Sets are fundamental, not points. — fishfry

Indulge me in an analogy.

I see the Matrix (pictures):


You see 'Digital Rain' (sets):


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. 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.

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.

Lost me again. In a metric space the distance between two points is 0 if and only if they are the same point. — fishfry

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 |

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

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:

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

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.

Sorry what? We're doing Zeno now? I must pass on that. — fishfry

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)

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.

You are trying to invent something more powerful than contemporary math? — 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.

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 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.

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 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.

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

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.

Aside: Please note that I will have a house guest for several days, which may cause my responses to be slower than usual.

A metric space is typically just called a metric space. — fishfry

Point taken.

It makes no sense to talk about "outside" the space till we say what set that is — fishfry

I need to bring this one picture back.


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)

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.

I have no idea what the "universal metric" is. You have not communicated that to me. — 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.

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.

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. 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.

You're pointing out that some topological spaces aren't metrizable. Right? — fishfry

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 can put the discrete metric on any space of points whatsoever. — fishfry

Interesting! Let's treat the Discrete Metric as a trivial metric, and by Universal Metric I'm considering only non-trivial metric.

There's a whole SEP article on holes. Deep stuff. — fishfry

Wow, it's a deeper topic than I imagined.

I did not understand the photos. — 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.

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:

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.
• 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.

I've observed that orthodox mathematics predominantly favors the bottom-up approach. However, my informal exploration of the top-down method has revealed a perspective where everything seems to fit together perfectly, without any apparent disadvantages, paradoxes, or unresolved issues compared to the bottom-up view. I'd like to share this perspective with you, 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).

I'm still concerned about that screwdriver ... — 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.

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

Yes, that's right.

Ok, but "universal elastic ruler?" That part I don't get. — fishfry

Instead of saying that there cannot exist a "Unversal Elastic Ruler" what if I say there cannot exist a "Universal Metric"?

Emergent objects become actualized? Bit vague for me. — 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.

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

If you return to my photographs, 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.

Not much disagreement, only confusion about where this is all going. — 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.

If it's not obvious, I want you to know that I really appreciate you sticking with me on this.

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.


I think what I'm trying to say is the following:

1) Topological spaces have no sensible notion of distance.
2) Topological metric spaces have a sensible notion of distance.
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).
4) If you lived inside a topological metric space, you'd perceive it as a metric space, 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.
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'.
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?

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. 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. 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.


Is there disagreement or confusion on any of these points?

It is impossible to prove anything mathematically using physical constructions. — fishfry

I know that, and I explicitly stated that the photographs are not perfect representations of the abstract objects which I'm actually talking about.

That I would be engaging with someone too obsessed for their own good. I would feel that I need to tread cautiously. — fishfry

I'm certainly not too obsessed with this. This topic has sat dormant in the back corner of my mind for years before it resurfaced with my unexpected hanging paradox paradox post a few weeks back. Should this discussion not lead anywhere significant, it'll probably return to its quiet corner. Nevertheless, I'm keen to hear your candid thoughts. Whether they nudge me towards new insights or help me lay these concepts to rest, I am welcoming of both possibilities.

Now I do want to try to give this a fair reading. — 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.

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. My goal is to present my ideas with such simplicity and clarity that you’ll effortlessly grasp them, swiftly perceiving their evident truth. So I hope that quick instead of short is acceptable.

So here goes…

PART 1 - The Elastic Ruler

THE MATERIALS
Everything that follows takes place within my abstract sandbox. I intend to construct a ruler in an unconventional manner, starting with an abstract elastic band (i.e. a k-topological object). The graphics that follow are an imperfect representation of what's truly happening my sandbox. For example, my abstract elastic band is a one-dimensional object with the property of perfect elasticity (i.e. it can stretch infinitely far), unlike its physical counterpart. It's crucial that you not ascribe physical characteristics to elements within my sandbox. So, resist the urge to suggest that my elastic band is made up of indivisible atoms or exists amidst a quantum foam.



THE TERMINOLOGY
I am going to cut the elastic band a couple of times. At this point we can begin to label regions.

• a-The void left of the band
• b-The void corresponding to the first cut of the band
• c-The void corresponding to the second cut of the band
• d-The void right of the band
• (a,b)-The band between a and b
• (b,c)-The band between b and c
• (c,d)-The band between c and d

Notice that both the void and the elastic band segments display a sense of duality in that they are defined in relation to one another. They are both important.

While we must remember that we're talking about cuts to an elastic band, the gap size is unimportant. As such it is much cleaner to make the cuts 0-width in illustrations as depicted below.



Let us use the following terminology:

• The cuts - 'k-points'
• The elastic bands - 'k-lines' or, more generally, 'k-continua'
• The labels of cuts (e.g. b) - 'k-numbers'
• The labels of k-lines (e.g. b,c) - 'k-intervals'

THE RULER
I will take the elastic band and cut it 3 times, adding k-number and k-interval labels as depicted below.

And there we have it, an elastic band ruler!

USING THE RULER
Test one: Measuring a pen in my sandbox. Conclusion, pen is 1 unit long.


Test two: Measuring a pen again in my sandbox with ruler stretched. Conclusion, pen is 0.5 units long.


Test three: Measuring a screwdriver in my sandbox with ruler slightly stretched. Conclusion, screwdriver is 1 unit long.


Test four: Measuring a screwdriver in my sandbox with ruler stretched more. Conclusion, screwdriver is 0.5 units long.


Employing this ruler as demonstrated could yield highly questionable outcomes, like equating 0.5 with 1, suggesting all objects are of the same size, or that an object's size is not consistent with itself, among others. This would appear to render the ruler quite ineffective, wouldn't it? Yet, let's explore the subsequent experiment.

Test five: Achilles and the tortoise live on the elastic band. Achilles is at (0,0.5) and the tortoise is at 0.5.



As the elastic stretches, the positions relative to one another remain constant, thus allowing the ruler to accurately depict their placement.

CONCLUSION
1) Elastic rulers are useful only when applied internally (i.e. locally).
2) Elastic rulers are not useful, and in fact misleading, when applied externally (i.e. globally).
3) IF it can be proved that there will always be abstract objects that exist outside of the ruler (similar to how there is no set of all sets), then it would follow that the existence of a Universal Elastic Ruler (one that can measure everything) is not possible.

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.

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.

you didn't say a thing yet. — fishfry

I agree, I just wanted one post to set the stage before I get into it...

