would be right to ask you to defined the undefined terminology in those definitions, and again until (1) You finally fail to reach primitives or (2) You end up in a circle or (3) You do reach primitives. — TonesInDeepFreeze

Yes, but readers often offer a certain level of leniency to keep the conversation flowing smoothly. For example, you stated that 'is a continuum' is an adjective, which it isn’t. I didn’t want to nitpick that because, as a reader, I can easily fill in the gaps and understand what you meant.

As I understand, you reject using infinite sets. But you say that we encounter continua. So continua are finite? — TonesInDeepFreeze

Yes, that's the first part of my definition - continua are finite objects. I believe all objects (mathematical and physical) are finite. At the same time I believe that the rigorous mathematical foundations of set theory, including all the work of Cantor, are valid, and that my only qualm with set theory is the philosophy underlying it (centered around actual infinities). But perhaps we should focus on continua for this discussion...

So, I don't know why ChatGPT "struggled to make sense of it" for you. — TonesInDeepFreeze

The following is for your amusement only. It's not worth our time to discuss the validity of this response.

As of a recent amendment to site guidelines, using ChatGPT as an academic source is grounds for a warning. You're not being warned for it now, I just thought I'd bring it to your attention, so that you can avoid doing it again. — fdrake

:up:

Ostensive indications and understandings not from explicit definition but rather from gleaning in context are fine and useful, mainly at the stage of basic intuitions. — TonesInDeepFreeze

Consider the following as an ostensive definition. Is this not basic?

I will be interested in what younger and more agile brains make of this. :chin:

• fundamental objects: indivisible wholes that serve as the basis for constructing composite objects (which are composed of other fundamental objects) through partitioning.
• by its very nature: the characteristic of concern is fundamental and inseparable from the object or concept itself
• multiple vertices representing the same node: multiple vertices that mistakenly represent the same object (are consolidated into a single vertex).
• broken down into finer and finer elements: Used as a basis for constructing a composite continua composed of more and more objects.
• change the continuity: When a continuum is used as the basis for constructing a composite object, there is a disagreement in meeting the continuity requirements between the original continuum and the composite object constructed based on it.
• acts: procedure
• algorithm describes: Outlines the process of
• The cardinality of an algorithm is determined by comparing it to other algorithms whose cardinalities are already known: I'm trying to capture the essense of Cantor's diagonal argument.
• continua one partition at a time: Continua are fundamental and require little explanation. What matters is how they serve as the basis for constructing composite continua, and how this process connects to established mathematics. This process involves partitioning, which is what I've been describing.
• connected: Hopefully my illustrations in my last post make clear that I mean connected in the exact same sense as it is mean in mathematical graphs.

I will be interested in what younger and more agile brains make of this. :chin: — jgill

Me too. I often think about giving this up and just spending my free time like most people do. I'm truly open to being pushed in either direction.

I've never been able to see where it is you are going. Maybe it's just me, old and weary of days. — jgill

It's not just you. My ideas are half-baked and I'm not great at communicating them in a way digestible to mathematicians. I appreciate you following along even though you don't see where it is I am going with this.

misinterpretation of their meaning — keystone

What misinterpretations of the meanings of foundational mathematics? What writings by mathematicians or philosophers are you referring to?

I think it makes sense to focus on the topics that are absolutely essential for understanding continua — keystone

The mathematical definition is given in topology. How could the actual mathematical definition not be at the very heart of comparing the mathematical definition with alternative definitions? It seems to me that you're rationalizing your unwillingness to inform yourself on the subject.

I suspect I'd have to read at least one textbook to truly understand what you mean by "A continuum is a decomposition of a hyperspace module with sidewise-partitionable step-wise radii." — keystone

What? You didn't immediately apprehend that was a spoof? But a spoof with a point; a point that you missed even though I mentioned it explicitly.

Thus you went past my point, which is: If you don't understand the actual definition I gave, then you can ask me. It's curious that you wish to engage people with your notions, to provide them with, at best, vague definitions that they should take time and energy to understand or persist asking for explanations to then also take time and energy to understand, but you haven't the slightest inclination to even glance over a mathematical definition given to you by a person who has, at extreme length and in extreme detail (in at least two other threads) engaged your notions. Why is that? Could it be in your personal characteristics? (Some variation of being so overly infatuated with your own mind that there's little intellectual juice left in you to bother learning much about the mathematics that other people have given lives of intellectual labor to?) But I should hush myself lest one of the Pecksniffian pearl clutchers here accuses me of "ad hominen".

you stated that 'is a continuum' is an adjective, which it isn’t. — keystone

What? You started your post by agreeing that it is an adjective. It is an adjective, a predicate in this case.

You said you agreed but then you disagree. So I doubt you got the point at all:

'the continuum' is a noun phrase that designates a particular mathematical object, viz. the set of real numbers.

'is a continuum', formally, is an adjective phrase, a predicate, that designates a property of certain mathematical objects, viz. the property of being a compact, connected, Hausdorff topology (or, put another way, being a compact, connected, Hausdorff set relative to a topology on that set).

'is a continuum', informally, is an adjective phrase, a predicate, that designates a property of certain things, e.g. a rainbow.

But in this discussion, we see people refer to both 'the continuum' and 'continua', so we should be careful not to conflate those terms.

my only qualm with set theory is the philosophy underlying it (centered around actual infinities). — keystone

So you have no objection to the axiom of infinity itself, only with philosophizing that there exist "actual" infinite sets? And what do you mean by "actual"? If one views mathematical sets to be mathematically actual but one does not opine as to whether there are physically actual sets, is that okay with you? If one holds that abstractly there are infinite sets but one does not opine that physically there are infinite sets, is that okay with you?

