I take it you're not a fan of analogies. — keystone

I like analogies fine. I don't understand any of yours. I thought we were making progress on at least having the same conversation when we were traversing the unit interval. Instead of engaging you're changing the subject.

You're taking that as fundamental? — fishfry

I'm sure there are other ways to define the ordering of rational numbers, that's just my favorite.

and then you declined to respond when I asked you twice how you get from (0, .5) to .5 without invoking a limiting process. Then you changed the subject. — fishfry

I thought I twice answered your question. Let me try again. What you don't seem to appreciate is that with the top-down view we begin with the journey already complete so halving the journey is no problem. If we already got to 1, then getting to 0.5 is no problem. You can't seem to get your mind out of the bottom-up view where we construct the journey from points, which indeed requires limits.

I thought we were making progress on at least having the same conversation when we were traversing the unit interval. — fishfry

I think we're making progress. 2 steps forward, 1 step back.

I'm sure there are other ways to define the ordering of rational numbers, that's just my favorite. — keystone

So you believe in the rational numbers? But then the reals are easily constructed from the rationals as Dedekind cuts or equivalence classes of Cauchy sequences. If you believe in the rationals you have to believe in the reals.

I thought I twice answered your question. Let me try again. What you don't seem to appreciate is that with the top-down view we begin with the journey already complete so halving the journey is no problem. If we already got to 1, then getting to 0.5 is no problem. You can't seem to get your mind out of the bottom-up view where we construct the journey from points, which indeed requires limits. — keystone

You are the one who started at 0, then got to (0, .5), and then magically completed a limiting process to get to .5. I ask again, how is that accomplished?

You are the one who started at 0, remember?

So you believe in the rational numbers? But then the reals are easily constructed from the rationals as Dedekind cuts or equivalence classes of Cauchy sequences. If you believe in the rationals you have to believe in the reals. — fishfry

I'm not quite sure what you mean by "believe in the rational numbers." From a top-down perspective, there's no need to assert the existence of either R or Q, especially since all the subsets within the enclosing 'set' are finite. If you suggest that this enclosing 'set' is infinite, then we must rethink our definition of what an 'enclosing set' actually is in this context. I was hoping to put this particular discussion aside for now, as it will likely divert attention from our main focus.

Regarding Dedekind cuts, they involve splitting the infinite set of rational numbers into two subsets. This presupposes both the existence of an infinite entity (Q) and the completion of an infinite process (the split). If one rejects the concept of actual infinity, then it's questionable whether real numbers necessarily follow from rational numbers.

However, the discussion about actual infinity and the nature of real numbers could go on endlessly. I acknowledge that these concepts are crucial for a bottom-up approach, but can we instead focus on seeing how far a top-down perspective—devoid of actual infinities and traditional real numbers—can lead us? In the top-down view, reals hold a special role, just not as conventional numbers.

You are the one who started at 0, then got to (0, .5), and then magically completed a limiting process to get to .5. I ask again, how is that accomplished?
You are the one who started at 0, remember? — fishfry

I believe the confusion arises from the dual meanings of "start" due to there being two timelines: (1) my timeline as the creator of the story and (2) the timeline of the man running from 0 to 1 within the story.

On my timeline, I start by constructing the entire narrative of him running from 0 to 1. The journey is complete from the start. I can make additional cuts to, for example, see him at 0.5. Regardless of what I do, the journey is always complete.
On the running man's timeline, he experiences himself starting at 0, travelling towards 1, and later arriving at 1.

I think you're trying to build his journey on his timeline, one point at a time. The runner would indeed believe that limits are required for him to advance to 0.5. I want you to look at it from my timeline (outside of his world), where the journey is already complete. If I want to see where he is at 0.5 I just cut his complete journey in half. Does that clarify things?

Unlike supertasks, no magic is required to complete the journey with the top-down view. Assuming you accept the Peano Axioms as a conventional framework, you're familiar with the concept of succession, which defines progression from 1 to 2 to 3, and so on. This is essentially what I'm applying as well; on the runner's timeline he progresses in succession from 0 to (0,0.5) to 0.5, and so on. Please take note, this particular succession from 0 to 0.5 involves only 2 steps. No limit is required, just as no limits are employed with the Peano Axioms.

d([2,3],[1,4])=0 ? [2,3] not equal to [1,4] — jgill

Returning to your example, (2,3) and (1,4) cannot both be elements of a continuous set so the set you are considering is not included in the enclosing set — keystone

You speak of a metric space. Precisely what are the "points" in such a space? Then explain the metric you have created giving "distances" between these points.

You speak of a metric space. Precisely what are the "points" in such a space? Then explain the metric you have created giving "distances" between these points. — jgill

I'm not suggesting that a single metric space can represent all continuous 1D systems. Rather, for each continuous 1D system we generate using a top-down approach, we can define a finite set and a corresponding metric to describe it.

The now familiar example is the unit line with points at 0, 0.5, and 1. This system is composed of the following 5 intervals:

Interval 1: [0,0]
Interval 2: (0,0.5)
Interval 3: [0.5,0.5]
Interval 4: (0.5,1)
Interval 5: [1,1]

For this system, there are 3 points: 0, 0.5, and 1.

As for the corresponding metric space, the set consists of the following 5 'points':
Ordered pair 1: (0,0)
Ordered pair 2: (0,0.5)
Ordered pair 3: (0.5,0.5)
Ordered pair 4: (0.5,1)
Ordered pair 5: (1,1)

The metric is d((a1,b1),(a2,b2)) = | (a1+b1)/2 - (a2+b2)/2 |
This metric corresponds to the distance between the midpoint of two intervals.

I've mentioned this before though...

I'm not quite sure what you mean by "believe in the rational numbers." — keystone

You confused me a while back. You said you don't believe in the real numbers [or some similar wording].

So I asked you, what are those symbols 0, .5, 1, and so forth? If they're not real numbers, what are they?

