Yes, but it is not difficult to abandon the (pseudo-physical) lamp for a purely abstract version, which does not have the same problems.

The point I've been arguing since the beginning of the thread, is that if we abandon the empirical, and adhere strictly to the prescribed, purely abstract version, then nothing indicates that 60 seconds will pass. You see in both the op, and the lamp example, 'there will be a condition at 60 seconds' is an unwarranted conclusion simply added on, and not derived from the initial premises. The initial premises, being the described activity do not allow that 60 seconds will pass. This conclusion, 'there will be a condition at 60 seconds' is pulled from empirical evidence, and is completely inconsistent with the prescribed purely abstract version.

We need infinite divisibility for the same sort of reason that we need infinite numbers. The infinite numbers guarantee that we can count anything. Infinite divisibility guarantees that we can measure anything (that is measurable at all). Limitations on either are physical.

No we don't need infinite divisibility, for the same sort of reason that we need infinite numbers, for the reasons I described. Any thing which is to be divided has its divisibility determined by the sort of thing that it is. And each type of thing is divisible in different ways. And, the way that the thing is divisible must be determined prior to division, or else we attempt to do the impossible. There is nothing that is divisible infinitely, therefore this ideal needs to be excluded as necessarily an attempt to do the impossible.

This is not the same as infinite numbers. The countability of a multitude cannot be determined beforehand, as the divisibility of a thing must be determined beforehand. So, we need infinite numbers because we do not know how large the multitude will be until we count, but we do not need infinite divisibility because it is impossible to divide something prior to knowing its divisibility, and the possibility of infinite divisibility is already excluded as actually impossible.

That is a fundamental feature of the difference between a unity and a multitude. If a unity is composed of a number of parts, then the number of parts is necessarily finite. To have infinite parts would violate the finitude of the boundaries implied by the concept "unity". But the concept of "multitude" has no such implied boundaries, principles of limitations. Therefore a "unity" is always limited in its divisibility, limited by the principles which make it a unity rather than just a multitude (I cannot say "multitude of parts" because "part" implies a whole), and a multitude is not necessarily a whole, so its countability is not necessarily limited.

I think you are being misled by the temptation to take the divisibility of "medium-sized dry goods" as the paradigm of divisibility.

No, I am talking about the divisibility of anything, in an absolute sense. There is no such thing as a unity which has absolutely zero limitations on its divisibility. That is a fundamental feature of what it mean to be a unity.

The colour of something isn't divisible at all.

Colour is very much divisible. It is a collection of distinct wavelengths, and I believe it is divided by the harmonic principles of the Fourier transform.
There is no doubt that it is easy to do that. But it seems that people disagree about whether the scenario makes sense or is incoherent and even if they do agree, they still disagree about why.

It's easy to see why it's incoherent. Start out with the concept of infinite. We can easily see why it is beneficial to allow for numbers to be infinite. This allows that there is no limit to our capacity to count any quantity, or to measure any size of thing, because we can always have a large number. In the case of division though, we may assume that infinite divisibility would allow us to divide anything anyway, but this is really incoherent. That is because division implies, or requires logically, that there is something, an object of some sort, to be divided, and its divisibility will always be dependent on the sort of thing that it is. An object, or thing is a unity of some type, and as such there is always limits to its divisibility, whatever unifies also determines divisibility.

To propose a thing which is infinitely divisible is an incoherent proposition, because as a thing, it is already necessarily limited in its divisibility. This is the issue of the proposition of a finite object being infinitely divisible. That is incoherent because the finiteness of the object limits its divisibility. In whatever way it is finite, its divisibility is limited accordingly.

I agree that this isn't really about anything empirical, but it sort of seems to be.

This is the trick of the whole thing. It really is about empirical things. These empirical things are space and time, each of these is known through experience. Then we take these empirical things and pretend that they are absolutely abstract, purely ideal, and stipulate ideal principles like infinite divisibility. Then, someone creates a scenario, like the lamp or the op, which utilizes this purely ideal feature of infinite divisibility. Now we do not properly separate the purely ideal from the empirical, in our minds, so that "empirical time" interferes, and we say that 60 seconds must pass, it has to because experience tells us that it will. But that is allowing "time" to be an empirical thing.

In fact, one could simulate the on/off lamp so that at a certain rate you would see what appears to be a constant light.

The problem though, is that in the prescribed scenario there is no such thing as "a certain rate". The rate is not constant, but rapidly increasing. The only constant is the rate of increase. That rate of increase is what I say is incomprehensible and incoherent.
That's true, but seems to be a purely physical limitation. It raises the question whether that means it is really on or off, or a some sort of in-between state. Fluorescent lights flicker on and off all the time (at least if they are running on AC, and we just say they are on. And it is true that for practical purposes there is no relevant difference between that light and sunlight or candle-light.

Jgill talked about how the lamp would "appear", and this implies a sense observation, and empirical judgement. The point I made is that the description describes something far beyond our capacity to sense, so it is incoherent to talk about how this described thing would "appear".

