"Not defined" does not mean that you are free to choose the result. — SolarWind

Yes it does. If I define the first three elements of a sequence, like 3, 12, 84, what number comes next? Mathematically, it can be any number at all.

Likewise if I define a function at 1, 2, 3, 4, ..., and I want to also define it at $\omega$, a symbolic point AFTER all the natural numbers, I can define it to be anything I want. Like 42. Or like Cinderella's coach, which is a fine, beautiful coach at 1 second before midnight, 1/2 second before midnight, 1/4 second before midnight, etc., yet turns into a pumpkin at the stroke of midnight.

Which solution has n = n+1? — SolarWind

Not the same kind of undefined. Here, it can't be defined. But the state at the bottom of the stairs can be anything at all.

So there's a distinction between something that can't be defined because it's impossible, and something that simply hasn't yet been defined, and that can then be defined as anything at all.

Certainly not 42. — SolarWind

Don't see why not. When you first heard the Cinderella story, did you make the same objection to the coach turning into a pumpkin at the stroke of midnight?

I don't want you to go easy on me. I pride myself in my ability to correct my trajectory in the face of new evidence/feedback. — keystone

Standing by for something specific. And if it's not too much to ask, can you keep it short? I myself tend to write long-assed posts. I should take my own advice.

If we could improve equality, is the question below what needs to happen? — Rob J Kennedy

Make everyone an impoverished slave and feed them all the same bowl of gruel everyday.

That's the problem with "equality." If you have a system that allows everyone to thrive at their own level of ability and ambition, you'll get lots of great art, science, and wealth. Lots of excellence among the excellent. You'll also get lots of inequality. And if you hammer down every nail that stands up, you'll get all the equality you want ... good and hard.

Suppose Icarus writes the number of the step on a piece of paper with each step, erasing the previous number. What number will be on the paper at the end? — SolarWind

42. Can you argue otherwise? The final state is not defined. It can be anything we like.

Take the scenario here:

After 30 seconds a white square turns red, after a further 15 seconds it turns blue, after a further 7.5 seconds it turns back to white, and so on.

We can sum the geometric series to determine that the limit is 60 seconds. The claim some make is that this then proves that this infinite sequence of events can be completed in 60 seconds.

However, then we ask: what colour is the square when this infinite sequence of events is completed? — Michael

This is just the lamp story with three states. The answer is that the sequence 0, 1, 2, 0, 1, 2, 0, 1, 2, ... has no natural completion or limit. So if we want to define its state "after" the natural numbers, we can say it's anything we want. I like the Cinderella analogy. The square turns into a pumpkin at midnight. That's no less realistic than the square story.

As per the setup, the square can only be red, white, or blue, and so the answer must be red, white, or blue. — Michael

Why? After all in the sequence 1/2, 3/4, 7/8, ... the "setup" is that each element of the sequence is a rational number strictly less than 1.

But the limit of the sequence is 1. This illustrates a general mathematical principle:

Taking limits does not necessarily preserve all properties of a sequence.

All of 1/2, 1/3, 1/4, 1/5, ... are strictly positive. But the limit of the sequence is 0, which is not positive. (I'm positive!)

You have to be very careful not to fall into the trap of assuming that a limit must preserve all the properties of the elements of the sequence that approaches it. You have made that mistake.

However, as per the setup it will never stay on any particular colour; it will always turn red some time after white, turn blue some time after red, and turn white some time after blue, and so the answer cannot be red, white, or blue. This is a contradiction. — Michael

It's not a contradiction. It's the straightforward observation that the sequence 0, 1, 2, 0, 1, 2, ... has no sensible limit. So if you wish to define a final state, you can make it anything you like. I choose pumpkin.

Remember, Cinderella's coach is a coach at 1 second before midnight; at 1/2 second before midniht; at 1/4 second before midnight; and so forth. Yet at the stroke of midnight, the coach turns into a pumpkin.

That story makes exactly as much sense as Thompson's lamp. Except that with Cinderella, we introduced a discontinuity. Where as with the lamp, and with your three-state lamp, there is no possible way to define the limiting state in such a way as to preserve continuity.

The conclusion, then, is that an infinite sequence of events cannot be completed, — Michael

An infinite sequence of events can have a limit. I assume you agree that 1/2, 3/4, ... has the limit 1. We can think of 1 as the "completion" of the sequence. It's reached not by a "final step," but rathe by the limiting process itself.

and the fact that we can sum the geometric series is a red herring. — Michael

No, it's the heart of the matter. .999... = 1 even though there's no "last 9." The limiting process is real. It's important. It exists.

To resolve the fact that we can sum the geometric series with the fact that an infinite sequence of events cannot be completed we must accept that it is metaphysically impossible for an infinite sequence of events to follow a geometric series: we must accept that it is metaphysically impossible for time to be infinitely divisible. — Michael

That is flat out false and does not follow at all.

We can "complete" the sequence .9, .99, .999, .9999, ... with the number 1, which is reached via a limiting process.

we must accept that it is metaphysically impossible for an infinite sequence of events to follow a geometric series: — Michael

Sorry, what? You don't believe that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1? You don't believe in calculus? You are arguing a finitist or ultrafinitist position? What do you mean?

Of course if you mean real world events, I quite agree. But your three-state lamp is not a real world event, it violates several laws of classical and quantum physics, just as Thompson's two-state lamp does.

