There are no new original records of Zeno's paradoxes so they are not new ideas. However, I think that Zeno's paradoxes remain unsolved, and I have an original perspective that resolves these and many other paradoxes in a way that they no longer seem contradictory. — keystone
I sense you can tell I'm enthusiastic about this viewpoint, but it seems you aren't interested in delving into or critiquing it. — keystone
Perhaps after considerable reflection, you've already formed your opinion on these issues and don't find additional discussion worthwhile. — keystone
So your point was that if everyone older than you dies, you'd win the argument?
Your use of Planck's quote makes not a lick of sense. He was talking about older scientists not being able to get on board with radical new ideas accepted by younger ones. But there's no radically new theory of Zeno that old scientists are rejecting, except for your own personal theory, which as far as I can tell you have not clearly articulated. So it's a failed analogy. — fishfry
I've been sharing aspects of my perspective here (but I feel like you never read it, perhaps because it seemed tangential), and other details have emerged in the Staircase thread. Nevertheless, I haven't presented it as a complete picture. Should we continue such a discussion in this thread, which has become like our private chat room, or would you like me to start a new thread?I'd be happy to critique your idea if you stated it clearly. — fishfry
Fine. What matters is that you're being very generous with your time to me and I offended you. I don't want to waste the time I have with you arguing over this. Again I'm sorry and I grant that you're entirely right on this. I hope we can put to rest this specific topic. — keystone
I've been sharing aspects of my perspective here (but I feel like you never read it, perhaps because it seemed tangential), and other details have emerged in the Staircase thread. — keystone
Nevertheless, I haven't presented it as a complete picture. — keystone
Should we continue such a discussion in this thread, which has become like our private chat room, or would you like me to start a new thread? — keystone
Agreed. Okay, let's begin!This thread's fine. The Staircase thread's hopeless, way too many side issues. It's nice and peaceful in here. — fishfry
Agreed. Okay, let's begin!This thread's fine. The Staircase thread's hopeless, way too many side issues. It's nice and peaceful in here. — fishfry
Even if you believe that the foundations of mathematics and our understanding of continua is rock solid, — keystone
you must acknowledge that it confounds many people. — keystone
Take, for instance, the difficulty in convincing a child that 0.999... equals 1, or the prominance of Cantor cranks. [/quotet]
A byproduct of bad education. Not something I can personally remedy.
— keystone
By contrast, I believe children would grasp my concept more easily because it is fundamentally simple, albeit it requires adopting a different viewpoint towards the foundations of math. To use an analogy, my perspective is less like a target that's difficult to hit and more like one that's difficult to spot. — keystone
Why I believe it's important
The validity of my ideas is still up for evaluation, but if they prove to be correct, deep truths often end up having practical relevance, even if their complete implications are not immediately apparent. Nevertheless, I am convinced that my theories could enhance mathematics education, resolve many paradoxes, and shape our understanding of reality, particularly in the context of physics. Ironically, coming from an engineer, I don't anticipate any significant impact on applied mathematics, as practitioners in such fields typically do not focus on the foundational aspects of math. I also want to clarify that my work is not meant to suggest that previous efforts by mathematicians were wasted. — keystone
How I'm going to share my ideas
I understand that for an idea to gain acceptance in the mathematical community, it needs to be formalized. I'm just not there. I don't have a formal paper to share with you, but instead, I plan to share my ideas gradually, in a manner akin to our ongoing discussions. Just as we can introduce children to the basic concepts of Cartesian coordinate systems without heavy formalities, I hope you can allow me the same flexibility in explaining my ideas with a similar level of informality. — keystone
Mathematical terminology often comes with preconceived notions; for instance, mentioning a continuum might lead you to assume I am discussing real numbers. — keystone
To avoid these assumptions and start with a clean slate, I'll be using a 'k-' prefix in front of familiar terms (like k-points, k-curves, k-continua, etc.). — keystone
By the end of our discussions, I hope you'll not only find my approach more appealing but also recognize that it aligns with the mathematics that applied mathematicians have been practicing all along. At that point, it may be justified to remove the 'k-' prefix. — keystone