Something flashing on and off at a constant rate is not comparable, because the description is of a rapidly increasing rate. And the rate increases so rapidly that the prescribed rate becomes incoherent even to the mind, as well as the senses. This is just an example of how easy it is to say something, or even describe a fictional scenario, which appears to make sense, but is actually incoherent.
• Wittgenstein and How it Elicits Asshole Tendencies.
Do you mean "evolved" in terms of man's ability to use language overall, or in terms of how individual languages evolve?

I would say both, as the one seems to just be a more specific example of the other, which is more general.

I agree with what you're saying to some degree, but it's also the case that various metaphysical traditions: Platonism, atomism, Aristotleanism, etc. are all significantly older than any of the languages people on this forum are likely to speak as their native language. So there has been plenty of time to "work out the kinks," if it was easy to do so.

I don't see the point. The evolutionary force of change, brought about by the common usage of billions of people is much stronger than the force of a few metaphysicians. This makes it impossible for metaphysicians to "work out the kinks", because the kinks are being created at a rate much faster than anyone could have a hope of working them out.

Furthermore, the evidence of history shows, that controls over language use are not well received by the common people, and attempts at this will backfire. Look at the Catholic Church's attempt to control heresy, by controlling language use, The Inquisition.

Probably more relevant to the linguistic turn's hopes is that, for over a millennia, philosophers and theologians actually did use a dead language whose function was primarily to discuss these sorts of issues (outside of the liturgy obviously).

I don't see where you derive this idea. There was never a language whose primary function was to discuss theology and metaphysics. Latin's primary use was never simply theology and metaphysics. Now its use in religion is simply ceremonial, symbolic. In its late stage of actual usage, it was the language of all science and higher education. It had that role because the institutions of educational material were constructed with that language. Such institutions maintain tradition and are late to be affected by evolutionary change.

. You have a thousand year stretch of philosophers using a language that had been denuded of its "everyday" implications, supported by vast and elaborate lexicon of technical terminology worked out within that time period.

This is not at all true. It implies that higher education has no "everyday" implications. That of course is false, as higher education is a major driving force of evolutionary change. Changes at the higher levels of education trickle down to the less well educated, and the word usage gets altered on the way, because of the difference in understanding.

Yet this clearly didn't resolve all the issues vis-á-vis metaphysical questions—questions that appear to be at least as old as the written word itself, and which will seemingly always fascinate us.

Of course all the metaphysical questions have not been resolved. If that were the case, knowledge would be absolutely complete. The problems of quantum physics, wave-particle duality, entanglement, etc., and the problems of cosmology, dark energy, dark matter, etc., demonstrate that knowledge is far from complete, and many metaphysical questions remain unsolved.

I don't understand what you are arguing. Language does not resolve metaphysical problems, it is simply a tool used by the human beings who work to do this. When human beings are uninspired toward addressing such problems, directing their attention in other ways instead, and using language toward those other endeavours, it is incorrect to blame the failure of solving those metaphysical problems on the language. Clearly, when human beings have no interest in solving metaphysical problems, the failure of solving these problems is not to be blamed on the available tools.

Human beings are quite innovative, and are very capable of formulating, adapting, and shaping tools to suit there purposes. So if the human population was inclined toward solving specific metaphysical problems, they would adapted the tools necessary for this task, as they have done in science. The reality is that very few are inclined in this way, so the tools do not get produced.

The Latin era sort of seems like a gigantic natural experiment to see if the problems of philosophy can be fixed by moving away from everyday language. There is an irony in the fact that the medieval period is often singled as an exemplar period "bad philosophy" vis-á-vis the linguistic turn given the language philosophy was done in at the time.

You've got this backward. You are not looking at the proper chronological order, looking backward from now instead of from the ancient times toward now. What is the case is that "everyday language" moved away from Latin, not vise versa. There was, historically, a very close relationship between everyday language and Latin, and even further back in time, Latin was everyday language for many. However, as the written language, replacing Greek, Latin always held a place of authority, being "the memory" of the people. When people started questioning the authority (and this was deemed heresy), and those acting in the position of having authority responded with enforcement rather than allowing freedom, then everyday language rapidly moved away from Latin.

Is there another way to study and critique metaphysical and epistemological issues, or is language indispensable for the task?

I don't think there is any other way. But the issue is as I mentioned above. Language changes and evolves according to usage, and the usage is determined by the aims (intentions) of the users. Primary usage is the billions of mundane communicative everyday expressions. Secondary usage is business, legal and political. Tertiary is higher education. Metaphysics and epistemology are far down on the list of importance. Therefore language in its natural form, is fundamentally not well suited for these purposes.