I've taken calculus and I understand what limits are. By definition, a limit is not reached, it is approached. The sequence of steps maps to a mathematical series that approaches, but never reaches 1. The sequence of steps is actually unending (that is how infinity is manifested in this thought experiment)- there is no last term. — Relativist

I did not get a mention for this post, does that happen sometimes? Maybe I just missed it.

As I have been explaining in this thread, you can conceptually adjoin the limit of a sequence to the sequence, as in 1/2, 3/4, 7/8, ..., 1. This is a perfectly valid mathematical idea. This is a representation of the ordinal $\omega + 1$. In this case, 1 is indeed the "last term," although to be fair, you can no longer call this a sequence, since a sequence by definition is order-isomorphic to the natural numbers.

However, the clock does reach 1. At time 1, the stairway descent must have ended, because the descent occurs entirely before time 1. The descent is not a mathematical process (even though it can be mapped to a mathematical series), it is a sequence of movements from one step to the next. No movements are occurring AT time 1. If the descent has ended at this time, how can there NOT have been a final step? — Relativist

You can model this situation with $\omega + 1$, as I've tried to explain a number of times.

After all, if we work in the close unit interval [0,1], the sequence 1/2, 3/4, 7/8, ... never ends, yet there's its limit right there at the right end of the interval. We "get there" through a limiting process. There is no last step if your steps are required to be discrete. But we can also take limits. Limits aren't steps, but that's a semantic quibble. We can adjoin 1 to 1/2, 3/4, 7/8, ... to form the $\omega + 1$ "extended sequence" if you want to call it that, 1/2, 3/4, 7/8, ..., $\omega$.

I don't know if this will help, but at least I can motivate the legitimacy of the ordinal concept by linking the wiki page on ordinal numbers.

But there's an easier way to think of it. We're just adjoining a formal symbol at the end of the natural numbers:

1, 2, 3, 4, ..., $\omega$. It's just a formal symbol, means nothing at all. But we can define it in such a way that it's the upper limit of 1, 2, 3, ... in exactly the same way that 1 is the upper limit of 1/2, 3/4, ...

Yes there is no "last step" but there is in fact a limit.

By definition, a limit is not reached, it is approached. — Relativist

That is sadly a misunderstanding very common among calculus students. So lot of smart people, physicists and engineers and other scientists, have this belief.

In fact a limit IS reached. A limit is exact, it's not merely approached or approximated. It is literally reached.

It's not reached by a single step. Rather, it's reached by the limiting process itself.

Anyway, I don't want to write another long post. My first real post will come tomorrow...I got consumed by the Staircase post this evening... — keystone

I await your next missive with both curiosity and trepidation. Is it wrong for me to encourage you on the one hand, then give you a hard time the next? I'm conflicted.

Even if you believe that the foundations of mathematics and our understanding of continua is rock solid, — keystone

I have never expressed, nor do I presently hold either of those beliefs.

you must acknowledge that it confounds many people. — keystone

I cannot take responsibility for the execrable state of math education, or frankly education in general these days.

Take, for instance, the difficulty in convincing a child that 0.999... equals 1, or the prominance of Cantor cranks. [/quotet]

A byproduct of bad education. Not something I can personally remedy.
— keystone

By contrast, I believe children would grasp my concept more easily because it is fundamentally simple, albeit it requires adopting a different viewpoint towards the foundations of math. To use an analogy, my perspective is less like a target that's difficult to hit and more like one that's difficult to spot. — keystone

Lotta fluff so far. "Where's the beef?" (*)

Why I believe it's important
The validity of my ideas is still up for evaluation, but if they prove to be correct, deep truths often end up having practical relevance, even if their complete implications are not immediately apparent. Nevertheless, I am convinced that my theories could enhance mathematics education, resolve many paradoxes, and shape our understanding of reality, particularly in the context of physics. Ironically, coming from an engineer, I don't anticipate any significant impact on applied mathematics, as practitioners in such fields typically do not focus on the foundational aspects of math. I also want to clarify that my work is not meant to suggest that previous efforts by mathematicians were wasted. — keystone

More marketing fluff. I'm regretting this already.

How I'm going to share my ideas
I understand that for an idea to gain acceptance in the mathematical community, it needs to be formalized. I'm just not there. I don't have a formal paper to share with you, but instead, I plan to share my ideas gradually, in a manner akin to our ongoing discussions. Just as we can introduce children to the basic concepts of Cartesian coordinate systems without heavy formalities, I hope you can allow me the same flexibility in explaining my ideas with a similar level of informality. — keystone

I fear that you're going to wave your hands and present a lot of mathematically naive ideas, and I'm going to find myself back in the .999... wars and all the rest of it. And I'm frustrated that you wrote so many words here without saying anything at all.

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

If not, then what?

To avoid these assumptions and start with a clean slate, I'll be using a 'k-' prefix in front of familiar terms (like k-points, k-curves, k-continua, etc.). — keystone

You're going to rework the whole of mathematics? I'm getting a sinking feeling.

By the end of our discussions, I hope you'll not only find my approach more appealing but also recognize that it aligns with the mathematics that applied mathematicians have been practicing all along. At that point, it may be justified to remove the 'k-' prefix. — keystone

I regret encouraging this conversation.

Thoughts? — keystone

I apologize for the negativity, but you didn't say a thing yet. And you apparently have some kind of grand unified theory of math that you're going to wave your hands at while I attempt to be open-minded.

I am open-minded, and want to hear your ideas, but you don't seem to be interested in presenting them. I was hoping for a paragraph, but you gave me what looks to be the introduction to a very long and very frustrating exposition.