Thoughts? — keystone
I agree, I just wanted one post to set the stage before I get into it...you didn't say a thing yet. — fishfry
Well, how much beef can one actually put in a paragraph? Have you ever sunk your teeth into an abstract?If you have a paragraph or two that I can sink my teeth into — fishfry
Anyway, I don't want to write another long post. My first real post will come tomorrow...I got consumed by the Staircase post this evening... — keystone
I don't want you to go easy on me. I pride myself in my ability to correct my trajectory in the face of new evidence/feedback. — keystone
And if it's not too much to ask, can you keep it short? — fishfry
Should I abbreviate my explanation, you might resort to conventional thinking to bridge the gap, which could lead to misunderstanding. — keystone
Pass the popcorn, please. I am sitting in the bleachers watching with interest. :chin: — jgill
It is impossible to prove anything mathematically using physical constructions. — fishfry
That I would be engaging with someone too obsessed for their own good. I would feel that I need to tread cautiously. — fishfry
I would really appreciate that. I don't plan to have many photographs in my subsequent posts. This was just my way of laying the groundwork.Now I do want to try to give this a fair reading. — fishfry
But it is no mirage. The same difficulty arises in math with limits and with the repeating value of constructs like Pi. These are NOT mirages. They are actual and demonstrable within reality. So much of reality answers to the limit functions that their utility and probable inclusion as meaningful and dependable is a great practice. If you wish to dismiss them, I must report that you'd need some fairly compelling, next to miraculous new ways of looking at the entire universe in order to approach success.You wish to speak and reason in the realm of actual infinities when you cannot do such a thing. Reasoning fails there. So your tool of reasoning is the wrong tool. Well done.
— Chet Hawkins
I don't think you understand my position. I'm playing in the "paradise" which Cantor created (involving infinite sets) not because I believe in it but because I want to convince others that it's a mirage (at least in my view). — keystone
So YES, it is intriguing and also impossible. As for your second sentence, no, not at all. Unless you misstated what you were trying to say, all regular shapes of equal sides are easily of finite volume at any n where n = length of a side. {picks up D&D dice to prove it. Yup, finite volume. }The sides should be of the same length.
— Chet Hawkins
Isn't the concept of an infinite-sided die that could fit in your hand intriguing? It’s impossible to construct a die with finite volume if you insist that all sides must be of equal length. — keystone
Belief in such a concept has no relationship to stating that the concept is not useful in the example given. In REALITY the infinity goes, BOTH, MULTIPLE, OR ALL; ways at the same time. So, offering examples that do not match reality is ... a mirage ... which I thought was what you were trying to avoid or point out.And since infinity extends in both directions, or all directions, and not just one direction your arbitrary single bound of natural numbers is yet another nonsensical limit that does not help in any way.
— Chet Hawkins
Do you not believe in natural numbers being bounded by 0 (or 1) on one end? And regarding time, isn't it widely believed that time had a beginning (meaning one boundary of time is t=0)? — keystone
Are you with me? I know this seems extremely basic (and perhaps inconsequential), but I'm laying the groundwork for a more consequential idea so I hope you stick with me. — keystone
I would really appreciate that. I don't plan to have many photographs in my subsequent posts. This was just my way of laying the groundwork. — keystone
I have no idea what your point is... — fishfry
At this stage, I'm making such minor points that perhaps you are confused why it took me so many words (and pictures) to express it. If that is the case, my apologies. — keystone
I think what I'm trying to say is the following:
1) Topological spaces have no sensible notion of distance. — keystone
2) Topological metric spaces have a sensible notion of distance. — keystone
3) If you lived outside a topological metric space, you wouldn't be able to use it as a measuring tool on external objects (i.e. the metric qualities of the space are not applicable to objects outside of the topological metric space). — keystone
4) If you lived inside a topological metric space, you'd perceive it as a metric space,[/quoute]
Little unclear. Who is the perceiver? How do they perceive they're in a metric space? I suppose by applying the basic definition that there exists a distance function satisfying the usual requirements. In which case an internal perceiver and an external perceiver would use exactly the same method of determining that a space is a metric space.