Since natural language is not well suited to these purposes, yet language is the tool which must be used, then we can conclude that a special form of language needs to be designed for this purpose. And, there is nothing absurd about designing a special form of language for a specific purpose, mathematics is an example of such a specialized form of language. It's formulated for the endeavours of empirical science. Also we commonly create languages for purposes, in computer science.
• Wittgenstein and How it Elicits Asshole Tendencies.
But it seems just as plausible that language evolved in such a way as to be vague and confusing precisely because it's being generated by people facing a world full of vague and confusing metaphysical and epistemological conundrums.

Language evolved to be efficient for the purpose of mundane communication, that's why it's vague and ambiguous. We learn the minimum number of words required to make ourselves understood in a maximum number of different circumstances. Accordingly, it's not well suited for metaphysical and epistemological problems, and it's confusing when applied in this way.
• Wittgenstein and How it Elicits Asshole Tendencies.
This is more of a meta-thread on HOW PEOPLE debate Wittgenstein..

You could look to @Banno as an example. Banno has argued an interpretation of Wittgenstein, supporting that interpretation with an appeal to authority, Wikipedia. Later, in a completely different context, Banno bragged, I wrote that Wikipedia page. Hahaha, good one, Banno.

And, I might add, that I don't think such shenanigans are exclusive to Banno, or discussions of Wittgenstein in general.
No, it isn't the same as being stopped. Being stopped is an everyday occurrence. Infinite speed, is, as you say, unintelligible. If that's what underpins the supertasks, it makes sense of the narratives - apart from the fact that it doesn't answer the question whether the lamp is on or off.

I think that if the lamp is going on and off at an infinite rate, then it's not correct to say that it would be on at any particular time, or off at any particular time, because it is going on and off at a rate faster than our ability to determine a particular time.

If one watches the lamp in a dark room, at some point it will appear to be on continuously.

We're talking about a time duration which is far beyond the distinctions which could be made by the human eye. This would be what is occurring in a tiny fraction of a second. It doesn't make any sense to talk about how the lamp would appear in this time.
If I recite the first number after 30 seconds, the second after 15 seconds, and so on, then I have recited them all and so stopped after 60 seconds, even though there is no largest number for me to stop on.

I have to disagree. What you describe is a rate of acceleration which would produce an infinite speed. The rate at which you recite the numbers becomes infinite before 60 seconds passes. And, despite the fact that infinite speed is in some sense unintelligible, it is clearly not at all the same as being stopped.
If (2) is true then we can stop without stopping on some finite number.

How do you make this conclusion?
Not in those words. "Does not allow for a minute to pass", like somehow the way a thing is described has any effect at all on the actual thing.

Let me remind you, the "thing" being described here, in the op is a fictitious scenario. It is one hundred percent dependent on the description, just like a counterfactual. We might say that "the factual situation" is that a minute will pass, but the counterfactual described by the op does not allow for a minute to pass. You seem to be unable to provide the required separation between these two, thinking that the factual and the counterfactual may coexist in the same possible world.

Anyway, I see nothing in any of the supertask descriptions that in any way inhibits the passage of time (all assuming that time is something that passes of course).

Right, as I said there is nothing in the op to inhibit the passing of time, in fact the passing of time is an essential part, it is a constant. However, the premises of the op restrict the passing of time such that 60 seconds will not pass.

Ah, it slows, but never to zero. That's the difference between my wording and yours. Equally bunk of course. It isn't even meaningful to talk about the rate of time flow since there are no units for it. The OP makes zero mention of any alteration of the rate of flow of time.

There is nothing in the op to indicate that the passing of time slows. That is an incorrect interpretation. As you say, it isn't meaningful to talk about the rate of time in this scenario. What happens is that the speed of the person descending the staircase increases. And, as the speed increases, there is no limit to the acceleration indicated. The velocity is allowed to increase without limit. Even if we considered "infinite velocity" is a limited (which is of course contradictory), and assume that limit could be reached, this would still not imply "no time is passing". It would only make the spatial-temporal relationship unintelligible due to that contradiction.

This is actually very similar to the perspective of special relativity theory, which uses the speed of light as the limit, rather than infinite speed. This avoids contradiction but ti still renders the spatial-temporal relationship as unintelligible at the speed of light. From the perspective of the thing moving that fast, it appears like no time is passing, yet time is still passing. It's just a twisted way of making the passage of time relative to the moving thing for the sake of the theory. But there is no relativity theory stated in the op, nor any other frame of reference, so there is nothing to indicate a stopping, or even a slowing of time. The frame of reference which you keep referring to, in which 60 seconds passes, is excluded as incompatible with the described acceleration. The described acceleration is purely fictional though, like a counterfactual.
Socrates (as presented by Plato) considered himself wiser than anyone else because he knew he didn't know anything, which doesn't seem to leave much room for anyone else (at least in Athens) to be a philosopher. However, his dialogues with sophists do not show Socrates treating them disrespectfully and this is something of a puzzle. The orthodox interpretation regards Socrates' respect as ironic. Maybe it is. But maybe Plato's practice was a bit less dismissive than all this implies.