If you have a paragraph or two that I can sink my teeth into — fishfry

Well, how much beef can one actually put in a paragraph? Have you ever sunk your teeth into an abstract?

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...

I suppose that if Zeno actually accepts his (unreasonable) conclusions, then you get something like just that one state. — noAxioms

Exactly. Let’s deconstruct the argument:

(1) We accept Zeno's premise as valid, asserting that in a presentist world where only a single state exists, motion is impossible.
(2) We assume that the world functions according to presentist beliefs.
(3) Our experiences clearly indicate that motion is possible.

These three assertions cannot all be true simultaneously. It’s unlikely that anyone, including Zeno, would dispute (3). You find (1) to be unreasonable. However, consider the possibility that (2) is incorrect.

Here's an alternative approach:

(1) We accept Zeno's premise as valid, asserting that in a presentist world where only a block universe exists, change of the block is impossible.
(2) We assume that the world functions according to eternalists beliefs.
(3) Our singular, consistent historical experience gives us no reason to believe that the block universe is subject to change.

In this scenario, all three points could indeed hold true, suggesting that an eternalist viewpoint might be more suitable. However, quantum mechanics challenges point (3), necessitating a more nuanced argument. Despite this, an eternalist framework—albeit with some adjustments to incorporate the quantum aspects of our universe—appears to be the most rational choice.

Not sure of the difference. If I cut a string, I don't get points, I get shorter strings. — noAxioms

The cuts themselves are the points (think Dedekind cuts).

You can under some interpretations. — noAxioms

One can observe a superposition directly? Please share a link.

Zeno's arguments are of the form (quoted from the Supertask Wiki page):
"1 Motion is a supertask, because the completion of motion over any set distance involves an infinite number of steps
2 Supertasks are impossible
3 Therefore, motion is impossible"

If motion is discreet, then premise 1 is demonstrably wrong. If it isn't, then premise 2 is demonstrably wrong, unless one just begs the conclusion and adopts the 'photo' interpretation. — noAxioms

What I aim to demonstrate is that there is a scenario where local motion is possible and continuous without involving supertasks. This occurs in a block universe where the block itself remains unchanged (i.e., no global motion), yet the entities within it experience change (i.e., local motion).

Necessary only if the first premise is to be accepted. — noAxioms

If the universe is discrete, then Zeno's paradoxes cannot occur as he described them. What I'm suggesting is that in a continuous universe, the scenarios depicted in Zeno's paradoxes can indeed unfold precisely as he described them, without necessitating the completion of supertasks.

I'm asserting that an infinite process is necessarily never completed - by definition. — Relativist

Good. Then we're on the same page!

the process of counting steps is not completable — Relativist

Are you suggesting that supertasks cannot be completed?

The process is:
1) Have fuzzy intuitions;
2) Study some math;
3) Develop far better intuitions. — fishfry

Agreed, but most importantly: (4) apply those intuitions to (the original) experiments.

It's only a finite number of steps back, even from infinity. — fishfry

I like where you're going with this. To navigate between the staircase and omega (and back), one must leap over infinite steps. This concept becomes more palatable if we consider that the steps become progressively smaller towards the bottom. However, let me try to rephrase your perspective: Icarus requires a finite number of strides to reach the bottom and a finite number to return to the top, thus avoiding any supertask. When Icarus adds 1/2, then 1/4, then 1/8, he gets bored and chooses to make a final leap. On his final leap, instead of adding an infinite series of smaller terms, he simply adds another 1/8 and reaches omega, where his calculator displays exactly 1. In this case, the infinity in the paradox describes the steps which he potentially could have traversed (and seen), not what he actually did (and saw). Since he never actually observed all steps, he is in no position to confirm that there were actually infinite steps...but there could have been...potentially. Paradox solved?

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!

It's elementary
Even if you believe that the foundations of mathematics and our understanding of continua is rock solid, you must acknowledge that it confounds many people. Take, for instance, the difficulty in convincing a child that 0.999... equals 1, or the prominance of Cantor cranks. 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.

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.

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.

Mathematical terminology often comes with preconceived notions; for instance, mentioning a continuum might lead you to assume I am discussing real numbers. 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.). 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.

Thoughts?

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!

It's elementary
Even if you believe that the foundations of mathematics and our understanding of continua is rock solid, you must acknowledge that it confounds many people. Take, for instance, the difficulty in convincing a child that 0.999... equals 1, or the prominance of Cantor cranks. 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.

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.

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.

Mathematical terminology often comes with preconceived notions; for instance, mentioning a continuum might lead you to assume I am discussing real numbers. 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.). 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.

Thoughts?

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

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.

I'd be happy to critique your idea if you stated it clearly. — 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?

STAIRCASE PARADOX

Can you see that? It's actually the exact same example as 1, 2, 3, 4, ... ω
. Any step back takes you to a number that is only finitely many steps from the beginning. You don't need infinitely long legs. In fact your legs can be arbitrarily small. Any step backward (or up the stairs) necessarily jumps over all but finitely elements of the sequence. — fishfry

I see your point, and I appreciate your analogy with the [0,1] interval. However, you need to clarify what happens in the narrative. The purpose of this narrative is to ensure that one cannot simply retreat behind formalisms. This mathematical observation doesn't change the reality that Icarus would need to jump over infinite steps. If you're suggesting he doesn’t have infinitely long legs, then perhaps he possesses infinitely powerful legs that enable him to leap over infinite steps. This might explain how he returns to the top, but it essentially sweeps the infinite staircase under the rug.

I've bethought myself and realized that the step numbers will only align if the number of steps is odd. If it is even, they won't be such a point. — Ludwig V

This brings us to another paradox - Thomson's Lamp - in that the last step can neither be even nor odd.

So the staircase down defines the staircase up. — Ludwig V

Now explain how your algorithm works for infinite stairs.

So why don't you just link me to the reading materials that would lead me to believe that the supertask you described in your op is possible to complete? That specific supertask, not supertasks in general. Let's not beat around the bush, let's get right to it. — flannel jesus

Instead, please present any supertask you consider viable, and I will demonstrate its connection to Icarus descending the staircase. For instance, do you agree that the sum of the infinite series 1/2 + 1/4 + 1/8 + 1/16 + ... equals exactly 1?

Once you decide to make this supertask accomplishable by *a human mind*, then you run into brand new problems that don't exist in a purely mathematical context. — flannel jesus

I'm unclear on whether you're disputing the existence of supertasks or merely the ability of humans to perform them. Do you believe it's conceivable for anyone physical or abstract, perhaps even a divine being like God, to accomplish a supertask?