Engineering claringly uses math as if it applies to reality. — Gregory

Claringly, not a word I know. Typo for something else? Clearly? Clarity?

In any event, many disciplines from physics to engineering to biology to economics to baseball statistics use math as a tool. Math itself is studied by mathematicians for its own sake, without regard for utility. But I think you are conflating using math as a tool to study or model an aspect of reality, with reality itself. An error you will repeat below and that I'll try to place in context.

You seem to be saying there is nothing contradictory about continuums or that there would only be such only if they were in the real world. — Gregory

If standard set theory, ZFC, is consistent, then there is nothing contradictory about the mathematical real numbers; as they are a construction within ZFC.

As Gödel showed, ZFC can not prove its own consistency. So we can never be sure if ZFC is consistent, without assuming the consistency of even stronger axiomatic systems. But if ZFC is consistent, then the real numbers are not contradictory. That's as far as I can go. I can't tell you for sure if ZFC is consistent, and I can't claim with certainty that there isn't some terrible contradiction lurking within our conception of the mathematical real numbers.

I have no idea what's true in an absolute sense about the real world, nor does anyone else. We have some fabulous mathematical models of the real world that predict the outcomes of experiments to ten or twelve decimal places (the magnetic moment of the electron being a famous example). But that just says that we have a pretty good mathematical model. It doesn't tell us what nature is all the way down.

Science is about models, not ultimate truth. I think that's pretty well understood these days, but also widely misunderstood.

So then there is something about physical matter that in its properties is not entirely mathematical as we understand that. — Gregory

Other way 'round, I would say. There is something about mathematics that's not necessarily entirely physical. Mathematics has a true continuum. It's unknown whether any such thing exists in nature.

Mathematics has infinite sets; the natural numbers {0, 1, 2, 3, 4, ...} being an example intuitively plausible to almost everyone. You can "always add one more." But there are no infinite collections in nature, as far as we know.

So mathematics has many objects, ideas, and gadgets, that as far as we know, have no correlate or instantiation in nature.

That may be true, although I would like to hear reasons why some day. Where do we draw the line when applying math to matter? How do we know we've gone too far? — Gregory

We apply math to nature to the extent that it's useful. Riemannian geometry is just the thing for general relativity. Functional analysis of Hilbert spaces is just the thing for quantum physics.

We've "gone too far" when we start believing that our mathematical theories ARE reality, as opposed to merely MODELING reality. As you are consistently doing, and as you are about to do in your next paragraph.

String theory vs loop quantum gravity. One has little points that are really strings (1 dimension in 0 dimension?) And the other discrete space. The biggest question in physics (quantum gravity) wants to settle the question of the continuum. They don't want to just throw their hands up — Gregory

This is exactly wrong. In fact you mentioned this wrong idea a while back, in a post that was filled with so many misconceptions and errors (IMO of course, nothing personal) that I didn't bother to answer it, lest I appear to be piling on. Perhaps I should get to it.

String theory and LQG are competing mathematical models, I repeat, mathematical models, that are proposed to be able to predict the results of experiments that we can carry out with our historically contingent experimental apparati.

I don't think anyone claims that they are competing metaphysical theories, claims that nature is "really" that way. Or if they do, they are making a category error, confusing physics for metaphysics.

There is not a dispute between string theory and LQG as to how nature "is." There are two competing ideas for how nature should best be modeled.

Or to put this another way: To the extent that physicists argue about the best mathematical model of the world, they are doing physics. To the extent that they argue about how the world really "is," they are doing metaphysics.

Physics is not metaphysics. Physics is about models. Metaphysics is about the (possibly unknowable by we finite, fallible humans) way things really are. Plato's cave and all that. All we can see is shadows.

I hope this point is clear. Physics is about building models that explain the experiments we can do, up to the limits of precision that we can measure. It is NOT any longer the theory of what is "true" in any absolute sense. This is summed up in the famous phrase, shut up and calculate. Meaning, don't ask what's truly going on. Just use the theory to analyze and predict the results of experiments.

We should remember not to confuse the map with the territory.

Are you mixed up? You said ChatGPT struggled to make sense of my actual definition, not my parodic definition. When I put my actual definition to ChatGPT, it replied as I posted. Meanwhile, it seems that the point of my parody went past you, even though I said what the point is. And ChatGPT listed possible purposes of that whacky definition but overlooked parody or put-on as one of them.

AI is remarkable. The ability, on the fly, to compose text, to code programs, to create images, to make spreadsheets, etc. is amazing. But it's not intelligence. It can't discern truth from falsehood even in a million different simple cases. It fails often to prove even simple mathematical theorems. Etc. And it can't explain its own errors. It's a source of gross misinformation and misunderstanding. It is to be vigilantly mistrusted. Humankind doesn't deserve to degrade itself by blind acceptance of every inferior and dangerous stage of technology. And AI doesn't have a sense of humor at a level greater than a five-year-old.

using ChatGPT as an academic source is grounds for a warning. — fdrake

Should be Wikipedia too.*

* Actually, I don't favor censoring references to the output of AI bots or publicly edited encyclopedias. And Wikipedia is magnitudes better than AI bots (which is not saying much for Wikipedia). But that doesn't mean I shouldn't say how dangerously unreliable they are.

My ideas are half-baked and I'm not great at communicating them in a way digestible to mathematicians. — keystone

Who do you think they are comprehensible to, other than yourself?

I'll look at this later, if my time, patience and supply of snacks is adequate.