I believe Socrates (as portrayed by Plato) had great respect for the sophists. They displayed power and influence, and this piqued his interest. In Plato's dialogues, Socrates holds lengthy discussions with some sophists, and this would not be possible without the appropriate respect. On the other hand, I also believe that since the sophists presented themselves in a conceited way, as filled with a sort of complete or perfect knowledge, this produced a challenge in Socrates, to demonstrate their faults and weaknesses. Because Socrates had some degree of success in this personal challenge, Plato developed a level of disdain for them.

Prior to Socrates I believe that sophists were generally well respected, and this is evidenced by the power of their rhetoric. Socrates revealed the subjectivity of rhetoric, leaving the character of the sophists who employed it, exposed. The principal sophists who were exposed in this way, were the the politicians of Athens. But Socrates carried on toward exposing those in the even higher level, more exclusive schools of logic (I don't agree with you that there was no concept of "logic" at this time) like the Pythagoreans and Eleatics, and this allowed Plato to class them as sophists. This is where Zeno fits in. And Socrates is portrayed by Plato as having great respect and curiosity for the lofty principles held by these prestigious schools. Nevertheless, despite great respect for the individuals, he sees that there must be flaws in the principles, and therefore proceeds with his personal challenge of engaging the individuals to defend, and ultimately reveal those faults.

I think that the important point is the use of valid reasoning with unsound premises. This is how Aristotle attacked Zeno's paradoxes. But Aristotle didn't have a good understanding of the nature of knowledge, and the effects of faulty premises. He claimed that logic leads us from premises of greater certainty, to conclusions of lesser certainty, when in reality the opposite is true. Uncertainty in the premises is what introduces uncertainty into the conclusions. And the problem is that many premises are intuitive notions simply taken for granted, such as in the Achilles, the premise that the faster must first reach the place where the slower is, prior to passing. In reality, the faster passes the slower without ever sharing the same place.

The Aristotelian view of knowledge is still common place. You'll see that many here at TPF argue that there are fundamental principles, 'bedrock propositions' or something like that, which are beyond doubt, and support the whole structure of knowledge. In reality, those fundamental principles are the least certain because they are taken for granted, lying at the base of conscious thinking, bordering on subconscious knowledge. Those highly fallible intuitions are the ones most needing the skeptic's doubt, but it takes someone like Zeno to demonstrate this.

Says the proponent that time stops.

Huh? I said that time stops? I don't think so. I said that in the scenario of the op, 60 seconds will never pass. But clearly time does not stop. In that scenario, time keeps passing in smaller and smaller increments, such that there is never enough to reach 60 seconds, but time never stops. The claim that 60 seconds must pass or else time will stop, is derived from different premises which are inconsistent with the described scenario.

I suspect Zeno believed his premise to be false...

That's what I was arguing as well, but Ludwig produced references to show that this might not be the case.
No evidence of your interpretation here.

A few quotes with no real context, does little. Anyway, it's off topic, and really sort of pointless to argue a subject like this. You have your opinion based on how you understand Plato, and I have mine. Due to the reality of ambiguity, i don't think there is a correct opinion here.

Fair enough, but to go on, as Plato does, to accuse the sophists of deliberate deception or wilful blindness is completely unjustified (except when, as in the Protagoras,(?) Gorgias (?) someone boasts about doing so – though it doesn’t follow that everyone that Plato accuses of rhetoric and sophistry did so boast.).

The problem is that "sophist" was a word with a very wide range of application at that time. In the most general sense, you'll see Aristotle use it to refer to someone who uses logic to prove the absurd. Zeno might be a sophist in this sense. But also "sophist" referred to people like Protagoras and Gorgias, for their use of rhetoric. And "sophist" also referred to those who had schools and charged money to teach virtue. So there was a range of meaning, but "rhetoric" seems to be the essential aspect, and this is a mode of persuasion which is not necessarily logical. Accordingly, "sophist" has bad connotations, but as Plato demonstrates in "The Sophist", it's very difficult to distinguish a philosopher from a sophist. It appears like either the sophist is a type of philosopher, or a philosopher is a type of sophist.

But accepting that connection is a long way from accepting that he had any doubts about the validity of his conclusions.

The issue is not the validity of the conclusions, it's the soundness. Take the Achilles for example, with two principle premises. First, to overtake the slower, the faster has to get to where the slower was. Second, in that time, the slower will move further ahead. So the faster does not overtake the slower, and this may repeat if the faster is still trying. It appears valid to me, so if we want to refute it we need to look at the premises, as Aristotle did. But when we try to understand how the premises are wrong, then there is disagreement amongst us, because we really can't demonstrate exactly what the premises ought to be replaced with.

,
That there is no first number to recite is the very reason that it is logically impossible to begin reciting them in reverse and it astonishes me that not only can't you accept this but you twist it around and claim that it not having a first number is the reason that it can begin without a first number.

NoAxioms has a habit of making astonishing claims, then instead of recognizing the incorrectness, arguing some twisted principles. Like above, noAxioms insisted Zeno did not conclude that the faster runner could not overtake the slower, then refused to recognize my references, insisting they were in some way improper.