That's why I asked you if you believe in the rational numbers. If you do, then you have to also believe in the reals, since the reals are constructed from the rationals. If you don't, then again I ask you what are 0, .5, and 1?

From a top-down perspective, there's no need to assert the existence of either R or Q, especially since all the subsets within the enclosing 'set' are finite. — keystone

You have been freely using the symbols 0, .5, and 1. If they are not real, and they are not rational, then I don't know what those symbols mean. Can you define them?

"... all the subsets within the enclosing 'set' are finite"???? Means what? Lost me there.

If you suggest that this enclosing 'set' is infinite, then we must rethink our definition of what an 'enclosing set' actually is in this context. I was hoping to put this particular discussion aside for now, as it will likely divert attention from our main focus. — keystone

You're the one with some notion of enclosing set. A metric space is a set with a distance function. If it lives in a larger ambient set, then you have to say what that is. You started a long time ago saying something like "the metric doesn't apply outside the metric space." Ok that's true, but what is outside? You have to say what that is.

Regarding Dedekind cuts, they involve splitting the infinite set of rational numbers into two subsets. This presupposes both the existence of an infinite entity (Q) and the completion of an infinite process (the split). If one rejects the concept of actual infinity, then it's questionable whether real numbers necessarily follow from rational numbers. — keystone

Ok fine. You reject the real numbers. You already said that.

So I asked you, do you believe in the rational numbers. And you asked me what I mean by that!

If you use symbols like 0, .5, and 1, you have to say what they are.

So, do you believe in the rational numbers? Is that the number system we're working in?

In which case I have to echo @jgill's excellent question as to whether you accept intervals like [pi, pi + 1], and if not, why not.

However, the discussion about actual infinity and the nature of real numbers could go on endlessly. — keystone

You could bring it to a quick conclusion by saying, "Yes, we are working in the rational numbers."

But you won't even say that! Leaving me totally confused.

I acknowledge that these concepts are crucial for a bottom-up approach, but can we instead focus on seeing how far a top-down perspective—devoid of actual infinities and traditional real numbers—can lead us? In the top-down view, reals hold a special role, just not as conventional numbers. — keystone

Sure. Then what are these funny symbols 0, .5, and 1 that you keep on using? What do your interval notations denote?

If you're working in the rationals that's fine, but when I asked you about it, you asked me what I meant by the question.

You are the one who started at 0, then got to (0, .5), and then magically completed a limiting process to get to .5. I ask again, how is that accomplished?
You are the one who started at 0, remember?
— fishfry

I believe the confusion arises from the dual meanings of "start" due to there being two timelines: (1) my timeline as the creator of the story and (2) the timeline of the man running from 0 to 1 within the story. — keystone

The stories are very unhelpful to me. As are timelines.

On my timeline, I start by constructing the entire narrative of him running from 0 to 1. — keystone

What is this '0'? What is this '1'? Define your terms.

The journey is complete from the start. I can make additional cuts to, for example, see him at 0.5. Regardless of what I do, the journey is always complete.
On the running man's timeline, he experiences himself starting at 0, travelling towards 1, and later arriving at 1. — keystone

You are using these funny symbols. I know the usual standard mathematical meaning of those symbols, but you have rejected them in favor of your "top down" idea. So what are these symbols? What if we called them "fish" and "bazooka?" Then nothing at all would be clear, but your logic error would be more obvious

You want to reject standard mathematics but freely use symbols like 0, .5, and 1, without defining them.

Do you see the problem?

I think you're trying to build his journey on his timeline, one point at a time. The runner would indeed believe that limits are required for him to advance to 0.5. I want you to look at it from my timeline (outside of his world), where the journey is already complete. If I want to see where he is at 0.5 I just cut his complete journey in half. Does that clarify things? — keystone

No, since I don't know what 0.5 and "half" mean, in the absence of standard bottom-up math.

Do you see your circularity problem? You want to start by rejecting standard math, but then you won't tell me what these symbols mean in your system.

Unlike supertasks, no magic is required to complete the journey with the top-down view. Assuming you accept the Peano Axioms as a conventional framework, — keystone

Ah. That's quite a lot already, for someone claiming to reject infinite processes and standard bottom-up math.

So you are willing to start with the Peano axioms? Is that your starting place? Then I know what 0 and 1 are, but I'm still not sure about this 0.5 thing.

you're familiar with the concept of succession, which defines progression from 1 to 2 to 3, and so on. This is essentially what I'm applying as well; on the runner's timeline he progresses in succession from 0 to (0,0.5) to 0.5, — keystone

0.5 is not defined by the Peano axioms. What is it?

and so on. Please take note, this particular succession from 0 to 0.5 involves only 2 steps. No limit is required, just as no limits are employed with the Peano Axioms. — keystone

No idea what 0.5 is. But at least after all this you agreed to stipulate the Peano axioms. That's a start. A start from classical, bottom-up math.

I'll save you some trouble and show you how to build out the rational numbers from the Peano axioms. You extend the natural numbers to the integers, then you do a construction called the field of fractions of an integral domain.

I'm not entirely sure if that construction is legit in Peano without the axiom of infinity, but I can live with it.

So after all this, I think you are working in the rational numbers, and 0.5 has its usual meaning. Is that right?

I can live with that. Although the rational numbers are tragically deficient as a continuum. You know that, right? They're full of holes. They're not continuous in the intuitive sense.

So, do you believe in the rational numbers? Is that the number system we're working in? — fishfry

You may not realize it but you are asking a loaded question. I believe in 'rational numbers' but not 'the rational numbers'. The difference is that 'the rational numbers' corresponds to Q, the complete set of rational numbers. With the top-down view, such completeness isn't essential (rather, consistency is the aim of the top-down approach). When constructing my metric spaces, I find that I only need to traverse a certain depth in the Stern-Brocot tree to encompass all the rational numbers I require.

