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? — keystone

I think this is all going south.

I don't think you're truly entertaining my propositions. — keystone

I don't find your propositions entertaining.

[Hey man sorry, with a straight line like that I could not resist!]

Did you understand what I was saying? — keystone

About the real numbers? It just seemed like a change of topic. I didn't feel like engaging on the subject.

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. — keystone

Noncomputable numbers directly bear on your idea of using computations to define real numbers. Most real numbers can not be so characterized.

I am not sure why you feel I'm obligated to follow you far afield from the OP. I only jumped in because you were speculating on a uniform probability measure on the natural numbers, and I just happen to know the factoid that there can't be one. I thought I was helping.

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. — keystone

I don't know much about the Stern-Brocot tree. You can get the same result using the completed infinite binary tree, in which there are countably many nodes, yet the real numbers are encoded as paths through the tree.

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. — keystone

Arguing finitism is a lot different than what you presented in the OP. I just don't care to follow all the changes of subject.

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. — keystone

Why are you throwing this out there? I'm not going to respond. I don't have to respond. Let me just note that several dozen other currently active members of the forum didn't respond either. Why aren't you unhappy with them?

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. — keystone

I suggested counting measure since it bears on the OP. Can't imagine what opening his eyes has to to with this. I'm feeling a little backed into a corner and no longer enjoying this.

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

I was enjoying this a lot more when you wrote ...

— keystone

I'm really enjoying our discussion and finding it incredibly beneficial. Thank you for your patience and the knowledge you share. I feel very lucky to have you sticking around. — keystone

I'm sorry I can't be more helpful. You have many questions. I can't respond to them all.

As far as my remark about pop quantum, I meant it. You just pulled superposition out of the air. It has nothing to do with any of this.

Clearly my responses are making you unhappy and that's not my intent. But I don't have the inclination to discuss all of the varied subjects you brought up. Really, I said everything I had to say in my first post.

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. — keystone

What makes you think that wasn't determined at the moment of the big bang?

Quantum events have certain probabilities of being observed in certain states. We don't know for sure whether they're actually deterministic or not.

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. — keystone

For all you know, this procedure is entirely deterministic. Just as flipping a fair coin is random for practical purposes, but is clearly deterministic if we only knew the physical variables precisely. The same might be true for subatomic events. We don't know.

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. — keystone

Return infinity? Supertasks? I can't respond to any of this.

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. — keystone

You've lost me again.

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. — keystone

It's a conceptual thought experiment. If you don't like the metaphor, forget it.

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. — keystone

You can't prove that there is any such thing as a true RNG.

In mathematics, there is a tendency to treat the output (1) as the fundamental element. — keystone

I've already agreed that it's perfectly fine to identify a computable real number with the algorithm, or the set of all algorithms, that generate it. But that misses all the noncomputable real numbers.

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. — keystone

Ok, you're arguing a constructivist viewpoint. Interesting subject, but far removed from the subject at hand.

But you have me so confused.

On the one hand, you are asking about a totally abstract thought experiment where God rolls an infinite-sided die.

And in the next breath, you say you're only concerned with the "tangible" aspects of math. Which surely precludes God rolling dice, right?

@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: 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. — keystone

Likewise, and thanks.

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. — keystone

I did suggest I was trying to wrap it up. And then I asked you a question! "Do I contradict myself? Very well then I contradict myself, (I am large, I contain multitudes.)" -- Walt Whitman

So even though I'm trying to wrap this up, I do have a question.

You said you were interested in the "tangible" aspects of mathematics.

But your original question is totally intangible, concerning God rolling an infinite-sided die.

So: tangible or abstract?

ps -- I just had a thought. It's a semantic solution.

If instead of choosing a random number, what if we just choose an arbitrary one? That conveys the same conceptual scenario, but without invoking all the mathematical and philosophical context of randomness.

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.

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

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.

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. — keystone

I see your point.

4) Abstract and impossible - such as a four-sided triangle. — keystone

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.

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. — keystone

Agree.

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. — keystone

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.

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

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. — keystone

I don't share your enthusiasm for logical paradoxes as the fulcrum for the next scientific revolution, I do agree with your point.

Again I don't like four-sides triangles or married bachelors as examples, because those are only based on the meaning of the words. Like jumbo shrimp, or Kosher pork.

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. — keystone

4.Abstract and Impossible - Typically, those who challenge the established norms here are labeled as cranks. — keystone

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.

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. — keystone

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

With this in mind, we seem to disagree on whether the paradox I propose is abstract and impossible or abstract and possible. — keystone

I'm not really on board with your terminology, so I can't agree or disagree.