Continually halfing the time it takes to perform the subsequent step does not just contradict the physical laws of our world but is a metaphysical impossibility. With these paradoxes we shouldn't be looking for some answer that is consistent with the premises but should accept that they prove that the premises are flawed. — Michael

Reading your posts gives me a sense of calm. :D

ou have provided no propositions or premises whatsoever, to conclude that 60 seconds may actually elapse. — Metaphysician Undercover

I said he "reached the bottom of it in just a minute." Thus, the premises address both the completion of the supertask and the passing of a minute. It seems you are challenging the incorrect premise.

There's nothing contradictory with the EXISTENCE of an actual infinite, but it's not accepted that an infinity can be traversed in a supertask. — Relativist

I would contend that all of the infinity paradoxes clearly illustrate contradictions inherent in the concept of actual infinity. Furthermore, I would argue that every definition of real numbers inherently suggests that supertasks are completable.

So a complete (i.e. well-defined) mapping shouldn't be conflated with a completed PROCESS. — Relativist

We can also map the steps to the elapsed time (1 → 0.5, 2 → 0.75, 3 → 0.875, etc.). If we conclude that a full minute has elapsed, doesn't this imply that he has traversed all the steps?

Analogously, a limit entails an abstract operation applying to a mathematical series and shouldn't be conflated with a consecutive process. — Relativist

Why not?

ZENO'S PARADOX

I don't think what you describe can be validly categorized under the term 'presentism'. — noAxioms

You're correct that presentists don't explicitly hold this belief. However, what Zeno's Paradoxes demonstrate is that if their ideas are taken to their logical conclusion, this belief is implicitly suggested.

There is no 'past, present. future' defined under eternalism. All events share equal ontology. The view differs fundamentally from presentism only in that the latter posits a preferred location in time, relative to which those words have meaning. — noAxioms

Instead of presentism vs. eternalism, let's talk about the photo vs. movie reel. For the photo and every frame of the movie reel the characters believe they're in the present. So if you're saying that the experience of the present has nothing to do with Zeno's Paradox, then I agree with you. But there is a very significant difference between a photo and a movie reel.

Irrelevant, but I prefer the one that doesn't posit the additional thing for which there is zero empirical evidence. This is my rational side making that statement. — noAxioms

Reconciling general relativity with presentism is quite challenging. Therefore, if empirical evidence influences your thinking, eternalism might be a more suitable perspective to adopt. Plus, adopting eternalism helps to render Zeno's Paradoxes largely non-paradoxical.

a attempted demonstration that a nonzero thing cannot be the sum of zeroes, a sort of analysis of discreet vs continuous. — noAxioms

You're approaching this with a whole-from-parts mindset, where you aim to construct everything from smaller components. Thus, you believe the only options are to assemble a continuous line from infinite points or from discrete line segments. Consider reversing this perspective: adopt a parts-from-whole approach. Start with a single continuous line and then, as if it were a string, cut it to create discrete points (which correspond to the gaps). I encourage you to explore this mindset; I'm eager to discuss it more with you.

While my explanation might differ from how Zeno would phrase it, I believe it aligns with his philosophical approach. He is quoted to have said “My writing is an answer to the partisans of the many and it returns their attack with interest, with a view to showing that the hypothesis of the many, if examined sufficiently in detail, leads to even more ridiculous results than the hypothesis of the One.”

But he cannot indicate a time that isn't represented by such a point, so I don't think he's shown this. — noAxioms

However, you're working under the assumption that a timeline consists only of discrete points in time. You cannot directly observe a particle in a superposition state, but this doesn't mean that superposition states are merely fictional. I bring in QM, not to sound fancy, but there is an analogy here between observed states (which are like points) and the unobserved a wavefunction (comparable to a line) that lies between them.

The block universe can still be interpreted as discreet or not, just like the presentist view. — noAxioms

I believe you are discussing whether time is discrete or continuous. In the context of Zeno's Paradoxes, it's necessary to consider space and time as continuous (as you later noted). I'm not sure what you're referring to with time being continuous or discrete from a presentist perspective, especially since Zeno's arguments suggest that time does not progress in a presentist's view of the world.

You do if it is discreet. A physical string is very much discreet — noAxioms

I explicitly wrote abstract string.

Nonsense. It says no such thing. — noAxioms

Perhaps it's not my place to speak for others, but let’s say that adopting an eternalist perspective allows someone to reframe the impossibility of supertasks, turning it's non-existence from having unacceptable consequences to acceptable consequences.

This also seems irrelevant since none of his paradoxes seem to reference observation or comprehension. — noAxioms

Additionally, none of the paradoxes explicitly rule out this as a possible solution.

Surely it would take forever to comprehend the counting from 1 on up. Michael's digital counter runs into this: the positing of something attempting to measure the number of steps at a place where the thing being measured is singular. — noAxioms

If there is a continuous film reel capturing the ticking counter, the limits of observation dictate that there are just some frames that we cannot see. They're blacked out. In fact, I would argue that we can only ever observe countably many frames so in fact, most of the frames remain unobserved (in a superposition of sorts). This allows the story to advance and avoids singularities.

And so conversely, if an infinite task may not be completed in a finite amount of time then we must agree that time is not infinitely divisible. — Michael

This only applies if you adhere to a whole-from-parts construction approach. As I mentioned in my discussion with NoAxioms, a seldom considered alternative is that the universe is constructed parts-from-whole. I really hope you will engage with me on this possibility.

THOMSON'S LAMP

Depends on if the calculator is required to follow the mathematical theory of convergent infinite series. If yes, 1, If no, then it can be anything at all. — fishfry

In this scenario, the calculator isn't equipped to perform calculus; it's a basic model tasked with adding each term of the infinite series. While mathematical theory predicts that at 60 seconds, it will display 1, it's true that the narrative does not specify what should appear at that moment. I am even welcoming of the idea that it turns into a black hole at 60 seconds. Nevertheless, isn't it concerning to you that there's a discrepancy between mathematical theory and your intuition? I completely agree that freshman calculus is invaluable, and I'm not suggesting that infinite series or any aspect of calculus are without merit. I use aspects of it everyday. Instead, I propose a new interpretation of what these infinite series represent. The story of the calculator isn't really about what it displays at 60 seconds; it's about the approach to 60 seconds. Likewise, I suggest that infinite series don't actually sum up to a specific number, but rather they outline a continuous, unbounded process. We don't have to assert that there's a least upper bound to this process.