To clarify, I don't believe in the existence of a complete Stern-Brocot tree. Instead, I believe in the existence of the algorithm capable of generating the tree to any arbitrary depth, although not infinitely. No one has ever encountered the entire tree; rather, we've only interacted with the algorithm and finite trees that it creates. Henceforth, let's refer to it as the Stern-Brocot Algorithm to eliminate ambiguity.

Equipped with the Stern-Brocot Algorithm, the mathematical symbols of rational numbers retain their conventional meanings. If we could execute the Stern-Brocot Algorithm to its limiting conclusion and produce the entire tree, there would theoretically exist a 'row-omega' containing the real numbers. This implies that, theoretically, real numbers necessarily follow from the rational numbers and the Stern-Brocot Algorithm. However, it's evident that running the Stern-Brocot Algorithm to completion is impossible. Consequently, the existence of real numbers doesn't necessarily follow from the existence of rational numbers.

Again, I have a strong affinity for the Stern-Brocot Algorithm, but I don't assert that it's the exclusive method to assign meaning to rationals.

You're the one with some notion of enclosing set. A metric space is a set with a distance function. If it lives in a larger ambient set, then you have to say what that is. — fishfry

The difference lies in our perspectives on the existence of mathematical objects. I assume you are with the bottom-up majority who adhere to the belief that all mathematical entities actually exist, accessible when required, and that these objects fit neatly into sets. In contrast, my perspective maintains that no mathematical object inherently exists; it only manifests when a mind conceives of it. Therefore, if no mind currently contemplates the number 42, it does not exist in actuality; it merely holds the potential for existence.

Regarding the enclosing set, I don't subscribe to the notion of its inherent existence. Instead, I endorse an algorithm capable of generating sets to have arbitrarily many elements, albeit not infinite. If you run this algorithm long enough, it will generate the set we're looking for to define our metric space.

In which case I have to echo jgill's excellent question as to whether you accept intervals like [pi, pi + 1], and if not, why not. — fishfry

I refuse to regard pi as a boundary for my intervals because it cannot be generated using the Stern-Brocot Algorithm. Pi does hold significance in my perspective, but I think it's more appropriate to delve into that explanation if/once we move on to two dimensions.

So you are willing to start with the Peano axioms? Is that your starting place?...But at least after all this you agreed to stipulate the Peano axioms. That's a start. A start from classical, bottom-up math. — fishfry

I only referred to the Peano Axioms to point out the concept of succession. When viewed from the top-down perspective, numbers are not constructed from the naturals (I agree, that would imply a classical, bottom-up math start). Natural numbers are only distinctive in that they are positioned on the right-most branch of any tree created with Stern-Brocot Algorithm, which indeed makes them quite unique.

Although the rational numbers are tragically deficient as a continuum. You know that, right? They're full of holes. They're not continuous in the intuitive sense. — fishfry

I agree that rational numbers alone cannot model a continuum. With the top-down view, this is equivalent to saying that points alone cannot model a continuum. And that's why I'm starting with a continuum (i.e. using intervals rather than numbers). It's much easier to get points from a continuum than it is to get a continuum from points.

I have a strong affinity for the Stern-Brocot Algorithm — keystone

This is perhaps the second time this oddity from number theory has cropped up on this forum. I knew a tiny bit about it since it can involve elementary continued fraction theory. How did this become so important to you?

Is this a legitimate "path" ?. A linear arrangement of points and intervals.

Ordered pair 1: (3,3)
Ordered pair 2: (3,3.7)
Ordered pair 3: (3.7,3.7)
Ordered pair 4: (3.7,4.2)
Ordered pair 5: (4.2,4.2)
Ordered pair 6: (4.2,5.1)
Ordered pair 7: (5.1,5.1)

So, a "metric space" for this path consists of "points" (a,b) within this structure. For example, d((3,3.7),(4.2,5.1))=|(3+3.7)/2 - (4.2+5.1)/2| = 1.3. So you do not compare "points" from one path to another. Altering the path, even slightly, places it in another metric space. But a ms could be a subspace of a bigger ms. Just talking to myself, here.

You spoke earlier of an "elastic band". where does that come into the picture? Especially with regard to metric spaces? Can a path be circular?

How did this become so important to you? — jgill

I learned of it and, as I delved deeper, my amazement grew. For instance, from a paper by Niqui, I learned that the Stern-Brocot tree can serve as a basis for performing arithmetic on rational numbers. Although this method is far too inefficient and complex to be taught to children, the mere possibility of its application reinforces the idea that the Stern-Brocot tree has capabilities that are often underestimated.

Is this a legitimate "path" ? — jgill

I'll write the legitimate path from point 3 to point 5.1 with interval notation:
[3,3] U (3,3.7) U [3.7,3.7] U (3.7,4.2) U [4.2,4.2] U (4.2,5.1) U [5.1,5.1]

So you do not compare "points" from one path to another. Altering the path, even slightly, places it in another metric space. But a ms could be a subspace of a bigger ms. Just talking to myself, here. — jgill

Exactly.

You spoke earlier of an "elastic band". where does that come into the picture? Especially with regard to metric spaces? Can a path be circular? — jgill

I initially used non-technical language due to a lack of knowledge. Through this discussion, I've learned that my actual interest lies in metric spaces. To illustrate, imagine a point as a pin that anchors a 1D continuum in place. From a bottom-up perspective, the continuum, being pinned throughout, appears rigid and static. However, from a top-down perspective, there are always gaps between the pins where the continuum can deform freely. In this way, the metric space is topological and resembles what I initially described as elastic.

Here's another perspective. Consider the following question: Where is the point 0.25 in the image below? Is it at position a, b, c, or d? As I've been suggesting, until a cut is made, there is no actual point 0.25. This isn't a rigid continuum where one could say point 0.25 definitely sits at position c, the middle 0 and 0.5 in the visual representation. All that can be stated is that the interval (0,0.5) contains the potential for point 0.25. I would even be open to saying that that interval contains the potential for aleph-1 points.

Can a path be circular? — jgill

Certainly. I think that every object or concept in the bottom-up view has a counterpart in the top-down approach. It typically just needs some reimagining, often involving the transformation of an actually infinite object into a potentially infinite process.