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.

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. — keystone

It's a cute problem, but as I indicated originally, it has a mathematical answer, which is that there's no uniform probability on the natural numbers.

I don't think it has any sigificance beyond that, but of course that's a matter of opinion and not fact, so we can agree to disagree on that.

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 ... — keystone

What you call are abstract and impossible are just word meanings like married bachelor. There's nothing of real interest.

Liar's Paradox, — keystone

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

the Dartboard Paradox, — keystone

This 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.

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.

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

and Zeno's Paradox. — keystone

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.

Yet, I find myself struggling to even convince you that the paradox, which appears possible from a conventional standpoint, is actually abstract and impossible. — keystone

Well I've already noted that your definition of abstract/impossible is only about word games like married bachelor. I would say your definition of abstract/impossible is not fully thought out.

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.

I've seen all the paradoxes but they don't hold central interest for me.

What do you think about this? — keystone

I agree with you that some of these seeming paradoxes might be the key to future insights. But surely not semantic jokes like married bachelors or four-sided triangles. Those aren't paradoxes and they're not of interest at all IMO.

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. — keystone

I saw the other thread, that looks like a variant of Thompson's lamp or any of several other similar puzzles. In Thompson's lamp, the final state is not defined so you can make it anything you want it to be. It's not as interesting to me as it is to others I suppose.

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. — keystone

My point is that "arbitrary" works just as well, without carrying all of the context of randomness. But if it's not helpful, then nevermind on that.

Bottom line I agree with you that things that seem useless today, like transfinite cardinals, may someday be useful to physics, as non-Euclidean geometry became.

And I agree that the dartboard paradox shows (to me) that the physical world is highly unlikely to be accurately modeled by the mathematical real numbers.

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.

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).

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...

What thread are we on? Did you combine this one with the other one? Maybe you could ask a mod to do that for you.

You're suggesting that the issue lies in the impossibility of a minute passing? — keystone

I wasn't saying that this is "the issue", only that it is the logical outcome. For Icarus a minute cannot pass because he always has steps to cover first, just like Achilles cannot pass the tortoise for the same reason. Maybe if we call the staircase a line, and the steps are "points" it would make more sense to you. No matter where Icarus stands on the prescribed line, he has to cover an infinite number of points before a minute can pass. And to traverse each point requires a non-zero amount of time. Therefore no matter where Icarus is on the line (stairs), there will always be time left before a minute passes. A minute cannot pass, and Icarus' journey cannot end.

No end to the staircase but the end is reached - Yes, this is the very issue I'm trying to highlight. — keystone

No, the end is not reached, as explained above. This is what Andrewk neatly explained in the other thread. That a minute will pass, and the end will be reached, is a presumption outside the prescribed scenario. You are assuming that from some other principles.

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!

moving this post about omega sequences to the stairway thread, where it belongs

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!

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

Oh YOU messed the threads up? I apologize to the moderators, whose names I have taken in vain. :-) said jokingly of course

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.

Your point is valid, for brevity I didn't explicitly state that the first instant he passes the stairs he arrives on the ground. — keystone

Before making such a statement, we'd need to define what we mean by "the ground". Very difficult, because it needs to be a specific point that is infinitely far below the top of the stairs.

I think we could construct an imagined world where we could make such a definition, using a concept like the first infinite ordinal ω, as described here.
The ordinals deal only with whole numbers, whereas we want fractions too, as we're measuring distance. I expect we could extend the ordinals to include fractions, simply by interpolating between successive ordinals. But there may be an obstacle I'm missing.

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.

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.

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

Oh well then now I'm thoroughly confused. It's fitting to be down a rabbit hole, given the nature of the topic.

ps -- Oh I see what happened. I posted a response about omega sequences that should have been over in the stairway thread. I moved it over there.

https://thephilosophyforum.com/discussion/comment/898761

I'll try to get to your other points later.

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.

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.

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. — keystone

I'm perfectly happy to continue the conversation. 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.

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

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. :)

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 — keystone

Believe so.

Might? As in there is still a chance? — keystone

No chance.

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

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

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.

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.

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. — keystone

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

Zeno lived 2700 years ago (5th century BCE according to SEP). So your argument fails.

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

But in what way could any living human be too old to understand Zeno's ancient paradoxes? Your analogy is totally flawed.

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

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

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.

I apologize if it seemed like I was implying anything about wishing for your death; that was not my intention at all. — keystone

Ok fine.

My main point was about the acceptance of new ideas, — keystone

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

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.