That's the problem with all these puzzles. — fishfry

Your argument that the paradox is nonphysical is a red herring. This narrative takes place in the abstract realm, and unless you can pinpoint a contradiction within that context, we should consider it as abstract and possible and acknowledge its validity. Perhaps you lean towards theoretical perspectives, but it's important not to undermine the significance of thought experiments. They have arguably been among the most influential types of experiments conducted by humans.

In what sense do you regard Zeno's paradoxes as new ideas? That doesn't make sense. — fishfry

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. I sense you can tell I'm enthusiastic about this viewpoint, but it seems you aren't interested in delving into or critiquing it. Perhaps after considerable reflection, you've already formed your opinion on these issues and don't find additional discussion worthwhile. That's completely acceptable.

I'll assume that your wish for my death did not come out the way you meant it. Way over the line. — fishfry

I apologize if it seemed like I was implying anything about wishing for your death; that was not my intention at all. I specifically expressed a desire for you to have a long life. My main point was about the acceptance of new ideas, highlighting that they often gain traction because a new, possibly more open-minded audience emerges over time. The longevity of those holding old beliefs isn't the crucial factor.

Your argument is that Zeno's paradox is so new and revolutionary that I'm too old to see it? — fishfry

Zeno was significant, but the concepts and solutions I'm advocating are not entirely his ideas.

What perspective do I have and why on earth are you going on about it like this? — fishfry

I believed we agreed to confine the Zeno discussion to the Staircase thread, which is why I was vague here. However, I offered detailed criticisms of your perspective in that other thread. I'm not trying to be cagey.

STAIRCASE PARADOX

The "ground", thus defined, is a point that cannot be reached from the stairs, being infinitely far below it. Similarly, you cannot reach the stairs from that point, as every stair is infinitely far above it. That's why the man on the "ground" can't see any stairs as described in the OP story. They are all too far away above him. By making such a definition, we are essentially dividing our thought-experiment-world into two parts, neither of which can reach the other. — andrewk

You are effectively arguing that supertasks cannot be completed since if he cannot reach the ground, he must still be on the stairs.

No chance. — fishfry

Max Planck once said "a new scientific truth does not triumph by convincing its opponents and making them see the light, but rather because its opponents eventually die, and a new generation grows up that is familiar with it." Certainly, I hope you have a long and fulfilling life, but your response brought this quote to mind.

Didn't do any good, nobody understood a word I said. — fishfry

I think I understand what you said; I just have some issues with your perspective.

STAIRCASE PARADOX

It's always only a finite number of steps from infinity back to zero — fishfry

This brings to mind Sagan's quote "extraordinary claims require extraordinary evidence." We start with an extraordinary premise—the existence of infinite stairs and supertasks—and to resolve it, we resort to an equally extraordinary solution: he has infinitely long legs, enabling him to ascend to the top in just one stride. This doesn't strike me as a satisfactory resolution.

You described it as endless, and yet claim he reached the end... The "paradox" is just you choosing to invent a story with contradictory concepts. — flannel jesus

What you seem to overlook is that I'm beginning with a premise widely accepted within the mathematical community: the existence of actually infinite objects (like these infinite stairs or the set, N) and the completion of actually infinite operations (such as traversing the stairs or calculating the sum of an infinite series). If you do not accept the concepts of infinite sets or supertasks, then this paradox is not aimed at you. If you claim that an old woman is 2 years old, then you're not basing your argument on any widely accepted concepts of age.

But if a staircase down can be created by our, or your, say-so, another one, going up, can be created in the same way. — Ludwig V

If there is a parallel staircase where the steps start at 1 and increase as you go up, then there must be a point where the step numbers on both staircases align. What would that step number be?

But the end is not reached. — Metaphysician Undercover

Then your argument should be that supertasks are impossible, not that 60 seconds cannot elapse.

ZENO'S PARADOX

I suggested that movement was discrete, not that space was discrete — Michael

Consider linear motion. If you plot position against time, are you suggesting that the resulting curve, when examined closely, appears stairstepped rather than smooth? If that's the case, what would be the width of these incremental steps? This presents the same issue, as I could always plot a more accurate curve of motion using even smaller incremental steps.

I wouldn't say that. — Metaphysician Undercover

This response does not adequately address my reinterpretation of Zeno's ideas.

I don't see how Zeno's paradoxes work any differently under presentism than under eternalism. — noAxioms

Zeno contends that change is impossible, leading to stark implications depending on one's philosophical stance on time. Under presentism, this translates to an unchanging, static present—life as nothing more than a photograph. In contrast, the eternalist perspective views this as a static block universe, a continuous timeline that encompasses past, present, and future—akin to a film strip. Which view do you think is more reasonable? Of course, this raises profound questions, such as why we experience time's flow, but that discussion is for another thread.

Let me reframe Zeno's argument in different terms more relatable to a modern audience. Consider whether it is easier to draw a one-dimensional line by assembling zero-dimensional points consecutively or to cut a string (akin to dividing a line into segments). Zeno would argue that the first option is impossible: a timeline cannot be constructed from mere points in time. Instead, modern Zeno would suggest that the entire timeline already exists as a block universe, and our experience is merely about observing different parts of it, similar to making cuts in a string. However, there's a twist: abstract strings, like time, are infinitely divisible. No matter how many cuts we make (one after another), we never actually reduce the string to mere points. Each cut still leaves a segment of string, however minuscule. This introduces new challenges (for which there are answers) but as it relates to the discussion at hand, the eternalist perspective reframes the impossibility of supertasks from an unacceptable notion—that motion itself is impossible—to a more acceptable one—that observing every instant in history is impossible. This essentially echoes Aristotle's proposal, but it is only in the quantum era that such a solution becomes truly acceptable.

For only two of the three following premises can be true of a sequence: i) The length of the sequence is infinite. ii) The sequence is countable iii) The sequence is exhaustible — sime

The issue arises if Achilles toggles Thomson's Lamp with each stride, leading to a contradiction: his feet suggest that the sequence is exhaustible, but his hand indicates it is not.