You may not realize it but you are asking a loaded question. I believe in 'rational numbers' but not 'the rational numbers'. — keystone

Um ... ok ... I think ...

The difference is that 'the rational numbers' corresponds to Q, the complete set of rational numbers. With the top-down view, such completeness isn't essential (rather, consistency is the aim of the top-down approach). When constructing my metric spaces, I find that I only need to traverse a certain depth in the Stern-Brocot tree to encompass all the rational numbers I require. — keystone

Ok whatever. You accept some rational numbers. Not much of a continuum you have there. You understand that, right?

To clarify, I don't believe in the existence of a complete Stern-Brocot tree. Instead, I believe in the existence of the algorithm capable of generating the tree to any arbitrary depth, although not infinitely. No one has ever encountered the entire tree; rather, we've only interacted with the algorithm and finite trees that it creates. Henceforth, let's refer to it as the Stern-Brocot Algorithm to eliminate ambiguity. — keystone

There's no difference between an algorithm and the number it generates. 1/3 = .3333..., an infinite decimal, but 1/3 has a finite representation, namely 1/3

Pi has a finite representation. All the computable real numbers do.

Equipped with the Stern-Brocot Algorithm, the mathematical symbols of rational numbers retain their conventional meanings. If we could execute the Stern-Brocot Algorithm to its limiting conclusion and produce the entire tree, there would theoretically exist a 'row-omega' containing the real numbers.
This implies that, theoretically, real numbers necessarily follow from the rational numbers and the Stern-Brocot Algorithm. However, it's evident that running the Stern-Brocot Algorithm to completion is impossible. Consequently, the existence of real numbers doesn't necessarily follow from the existence of rational numbers. — keystone

I already understand that your mathematical ontology includes some but not all rational numbers. Can we move past this please?

Again, I have a strong affinity for the Stern-Brocot Algorithm — keystone

Fine whatever. Enough already.

, but I don't assert that it's the exclusive method to assign meaning to rationals. — keystone

Can we move on please?

The difference lies in our perspectives on the existence of mathematical objects. I assume you are with the bottom-up majority who adhere to the belief that all mathematical entities actually exist, accessible when required, and that these objects fit neatly into sets. — keystone

A view that has near universal mindshare, but ok, I'm a brainwashed mathematical sheep if you like.

In contrast, my perspective maintains that no mathematical object inherently exists; it only manifests when a mind conceives of it. Therefore, if no mind currently contemplates the number 42, it does not exist in actuality; it merely holds the potential for existence. — keystone

Ah. Perhaps you would enjoy intuitionism.

In the philosophy of mathematics, intuitionism, or neointuitionism (opposed to preintuitionism), is an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality.[/url]
— Wiki

Regarding the enclosing set, I don't subscribe to the notion of its inherent existence. Instead, I endorse an algorithm capable of generating sets to have arbitrarily many elements, albeit not infinite. If you run this algorithm long enough, it will generate the set we're looking for to define our metric space. — keystone

I think you are an intuitionist. I think that's what you're getting at. Can you read the Wiki link and tell me if that's what you're getting at?

Intuitionism is closely related to constructivism, the idea that mathematical objects only exist if there's an algorithm or procedure to construct them. Intuitionism is like constructivism with an extra bit of mysticism that I can never quite grasp.

I refuse to regard pi as a boundary for my intervals because it cannot be generated using the Stern-Brocot Algorithm. Pi does hold significance in my perspective, but I think it's more appropriate to delve into that explanation if/once we move on to two dimensions. — keystone

You reject the algorithm given by the Leibniz series pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...?

I only referred to the Peano Axioms to point out the concept of succession. When viewed from the top-down perspective, numbers are not constructed from the naturals (I agree, that would imply a classical, bottom-up math start). Natural numbers are only distinctive in that they are positioned on the right-most branch of any tree created with Stern-Brocot Algorithm, which indeed makes them quite unique. — keystone

Ok. My eyes glaze a little more every time you mention the S-B tree, I have no idea why this idea has such a hold on you.

I agree that rational numbers alone cannot model a continuum. With the top-down view, this is equivalent to saying that points alone cannot model a continuum. And that's why I'm starting with a continuum (i.e. using intervals rather than numbers). It's much easier to get points from a continuum than it is to get a continuum from points. — keystone

If you have a continuum but disbelieve even in the set of rationals, the burden is on you to construct o define a continuum.

Ok. My eyes glaze a little more every time you mention the S-B tree — fishfry

Ditto.

the metric space is topological — keystone

So, a continuous deformation takes path A to path B, but inside the ms of path A? Or a new ms of path B? You might illustrate this. I'm curious about these continuous deformations in the contexts of your ideas. A topology, on the other hand . . .

Formally, let X be a set and let τ be a family of subsets of X. Then τ is called a topology on X if:
Both the empty set and X are elements of τ. Any union of elements of τ is an element of τ. Any intersection of finitely many elements of τ is an element of τ.
If τ is a topology on X, then the pair (X, τ) is called a topological space.

Wikipedia

Intuitionism is closely related to constructivism, the idea that mathematical objects only exist if there's an algorithm or procedure to construct them. Intuitionism is like constructivism with an extra bit of mysticism that I can never quite grasp. — fishfry

On those very rare occasions in which the subject arises I have felt the two to be more or less alike. But, here is what Wiki has to say:

Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity. Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics.

You accept some rational numbers. Not much of a continuum you have there. You understand that, right? — fishfry

I concur that rational numbers alone, represented as points, are insufficient for constructing a continuum. That's not the argument I'm making. You keep thinking I'm trying to build a continuum. No, I'm starting with a continuum, defined by the interval notation we have discussed, and working my way down to create points.