If you have a paragraph or two that I can sink my teeth into, by all means present it.

If what you've got is as vague as this post, then I humbly apologize for encouraging you to aim this at me. On the one hand I'm curious as to what you are talking about, and on the other, well ... you just haven't said anything but you're promising to say way too much.

Can you boil down what you want to say in a couple of clear paragraphs? Without the marketing about how it's revolutionary and will be understandable to children?

I do welcome your thoughts, but I encourage you to get to the point and try not to turn my curiosity into frustration. Clearly this post sent me over that line.

Here is my response to this post in a nutshell.

(*)
https://www.youtube.com/watch?v=Ug75diEyiA0

The law of non-contradiction. An infinite series of processes entails never completing, but at points of time that occur after the delinieated interval - the task is necessarily completed. — Relativist

You've just described the ordinal $\omega + 1$, which has as one representation the sequence 1, 2, 3, 4, 5, ... , and another more familiar representation as 1/2, 3/4, 7/8, ..., 1.

These are perfectly rigorously defined and logically consistent mathematical objects (assuming ZF is consistent of course].

They're just limits. There is no mathematical mystery. People just get confused when you start making up fictional entities like switching circuits that change state in arbitrarily small amounts of time.

You're not (ahem) a .999... = 1 denier, are you? That's one of the standard crank arguments, that the process of adding the next 9 is "never completed." It's a fallacious argument. There is no temporal process of adding 9's. Rather, you have a mathematical function that assigns to each natural number the digit 9. That's a completed process (or function, more accurately) once you accept the axiom of infinity. When you interpret the string of 9's as a sum 9/10 + 9/100 + ..., the sum of the series is 1, by the definition of the limit of a convergent infinite series.

The "never ends" argument is simply mathematical ignorance. The fast-switch circuit is as realistic as Cinderella's coach that turns into a pumpkin at the stroke of midnight.

At 1/2 second before midnight it's a coach. At 1/4 second before midnight it's a coach. Dot dot dot. At midnight it's a pumpkin. How does that happen? It's a fairy tale. For some reason, philosophers recognize Cinderella as a fairy tale (scrub enough floors and you'll attract the devotion of a handsome prince with a foot fetish); yet these same philosophers take Thompson's lamp seriously. I can't account for this cognitive error.

The lesson is that the defined supertask (the fictional, physical process) is logically impossible, — Relativist

The lamp and staircase scenarios are physically impossible. What law of logic makes them logically impossible?

if a physical process ends, there has to be a final step. — Relativist

There is no physical process. There's a fictional process that doesn't obey the known laws of physics.

In what sense does anyone think the staircase or the lamp are physical processes?

Question: Cinderella's coach turns into a pumpkin at the stroke of midnight.

Is that transition a physical process? In what world?

:up: Amen :roll: — jgill

Nice to see you again @jgill, and thanks.

We can determine whether or not something entails a contradiction. If time is infinitely divisible then supertasks are possible. Supertasks entail a contradiction. Therefore, time being infinitely divisible entails a contradiction. — Michael

This is an interesting argument. I have some issues with it.

First, I should note that infinite divisibility is a weak condition. The rational numbers are infinitely divisible, but they are not a continuum. They are full of holes, such as the point where sqrt(2) should be.

The question often argued is whether physical spacetime is a continuum in the sense of the uncountably infinite, Cauchy-complete real numbers. But you have strengthened the claim to saying it's not even a countably infinite non-continuum like the rationals. So even if you're right, your claim is too strong to be right. That's a meta-argument, not an argument. But claiming spacetime isn't even like the rationals is much stronger than claiming it's not like the reals. [This is all beside the point, but I wanted to make the point that infinite divisibility is not enough to make something a continuum].

Now to the argument.

"If time is infinitely divisible then supertasks are possible."

By this I take it that you mean that if we take, say, the rationals in the unit interval to model one second of time, we could do something in [0,1/2) and something else in [1/2, 3/4) and so forth, and thereby do infinitely many things in one second, which is the definition of a supertask. Have I got your argument right?

So yes, I agree that if time is dense -- that's the math term for the property that there's always a third thing between any two distinct things -- then supertasks are possible. I'd never thought of that argument before and it's pretty good. Although for all we know, there could be some law of nature that the smaller the time interval, the longer things take to happen, wrecking your supertask. You can't rule that out. Just like objects gaining mass as their velocity approaches the speed of light. Strained analogy but I hope you see what I'm getting at.

"Supertasks entail a contradiction."

What contradiction is that? You just convinced me that if time is like the rational numbers (dense but full of holes) supertasks are possible. Then you claim supertasks entail a contradiction, but I'm not sure what contradiction that is.

So your argument's incomplete here, and if you did explain this elsewhere in the thread, I apologize for having missed it.

I have another concern, which is that in our current theory of physics, we can not reason sensibly about intervals of time below the Planck time. So you are making an argument that can never, even in theory (pending the next revolution in physics) be observed, measured, or confirmed by experiment.

That's what we call speculation. Like the cosmological theory of eternal inflation, in which the universe had a definite beginning but exists infinitely far into the future. That's not physics, that's mathematical metaphysics. Science fiction with equations. I do think your idea has a problem in this area. You can't actually reason below Planck scale.

You can argue that reality allows for the possibility of contradictions if you want, but most of us would say that it is reasonable to assert that it doesn't. — Michael

On the contrary, most people would agree that life is full of contradictions. I love you and I hate you. We should clean up the environment but that raises the cost of energy for the poor. (Ok that's a tradeoff and not a logical contradiction, but it's still a situation where two virtues are in conflict). I am large, I contain multitudes. (That diet's not working). The electron is a particle. It's a wave. No, it's an excitation in a quantum field. That's the latest attempt to resolve the contradictions in physics.