They're clearly being confused by maths. They think that because a geometric series of time intervals can have a finite sum and because this geometric series has the same cardinality as the natural numbers then it is possible to recite the natural numbers in finite time. Their conclusion is a non sequitur, and this is obvious when we consider the case of reciting the natural numbers (or any infinite sequence) in reverse.

This is the problem with mathematical axioms in general. As fishfry said, I can't really count the natural numbers, but I can state an axiom that the natural numbers are countable, and this counts as me having counted the natural numbers. So mathematicians really need to be careful to distinguish between the fantasy world they create with their axioms, and the true nature of what is "logically possible". Just because it can be stated as an axiom does not mean that it is logically possible. And when the axiom claims that something which is by definition impossible (to count all the natural numbers), is possible, then there is contradiction, therefore incoherency, inherent within that axiom. But would a mathematician accept the reality of an axiom which is self-contradicting?

There is a far more fundamental problem, and they're just ignoring it.

The problem is the age-old incompatibility between being and becoming. Logic, and this includes mathematics, applies naturally to "what is", "being". But "becoming" has aspects which appear to escape logic, what lies between this and that, one and two, etc., and therefore it seems to be illogical. If we apply the logic of being, to the reality of becoming, we find paradoxes as Zeno demonstrated.

This implies that "becoming" requires a different form of logic. That's what Aristotle laid out with his definitions of "potential", and "matter", as the aspects of reality which violate the law of excluded middle. In modern times, much progress has been made with modal logic, and probabilities. But the truth is that these aspects of reality, those which are understood through probability, remain fundamentally illogical, and the so-called "knowledge" which is derived creates an illusion of understanding.

I'm not sure it is possible to articulate what people who have not thought about the question think the answer to it is.

It's simple, talk to people, ask them. Then you'll see that it's more than just a matter of thinking about the question, it is a matter of making the effort to educate oneself. Metaphysics is not apprehended as a valuable subject.

I don't think we have anything near the evidence required to divine Zeno's motives. We don't even have his articulation of the argument.

Well, there is a lot of information available from Plato. In works like "The Sophist" and "The Parmenides", he takes a very critical look at the motives of some of the Eleatics, Zeno in particular. Specifically, in "The Sophist" he attempts the very difficult task of developing a distinction between philosophy and sophistry, even the sophist is engaged in philosophy.

But you don't know that he recognised what is so very clear to you, that the argument was ridiculous, or that he had "apprehended the faults in that ontology", though I admit that if he had understood what you understand, he might well have been poking fun at it. Still, other people since then have poked plenty of fun at it. But that's not a substitute for understanding the argument.

It's very clear from the discussion at the time, Plato and Aristotle, that Zeno knew he was using logic to produce absurd conclusions. There should be no doubt in your mind about that. He did not pretend to believe what he had proven, that motion is impossible, that the faster runner could never overtake the slower, etc..
I think that's perfect. It's the conjunction of mathematics and - what can I say? - the everyday world.
What's difficult is the decision which is to give way - mathematics or the everyday world. Zeno was perfectly clear, but some people seem to disagree with him.

The difficult thing is that many human beings are like naive realists, and they think that our sense perceptions of "the everyday world" are a direct copy of the way an independent world would be. From this perspective, we cannot look toward the everyday world to be what needs to give way. But from a more philosophical perspective, we know that sense perception doesn't really show us the way the world is.

So to be prudent, I'd say both sides need to allow give and take. This may be like the ancient division between Parmenides with being and not being, and Heraclitus with becoming. Plato described how the two seemed to be fundamentally incompatible, and Aristotle provided principles whereby they both could coexist as different aspects of reality.

That suggests that we do know roughly how things move. I don't think that's what at stake in Zeno's thinking. His conclusion was that all motion is illusory. The only alternative for him was stasis. But I guess we can do better now.

That was Zeno's conclusion, from his paradoxes, that motion is impossible. But I do not think that this was what he was sincerely trying to prove. Clearly he could observe motion, and he would know that this would be considered a ridiculous proof. So I think his arguments were designed to show that there is incompatibility between motion as observed, and motion according to the principles of logic applied to it. Zeno came from the Eleatic school, so the first principle was "being", stasis, but what he was demonstrating was that this principle was insufficient to understand reality. That's why Socrates and Plato took interest in the sophistry of the Eleatics. The Eleatics could employ logic to prove absurd things, and this showed the gap between the "becoming" of the physical world, and the "being" of the Eleatics and Pythagorean idealism. So I think that Zeno, even though he came from the Eleatic school, was apprehending the faults in that ontology, and was sort of poking fun at it.

That's apparently what somebody else reported about what Aristotle reported. I've seen it conveyed about 20 different ways.