There's no difference between an algorithm and the number it generates. 1/3 = .3333..., an infinite decimal, but 1/3 has a finite representation, namely 1/3 — fishfry

Oh no, the classic debate about whether 0.9=1. I know you dislike the S-B tree but it makes the top-down and bottom-up views very clear. Maybe use some eyedrops? :P



Top-down view: 0.9 corresponds to a journey down the tree, symbolized by the string "LR". In contrast, 1 corresponds to a specific node at the top of the tree, symbolized simply by "[]". These are fundamentally different concepts: one is a potentially infinite journey with no final destination (since there is no bottom of the tree), and the other is a definitive destination. From the top-down perspective, 0.9 does not equal 1.

Bottom-up view: Using a supertask, the creation of the tree is completed and we are able to go the limit to observe the bottom row of the tree: 'row-omega'. In this case, the journey "LR" does indeed arrive at a destination (at row-omega), and that destination is precisely 1. When working with row-omega (i.e. real numbers), the journey is indistinguishable from the destination. From the bottom-up perspective, 0.9 equals 1. (Or does it equal a pumpkin?)

I'm pretty sure that you won't like my depiction of the bottom-up view as I frame it in a way that make's it clearly problematic. I'm fine with not investing further on this specific topic at this time as it really will just be a distraction from the main topic.

A view that has near universal mindshare, but ok, I'm a brainwashed mathematical sheep if you like. — fishfry

I'm not questioning the mathematics itself, but rather the philosophical underpinnings of the mathematics. For instance, I recognize Cantor's remarkable contributions to math, even though I personally do not subscribe to the concept of infinite sets. His contributions have a valuable top-down interpretation.

I think you are an intuitionist. — fishfry

You make a good point. However, I'm not sure about the details of the constructivist approach - my impression is that a typical intuitionist would say that the number 42 permanently exists once we've intuited it. So while I'm hesitant to label myself hastily, I do think that broadly speaking I fit into this camp.

You reject the algorithm given by the Leibniz series pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...? — fishfry

I totally accept and am in awe with the algorithm. I just don't think the algorithm can be run to completion to return a number. I also don't think it has to be run to completion to be valuable.

If you have a continuum but disbelieve even in the set of rationals, the burden is on you to construct o define a continuum. — fishfry

I agree, but isn't that what I've been doing all along? Doesn't [0,0] U (0,0.5) U [0.5,0.5] U (0.5,1) U [1,1] define a continuum? Maybe it would be valuable if you detail what you think a continuum must be. For example, will you only accept the definition if it is composed solely of points (and no intervals)?

Can we move on please? — fishfry

I'd like to move forward since we haven't yet reached the most interesting topics, but if you believe that I'm not defining a continuum, then there's no point in proceeding further.

So, a continuous deformation takes path A to path B, but inside the ms of path A? Or a new ms of path B? You might illustrate this. I'm curious about these continuous deformations in the contexts of your ideas. — jgill

Continuous deformations cannot alter topology (i.e. convert the path to a different path). Let's illustrate with an example: imagine a class of 100 students tasked with 2D graphing y = x^2 within the domain of [-10,10] and range of [-10, 10], including grid lines every 1 unit. Suppose you collect all the completed drawings and stack them. What are the chances that every graph aligns perfectly? Essentially zero. Some graphs might be larger, others smaller, some with lines drawn with rulers, others not. However, assuming there are no errors, each graph conveys the same underlying information. You could, theoretically, continuously deform one student's graph to match another's. It is in this sense that I mean they are all topologically equivalent. What would mess things up is if someone added an extra line, e.g. y=x. Then it is a different graph with a different topology described with a different metric space.

Do you think I'm using the term topological incorrectly?

Also, as an aside, think about what the students actually did - they started off with a blank piece of paper (a continuum) and drew lines on them (cuts). They didn't use pointillism (points) to create the graph (continuum). I believe that, in many senses, we've been approaching things top-down all along.

By the way, I just want to say that I truly appreciate our interactions on this thread. You are genuinely trying to understand what I'm saying and giving good feedback. Thanks!

Do you think I'm using the term topological incorrectly? — keystone

Well, if you were to avoid both metric spaces and variations of the word "topology" it might mitigate what seems to be a questionable attempt to employ legitimate mathematical notions within a somewhat murky mix of ideas. However, I applaud your enthusiasm. I used to teach point set topology and metric spaces, so I am biased toward their traditional interpretations. In any event at some point you must present a clear and detailed description of your ideas that mathematicians might have reservations about but can follow the logic.

What would be a homeomorphism of [0,0]U(0,.3)U[.3,.3]U(.3,.5)U[.5,.5] ?

The late George Simmons of Colorado College wrote a marvelous book many years ago, perhaps the finest introduction to modern analysis and topology ever written: Introduction to Topology and Modern Analysis. Slowly work your way through this book and you will see why we ask so many questions. And don't mix philosophy of mathematics with the real deal. Just an I idea.

What would be a homeomorphism of [0,0]U(0,.3)U[.3,.3]U(.3,.5)U[.5,.5] ? — jgill

I'm not saying that [0,0]U(0,.3)U[.3,.3]U(.3,.5)U[.5,.5] is homeomorphic to anything. Rather, that interval description describes paths which can be transformed into each other via stretching and compressing, such as the following 3 paths:



If you are suggesting that perhaps I shouldn't say they're homeomorphic because I haven't formally defined how they are 'essentially the same' then technically you're right, but might using that term help convey the general idea in this informal discussion? After all, it seems easier to say homeomorphic than it is to repeatedly say 'paths that can be transformed into each other via bending, stretching, and compressing, without any cutting or gluing.'

Well, if you were to avoid both metric spaces and variations of the word "topology" it might mitigate what seems to be a questionable attempt to employ legitimate mathematical notions within a somewhat murky mix of ideas. — jgill

I acknowledge that I might be using the term "topology" somewhat loosely, but based on the above comment, it seems to fit what I'm trying to describe, doesn't it? Also, am I not using metric spaces correctly?