As we go through our daily lives we are faced with one contradiction after another. And when we study physics, we see contradictions and impossibilities at the most fundamental nature of reality.

I do not believe you can convince me that nature isn't self-contradictory. Why shouldn't it be? What law of nature says that nature must satisfy Aristotelian logic?

tl;dr: Well those are my thoughts. Interesting argument though. If time is modeled by the rational numbers, supertasks are possible. I will give that some more thought. But again: why do supertasks entail a contradiction? That's the weak part of the argument I think. That, and the Planck scale issues.

I would propose a parametric curve on the ball path, and, for fantasy sake, by whatever mechanism, the plate knows at what part of the parabola the ball is at, defining the counter. As time goes on, the revolution gets smaller and smaller. Eventually the ball will completely rest on the table, which is 0: — Lionino

Cute. In this case I agree that it's natural, in the sense of preferring continuity, to say that the final (ie limiting) state is resting on the table.

Yes, but check the solution at https://plato.stanford.edu/entries/spacetime-supertasks/#MissLimiThomLamp — Lionino

Will check it out, thanks.

The paradox is this:

1.The bottom of the stairs is reached at the 1 minute mark.
2.Reaching the bottom of the stairs entails taking a final step.
3. Therefore there is a final step
4.The steps are countably infinite (1:1 with the natural numbers)
5. There is no final (largest) natural number.
6.Therefore there is no final step

#3 & #6 are a contradiction. — Relativist

Thanks for clarifying that for me.

I don't see a paradox. All I see is a lack of understanding of mathematical limits.

Consider the sequence 1/2, 3/4, 7/8, 15/16, ...

Mathematically, this sequence as a limit of 1.

The sequence never "reaches" 1; nor is there a last step. Neither of these statements is controversial once you understand what a limit is. Sadly, most people have never taken calculus; and most students who take calculus never really learn what a limit is. The subject isn't taught properly till a math major class in real analysis. So almost everyone in the world is ignorant of the mathematical theory of limits, and is therefore vulnerable to confusions about "reaching" and "last steps."

It's perfectly clear that if you start at 1 and move leftward on the number line, you necessarily skip over all but finitely many elements of the sequence, so that it's always only finitely many steps back from 1 to the start of the sequence.

When you dress the story up with fictional staircases and physics-violating lamps, people get confused.

But there is no confusion. 1 is the limit of the sequence, but the sequence never "reaches" 1 nor is there a last step. The definition of a limit is logically rigorous and unambiguous. The fictional staircases and lightbulbs only have the purpose of confusing people.

Finally, we can consider the sequence 1, 2, 3, ... which never reaches infinity nor does it have a last step. But we can place an arbitrary symbol at the end, usually called $\omega$, so that the sequence looks like this:

1, 2, 3, 4, ..., $\omega$

Once again there is no "reaching" and no last step, but it's mathematically legitimate to say that $\omega$ is the limit of the sequence. And we see that if you start at $\omega$ and take any step back, you will land on a natural number, and it's always only finitely many steps backward from $\omega$ to 1.

In fact these two augmented sequences 1/2, 3/4, 7/8, ..., 1 and 1, 2, 3, ..., $\omega$ are order-isomorphic.

So there's just no paradox. There is only taking perfectly well-understood mathematical facts and dressing them up with physics-contradicting staircases and lightbulbs so as to confuse people.

In the case of the lamp, we have a sequence 0, 1, 0, 1, ... that has no limit. No matter what you define as the final state of the lamp (the state at $\omega$), you can't make the sequence continuous. So I say the lamp turns into a pumpkin at midnight, just as Cinderella's coach did. Since the lamp is entirely physical, and its switching circuitry violates the known laws of physics, that's as sensible as any other solution.

The staircase story has a perfectly natural solution, though. At each step, the walker is present on that step. So the corresponding sequence is 1, 1, 1, 1, ... So if we define the limiting state is 1, we have made the walker's sequence continuous. That's a natural solution.

We could say that the limiting state of the walker is "not downstairs," but that would make his path discontinuous. There's a clear preference for the continuous solution.

There's no way to make 0, 1, 0, 1, ... so the pumpkin is as reasonable as anything else.

0 may be defined as Robinson's h — alan1000

You seem to be under the impression that Robinson's hyperreals define a unique smallest positive infinitesimal. This is false. If h is a positive infinitesimal, then 0 < h/2 < h.

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. — keystone

The formalisms are wonderfully clarifying of one's formerly fuzzy intuitions.

For example the idea of stepping back from the bottom. It's only a finite number of steps back, even from infinity. Absolutely nobody has that intuition at first. Once one has studied the ordinal numbers, it's literally a theorem that it's always only finitely many steps back from a transfinite ordinal. And once you understand why that is, you now have a better intuition.

The process is:

1) Have fuzzy intuitions;

2) Study some math;

3) Develop far better intuitions.

Some may think of that as "retreating behind formalisms." I think of it as developing better intuitions about the real numbers, infinite processes, and so forth.

This mathematical observation doesn't change the reality that Icarus would need to jump over infinite steps. — keystone

You can not use the word "reality" in this context. In reality there is no such staircase. This is an abstract conceptual thought experiment. It has a mathematical answer. If you are at 1 and you take even the tiniest step backward, you necessarily jump over all but finitely many elements of any sequence that approaches 1. That's a better intuition than the pre-mathematical intuition.

