Imagine the nerve of somebody demanding fair treatment for all kinds of people, even the designated victims! Appalling, innit? — Vera Mont
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
That's not quite what I'm saying. The process described by the op has no limit. — Metaphysician Undercover
That should be clear to you. It starts with a first step which takes a duration of time to complete. Then the process carries on with further steps, each step taking half as much time as the prior. The continuity of time is assumed to be infinitely divisible, so the stepping process can continue indefinitely without a limit. Clearly there is no limit to that described process — Metaphysician Undercover
I think what's confusing you into thinking that there is a limit, is that if the first increment of time is known, then mathematicians can apply a formula to determine the lowest total amount of time which the process can never surpass. Notice that this so-called "limit" does not actually limit the process in any way. The process carries on, unlimited, despite the fact that the mathematician can determine that lowest total amount of time which it is impossible for the process to surpass. — Metaphysician Undercover
Clearly, the supposed "limit" is something determined by, and imposed by, the mathematician. — Metaphysician Undercover
To understand this, imagine the very same process, with an unspecified duration of time for the first step. The first step takes an amount of time, and each following step takes half as much time as before. In this case, can you see that the mathematician cannot determine "the limit"? All we can say is that the total cannot be more than double the first duration. But that's not a limit to the process at all. It's just a descriptive statement about the process. It is a fact which is implied by an interpretation of the described process. As an implied fact, it is a logical conclusion made by the interpreter, it is "not inherent to the sequence", but implied by it. — Metaphysician Undercover
That it is not inherent, but implied, can be understood from the fact that principles other than those stated in the description of the process, must be applied to determine the so-called "limit". — Metaphysician Undercover
We must subvert our tendency to compete — Benj96
This appears to be the case. — Vera Mont
No. What Trump says and does and what the Supreme Court says and does are not the same. — Fooloso4
You're pointing to the limit of a mathematical series. A step-by-step process does not reach anything. There is no step that ends at, or after, the one-minute mark. Calculating the limit does not alter that mathematical fact. — Relativist
I also think you are misinterpreting the meaning of limit. — Relativist
This article describes it this way:
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value...
The formal definition intuitively means that eventually, all elements of the sequence get arbitrarily close to the limit, since the absolute value |an − L| is the distance between an and L. — Relativist
You just said to me that one second of time can't pass; and this, I reject. Am I understanding you correctly?
— fishfry
No, I didn't. I said the stair-stepping PROCESS doesn't reach the 1 second mark. Are you suggesting it does? — Relativist
There is no limiting process in the premises of the op, nor in what is described by ↪Relativist . The "limiting process" is a separate process which a person will utilize to determine the limit which the described activity approaches. Therefore it is the person calculating the limit who reaches the limit (determines it through the calculation), not the described activity which reaches the limited. — Metaphysician Undercover
This isn't the sense of "counting" I'm using. The sense I'm using is "the act of reciting numbers in ascending order". I say "1" then I say "2" then I say "3", etc. — Michael
P1. It takes me 30 seconds to recite the first natural number, 15 seconds to recite the second natural number, 7.5 seconds to recite the third natural number, and so on ad infinitum.
P2. 30 + 15 + 7.5 + ... = 60
C1. The sequence of operations1 described in P1 ends at 60 seconds without ending on some final natural number.
But given that ad infinitum means "without end", claiming that the sequence of operations described in P1 ends is a contradiction, and claiming that it ends without ending on some final operation is a cop out, and even a contradiction. What else does "the sequence of operations ends" mean if not "the final operation in the sequence is performed"?
So C1 is a contradiction. Therefore, as a proof by contradiction:
C2. P1 or P2 is false.
C3. P2 is necessarily true.
C4. Therefore, P1 is necessarily false.
And note that C4 doesn't entail that it is metaphysically impossible to recite the natural numbers ad infinitum; it only entails that it is metaphysically impossible to reduce the time between each recitation ad infinitum. — Michael
What is it about 'physical' that makes this difference? Everybody just says 'it does', but I obviously can physically move from here to there, so the claim above seems pretty unreasonable, like physics is somehow exempt from mathematics (or logic in Relativist's case) or something. — noAxioms
You italicize 'according to present physics', like your argument is that there's some basic flaw in current physics that precludes supertasks. How so? — noAxioms
I mean, I can claim that there are no physical supertasks, but only by presuming say some QM interpretation for which there is zero evidence, one that denies physical continuity of space and time. — noAxioms
By definition a supertask, physical or otherwise, is completed. If it can't, it's not a supertask. — noAxioms
I agree — Benj96
A healthy society can have universal healthcare — Benj96
Which has no bearing on what I'm arguing. — Michael
I'm not talking about infinite sets and transfinite ordinals. I'm talking about an infinite succession of acts. If you can't understand what supertasks actually are then this discussion can't continue. — Michael
Here's a definition for you: "a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time".