In any event at some point you must present a clear and detailed description of your ideas that mathematicians might have reservations about but can follow the logic. — jgill

Agreed. At this point I'm presenting a very informal idea and you and fryfish are having a really tough time following. This is the least desirable scenario for me. Here's how I see this possibly turning out:

• 1) You can follow it and you think it's wrong - Great. I won't waste any more time on it.
• 2) You can follow it and you think I'm onto something - Great. I'll invest the needed time and money to make it better and hopefully learn a few tips from the discussion.
• 3) You can't follow it so you have no idea what to think - Bummer, I don't know what to do now.

I know you guys don't owe me anything, but I'll be lucky if you guys stick with me long enough for this to conclude in one of the first two scenarios. I have a history of working in isolation for extended of periods of time before getting feedback so this conversation is refreshing.

Slowly work your way through this book and you will see why we ask so many questions. — jgill

If you leave this thread soon, I'll still appreciate your suggestion. However, if you stay longer and gain a deeper understanding of my needs, I'll be even more grateful. Perhaps by the end, you might even suggest that I need a mathematics degree, not just a single textbook.

And don't mix philosophy of mathematics with the real deal. — jgill

I don't understand why you would say this. Some of the greatest mathematicians were philosophers and vice versa. There's a lot of overlap.

Rather, that interval description describes paths which can be transformed into each other via stretching and compressing, such as the following 3 paths: — keystone

It looks like you simply move the point [.3,.3] down the line segment to different (faulty) positions.
How does this affect your metric?

you and fryfish are having a really tough time — keystone

:cool:

And don't mix philosophy of mathematics with the real deal. — jgill


I don't understand why you would say this — keystone

Very very few contemporary mathematicians give a fig leaf about Platonic vs non-Platonic arguments or similar discussions about whether math is embedded in nature or in the mind.

On those very rare occasions in which the subject arises I have felt the two to be more or less alike. But, here is what Wiki has to say:

Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity. Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics. — jgill

Right. Constructivism is purely technical. Intuitionism is constructivism plus some kind of psychological motive or mystic woo. That's my understanding.

You accept some rational numbers. Not much of a continuum you have there. You understand that, right?
— fishfry
I concur that rational numbers alone, represented as points, are insufficient for constructing a continuum. That's not the argument I'm making. You keep thinking I'm trying to build a continuum. No, I'm starting with a continuum, defined by the interval notation we have discussed, and working my way down to create points. — keystone

But you haven't got a continuum if your intervals contain only rational numbers

How can you say you exclude the real numbers, then write down an interval and call it a continuum?

There's no difference between an algorithm and the number it generates. 1/3 = .3333..., an infinite decimal, but 1/3 has a finite representation, namely 1/3
— fishfry

Oh no, the classic debate about whether 0.9=1. — keystone

No that is not what I said at all and it has nothing to do with that.

I know you dislike the S-B tree but it makes the top-down and bottom-up views very clear. Maybe use some eyedrops? :P — keystone

Just gonna skip it. Can't relate, don't see its relevance. I'm more focussed on what you just said: that you are "starting with a continuum" that does not include the real numbers.

I'm afraid I can't comprehend that at all.

Bottom-up view: Using a supertask, — keystone

Time is not an aspect of the tree, there is no supertask.

I'm pretty sure that you won't like my depiction of the bottom-up view as I frame it in a way that make's it clearly problematic. I'm fine with not investing further on this specific topic at this time as it really will just be a distraction from the main topic. — keystone

I don't even dislike it. I don't get the relevance of the entire subject. Tell me more about your continuum made up of only rational numbers. If we could get to the bottom of just one thing ...

I'm not questioning the mathematics itself, but rather the philosophical underpinnings of the mathematics. For instance, I recognize Cantor's remarkable contributions to math, even though I personally do not subscribe to the concept of infinite sets. His contributions have a valuable top-down interpretation. — keystone

You should renew your subscription :-)

I think you are an intuitionist.
— fishfry

You make a good point. However, I'm not sure about the details of the constructivist approach - my impression is that a typical intuitionist would say that the number 42 permanently exists once we've intuited it. So while I'm hesitant to label myself hastily, I do think that broadly speaking I fit into this camp. — keystone

I don't understand intuitionism, but you said mathematical objects come into existence via the imagination or acts of will of mathematicians (paraphrasing what you said earlier, sorry if I mis-stated it) and that reminded me of intuitionism.

You reject the algorithm given by the Leibniz series pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...?
— fishfry
I totally accept and am in awe with the algorithm. I just don't think the algorithm can be run to completion to return a number. I also don't think it has to be run to completion to be valuable. — keystone

Do you believe in the number 1/3 then?

If you have a continuum but disbelieve even in the set of rationals, the burden is on you to construct o define a continuum.
— fishfry

I agree, but isn't that what I've been doing all along? Doesn't [0,0] U (0,0.5) U [0.5,0.5] U (0.5,1) U [1,1] define a continuum? — keystone

Not if there are only rational numbers in the intervals. Do you understand this point? The rationals are full of holes. More holes than points in fact. Swiss cheese continuum.

Maybe it would be valuable if you detail what you think a continuum must be. For example, will you only accept the definition if it is composed solely of points (and no intervals)? — keystone

Me? The continuum is the real number line. Totally workable definition. Avoids all the philosophical overhead. But the nature of a continuum is pretty deep, way beyond my knowledge of philosophy.

I'd like to move forward since we haven't yet reached the most interesting topics — keystone

That's because you refuse to get there.

, but if you believe that I'm not defining a continuum, then there's no point in proceeding further. — keystone

You said you don't have any real numbers in your intervals, only rationals, and not even all the rationals.

Do you understand that your rational continuum is full of holes? At least tell me if you understand what I mean by that. In other words the rationals contain the points 1, 1.4, 1.41, 1.4142, ... but they don't contain the square root of 2. There's a hole there.

The reals are the completion of the rationals. The reals plug up all the holes in the rationals. That's why the reals are a continuum and the rationals aren't.

You are using interval notation but you are not including the reals. Moths ate your continuum.