You don't find it counterintuitive that moving slightly to the left of 1 jumps over all but finitely many elements of any sequence approaching 1 from the left, correct? That's clear to you I assume.

Well, that's the staircase. It's a better intuition, informed by a precise formalism.

The best I can do to meet you halfway here is to agree that I have had some mathematical training, and that I have had "improved" intuitions beaten into me by professors at some of our finest universities. But in truth, studying math clarifies all the fuzzy intuitions about the real numbers, infinite sequences and series, infinite processes, and so forth. The Thompson lamp is a sequence of alternating 0's and 1's and it's not defined at infinity. So the final state is anything you care to define. You can't make the series continuous no matter how you complete it.

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. — keystone

As the example of [0,1] shows, even the tiniest imaginable legs, the weakest possible legs, necessarily jump over infinitely many elements of any infinite sequence approaching 1 from the left.

There is no such thing as an infinite staircase, so this physical analogy confuses more than it enlightens.

I often make this point about Hilbert's hotel. You go online and people will argue that there's no such hotel, and how can there be room in an infinite hotel, and so forth. Many people end up more confused than they were before. In fact any infinite set may be placed into bijective correspondence with one of its proper subsets, and that's the "formalism" that clarifies the vague and unrealistic story. Hilbert, by the way, only mentioned the hotel once in his life, at a public lecture, and never wrote or spoke about it again. It's been blown up way out of its negligible importance, to the point where many people think it's a mathematical argument. It's not. /end rant

This might explain how he returns to the top, but it essentially sweeps the infinite staircase under the rug. — keystone

He has a magic carpet.

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. — keystone

It's perfectly valid. I already agreed that it's perfectly natural to consider that you are present at the bottom of the staircase after one minute, because that's the choice that makes the sequence continuous. I already said this. There is no paradox. As an abstract thought experiment it works out perfectly well.

In fact I don't even understand what the supposed paradox is. I do not believe your original exposition was sufficiently clear on this point.

The staircase is nicer than the Thompson lamp, which can not be made continuous by any choice of final state, since 0,1,0,1... does not converge.

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. — keystone

I haven't undermined the staircase story, I've agreed that it's perfectly sensible to assume the walker is present at the bottom, since that preserves the continuity of the sequence.

That's in contrast with the lamp, in which there is no possible final state that makes any more sense than any other.

Fine. What matters is that you're being very generous with your time to me and I offended you. I don't want to waste the time I have with you arguing over this. Again I'm sorry and I grant that you're entirely right on this. I hope we can put to rest this specific topic. — keystone

Apology completely accepted. I'm a little hypersensitive in general. No worries as they say.

I've been sharing aspects of my perspective here (but I feel like you never read it, perhaps because it seemed tangential), and other details have emerged in the Staircase thread. — keystone

I may have misunderstood a lot because I was focussed on the probability aspect. I'm not reading most of the posts in the Staircase thread and it's all over the map at this point.

Nevertheless, I haven't presented it as a complete picture. — keystone

Ok, that's fair. Would be happy to chat about your idea if you present it.

Should we continue such a discussion in this thread, which has become like our private chat room, or would you like me to start a new thread? — keystone

This thread's fine. The Staircase thread's hopeless, way too many side issues. It's nice and peaceful in here.

There are no new original records of Zeno's paradoxes so they are not new ideas. However, I think that Zeno's paradoxes remain unsolved, and I have an original perspective that resolves these and many other paradoxes in a way that they no longer seem contradictory. — keystone

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.

I sense you can tell I'm enthusiastic about this viewpoint, but it seems you aren't interested in delving into or critiquing it. — keystone

I'm pretty sure I haven't heard a clear statement of your idea. I could not repeat your idea back to you even to disagree with it. Perhaps I missed it. I'd be happy to critique your idea if you stated it clearly. [If you did state it clearly and I missed it, my apologies]. I could argue your thesis (if I knew what it was) all day, without ever having much interest in the subject. Well maybe not, but I'd kick it back and forth a little.

Maybe you can state your thesis so I know what revolution will ensue if everyone older than you would only die already.

Perhaps after considerable reflection, you've already formed your opinion on these issues and don't find additional discussion worthwhile. — keystone

I have no idea what point you are trying to make. You started this thread with a paradox of probability, which has a solution that's mathematically correct but not intuitively satisfying, namely that there's no uniform probability on the naturals.

Then you changed the subject to encompass many other ideas I consider irrelevant to the OP (quantum, etc) and I lost the ability to follow your thinking entirely. So yes, I don't have much interest in talking about Zeno's paradoxes of motion; but more to the point, I have no idea what is your grand new thesis that the old folks just don't get.

See https://thephilosophyforum.com/discussion/comment/898574 — Lionino

Ok I hadn't seen that before. Whatever shows at the end (if that even makes sense) it's certainly finite, since you're adding up finitely many finite numbers then resetting to 0. So the final total, if you can define such a thing, is 0 or some positive finite number. I don't believe such a series has a well-defined sum, since the sequence doesn't converge.

That is, the sequence is 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, ...

That sequence doesn't converge.

@michael then says, "If there is no answer then perhaps it suggests a metaphysically necessary smallest period of time."

I don't see how that follows at all. No mathematical thought experiment can determine the nature of reality. We can use math to model Euclidean geometry and non-Euclidean geometry, but math can never tell is which is true of the physical world. You can use math to model and approximate, but it is never metaphysically conclusive.