— keystone
where the topological qualities aren't obvious in everyday experiences. For instance, if our world were a topological metric space and everything, including the space, ourselves and our measuring tools, suddenly grew twice as big, we wouldn’t detect the change because all our measurements would scale up too. — keystone
5) If it is always possible for an object to exist outside of a topological metric space, it's notion of distance cannot be universally applied to all objects. I phrased this as, 'there cannot exist a Universal Elastic Ruler'. — keystone
6) I'm constructing a topological metric space from the ground up, rather than examining one that already exists in completion. So, in my example, it's a very crude ruler and there is no mention of real numbers. Does this qualify as a topological metric space? — keystone
Aside from the topological discussion, I also made the following point:
7) I'm treating continua as fundamental objects and points as emergent objects which become actualized when I make cuts. — keystone
I've adopted the 'k-' prefix to denote this distinction, as it's common to encounter the reverse belief - that points are fundamental objects and continua are created by assembling infinite points. — keystone
Perhaps you wouldn't characterize your viewpoint in these exact terms; you might regard points and continua as simply coexisting without one preceding the other. However, it's undeniable that the conventional approach primarily describes continua in terms of points rather than the reverse. — keystone
Is there disagreement or confusion on any of these points? — keystone
I understand that as a trained mathematician, you have the ability to articulate complex ideas clearly using descriptive language. I admire that skill, but as an engineer, my strengths lie more in visual thinking. This is particularly true with mathematics, where I sometimes struggle to express my thoughts precisely in words. Consequently, I tend to rely on illustrations to communicate my ideas. I ask for your patience and flexibility in trying to understand the essense of my message.I'm still concerned about that screwdriver ... — fishfry
Yes, that's right.Who is the perceiver? How do they perceive they're in a metric space? I suppose by applying the basic definition that there exists a distance function satisfying the usual requirements. In which case an internal perceiver and an external perceiver would use exactly the same method of determining that a space is a metric space. — keystone
Instead of saying that there cannot exist a "Unversal Elastic Ruler" what if I say there cannot exist a "Universal Metric"?Ok, but "universal elastic ruler?" That part I don't get. — fishfry
Think of it like this: a hole is an emergent property. To have a hole, you first need an object that can contain a hole. In this sense, the object is more fundamental. We begin with the object, which holds the potential for a hole. Then, once we make a cut, what we have is the same object, but now with an actual hole in it.Emergent objects become actualized? Bit vague for me. — fishfry
I've adopted the 'k-' prefix to denote this distinction, as it's common to encounter the reverse belief - that points are fundamental objects and continua are created by assembling infinite points.
— keystone
Losing me. — fishfry
Okay, this feels like progress. Let's iron out the points discussed above and then I'll give you more details on where this is going.Not much disagreement, only confusion about where this is all going. — fishfry
I understand that as a trained mathematician, you have the ability to articulate complex ideas clearly using descriptive language. I admire that skill, but as an engineer, my strengths lie more in visual thinking. This is particularly true with mathematics, where I sometimes struggle to express my thoughts precisely in words. Consequently, I tend to rely on illustrations to communicate my ideas. I ask for your patience and flexibility in trying to understand the essense of my message. — keystone
Instead of saying that there cannot exist a "Unversal Elastic Ruler" what if I say there cannot exist a "Universal Metric"? — keystone
Think of it like this: a hole is an emergent property. To have a hole, you first need an object that can contain a hole. In this sense, the object is more fundamental. We begin with the object, which holds the potential for a hole. Then, once we make a cut, what we have is the same object, but now with an actual hole in it. — keystone
I've adopted the 'k-' prefix to denote this distinction, as it's common to encounter the reverse belief - that points are fundamental objects and continua are created by assembling infinite points. — keystone
If you return to my photographs, — keystone
you will see that I start with a continous object and put cuts in it. I call those cuts points. Just as an object is more fundamental than the hole, with my view a continua is more fundamental than the cuts (i.e. points). I used k-continua and k-points instead of continua and points because I wanted to avoid a debate over what's more fundamental. In my sandbox the continua are more fundamental. If you want to grant me that, then perhaps we can set aside all this 'k-' terminology. — keystone
Okay, this feels like progress. Let's iron out the points discussed above and then I'll give you more details on where this is going.