SINGLE DIGIT COUNTER PARADOX

Assuming that paradoxes are metaphysically impossible then the counter is metaphysically impossible, and that suggests that it's metaphysically impossible for time to be infinitely divisible. — Michael

First, instead of using decimal, let's switch to binary, where the counter can only be 0 or 1. You suggest that quantum mechanics resolves this by introducing indivisible units, perhaps akin to Planck time. Looking to QM for inspiration is a good idea. However, the idea of Planck time doesn't hold up because in the abstract realm, we can always conceptualize a smaller increment. I propose that the correct solution is that at 60 seconds, the counter is in an unobserved state where its status fundamentally remains unknown. It could be either 0 or 1, so let's say it's in a state of (0 or 1). If we wish to steal technical terms from QM, we might refer to this state as being in superposition.

The paradox is that given the premise(s) what happens at the limit is undefined, and yet something must happen at the limit. This is a contradiction, therefore one or more of the premises must be false. — Michael

What if the undefined state is fundamentally unobservable? This raises the question similar to "If a tree falls in a forest and no one is around to hear it, does it make a sound?" The limitations I'm suggesting on observation should not be surprising to a generation that has grown up in the era of quantum mechanics.

It is metaphysically necessary that there is a limit to how fast something can change — Michael

Yet, it's impossible to determine what this limit might be. Would you argue that there is a limit to the slope of a line?

THOMSON'S LAMP

Why on earth must there be a behavior defined at the limit? — fishfry

Suppose that with each flick of the lamp, the lampholder adds another term to a cumulative total: first 1/2, then 1/4, then 1/8, and so forth. What does his calculator show at 60 seconds? Why on earth must we assert that it displays 1? After all, the narrative doesn't specify what his calculator must indicate at 60 seconds. It seems to me that you're contesting the very idea which you support - that infinite series can have definitive sums.

By the law of excluded middle and non-contradiction, after 60 seconds the lamp must be either on or off. — Michael

Yeah, that law needs updated. I propose "for every proposition, either this proposition or its negation can be measured to be true." This introduces the possibility of a third, unmeasured state—when we're not observing, the lamp could either be on or off, placing it in a state of being (on or off).

I'm perfectly happy to continue the conversation. — fishfry

Great. And if it seems like you're no longer making debatable points or asking questions, I'll take that as a hint that the conversation has reached its end. :D

I'm only saying that you might be disappointed if you hope to convert me to your degree of passion, even on items where I agree with your point of view. — fishfry

Might? As in there is still a chance? [said like a clueless teen not getting the hint from repeated rejections from his crush. Lol.]

I'm sure poor old Zeno is getting a sufficient workout in the staircase thread. — fishfry

Yeah, let's keep Zeno to that thread. I'm glad to see you couldn't resist joining in, though. :)

If you and I agree on something but I just don't allocate it the same percentage of my overall interest and passion as you do, that's ok, right? We basically agree on Zeno, I just don't give it much thought. I've given it some thought over the years. But I truly never cared about it in the sense that you do. And I hope you can make your peace with that, because you seemed to be saying that you wanted to convert me not only to your point of view, but also to your level of passion. And that may not be productive. — fishfry

Time is valuable, and it's perfectly fine for you to express that you're not interested in continuing our conversation; we can leave it at that. If you choose to end the discussion but also mention that you agree with me, that's a nice extra, though not necessary. Regarding converting you to my point of view, I do want to do that and will seize any opportunity that comes up. I thought that since you provided your resolution to Zeno's paradoxes that you invited further discussion, but it seems I may have misinterpreted your intentions.

ZENO'S PARADOX


Quantum Jump - Abstract space (as opposed to physical space) cannot be discrete because any minimum unit you propose can be halved. This is not an acceptable solution to Zeno's Paradox. I agree with you that Zeno's assumptions about motion are flawed, but you haven't offered an alternative premise that holds up. The whole point of his paradox was to highlight that the standard view of motion was flawed. Additionally, it's not definitively established that physical space is discrete. It's possible that only our measurement of space is discrete. This latter perspective is my belief which I'll expand on in a couple of paragraphs.


Zeno Non-sequitur fallacy - I agree that the conclusion doesn't logically follow from the premise, but that's only true if you interpret the paradox from a fresh perspective. From the traditional understanding of motion, the conclusion indeed seems to follow logically. This is precisely Zeno's point.


I wrote the following in a different thread but it's relevant here. Let's recast Zeno's ideas using contemporary terminology. In his era, the dominant philosophical view was presentism, which posits that only the present moment is real, and it unfolds sequentially, moment by moment. Zeno’s famous parables about Achilles' incremental pursuit are illustrative of (and an attack on) this presentist perspective. However, Zeno himself subscribed to the opposite belief, which we now call eternalism. This philosophy asserts that past, present, and future coexist as a single, unchanging "block universe." From a vantage point outside this block, everything would appear static; thus, in this comprehensive perspective, motion is impossible. One could argue that in his perspective, the only movement is in the gaze of God, and wherever God looks becomes the present (I use God here not to push a religious view, but for simplicity). The discreteness that are looking for is not in space but in measurement/observation. In other words, God's fundamentally cannot watch everything. This actually should come as no surprise since the Quantum Zeno Effect demonstrates that an observed system cannot evolve.

Zeno was remarkably prescient. The concept of eternalism and the block universe gained serious traction only after Einstein introduced theories that showed eternalism to be more consistent with the principles of relativity. Yet, the narrative is still unfolding, as the singularities in classical black holes demonstrated that relativity is not the ultimate explanation of physical reality. Enter QM and the importance of observation/measurement.

STAIRCASE PARADOX


A minute cannot pass - This scenario involves an infinitely large object (the staircase), an infinitely complex task (traversing the entire staircase), and the passage of one minute. You're suggesting that the issue lies in the impossibility of a minute passing? It seems you may have labeled the most logical and uncontroversial element in the paradox as illogical. If you think the problem has to do with Icarus's steps then frame your solution in that context.
No end to the staircase but the end is reached - Yes, this is the very issue I'm trying to highlight. And this has nothing to do with continuous acceleration or motion. Could it be that supertasks are impossible?
restricting ourselves to the physical world - The physical world is not the only realm that exists; there's also the abstract world, which operates under its own set of rules. For instance, in an abstract world I can define, it's perfectly valid to set the speed of light at 100 m/s. This isn't incorrect—it's simply a different premise. However, I do believe in a kind of symmetry where truths in the physical world often find parallels in the abstract world.


This is a paradox I've come up with myself. But as Michael has mentioned it's very similar to Thomson's lamp. Where do you see problems with it?


Focus first step up, not last step down- Unfortunately, the stairs are numbered in ascending order from the top down, so the first step up wouldn't be numbered 1.