Perhaps you can explain to me how an interval of rationals can be a continuum in your mind.

Bottom line: Define [0, 0.5]. Because I have no idea what you mean by that notation.

@keystone, One more thought that occurred to me, and I'm putting it in a separate post because it's short, and important.

Consider one of your rational intervals [0,1]. What is its length?

Well, if it were an interval of reals, it would have length 1. That's how length is defined in the real numbers.

But if it's an interval containing only rationals, then its length is 0. Why? Because of my very first post in this thread. It's countable additivity again. The length of a point is zero; and the length of an interval composed of countably many points is still 0.

That's another problem with your rational continuum. I said it's full of holes. And the problem is that all the length is stored in the holes!

It looks like you simply move the point [.3,.3] down the line segment to different (faulty) positions. How does this affect your metric? — jgill

Why is it faulty? That's the thing - such continuous deformations don't affect the metric. They don't affect anything at all. That's why I say those three paths are homeomorphic.

Very very few contemporary mathematicians give a fig leaf about Platonic vs non-Platonic arguments or similar discussions about whether math is embedded in nature or in the mind. — jgill

I believe that's where the issue lies. In physics, some who dismiss the quest to grasp the meaning of quantum mechanics simply say, 'shut up and calculate.' In my opinion, that approach is misguided.

But you haven't got a continuum if your intervals contain only rational numbers. — fishfry

Ok, this was an excellent post! I better understand your criticism. It lies in the fact that I'm using the term 'interval' in an unorthodox manner. I use the term interval to describe the objects (whatever they may be) lying between the upper and lower bounds.

Let's consider the interval (0,0.5).
From a bottom-up perspective, the objects within the interval are aleph-1 actual points.
From a top-down perspective, the object within the interval is a single continua. It doesn't contain the rational points between 0 and 0.5, it contains no points. However it holds the potential for rational number points between 0 and 0.5.

But the nature of a continuum is pretty deep, way beyond my knowledge of philosophy. — fishfry

It's only deep from a bottom-up perspective. From the top-down perspective it is elementary.

Do you believe in the number 1/3 then? — fishfry

I believe that I could use the Stern-Brocot algorithm to generate a 3 layer tree whose third layer will contain a node described by LL and having all the properties that we generally attribute to 1/3.

Consider one of your rational intervals [0,1]. What is its length? — fishfry

The length of continuum (a,b) is b-a. So consider the continuum defined by interval (0,0.3). It's length is 0.3 for all 3 paths depicted below because all 3 are homeomorphic.

It's length is 0.3 for all 3 paths depicted below because all 3 are homeomorphic. — keystone

Unfortunately, your approach is a muddled mixture of traditional ideas and speculative continua. I think you need to go back to the beginnings of your efforts and truly start with continua and develop an original approach to math in which points arise from these continua, avoiding the real line entirely at first. MU has written a similar notion about continua and points. Perhaps you can put some meat on the table.

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.

Please stop talking about the S-B algorithm. Both my colleague and I would rather not contemplate this thing. Leave the realm of real numbers at first.

Or, do your thing and persist until the thread dries up and vanishes. Good luck.

MU has written a similar notion about continua and points. Perhaps you can put some meat on the table. — jgill

I find myself agreeing with very little of what I've read from MU.

not referencing the real line or numbers. — jgill

avoiding the real line entirely at first — jgill

Please stop talking about the S-B algorithm. — jgill

do your thing and persist until the thread dries up and vanishes — jgill

Seriously, am I not supposed to mention numbers? Or lines? Or relevant mathematical concepts like the S-B algorithm? I'm not sure if this is meant as advice, or if it's just a polite way of telling me to be quiet.

Leave the realm of real numbers at first. — jgill

I'm not in the realm of real numbers. I'm not working with the real line. Could it be that your understanding of my view is more influenced by your preconceived notions than by what I'm communicating?

It looks like you simply move the point [.3,.3] down the line segment to different (faulty) positions. — jgill

You dodged my question about this. Why is it faulty?

I don't know. — jgill

...and yet you give advice as if you do. I think your advice might be a bit premature. It seems like you haven't fully grasped what I'm trying to say, and I haven't even reached the most important parts yet. Perhaps you're ready to move on from this discussion, and those were your final thoughts. If that's the case, that's perfectly fine. Thanks and I wish you all the best.

But you haven't got a continuum if your intervals contain only rational numbers.
— fishfry

Ok, this was an excellent post! — keystone

Yay! Thanks.

I better understand your criticism. It lies in the fact that I'm using the term 'interval' in an unorthodox manner. I use the term interval to describe the objects (whatever they may be) lying between the upper and lower bounds. — keystone

I have no problem with intervals. But intervals of rationals make terrible continua.

Let's consider the interval (0,0.5). — keystone

By this I take you to mean the set of some but not all rational numbers between those values, inclusive. Yes? I say some but not all because you have said that yourself.

From a bottom-up perspective, the objects within the interval are aleph-1 actual points. — keystone

Not if the interval contains only rationals. It depends on how you define your notation.

As a side remark, there are $2^{\aleph_0}$ real numbers, and the question of whether that happens to be equal to $\aleph_1$ is the continuum hypothesis.

From a top-down perspective, the object within the interval is a single continua. — keystone

How can it be if it contains only rationals? I have challenged you on this point several times already without your providing satisfactory explanation.

It doesn't contain the rational points between 0 and 0.5, it contains no points. — keystone

No points. It's empty?

However it holds the potential for rational number points between 0 and 0.5. — keystone

Even if I accept that, its length would be zero and it would be full of holes, hardly a continuum.

It's only deep from a bottom-up perspective. From the top-down perspective it is elementary. — keystone

So far you have expressed strange and unjustified ideas about continua. Such as that they are empty or have length 0.

Do you believe in the number 1/3 then?
— fishfry

I believe that I could use the Stern-Brocot algorithm to generate a 3 layer tree whose third layer will contain a node described by LL and having all the properties that we generally attribute to 1/3. — keystone

Um ... ok I guess that's a yes ...