If it's not obvious, I want you to know that I really appreciate you sticking with me on this. — keystone
No, I'm only talking about topological metric spaces. I'm pointing out that their metrics don't extend beyond their boundaries (meaning externally, they act like topological spaces without a metric), and internally, they have entirely geometric characteristics (meaning internally, they are indistinguishable from metric spaces without the topological aspects).You're pointing out that some topological spaces aren't metrizable. Right? — fishfry
Interesting! Let's treat the Discrete Metric as a trivial metric, and by Universal Metric I'm considering only non-trivial metric.You can put the discrete metric on any space of points whatsoever. — fishfry
Wow, it's a deeper topic than I imagined.There's a whole SEP article on holes. Deep stuff. — fishfry
It turns out the photos were more helpful to me than to you. You've helped me realize that what I'm actually discussing are metrics.I did not understand the photos. — fishfry
There are two primary methods for creating core mathematical artifacts:So far I've got the idea that you think objects are more fundamental than holes. I just don't see why you're telling me this. — fishfry
No, I'm only talking about topological metric spaces. — keystone
I'm pointing out that their metrics don't extend beyond their boundaries (meaning externally, they act like topological spaces without a metric), — keystone
and internally, they have entirely geometric characteristics (meaning internally, they are indistinguishable from metric spaces without the topological aspects). — keystone
Interesting! Let's treat the Discrete Metric as a trivial metric, and by Universal Metric I'm referring to a non-trivial universal metric. — keystone
There's a whole SEP article on holes. Deep stuff.
— fishfry
Wow, it's a deeper topic than I imagined. — keystone
It turns out the photos were more helpful to me than to you. You've helped me realize that what I'm actually discussing are metrics. — keystone
So far I've got the idea that you think objects are more fundamental than holes. I just don't see why you're telling me this.
— fishfry
There are two primary methods for creating core mathematical artifacts: — keystone
Bottom-up Approach:
Starts with tiny building blocks to assemble (or at least define) more complex mathematical objects.
Points are considered fundamental in this approach. — keystone
This method is akin to assemblage art, where separate elements are combined to form a whole.
Top-down Approach:
Begins with a larger, unified block and divides it to produce mathematical objects.
Continua are fundamental in this approach.
Similar to sculpting, where material is removed from a larger mass to reveal the desired form. — keystone
I've observed that orthodox mathematics predominantly favors the bottom-up approach. — keystone
However, my informal exploration of the top-down method has revealed — keystone
a perspective where everything seems to fit together perfectly, without any apparent disadvantages, paradoxes, or unresolved issues compared to the bottom-up view. — keystone
I'd like to share this perspective with you, — keystone
so you can either help identify any potential flaws (I don't want to waste my time on a dead end) or guide me further (for example, I've already learned from this discussion that I should be describing them as topological metric spaces rather than elastic rulers). — keystone
Point taken.A metric space is typically just called a metric space. — fishfry
I need to bring this one picture back.It makes no sense to talk about "outside" the space till we say what set that is — fishfry
You're right. Scratch the Universal Metric. If my metric is |x2-x1| I want to say that there is no Universal Set (within my sandbox) for which my metric yields 0 across the board. This is yet another trivial conclusion since we know that rational numbers alone cannot model a continuum.I have no idea what the "universal metric" is. You have not communicated that to me. — fishfry
Is it sets all the way down or do you eventually get to points? Anyway, you don't have to answer that question. I'm willing to agree that it doesn't matter which is more fundamental. What matters is what approach yields the most powerful math. Let's move on.Elements of sets are sometimes called points, but it's possible to do set theory without elements! — fishfry
I was hoping to get closure on the open topics first, but if you don't have any problems with this post then I think we're there. By the way, if you ever feel like my time is running out then please let me know and I'll plow through. But at the current pace I'm extracting a lot of value from our conversation.I don't get the top-down idea. 'Splain me please. — fishfry
Based on this picture, what I want to say is that Achilles can occupy any position on the continuous line, but, for this specific example where the ruler only has a few tick marks on it, I'm limited to describing his location using one of five specific intervals:
(0,0)
(0,0.5)
(0.5,0.5)
(0.5,1)
(1,1) — keystone
I believe what I want to do is define a 2D metric space on set S={(0,0),(0,0.5),(0.5,0.5),(0.5,1),(1,1)} where each element is an ordered pair (x1,x2).