I quoted that directly from Aristotle's Physics. I gave the page and lines, 239b, 14-17. Further, Aristotle compares it to the arrow paradox, and says "the 'Achilles' goes further in that it affirms that even the quickest runner in legendary tradition must fail in his pursuit of the slower"

This particular wording says 'never' and 'always', temporal terms implying that even when more than a minute has passed, (we're assuming a minute here), Achilles will still lag the tortoise.

The time length is irrelevant. The pursuer will "always" lag the pursued, for the reasons indicated. The pursuer must reach the point where the pursued was, and in the time that it takes to do that, the pursued will move further ahead.

The logic as worded here is invalid for that reason since the argument doesn't demonstrate any such thing.

The logic is invalid for what reason? There is no specic time periods mentioned.

. I've seen more valid ways of wording it (from Aristotle himself), in which case it simply becomes unsound.

I gave you Aristotle's wording. He rejects the arrow argument which demonstrates that motion is impossible, by saying that time does not consist of instants. So that is an attack on the premises of that paradox. He then says that the solution to the 'Achilles' "must be the same". But he doesn't show how time not consisting of instants would solve the Achilles paradox. The matter of instants appears irrelevant here, and the problem seems to be with the assumed nature of space, rather than time.

I just didn't like the fact that the quote didn't match the site linked.

I can assure you, the quotes are taken directly from the referenced sites. I just went back to check. Click the links and you will see.
You mean because they allow the convergent infinite series?
Mathematically? Physically? (I'm inclined to think you mean physically, because of your reference to fundamental particles.)

I meant, that they can mislead us when we apply the principles to the activities of the physical world. That's what Zeno's paradoxes show.

Is the direct spatial route not available because it contains a convergent regress?
What path does Achilles take? (I assume he is not a fundamental particle.)

What is evident, is that we do not know how things move, and the exact "path" through space, that things take, whether they are big planets, stars and galaxies, small fundamental particles, or anything in between.

I know the story. You seem to have reworded it for your purposes, since the quote you give does not come from that site, but the site also seems to be conveying the story in its own words, not as reported by Aristotle.

Here's what Aristotle reported:

The second is the so-called 'Achilles', and it amounts to this, that in a race the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. — Aristotle Physics 239b 14-17

How is this different from what I said? I gave a full explanation, as did the site I quoted. Aristotle just said "it amounts to this...", providing a shortened version, probably because the specifics were well known at that time.

Yes, and without justification, or at least without explicitly stating the additional premise that makes the conclusion valid.

I'm still waiting for you to explain how the conclusion is not justified, and why you think there is a requirement of an additional premise.
So are you going to conclude, with Zeno, that motion is impossible? or that Zeno is analyzing the situation in a misleading way?

Yes, Zeno is analyzing in a misleading way, but only because the axioms of continuity and infinite divisibility are themselves misleading. So Zeno simply demonstrates how standard conventions are actually misleading us.

And here we are. a couple of millennia later, still being misled by the same conventions. This is because we have not yet determined the natural points of divisibility. And so, fundamental particles take every possible path when they move from A to B, because the direct spatial route does not allow them to get ahead of the tortoise.
Anyway, I deny that Zeno in any way suggests that the overtaking will never take place. He just says that another step always follows any given step.

According to this reasoning, Achilles will never catch the tortoise, says Zeno. — Internet Encyclopedia of Philosophy

The paradox is like this. Both Achilles and the tortoise are moving, but the tortoise has a head start. So at t1 Achilles is at location A and the tortoise is at location B. At t2, Achilles reaches location B, but the tortoise has moved to location C. At t3, Achilles reaches location C, but the tortoise has moved to location D. As this procedure will carry on without end, Zeno concludes that the faster runner cannot overtake the slower.

Zeno Paradox 1: Achilles and the Tortoise
Achilles is a lightening fast runner, while the tortoise is very slow. And yet, when the tortoise gets a head start, it seems Achilles can never overtake the tortoise in a race. For Achilles will first have to run to the tortoise's starting point; meanwhile, the tortoise will have moved ahead. So Achilles must run to the tortoise's new location; meanwhile the tortoise will have moved ahead again. And it seems that Achilles will always be stuck in this situation.

Case closed, then.

I think so, but we'll have to see what noAxioms is talking about with the reference to a requirement for further premises. I think noAxioms looks at Zeno in a different way.
Great. Then show the logic that concludes this, without resort to another premise.

I don't see the need for any other premise. Achilles is moving, and described as doing this in a way in which he will always have to move further before he can overtake the tortoise. Since he will always have to move further before he will overtake the tortoise, we can conclude logically that he will never overtake the tortoise in that described activity. Why do you see the need for another premise?

I'm afraid that if you condescend to use ordinary arithmetic, one can predict exactly when Achilles will overtake the tortoise, given data about how fast each contestant moves and the size of the handicap.

Sure, but those mathematical principles are not the premises described by Zeno.
There is no first natural number to start with. It is logically impossible to have started reciting the natural numbers in descending order.