Consider one of your rational intervals [0,1]. What is its length?
— fishfry

The length of continuum (a,b) is b-a. So consider the continuum defined by interval (0,0.3). It's length is 0.3 for all 3 paths depicted below because all 3 are homeomorphic. — keystone

But that interval contains only rational numbers in your notation. Its length is zero by the countable additivity of measures.

You have not grappled with this problem yet.

In fact all the length of an interval is carried by the irrationals. There aren't enough rationals to have any length at all.

How can it be if it contains only rationals? I have challenged you on this point several times already without your providing satisfactory explanation. — fishfry

I fully understand your criticism. The problem is that you are missing my point (or perhaps I should say you are missing my 'continua').

Let's continue to work with the path defined as [0,0] U (0,0.5) U [0.5,0.5] U (0.5,1) U [1,1] as depicted below.


I say that (0,0.5) and (0.5,1) contain no points so you think I'm only working with three objects - the points as depicted below. The length of all points within my system is 0 so you think the objects I'm working with have zero length.


I say that (0,0.5) and (0.5,1) describe continua so I say I'm working with 5 objects as depicted below. The length of all points within my system is indeed 0 but the length of the continua within my system add up to 1.


I prefer working with such simple paths as described above but let's do the impossible and say that somehow I could cut my unit line aleph-0 times such that there is a point for each rational number between 0 and 1.

You say that the length of all these rational points adds up to 0. I agree.
You say that there are gaps between these points. I disagree. In between each pair of neighbouring points would lie an infinitesimally small continua. If I add up the lengths of all of these tiny continua it would add up to 1. These infinitesimally small continua are indivisible.

I'm not fond of discussing impossible scenarios as they tend to lead to incorrect conclusions. Indeed, rational points do not have neighbors, and continua are inherently divisible (unless we're treating points as 0D continua, in which case they are indivisible). Therefore, we shouldn't lend too much credence to this example, but I thought it was necessary to address your points more directly.

The problem is that you're not allowing continua to be valid objects in themselves. It is as if you are only allowing points to be valid objects.

there are 2^ℵ0 real numbers — fishfry

I accept this correction.

So I figured out a better way to talk about this instead of using metric spaces. Instead, it is better to use Graph Theory.

Consider the following path:


It can be described using the following graph:


where:
vertex 0 = [0,0]
vertex 1 = (0,0.5)
vertex 2 = [0.5,0.5]
vertex 3 = (0.5,1)
vertex 4 = [1,1]

To travel from vertex 0 to vertex 4 we simply walk the connected path. One nice thing about this view is that it's clear that no limits are required to walk these graphs.

I fully understand your criticism. The problem is that you are missing my point (or perhaps I should say you are missing my 'continua'). — keystone

I'm not missing your point, I'm challenging it. The length of your rational intervals is zero. That causes a problem for your argument.

Let's continue to work with the path defined as [0,0] U (0,0.5) U [0.5,0.5] U (0.5,1) U [1,1] as depicted below. — keystone

The length of that union is zero, if the intervals are restricted to rationals. Do you agree with that point?

I say that (0,0.5) and (0.5,1) contain no points so you think I'm only working with three objects - the points as depicted below. The length of all points within my system is 0 so you think the objects I'm working with have zero length. — keystone

No points. So they're all the empty set? I'm not supposed to push back on this?

I say that (0,0.5) and (0.5,1) describe continua so I say I'm working with 5 objects as depicted below. The length of all points within my system is indeed 0 but the length of the continua within my system add up to 1. — keystone

No countable additivity? What then? How do your segments add up to 1 if they only contain rationals.

I hate to be so pedantic about this but I don't understand how an interval of rationals can have a nonzero length.

I prefer working with such simple paths as described above but let's do the impossible and say that somehow I could cut my unit line aleph-0 times such that there is a point for each rational number between 0 and 1. — keystone

The length would still be zero. Countable additivity again.

And it's not a "unit line" if it only contains rationals. This is where your intuition is failing you.

You say that the length of all these rational points adds up to 0. I agree. — keystone

Ok good.

You say that there are gaps between these points. I disagree. In between each pair of neighbouring points would lie an infinitesimally small continua. — keystone

There are no "neighboring points." Bad intuition again. Between any two rational numbers are infinitely many more distinct rationals.

If I add up the lengths of all of these tiny continua it would add up to 1. These infinitesimally small continua are indivisible. — keystone

There are no "infinitely small continua." You're just making all this up out of bad intuitions about the nature of the real numbers.

I'm not fond of discussing impossible scenarios as they tend to lead to incorrect conclusions. Indeed, rational points do not have neighbors, and continua are inherently divisible (unless we're treating points as 0D continua, in which case they are indivisible). Therefore, we shouldn't lend too much credence to this example, but I thought it was necessary to address your points more directly. — keystone

Why on earth do you troll me into arguing with your points, then admitting that you agree with me in the first place?

The problem is that you're not allowing continua to be valid objects in themselves. — keystone

Of course I do. I very much believe in the continuum, which is pretty well modeled by the standard real numbers. I say pretty well because there are other models such as the constructive real line and the hyperreal line, but those lines are not Cauchy-complete. [The constructivists wave their hands at this with some technicalities].

It is as if you are only allowing points to be valid objects. — keystone

As opposed to what? You keep hypothesizing things that do not exist, like empty continua.

[==== your second post ====]

So I figured out a better way to talk about this instead of using metric spaces. Instead, it is better to use Graph Theory.

... [stuff omitted]

To travel from vertex 0 to vertex 4 we simply walk the connected path. One nice thing about this view is that it's clear that no limits are required to walk these graphs. — keystone

Ok.

I am at an utter loss as to what you have been getting at all this time. Can you get to the bottom line on all this? So far I get that your "continua" are either empty or have length 0. Or that they somehow have length 1, despite being composed of only rationals.