Non-standard numbers-I'm certain you're a strong mathematician, but I also feel like you're overcomplicating things. This reminds me of an Einstein quote: “If you can't explain it to a six year old, you don't understand it yourself.”

,
Only a potential infinity-My purpose in presenting this paradox is to underscore the problems associated with the concept of actual infinity.


Thomson's Lamp-Indeed, the Staircase Paradox shares significant similarities with Thomson's Lamp Paradox, particularly in that both scenarios lead to states considered invalid by conventional logic after one minute has elapsed. In the Staircase Paradox, we are left unsatisfied by claims that the staircase either exists or does not exist. Similarly, in Thomson's Lamp Paradox, we find it unsatisfactory to definitively say whether the lamp is on or off. The difference is that, supertasks aside, Thomson's Lamp is a critique of infinite series whereas the Staircase Paradox is a critique of N.


Trip from 0 to 1-I don't get it.


[0,60)-Your point is valid, for brevity I didn't explicitly state that the first instant he passes the stairs he arrives on the ground. However, as the poem indicates, my view is that at that instant, he actually arrives at a singularity, similar to what one might encounter at the center of a classical black hole.

PARADOX OF THE GODS


As Michael noted, your barrier paradox is Bernadete's Paradox of the Gods. I find this paradox intriguing. In the realm of physics, I think quantum tunneling offers a solution to this issue.

Okay, now I'm wondering if it was me...it is possible...and likely...

I've delete my message from here and posted it in the correct thread.

@Metaphysician Undercover, @fishfry, @andrewk, please move your related posts as well if that's not too much trouble.

EDIT: I think I made a mistake by incorrectly posting in this thread a message that was meant for another thread. I've since deleted that post.

Oh YOU messed the threads up? — fishfry

No it wasn't me. That was the Canadian in me saying sorry!

All of my responses were to messages on this thread! — keystone

This statement was incorrect. I said it not knowing that the threads got mixed up.

EDIT: I think I made a mistake by incorrectly posting in this thread a message that was meant for another thread. I've since deleted that post.

@fishfry

Oh you're right...this got messed up. Let me reach out to the moderators. Sorry!

EDIT: I think I made a mistake by incorrectly posting in this thread a message that was meant for another thread. I've since deleted that post.


All of my responses were to messages on this thread!

Zeno wasn't attempting to prove that motion itself is impossible; rather, he aimed to demonstrate that motion, as understood by the prevailing theories of his time, was impossible. — keystone

@fishfry: Let's recast Zeno's ideas using contemporary terminology. In his era, the dominant philosophical view was presentism, which posits that only the present moment is real, and it unfolds sequentially, moment by moment. Zeno’s famous parables about Achilles' incremental pursuit are illustrative of (and an attack on) this presentist perspective. However, Zeno himself subscribed to the opposite belief, which we now call eternalism. This philosophy asserts that past, present, and future coexist as a single, unchanging "block universe." From a vantage point outside this block, everything would appear static; thus, in this comprehensive perspective, motion is impossible. One could argue that in his perspective, the only movement is in the gaze of God, and wherever God looks becomes the present.

Zeno was remarkably prescient. The concept of eternalism and the block universe gained serious traction only after Einstein introduced theories that showed eternalism to be more consistent with the principles of relativity. Yet, the narrative is still unfolding, as the singularities in classical black holes demonstrated that relativity is not the ultimate explanation of physical reality. Enter QM...

Wow this was a good post. I understood everything you're saying and I agree with much of it. Even in parts where I disagree, we're still talking about the same thing. Thanks for this. — fishfry

Great! It does feel nice to feel heard.

This is different than the others. A four-sided triangle is impossible simply by virtue of the meaning of the words. I thought that since you called googolplex abstract and possible, then you would use the transfinite ordinals and cardinals as examples of abstract and impossible things.

Small quibble anyway. — fishfry

I didn't bring up transfinite numbers as examples of abstract impossibilities because I knew you might disagree. However, you're right that my initial example was trivial. Let's consider a non-trivial one: "This statement is false." This paradox challenges classical logic by appearing both true and false simultaneously. Yet, consider the profound influence it has had. This paradox sparked the development of numerous non-classical logics. Reflect on its siblings like "the set of all sets that do not include themselves" and "the formula with Gödel number ___ cannot be proved". Dismissing such a seemingly abstract and impossible statement would have deprived us of significant philosophical and mathematical advancements. And I believe the revolutionary impact of this paradox is far from over.

OMG my thoughts exactly. The analogy is non-Euclidean geometry, which was thought to be a mathematical curiosity with no practical use when discovered in the 1840s, and then becoming the mathematical formalism for Einstein's general relativity in 1915. — fishfry

I love this example.

My candidate for the next breakthrough like this is the transfinite cardinals, the higher infinite. Nothing more than a mathematical curiosity today, but in 200 years, who knows — fishfry

Fishfry called it here first :)

I don't share your enthusiasm for logical paradoxes as the fulcrum for the next scientific revolution — fishfry

I think you meant to say 'the next mathematical revolution'. Paradoxes, or singularities, have been and continue to be pivotal in sparking scientific revolutions.

But yes, the lack of enthusiasm applies to you and pretty much everyone else. Unfortunately, I lack the mathematical prowess needed to convince you to listen.

As a longtime student of crankology, I disagree. Alternative and novel ideas don't make one a crank. It's a certain lack of the logic gene or a certain basic misunderstanding of the nature of proof and logical argument that separates the cranks from the merely novel thinkers. — fishfry

You're generally correct, but there are exceptions like Norman Wildberger. I hope that one day AI can help cranks build a more compelling argument because I think they aren't completely mistaken.

Ok. I just don't know if the standard logical paradoxes are that important, but time will tell. — fishfry

It’s tempting to just snip off the loose thread and assume everything is fine. After all, how much damage could a small loose thread really do to your sweater, right? Abstract impossibilities are such rare gems I'm saddened that we don't value them.