Obviously, the described process has no start, that is implied by the description. So your conclusion that it is logically impossible to have started such a process is irrelevant, as what is already known. You need to show that such a process, one without a start, is logically impossible.

That's what "first cause" arguments attempt to do. They describe the temporal aspect of "a process", "a thing", or similar term, in such a way that it necessarily has a beginning and an end in time, then they produce a logical argument from that description. It's an attempt to bring the realm of material (physical, or temporal) reality to bear on the realm of logical possibility, by stating premises which are supposed to represent the essence of material (physical) reality, and restricting logic with them. Another example of a similar restriction is the law of identity, and the other two fundamental principles.
How does it start? That's easy. When the appropriate time comes, the number to be recited at that time is recited. That wasn't so hard, was it? It works for both scenarios, counting up or down.

This is not the issue. It clearly does not have a start. The question is whether it is logically necessary for such a task to have a start. This is argued in "first cause" arguments.

@Michael @fdrakeThe problem is that what is described is an activity, and the way that we understand activities is that they have a beginning. Activities are all physical. But if you remove that requirement of "physical", then the activity might be proceeding without a beginning, just like it could proceed without an end. It's the way that we look at the difference between past and future, which makes it difficult. If there is no such difference, then the past supertask must be logically possible, just like the future one.
Nah. That's an appeal to metaphysical or physical impossibility. Not logical impossibility!

It is logically impossible to have recited every natural number in descending order because it is logically impossible to even start such a task.

I think that's actually a very difficult issue to resolve. It's basically the same question as whether an infinite regress is logically possible. I believe the infinite regress actually is logically possible, and it requires a contradictory premise to make it impossible.

Correct, but a second unstated premise must be assumed in order to draw this conclusion, since without it, one can only say that the tortoise cannot be overtaken at any particular step.

If no particular step can overtake the tortoise, then the tortoise, by the described motion cannot be overtaken. Where's the need for another premise?

That second premise might well be that supertasks cannot be completed.

Following from the described premises, the supertask cannot be completed. It is logically implied that there is always further distance for Achilles to cover before overtaking the tortoise. Therefore the described task can never be completed. There is no further premise required, it is a logical conclusion, the described "supertask" cannot be completed.

That premise is indeed in contradiction with the first premise and empirical observation. At least one of the three is wrong.

The conclusion that Achilles cannot overtake the tortoise does contradict empirical evidence, that's the reason it's called a paradox. But I do not see how it contradicts "the first premise". Which premise would that be? The argument is valid, so how could it contradict a premise?
I beg to differ. That simply does not follow from the description. Zeno describes a physical completable supertask, which is only as possible as the soundness of his first premise.

I think you misunderstand Zeno's paradoxes. Zeno concluded that Achilles cannot overtake the tortoise. That is explicit. And therefore, it indicates that he is arguing that the supertask is not completed. The "supertask" of passing an infinite number of spatial divisions is never finished, therefore the faster runner never surpasses the slower.

The paradox is that physical evidence indicates that the faster runner always does overtake the slower, in reality, even though the logic proceeding from fundamental axioms proves that the faster overtaking the slower is a supertask which cannot be completed.

Due to the strength of the empirical evidence, we are led toward the conclusion that the fundamental axioms concerning the continuity of space and time, and the infinite divisibility of those continuums, must be faulty. Those axioms are the "unsound premises".

Declaring something to be impossible is a strong claim and requires strong evidence.

This is why we cannot simply accept the empirical evidence, and conclude that the supertask is a descriptive impossibility derived from faulty axioms. Empirical evidence is known to be unreliable. So, we need stronger principles to demonstrate the actual faults in the axioms.
Exactly so. I have correct my post. I meant valid and wrote 'sound' in haste. A simple application of modus ponens shows the lack of soundness of Zeno's conclusion iff empirical knowledge is given any weight.

The conflicting premise which would be used to disprove this, the limitations of divisibility
The conflicting premise seemed to be a denial of the completability of a supertask. He never suggests a limit to divisibility.

In Zeno's Achilles and the tortoise, empirical knowledge shows that Achilles will pass the tortoise. But empirical knowledge has problems like what Hume showed with the problem of induction. Because of this, empirical knowledge does not prove the supertask to be impossible.

That the supertask is not completable is not denied, that it is not completable is what actually leads to the problem. In Zeno' paradox Achilles never catches the tortoise because the supertask is never completed. By the premises of the op, Icarus cannot reach the bottom of the staircase, just like Achilles cannot reach the tortoise. So "the supertask" by the nature of what it means to be a supertask, cannot be completed.

The problem is that empirical evidence shows us that tasks will be completed, Achilles will pass the tortoise, and in the op 60 seconds will pass. This shows that the supertask as a fiction. However, due to the problem of induction, empirical evidence does not provide a proper proof. That is why I suggested we look at the divisibility of time as the means for providing a better proof.
To demonstrate the impossibility of Zeno's physical supertask, one must attack the premise, not the logic. The logic is sound, at least until he additionally posits the impossibility of the first premise, but that only gives rise to a direct contradiction, not a paradox.