But even taken on its own terms, I don't follow the reasoning. How does observing that the sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, 1, 2, 3, ... doesn't converge, imply anything at all about the nature of time?

By the way the Thompson's lamp sequence is 1, 0, 1, 0, 1, 0, ... and that doesn't converge either.

Speaking of extended real numbers, is there any useful application of it? — Lionino

It's used in calculus to talk about "limits at infinity" and "infinite limits." For example, the limit as x goes to infinity of 1/x is 0; and the limit of 1/x as x goes to 0 is infinity. You need a formal definition of infinity in order to make those statements rigorous.

Those usages are convenient but not necessary. We could talk about the limit of 1/x as "x gets arbitrarily large," but instead we just say, "as x goes to infinity," and everyone understands the meaning.

The extended reals are also used in measure theory (a generalization of length, area, volume, etc) so that, for example, we can sensibly say that the real line has length infinity.

If it is in a made-up universe where such counters are possible, and time is infinitely divisible, the counter should count to infinity after 30s. — Lionino

I don't follow. In calculus, the sum of an infinite series is defined. According to that definition, the sum of 1/2 + 1/4 + 1/8 + 1/16 + ... is 1.

But even if we reject that in the physical universe, or we reject it for other reasons (we're ultrafinitists, we're cranks, we hate math, we hate infinity, we simply don't care, etc) it's still perfectly clear that no matter how many finite number of terms you take, the sum is always less than 1. So I don't see how you can justify claiming that the sum should be infinity, even in a world where you reject the modern theory of convergent infinite series.

Let's say even, the counter counts 1 at 15 seconds, 2 at 22,5, 3 at 26,25 and so on. It seems it would converge to infinity at time 30s. — Lionino

Yes, agreed, if we allow the value $\infty$ as in the extended real numbers.

However what would the counter show at 60 seconds? Are we talking about aleph-0 and aleph 1 and so on? — Lionino

We'll never get past $\aleph_0$ by adding more finite numbers. Likewise $\aleph_0 + \aleph_0 + \aleph_0 + \dots$ where there are $\aleph_0$ terms, is still $\aleph_0$.

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.

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? — keystone

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.

That's the problem with all these puzzles. You take a situation that's mathematically straightforward, and you add in lights that flicker faster than the laws of physics would allow, and calculators to operate faster than the laws of physics would allow, and you try to reason sensibly on partial information. If a light can flicker faster than the laws of physics allow, what else can it do? What are the laws of physics in this made up universe?

Remember, Cinderella's coach turns into a pumpkin at the stroke of midnight. And for all we know, so does your hypothetical calculator. Because first, we already know that it doesn't follow the known laws of physics. And second, you've only told us what it does at each natural number step 1, 2, 3, ... You haven't told us what it does in the limit. So I say it turns into a pumpkin.

Can you prove me wrong? No, because the story's made up. In freshman calculus, the sum of that series is 1. But freshman calculus is just another made up story too. Just a highly useful one. There are no summable infinite series in the physical world. No physical computer can calculate the sum.

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. — keystone

I gave the example of the first transfinite ordinal $\omega$. Any step you talk backward from it lands you on a natural number, from which it's only finitely many steps back from zero.

This is a perfectly well known mathematical fact. See Asaf Karagila's answer here. It's always only finitely many steps back from any ordinal, even uncountable ones.

https://math.stackexchange.com/questions/3980267/infinite-strictly-decreasing-sequence-of-ordinals

Let me use the same example I gave earlier. In the closed unit interval [0,1], consider the infinite sequence 1/2, 3/4, 7/8, 15/16, ..., which has the limit 1.

Suppose we start at 1 and take a tiny tiny tiny step to the left, as small as we like, as long as we land on an element of our sequence. Then you can see that no matter how small a step you tak, you will land on some element of the sequence that is only finitely many steps away from the beginning of the sequence at 1/2. Can you see that? It's actually the exact same example as 1, 2, 3, 4, ... $\omega$. 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.

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?

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.

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

The lamp violates the laws of physics, so it's not a real lamp. It's only a metaphor for a mathematical puzzle. Why can't it turn into a pumpkin at midnight, like Cinderella's coach? What rule of the puzzle constrains a light, which can not physically exist, to be on or off at a time when its state is not defined?

Consider this mathematical variant.

In [0,1], the closed unit interval, we start with the sequence 1/2, 3/4, 7/8, ...

We have a function f on the sequence such that f(1/2) = 1; f(3/4) = 0, f(7/8) = 1, and so forth, alternating between 0 and 1.

What is the value of the function at 1? Well clearly, that value is not defined. It could be 0, 1, 47, or the Mormon Tabernacle Choir.

Until you tell me what is f(1), you're playing a silly game to ask me what it should be. It can be anything you like.

We're being asked what the lamp "does at 1", so you saying that we must be told what the lamp "does at 1" makes no sense. — Michael

It makes perfect sense, once you replace the lamp with a function on a sequence. The lamp is a red herring. No circuit could switch that fast. It's not a real lamp and there is no reason for it to be in any particular state where that state has not been specified. It's not a real lamp so it is not limited to be on or off.

Given the defined behaviour of the lamp, will the lamp be on or off after 60 seconds? — Michael

Is f(1) = 1 or 0 or 47?

If the answer is undefined, but if the lamp must be either on or off, then the behaviour is metaphysically impossible. — Michael

It's not a real lamp.