I have not realized earlier that you are not interested in the interesting question of choosing an arbitrary natural; but rather trying to link this to some kind of paradox. But the relation's a stretch. I still don't see the connections that you've tried to make with dice that roll forever (why gravity but no friction in your world?), quantum physics, and various other topics. — fishfry

I believe it's impossible to choose an arbitrary natural number in N. I understand that my earlier questions about the impact of observation seemed aggressive, so let me answer them instead and see if you have any comments. Before the dice stop rolling, Adam has a 50% chance of winning. Once Adam sees his roll, his chances drop to 0%. If Adam forgets his roll, his chances go back to 50%. If only God sees the roll, God knows Adam's chance of winning is 0%, but Adam still believes it's 50%. If God reveals that he saw the roll, both are aware that Adam's chance is 0%. It's pretty wild, isn't it? Even if we find a way to choose an arbitrary natural number in N, the situation remains just as bizarre. Declaring that there's no uniform probability on the natural numbers is not an answer. It's akin to dismissing "this statement is false" as an invalid statement that can be ignored because it doesn't fit into classical logic. You're snipping off the exposed part of the loose thread.

Much ink spilled over the years on this, but just not an interest of mine. Personal preference. — fishfry

Fair.

[The dartboard paradox] is a genuine paradox of interest. How does a collection of sizeless points make up a length or an area? We have mathematical formalisms but no real explanation. There's really nothing to be done about the basic paradox. — fishfry

I appreciate your acknowledgment that mathematical formalisms don't provide an explanation. However, I strongly disagree with the notion that nothing can be done about it. It just seems you might not be interested in an informal solution, and if that's your stance, I'm a little sad but it's a reasonable one to hold.

For what it's worth, Newton thought of lines as being paths of points through space, so there's no real paradox if you assume space is like the real numbers. Which it almost certainly isn't. — fishfry

I don't understand.

In fact I would venture to say that the ultimate nature of space or spacetime is nothing at all like the mathematical real numbers. — fishfry

I would venture to say that the ultimate nature of spacetime is very much like the objects that real numbers are intended to model (continua).

[Zeno's paradox:] Already resolved mathematically by the theory of infinite series, and physically by the fact that motion is possible. Also just not a major interest of mine. — fishfry

I strongly disagree on the topic of infinite series, but I won’t delve into it since it doesn’t seem to interest you. Zeno wasn't attempting to prove that motion itself is impossible; rather, he aimed to demonstrate that motion, as understood by the prevailing theories of his time, was impossible. This serves as a prime example of a concept once believed possible, which he identified as both a tangible impossibility and an abstract impossibility. The paradox remains unresolved to this day.

But what you have failed to convince me of is that "the paradox" -- which one of many that you've discussed?? -- is important, either in general or especially to me. — fishfry

I think we're both at fault here. I haven't explained my perspective well, and you haven't been entirely open to hearing it.

In Thompson's lamp, the final state is not defined so you can make it anything you want it to be. — fishfry

All of the major mathematical paradoxes today share a common theme: superposition. The liar's statement is (true or false), Thompson's Lamp is (on or off), the staircase (exists or doesn't), Icarus is (alive or dead), and the state of Adam's game is (win or lose). Unfortunately, I suspect you might dismiss this entire explanation as lacking substance.

But the other ones, Thompson's lamp and the staircase and so forth, arise from the fact that the final state is simply not defined. — fishfry

The universe must ultimately settle on a state because something has to occur. Are you suggesting that God simply flips a coin? All signs, including those from quantum physics, indicate that we need a new state for Thompson's Lamp upon completion of the supertask, one that goes beyond just being (on) or (off).

Here, instead of concluding that a minute cannot pass, as Zeno concluded that Achilles cannot pass the tortoise, keystone changes things up to say that after a minute has passed the infinite number of steps has been reached. — Metaphysician Undercover

Do you truly believe that Achilles is unable to surpass the tortoise? Do you think that Icarus's deeds influence the passage of time? Is there a concrete analogy in which your actions alter how time progresses for me?

Your poetry asserts this, but the reverse can be done There is simply no first step in the process, just like there wasn't a last step on the way down. The sum of the same series in reverse order is also 60 seconds. — noAxioms

How is it possible for him to ascend the stairs if there isn't a first step? Or do you think that he might not be able to fully descend the stairs?

If he would have traversed the staircase in Zeno like fashion, as specified, although he would have stepped on all the steps in a finite amount of time, there would be no definite position along the staircase that he was at immediately before he had arrived at his destination. — Pierre-Normand

What's your take on this? Do you believe he never finishes descending the stairs? If that's the case, then where would he be after one minute has passed?

Infinity minus one equals infinity
Would the above qualify as a paradox — kazan

If that statement is logically unacceptable, then it could be considered a paradox. However, many people today might not see an issue with it, so you would need to provide further explanation to convincingly demonstrate its paradoxical nature.

Can a paradox be conceived in the a&p realm? — kazan

Let me draw an analogy. Historically, our understanding of the world was believed to be tangible and possible. We thought we grasped the truth, whether through Newtonian mechanics or general relativity. Then, some thinkers pointed out inconsistencies in these prevailing views that defied explanation. What was once deemed tangible and possible turned out to be tangible and impossible. As a result, the model of the world was revised, and the new model was then assumed to be tangible and possible. Over the years, this process repeats, gradually bringing us closer to the truth.

I understand you're asking which of the following four scenarios interests me:

1) Tangible and possible - for example, a horse.
2) Tangible and impossible - such as a black hole as described by Relativity, with a singularity at the center.
3) Abstract and possible - like the number googolplex.
4) Abstract and impossible - such as a four-sided triangle.

Our physical universe, though entirely described by mathematics, appears to have circumvented singularities. Why not look to it for inspiration? In physics, breakthroughs often occur when one identifies something tangible and impossible and rethinks our understanding to shift it to tangible and possible. This approach has driven many major advancements in the frontiers of physics, which is why numerous eminent minds are engaged in quantum gravity research.

The next significant breakthrough in mathematics could occur when someone pinpoints what is currently abstract and impossible yet accepted within modern mathematics, and the community transforms it into something abstract and possible. The arithmetization of analysis is an excellent illustration of such a transformation. While I deeply appreciate the value of what is abstract and possible (acknowledging that mathematical truths are both beautiful and useful), much of it surpasses my grasp, so I can't personally revel in it. However, what really captures my interest is the pursuit of the abstract and impossible in mathematics. Personally, I view it as the most important, thrilling, and accessible area to engage in at the moment. Although most impossibilities in mathematics have been resolved (no serious mathematician is exploring four-sided triangles), I believe paradoxes like the ones we discuss suggest that some impossibilities still remain.

To summarize my interests:

1. Tangible and Possible - This is my day-to-day work as an engineer. I thoroughly enjoy the innovations that stem from exploring this domain, especially my computers.