While I will eventually explore higher dimensional spaces, for now, let's say that my sandbox is limited to sets of ordered pairs of rational numbers. — keystone
You're right. Scratch the Universal Metric. If my metric is |x2-x1| I want to say that there is no Universal Set (within my sandbox) for which my metric yields 0 across the board. This is yet another trivial conclusion since we know that rational numbers alone cannot model a continuum. — keystone
Elements of sets are sometimes called points, but it's possible to do set theory without elements!
— fishfry
Is it sets all the way down or do you eventually get to points? Anyway, you don't have to answer that question. I'm willing to agree that it doesn't matter which is more fundamental. What matters is what approach yields the most powerful math. Let's move on. — keystone
I don't get the top-down idea. 'Splain me please.
— fishfry
I was hoping to get closure on the open topics first, but if you don't have any problems with this post then I think we're there. [/quoote]
I don't understand what you are doing. Seems like random flailing.
— keystone
By the way, if you ever feel like my time is running out then please let me know and I'll plow through. But at the current pace I'm extracting a lot of value from our conversation. — keystone
Indulge me in an analogy.Sets are fundamental, not points. — fishfry
Okay, I lost you because I made a mistake. Let me try again:Lost me again. In a metric space the distance between two points is 0 if and only if they are the same point. — fishfry
And as I said, you will have trouble rigorously defining what you mean by outside of your metric space, unless you first say what the enclosing set is. So please do. — fishfry
If my metric is |x-y| I want to say that there is no Universal Set (within my sandbox) for which my metric yields 0 across the board. — keystone
Zeno greatly inspires me, yet from my viewpoint, his paradoxes serve merely as an aside. I assure you, the core thesis I'm proposing is much more significant than his paradoxes. But to save me from creating a new picture, please allow me to reuse the Achilles image below as I try again to explain the visuals.Sorry what? We're doing Zeno now? I must pass on that. — fishfry
Inconsistent systems allow for proving any statement, granting them infinite power. While debating the consistency of ZFC is beyond my current scope and ability, my goal is to develop a form of mathematics that not only achieves maximal power but also maintains consistency. Furthermore, I aim to show that this mathematical framework is entirely adequate for satisfying all our practical and theoretical needs.You are trying to invent something more powerful than contemporary math? — fishfry
I haven't studied his original work, so I can't say with certainty, but I don't believe I'm referring to Euclid's formulation.Sometimes a "point" in a function space can be a function. Sometimes a point is just a tuple of coordinates in Euclidean space. Points aren't fundamental. Perhaps you're thinking of Euclid's original formulation of geometry. — fishfry
I'm familiar with these methods. I believe there is a bottom-up and a top-down interpretation of them. I'm not satisfied with the orthodox bottom-up interpretation of them.For example we can define the real numbers internally, by building them up from the empty set to get the naturals, integers, rationals, and finally reals. — fishfry
I'm getting there, and your feedback has been instrumental in enhancing my understanding of this 'digital rain'. Up until now, my approach has primarily been visual.You seem to want to make points out of cuts in a line, but I don't see where you're going with that. — fishfry
Indulge me in an analogy.