The paradox is resolved by recognising that the premise is flawed. — Michael

The premises of the mathematical version are perfectly sensible, as is the answer I gave.

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.

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.

The answer to all those paradoxes is that you haven't defined what happens at the limit.
— fishfry

I think this is a misrepresentation. 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

No misrepresentation. And why must something happen at the limit? Take this mathematical example.

We work in the space (0,1), the open unit interval of real numbers. It excludes the endpoints.

We consider the sequence 1/2, 3/4, 7/8, 15/16, 31/32, ...

Clearly this sequence has the limit 1 ... except that 1 is not in our space. So this sequence has no limit. Such a sequence is called a Cauchy sequence. It's a sequence that should "morally" converge, whether its limit happens to be in the set of interest or not.

Say the Thompson lamp is turned on at 1/2, off at 3/4, on at 7/8, and so forth.

Why on earth must there be a behavior defined at the limit? In this case there is no limit because 1 is not in our set.

But now do the same thing, but in the closed unit interval [0,1], which does include its endpoints.

In this case the limit, 1, is defined and exists. But still, the behavior of the lamp is not defined at 1.

That's the point. There's no paradox. You've simply neglected to tell me what the lamp does at 1, and you're pretending this is a mystery. It's not a mystery. You simply didn't defined the lamp's state at 1.

Does this example better explain that the the "paradox" is simply that you're arguing over what is the state of the lamp, when the state of the lamp is undefined?

How about if we defined the state of the lamp as turning into a swordfish at 1. Then that's the answer. It's on, off, on, off, ... at each point of the sequence, and a swordfish in the limit. There is no contradiction and no mystery.

Also note that the sequence 1/2, 3/4, 7/8, ... is order-isomorphic to the sequence 1, 2, 3, 4, ...

And if we include the limit, then 1/2, 3/4, 7/8, ..., 1 is order-isomorphic to 1, 2, 3, 4, ... $\omega$ as I described earlier. From the standpoint of order theory, they have the same order.

People who have trouble imagining that we could reach a limit after counting the natural numbers, would have no trouble agreeing that 1 is the limit of 1/2, 3/4, 7/8, ... But those two situations are identical with respect to their order properties. Now we never "reach" a limit, which is another phrasing that confuses people. We don't reach the limit, but the limit exists.

I agree with this. The appearance of randomness is created by the system which analyzes, it is not a feature of the thing being analyzed. — Metaphysician Undercover

It's possible that sometimes it is.

For example if we flip a coin, that's only epistemically random, in the sense that it's a purely mechanical procedure that could be predicted by Newtonian physics, if we only knew all the variables precisely enough we could predict the flip. Yet since we can't, it's random in a practical sense. It's random only relative to what we can know. Hence epistemically random.

Compare to something that's ontologically random -- inherently random in and of itself.

Some people think quantum events are ontologically random.

But perhaps the low order bit of the femtosecond timestamp of the next neutrino to hit your detector was determined at the moment of the big bang. If it was, that would mean the entire universe is determined. But if not ... then there are things that are ontologically random.

It's an open question, but ontological randomness is at least logically possible, as far as we know.

Fair enough.
Nothing new of interest, comes to mind. Apart from adding negative l c omega with (to?) (positive) l c omega and getting the same answer as subtracting them,(still in the realms of arithmetic,) presumably zero? — kazan

Better not to try to subtract the endpoint infinities from each other, The result is undefined.

Never in memory, has "pure" mathematics been of such interest as now. Feel like you've open a window and there's a gale blowing in, here. — kazan

Thanks much.

Had more questions about l c omega, but will give them further thought first. And catch up with you elsewhere and later. Lounge perhaps in a few days? — kazan

I hear all the cool kids hang out in the lounge these days.

Yes,very helpful.Thank you for taking the time.
Which begs the question, (smile) how, if it's possible, would "the lower case omega" concept of "upper (lower?) limit" be applied to all integers? Surely, if it's possible,this could be useful in some areas of mathematics ( besides arithmetic ). — kazan

Another good question. Yes we could put "negative omega" at the leftward infinite end of the integers. There's not much use for it. The interesting aspect of omega is to keep going with the "add 1" game to get a whole infinite structure of higher ordinals continuing "to the right," in the positive direction. There's nothing new of interest that happens if you do the same thing on the left, it would just be a mirror image of the ordinal structure on the right.

Is minus one a natural number? And, is zero a natural number? Mathematicians' mathematics is not a strong suit for some. sad smile at one's own ignorance — kazan

Those are great questions.

Is minus one a natural number? — kazan

No. The natural numbers are the positive (or nonnegative, we'll talk about that in a moment) whole numbers like {0?], 1, 2, 3, 4, 5, ...

They're infinite (or endless if you prefer) in one direction.

The entire set of positive, 0, and negative whole numbers is called the integers.

The integers are infinite (or endless) in two directions:

..., -3, -2, -1, 0, 1, 2, 3, ...

The natural numbers are very basic and important, since on the one hand, they seem to be intuitive to every child -- you "just keep adding 1." On the other hand, they go on forever, giving us all a glimpse of infinity.

The integers, however, are much better for doing arithmetic. That's because the integers have "additive inverses." Given the number 5, in the natural numbers there is no number you can add to it to get 0.

But in the integers, there is: namely, -5. We say that the integers have additive inverses. And believe it or not, that happens to be really important in higher math.

So that's a long answer to a simple question but the bottom line is that the integers include the negative whole numbers, and the natural numbers don't.

And, is zero a natural number? — kazan

For some reason this question attracts a lot of controversy. People like to argue about it.

