I'm sure there are other ways to define the ordering of rational numbers, that's just my favorite.You're taking that as fundamental? — 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.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'm sure there are other ways to define the ordering of rational numbers, that's just my favorite. — keystone
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
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.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
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
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. — jgill
I'm not quite sure what you mean by "believe in the rational numbers." — keystone
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
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
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
However, the discussion about actual infinity and the nature of real numbers could go on endlessly. — keystone
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
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
On my timeline, I start by constructing the entire narrative of him running from 0 to 1. — keystone
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
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
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
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
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
So, do you believe in the rational numbers? Is that the number system we're working in? — fishfry
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
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
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
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 have a strong affinity for the Stern-Brocot Algorithm — keystone
How did this become so important to you? — jgill
I'll write the legitimate path from point 3 to point 5.1 with interval notation:Is this a legitimate "path" ? — jgill
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
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
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.Can a path be circular? — jgill
You may not realize it but you are asking a loaded question. I believe in 'rational numbers' but not 'the rational numbers'. — keystone
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
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
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
Again, I have a strong affinity for the Stern-Brocot Algorithm — keystone
, but I don't assert that it's the exclusive method to assign meaning to rationals. — keystone
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
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
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 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
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
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
Ditto.Ok. My eyes glaze a little more every time you mention the S-B tree — fishfry
the metric space is topological — keystone
WikipediaFormally, 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.
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
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.
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.You accept some rational numbers. Not much of a continuum you have there. You understand that, right? — fishfry
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
A view that has near universal mindshare, but ok, I'm a brainwashed mathematical sheep if you like. — fishfry
I think you are an intuitionist. — 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.You reject the algorithm given by the Leibniz series pi/4 = 1 - 1/3 + 1/5 - 1/7 + ...? — fishfry
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'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.Can we move on please? — fishfry
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
Do you think I'm using the term topological incorrectly? — keystone
What would be a homeomorphism of [0,0]U(0,.3)U[.3,.3]U(.3,.5)U[.5,.5] ? — jgill
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
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
Slowly work your way through this book and you will see why we ask so many questions. — jgill
And don't mix philosophy of mathematics with the real deal. — jgill
Rather, that interval description describes paths which can be transformed into each other via stretching and compressing, such as the following 3 paths: — keystone
you and fryfish are having a really tough time — keystone
And don't mix philosophy of mathematics with the real deal. — jgill
I don't understand why you would say this — keystone
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
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
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
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
Bottom-up view: Using a supertask, — keystone
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'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
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
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
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
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
I'd like to move forward since we haven't yet reached the most interesting topics — keystone
, but if you believe that I'm not defining a continuum, then there's no point in proceeding further. — 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? — jgill
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
But you haven't got a continuum if your intervals contain only rational numbers. — fishfry
But the nature of a continuum is pretty deep, way beyond my knowledge of philosophy. — fishfry
Do you believe in the number 1/3 then? — fishfry
Consider one of your rational intervals [0,1]. What is its length? — fishfry
It's length is 0.3 for all 3 paths depicted below because all 3 are homeomorphic. — keystone
MU has written a similar notion about continua and points. Perhaps you can put some meat on the table. — jgill
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
Leave the realm of real numbers at first. — jgill
It looks like you simply move the point [.3,.3] down the line segment to different (faulty) positions. — jgill
I don't know. — jgill
But you haven't got a continuum if your intervals contain only rational numbers.
— fishfry
Ok, this was an excellent post! — keystone
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
Let's consider the interval (0,0.5). — keystone
From a bottom-up perspective, the objects within the interval are aleph-1 actual points. — keystone
From a top-down perspective, the object within the interval is a single continua. — keystone
It doesn't contain the rational points between 0 and 0.5, it contains no points. — keystone
However it holds the potential for rational number points between 0 and 0.5. — keystone
It's only deep from a bottom-up perspective. From the top-down perspective it is elementary. — keystone
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
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
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 accept this correction.there are 2^ℵ0 real numbers — 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'). — keystone
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
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
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
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
You say that the length of all these rational points adds up to 0. I agree. — keystone
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
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
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
The problem is that you're not allowing continua to be valid objects in themselves. — keystone
It is as if you are only allowing points to be valid objects. — keystone
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
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