2. Tangible and Impossible - The physics community already excels in this area. They are actively working to resolve the impossibilities in their theories. Yet, there are still opportunities to influence through philosophical interpretations of quantum mechanics.

3. Abstract and Possible - Mathematicians excel in this field, continually advancing our understanding and capabilities.

4.Abstract and Impossible - Typically, those who challenge the established norms here are labeled as cranks. There is a significant opportunity for philosophers of mathematics to make strides in this area. This is where my interest lies, in exploring and potentially reshaping the abstract impossibilities that still exist in mathematics.

With this in mind, we seem to disagree on whether the paradox I propose is abstract and impossible or abstract and possible. It might be an exaggeration, but from my perspective, this disagreement translates to me seeing it as crucial, whereas you might view it as merely an interesting concept, but nothing more.

Additionally, I believe I have the beginnings of an idea that could transform it from abstract and impossible to abstract and possible. This concept also holds the potential to resolve many other persistent paradoxes, such as the Liar's Paradox, the Dartboard Paradox, and Zeno's Paradox. Yet, I find myself struggling to even convince you that the paradox, which appears possible from a conventional standpoint, is actually abstract and impossible.

What do you think about this?

Perhaps my next paradox will make a stronger impression. Even though this conversation might conclude, please keep in mind that I'm always open to picking it up again if you're interested.

If instead of choosing a random number, what if we just choose an arbitrary one? — fishfry

It appears that an arbitrary number would be relevant in discussing the potential outcomes of Adam's story before or after the event has occurred. However, for the story to progress as it unfolds, in Adam's 'present' a random number would need to be selected. Please correct me if I'm misunderstanding your point.

@fishfry: I just realized I might have misread the tone of your second-to-last post as suggesting we were wrapping up, even though your latest post raised new questions. I'll get back to those questions later, but I want to make it clear that I understand you're not obligated to continue this conversation.

@fishfry: While I would love to continue this conversation, it sounds like you see this as a good endpoint. I'm going to post a new paradox now that this conversation has ended. I hope to hear what you have to say about it. Sorry if I sounded rude at the end, that was not my intention. I recognize that you have been more than charitable with your time. You're a nice person. Thank you so much for this conversation.

@fishfry

For what it's worth, here's how I would construct a random number generator on N in our physical universe:

1) Employ a quantum event that has a 50% chance of yielding 1 and a 50% chance of yielding 0.
2) Assign the outcome to the first digit of a binary number—1 for a result of 1 and 0 for a result of 0.
3) Continue this process for each subsequent digit.

Two key observations:
1) There is one potential issue with this approach. It's remotely possible that the latter output could be an infinite sequence of 1's. If, hypothetically, this program could be executed as a supertask (completing in finite time), it might return infinity, which does not belong to the set of natural numbers.

2) The program never halts. If you stop it prematurely, you haven't encompassed all natural numbers. Since the program is intended never to halt, it avoids the theoretical problem of returning infinity, rendering the aforementioned flaw negligible.

If we're discussing fishbowls, I'd argue that when God reaches into the bowl and selects the top ticket, it's an unfair draw. He should shuffle the tickets first. However, when dealing with an infinite pile, the shuffle would never conclude. Let's set aside the fishbowl analogy and turn our focus to programming, which offers a more tangible approach to discussing random number generation on N.

Let's reframe this discussion in terms of my concepts of objects and processes:

1) The random number on N (i.e., the output of the RNG function) - an object that cannot feasibly exist.
2) The code defining the RNG function - a finite object that exists.
3) The process of executing the code to completion - an infinite process that cannot be completed.

In mathematics, there is a tendency to treat the output (1) as the fundamental element. However, I contend that the actual code (2) deserves our primary attention. This shift focuses on the tangible aspects of mathematical constructs rather than on abstract, unattainable outputs.

There are no infinite processes. You stick your hand into God's fishbowl and pull out a ticket and read the number. I don't understand why you're attacking the premises of your own problem. Conceptually, we pick an arbitrary natural number. That's very straightforward. You're just confusing yourself by going into all these different directions. — fishfry

Now you're introducing narrative elements into our discussion, mentioning God and fishbowls. If we assert that God can do anything, then we could just as easily conclude that God can define a uniform probability measure on N and leave it at that. However, there are limits to even what God can do. As a programmer would understand, creating a true random number generator is incredibly challenging. While theoretically, you might write such a program (using finite lines of code), in practice, it would run indefinitely without halting. Could God create a random number generator for N that actually stops? Or does his magic only work when we talk informally about fishbowls?

I don't think discussing the foundations of calculus is all that helpful either. I really think you have a lot of things in your mind and you're just tossing them out. — fishfry

I don't think you're truly entertaining my propositions. Did you understand what I was saying?

However, noncomputable real numbers exist, and they do not have algorithms. — fishfry

While I would really like to continue this tangential discussion, there's no point in addressing this (and other tangential) comments if you aren't going to read my responses simply because they don't directly relate to the original post.

The bit with the Stern-Brocot tree threw me for a loop. I have no idea where you were going with that. Wasn't there a thread about that on his board a while back? — fishfry

I would have appreciated your specific insights on this topic if you had engaged more sincerely in this tangential discussion.

Is your concern with the nature of the real numbers? That's really got nothing to do with the original post, which is trying to find a logical basis for Adam's strategy of always switching. — fishfry

My main concern revolves around the concept of completed infinities. R, N, and the process of generating a random number on N all inherently involve completed infinities. They are interrelated. Now, consider this 'paradox':

God created a married bachelor and declared he would kill the man at noon if he was married. Is the man alive at 12:01?

There are different ways to approach this paradox. One method is to seek a logical explanation for God's decision on whether or not to execute the man. Alternatively, and just as validly, one can challenge the premise itself. You are not allowing for this possibility, which seems unfair.

You know, there's a thing called the counting measure. — fishfry

This definitely aligns with Adam's reasoning. However, as you pointed out, the counting measure is not a probability measure, which I find problematic. Regarding the specific paradox, at what point would it be prudent for him to swap rolls with the serpent? Does this decision occur the moment he opens his eyes and makes an observation? What if he only pretends to open his eyes? What if he makes an observation but totally forgets what he observes? What if he keeps his eyes closed, but an ant sees his roll? What if God is watching? What if God sees the roll and informs Adam that he saw his roll but doesn't say what it was? Counting measure does not offer an answer to these questions.

Or will you instead chose not to answer these questions related to observation and simply say that pop quantum theory is not helpful here?