But I am here to tell you that it doesn't matter. Why is this?

Well, what's important about the natural numbers is their order. You have a linear, discrete procession of things. There is a first thing, then a next thing, then a next thing, and so on, forever.

If you call the first thing 0 or you call the first thing 1, it doesn't make any difference at all to the order structure of the sequence of "first, next, next, next, ..."

Suppose your friend thinks 0 is a natural number and you believe the natural numbers start with 1.

Then you just invent a new numbering system in which 1 means 0, 2 means 1, 3 means 2, and so forth.

You can tell your friend that you are starting with 0, but you just call it 1. And what they call 1, you call 2.

So you are right, and they are right. Why? Because the only interesting thing about the natural numbers is that (1) there's a first one; and (2) there's always a next one.

Those two rules generate the natural numbers, no matter what you call them!!

I hope this is taken to heart by someone. The answer to whether 0 is a natural number or not is that it absolutely doesn't matter. Just tell people what convention you're using and they'll be fine with it.

Now you might ask, what about arithmetic? 0 + anything = anything, but if you don't include 0 you don't have a number with that property.

And that's what I mentioned earlier. If you want to do arithmetic, the natural numbers are lousy anyway, they don't have additive inverses. So if you're doing arithmetic, you'll be working in the integers, not the naturals.

That's why it doesn't matter if 0 is a natural number. If you only care about order and "nextness," it doesn't matter what you call the first element.

And if you want to do arithmetic, you'll be using the integers anyway, which include zero.

Hope this was helpful. And again, these were great questions. Surprisingly deep. Thinking about various number systems and their properties like order or additive inverses is the basis of higher math.

I accidentally wrote this on the wrong thread so I'm moving it over here. I have some thoughts that may be of interest.

The staircase problem is called an omega-sequence paradox, a paradox that involves counting 1, 2, 3, ... and doing something at each step, then expecting the behavior to be defined in the limit. The answer to all those paradoxes is that you haven't defined what happens at the limit. You've told me what Thompson's lamp does at every finite $n \in \mathbb N$, but you have not told me what it does at the limit. Therefore the lamp could be on, it could be off, or it could have turned into the Mormon Tabernacle Choir. You haven't specified the behavior at the limit, so it can be anything you like.

There's a mathematical name for the upward limit of the natural numbers. It's $\omega$, lower-case Greek omega. You can think of it as a formal symbol that is greater than every other natural number, but that does appear in the sequence, as follows:

$0, 1, 2, 3, 4, \dots, \omega$.

You can think of $\omega$ as a "point at infinity." Or from a formalist view, it doesn't mean anything. It's just a symbol that satisfies $n \lt \omega$ for any natural number $n$, as well as all the usual meanings for 47 < googolplex and so forth.

This is a handy formalism. Now we can solve Thompson's lamp. The problem is that the state of the lamp is not defined at $\omega$. In other words you told me what the state is at every natural number, but not at $\omega$. That literally solves the paradox. It's no different than one of those "complete the sequence" questions. Mathematically, the next number can be anything you like.

In other words: You told me the state of the lamp at every finite number. You did not tell me the state at $\omega$. All confusion about Thompson's lamp is to realize that you just haven't told us the state at $\omega$. And there's no good reason to prefer one answer over the other.

The staircase paradox is a little more interesting, in the sense that you are present at each step 1, 2, 3, ... As before you can still define the behavior at $\omega$ to be either that you are there, or you aren't. But in this case, assuming you are there at $\omega$ is more natural, in the sense that the function that maps $\omega$ to "there or not there," is continuous if you're there.

What I mean is, at each successive step, the state of that step is "you are on it." Now the state at $\omega$ is undefined, but there is a natural way to define it; that is, to assume that your motion is continuous in some sense. So if you are there at every step, you are there at the bottom, the state of step number $\omega$.

So if you believe that your motion down the stairs is "continuous," however you define that in this context, then since continuity preserves limits, you are there at the bottom. But if you can't justify the assumption of continuity, then anything at all might happen at the bottom. You're there, you're not there, you're a sea slug at the bottom of the ocean. Since you haven't specified the value of the function at $\omega$, it can be anything you like.

One more note, and that is that you can indeed count backwards from $\omega$. But as you can see, any step that you take backwards necessarily jumps over all but finitely many numbers; so that it's always a finite number of steps back to zero, even from infinity.

Therefore if you are at the bottom of the stairs, you can just take a tiny tiny step up -- as small as you like -- you will necessarily skip over all but finitely many stair steps, and end up on some natural number like 47 or googolplex. Either way, it's still only finitely many more steps back to the top.

I should say that again, since this comes up so often. Even if you start at the point at infinity, it's always at most a finite number of steps back to zero.

This principle also applies to people making cosmological arguments about the impossibility of an infinite past, because it would take an infinite time to get to the present. Actually that's not true. If you put a point at negative infinity, it's only a finite number of steps to the present.

Finally, I'll mention that $\omega$ as I've used it is just a formal symbol; but in fact can be formally defined and shown to exist within set theory as the first transfinite ordinal number. And once you do that, you can keep on going into the wondrous and mysterious world of the ordinal numbers.

I'll leave you all with just one thought:

It's always only a finite number of steps from infinity back to zero, no matter how small a step you take. That's something a lot of people get wrong.

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.

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

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

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.

He just died. Surprised there was no mention here.

https://www.nytimes.com/2024/04/19/books/daniel-dennett-dead.html

RIP