The key parts are "sequence of operations" and "occur sequentially". — Michael
If I write the natural numbers in ascending order, one after the other, then it is metaphysically impossible for this to complete (let alone complete in finite time). This has nothing to do with what's physically possible and everything to do with logical coherency. — Michael
And it doesn't address the issue. — Michael
If I write the natural numbers in ascending order, one after the other, then this can never complete. — Michael
To claim that it can complete if we just write them fast enough, but also that when it does complete it did not complete with me writing some final natural number, is just nonsense, — Michael
and so supertasks are nonsense. — Michael
That we can sum an infinite series just does not prove supertasks. — Michael
No, I'm responding to you to explain that your reference to mathematical sets and mathematical limits does not address the issue with supertasks. — Michael
I've provided arguments, and examples such as Thomson's lamp that shows why. — Michael
Would you prefer the term "act"? It is metaphysically impossible for an infinite succession of acts to complete. — Michael
Have you even looked up supertasks? I don't know how you can confuse them with mathematical sets. — Michael
If there are an infinite number of whole numbers, and an infinite number of decimals in between any two whole numbers, and an infinite number of decimals in between any two decimals, does that mean that there are infinite infinities? — an-salad
And an infinite number of those infinities? And an infinite number of those infinities? And…(infinitely times. And that infinitely times. And that infinitely times. And…) … — an-salad
The task consists of a sequence of actions occurring at intervals of time that decrease by half at each step: 1/2 minute, 1/4, 1/8,.... It is logically impossible for this sequence of actions to reach the 1 minute mark (the point in time at which the descent is considered completed), it just gets increasingly close to it. — Relativist
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
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
↪jgill That's true, but that just makes it physically impossible. I think it's stronger: logically impossible. — Relativist
No. An infinite set is not an infinite sequence of events. An infinite sequence of events would be counting every member of an infinite set. It is metaphysically impossible to finish counting them. — Michael
That's not relevant to the claim I'm making. — Michael
I'm saying that if I have finished counting the members of some set then some member must be the final member I counted. — Michael
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
Because I'm arguing against the possibility of a supertask. You're the one who interjected with talk of mathematical limits. I'm simply responding to explain that this doesn't address the concern I have with supertasks. — Michael
I'm not saying that it's the same. I'm saying that as well as being a physical impossibility, supertasks are also a metaphysical impossibility. — Michael
No physical law can allow for an infinite sequence of events to be completed. — Michael
The very concept of an infinite sequence of events being completed leads to a contradiction. — Michael
To claim that it is metaphysically possible to have finished writing out an infinite number of natural numbers but also that there is no final natural number that I wrote is to talk nonsense. — Michael
If I finished writing out any number of natural numbers than there will be a final natural number and that natural number will be a finite number. This is a metaphysical necessity. — Michael
This is an example of a supertask:
I write down the first ten natural numbers after 30 seconds, the next ten natural numbers after 15 seconds, the next ten natural numbers after 7.5 seconds, and so on.
According to those who argue that supertasks are possible I can write out infinitely many natural numbers in 60 seconds.
Examples such as Thomson's lamp show that supertasks entail a contradiction. So even though it is true that 30 + 15 + 7.5 + ... = 60, it does not follow that the above supertask is possible.
It makes no sense to claim that I stopped writing out the natural numbers after 60 seconds but that there was no final natural number that I wrote. — Michael
I have not seen it demonstrated that anyone demands similarity of outcomes. — Vera Mont
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
Read the whole thing, but it did not counter my post any. Unless of course you didn't read my whole post and just assumed I said things I didn't, misunderstanding the context, and using the strawperson argument. — Echogem222
How can we define a hole as a type of nothing when empty space itself is considered a positive value? — Echogem222
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
What's that to do with equality or equity? Outcomes owe a whole lot to beginnings. It doesn't mean that everything (??) should be the same or that everyone should be the same, it means that everyone should have the same chance of a positive outcome. — Vera Mont
Right! It's not the sequence described in the scenario! There is a background temporal sequence, as the clock ticks, that reaches 1, but we aren't mapping the step counting to the ticks of the clock. The step-counting sequence occurs only at points of time <1. In real analysis, this is called a "right open interval" (i.e.it's open on the right= the endpoint is not included in the interval). 1 is the endpoint, but not included within this interval. — Relativist
The limit of the series is "reached" only in the sense that we can reach a mathematical answer. — Relativist
The physical process of sequentially counting steps, doesn't "reach" anything other than increasingly higher natural numbers. — Relativist
Deriving the limit just means we've identified where the sequential process leads. — Relativist
In this case, we've derived that the limit is infinity- but what does infinity correspond to in the scenario? — Relativist
The meaning is entailed by the fact there are infinitely many natural numbers, so it means the process continues without end. It can mean nothing else. — Relativist
There is a difference between saying that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1 and saying that one can write out every 1/2n in order. The latter is not just a physical impossibility but a metaphysical impossibility. — Michael
Some say that the latter is not a metaphysical impossibility because it is metaphysically possible for the speed with which we write each subsequent 1/2n to increase to infinity, and so that this infinite sequence of events (writing out every 1/2n) can complete (and in a finite amount of time). — Michael
Examples such as Thomson's lamp show that such supertasks entail a contradiction and so that we must reject the premise that it is metaphysically possible for the speed with which we write each subsequent 1/2n to increase to infinity. — Michael
If you want to say that supertasks are possible — Michael
but then have to make up some nonsense final state like "pumpkin" then I think this proves that your claim that supertasks are possible is nonsense and I have every reason to reject it. — Michael
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
Ontological randomness may be logically possible but it's philosophically repugnant. — Metaphysician Undercover
The problem being that if something is deemed as random, it is in that sense unintelligible. So if something is deemed as ontologically random, and it is considered to be unintelligible, then there is no will to attempt at figuring it out. — Metaphysician Undercover
Now the problem is that if something appears to be random there is no way of knowing whether it is epistemologically random, or ontologically random, because of the unintelligibility of it. — Metaphysician Undercover
So we won't know which until we figure it out, therefore we must assume it to be epistemologically random. — Metaphysician Undercover
And even if it is ontologically random, we will still never know that this is the case, so we will always have to assume that it is epistemologically random, and try to figure it out. The category of "ontological randomness" is absolutely useless. — Metaphysician Undercover
The idea that equality means that everyone is the same, or should be treated in the same way in all contexts is little more than political propaganda. No-one believes that. — Ludwig V