I see the Matrix (pictures): — keystone
Both perspectives accurately correspond to the simulation. So I agree that sets are fundamental, and I could even be convinced that digital rain is more fundamental than the Matrix. — keystone
But Let's not go there. I'm specifically talking about the (continuous version of the) Matrix where I believe continua are more fundamental than points. But I don't even want to debate this further, I'd rather show you what could be done with a Top-down approach and let you decide. — keystone
I bring up the Matrix because, I want you to know that I recognize the unique purity and precision of the digital rain, but there are times, especially in discussions on geometry, when it's more effective to visually interpret the geometry from within the Matrix. Please allow yourself to enter the Matrix, try to understand my visuals, just for a little while. End of Matrix analogy. — keystone
Okay, I lost you because I made a mistake. Let me try again:
Set: { (0,0) , (0,0.5) , (0.5,0.5) , (0.5,1) , (1,1) } where x1 and y1 in element (x1,y1) is a rational number
Metric: d((x1,y1),(x2,y2)) = | (x1+y1)/2 - (x2+y2)/2 | — keystone
Upon further consideration, I've decided to significantly restrict my focus to a smaller enclosing set. I am now interested only in what I want to call 'continuous sets' which are those sets where, when sorted primarily by the x-coordinate and secondarily by the y-coordinate, the y-coordinate of one element matches the x-coordinate of the subsequent element. For example, we'd have something like: — keystone
You're right, |x-y| doesn't qualify as a metric. Let me try again. Forget about Universal Set. Instead, I aim to define a Continuous Exact Set. A set is defined as an exact set if all elements satisfy |x-y|=0. I propose that within my enclosing set, the only Exact Set is the trivial set, containing just one element. Once again, this isn't a groundbreaking revelation; I am simply emphasizing that rational numbers by themselves are insufficient for modeling a continuum. — keystone
Zeno greatly inspires me, yet from my viewpoint, his paradoxes serve merely as an aside. I assure you, the core thesis I'm proposing is much more significant than his paradoxes. But to save me from creating a new picture, please allow me to reuse the Achilles image below as I try again to explain the visuals.
The story: Achilles travels on a continuous and direct path from 0 to 1.
The bottom-up view: During Achilles' journey he travels through infinite points, each point corresponding to a real number within the interval [0,1].
The top-down view: In this case, where there's only markings on the ground at 0, 0.5, and 1, I have to make some compromises. I'll pick the set defined above and describe his journey as follows:
(0,0) -> (0,0.5) -> (0.5,0.5) -> (0.5,1) -> (1,1) — keystone
In words what I'm saying is that he starts at 0, then he occupies the space between 0 and 0.5 for some time, then he is at 0.5, then he occupies the space between 0.5 and 1 for some time, and finally he arrives at 1. — keystone
Inconsistent systems allow for proving any statement, granting them infinite power. While debating the consistency of ZFC is beyond my current scope and ability, my goal is to develop a form of mathematics that not only achieves maximal power but also maintains consistency. Furthermore, I aim to show that this mathematical framework is entirely adequate for satisfying all our practical and theoretical needs. — keystone
I haven't studied his original work, so I can't say with certainty, but I don't believe I'm referring to Euclid's formulation. — keystone
I'm familiar with these methods. I believe there is a bottom-up and a top-down interpretation of them. I'm not satisfied with the orthodox bottom-up interpretation of them. — keystone
I'm getting there, and your feedback has been instrumental in enhancing my understanding of this 'digital rain'. Up until now, my approach has primarily been visual. — keystone
Aside: Please note that I will have a house guest for several days, which may cause my responses to be slower than usual. — keystone
I'll address your other comments later, but for now, let's concentrate on one particular issue. It seems that you're either unable or unwilling to acknowledge even the most basic points I've raised. I apologize if this appears to diverge from your interests, but focusing on the image below, can you see how the instructions on the left relate to the image on the right? (This is not a trick question)Wasted on me, hope you got something from it...No idea, eyes glazed long ago. — fishfry
It seems that you're either unable or unwilling to acknowledge even the most basic points I've raised. — keystone
I apologize if this appears to diverge from your interests, but focusing on the image below, can you see how the instructions on the left relate to the image on the right? (This is not a trick question) — keystone
Once again you leave me utterly baffled as to why you posted this. — fishfry
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