X is a true fact of motion. X is is a false fact of motion. Therefore either motion is impossible, or at least one of the premises is wrong.

That's almost right, the logic is valid, but not necessarily sound. Soundness requires true premises. Generally though, judgement of the premises is dependent on empirical knowledge, which all good philosophers know is unreliable. Therefore we have valid logic and if the premises are disproven they would be disproven by competing premises, and the judgement ought not be based on empirical knowledge..

The premise of infinite divisibility is provided from mathematical axioms. The conflicting premise which would be used to disprove this, the limitations of divisibility, cannot be derived from the unreliable empirical knowledge, and it has not yet been provided. So we need to defer judgement, until we can disprove the mathematical axioms in a more reliable way.
• Are posts on this forum, public information?
My posts are invaluable... so good luck trying to get money for them.
• Does Roundup (glyphosate) harm the human body?
In wide open flatland production, American grain fields are very large and spraying them before harvest would probably not be cost effective. That's probably true in Ukraine and Russia, too.BC

The practise may be more common in northern countries (Canada for example), where the drying conditions are not as reliable:
https://mbcropalliance.ca/directory/production-resources/staging-for-pre-harvest-glyphosate-application/
• Does Roundup (glyphosate) harm the human body?

Perhaps I was wrong to call it "common practise", but that was the information I was reading at the time. The degree of such usage has been debated, and there doesn't seem to be any hard statistics on it. I suppose any statistics would rely on the honesty of the farmers engaged in the activity, and negative press would influence their admissions. But, it is an approved practise. Check this:
https://extension.umn.edu/small-grains-harvest-and-storage/managing-wheat-harvest

In the following article, the representatives of wheat producers claim that use is not common, but do not deny that it is done:

"Glyphosate is typically applied with a ground rig, and a ground rig will only run the wheat down," said Brett Carver, Wheat Genetics Chair in Agriculture at Oklahoma State University. "In most U.S. wheat regions, it takes a situation of no-other-choice desperation to consider glyphosate as a harvest aid….certainly not the usual scenario."

And here's Anita Dille, Ph.D., a professor of weed ecology at Kansas State University.

"There's all sorts of research that goes on before information gets put onto a label as a legal recommendation," said Dr. Dille. "It starts with the companies. They've done the research. Then, it always goes to a contract research or university level, unbiased and independent kind of sources. Then, all that information goes together in a petition to the EPA (Environmental Protection Agency), before it can be put onto a label as a legal recommendation. All that is regulated. The label is a legal document that the growers have to go by."

Further:

"U.S. wheat producers do NOT routinely use Roundup®, or other formulations of glyphosate, for pre harvest applications," said Steve Joehl, Monsanto’s Industry Affairs Director for wheat. "Quite the contrary, it is the exception rather than the rule. You should be aware that Roundup has an approved label with the EPA for pre harvest use, in the event farmers desire to control perennial weeds, like Canadian thistle, prior to harvest; or for farmers in areas of short growing seasons where crop maturity can be delayed. When used according to labeled recommendations, it is a very safe application. But because Roundup is used in Roundup Ready crops of corn and soybeans, these perennial weeds infestations have been reduced and the practice has been reduced even more."
• Purpose: what is it, where does it come from?
he usual claim is omnipotence - God can do anything and everything, which if the author and creator of the universe we live in, he would pretty much have to be.

Why do you conclude this? Do you have absolute control over anything you created? Why do you think that God would have absolute control over the universe He created? It seems to me, that "creations", whether they are by human beings, some other creatures, or even God, are just not like that.

And if constrained, then not God

Why? What makes you think that God must be absolutely unconstrained? I think that if you took the time to read some theology, you would see that even if it is often said that God is "all mighty", and sometimes said that He is "omnipotent" these conceptions are qualified, and it is not very often meant that he is absolutely unconstrained. Consider for example that it is often said that God only does what is good, and he exercises complete self-control to only do good. Clearly this indicates a special sort of constraint, which we as human beings also share with God. However, since we do not really understand self-constraint, and therefore have not been able to perfect it, we tend to imagine it in a very strange paradoxical way. The complete and perfect self-constraint which God is often said to have, is portrayed as an absolute freedom to do anything. And this is paradoxical because even though there are many things God could do, He also cannot do them, because of His self-constraint.

As to any necessity for his reality - yours sounding like Anselm's - that is only a "proof" for those who already take that real existence as axiomatic.

I was not handing you that argument as a "proof". I was only trying to make it clear to you that if you want to talk about "God", then you need to talk about "God" as He is understood. I find this to be a common problem with the atheist approaches to God. The atheist commonly approaches God with the presupposition, that God is an imaginary, fictitious thing, not real. But this is not how God is understood in theology. This prejudice which the atheist holds is completely contrary and contradictory to how God is actually understood, so it prevents the atheist from having any understanding of God. Aquinas, for example, asserted that God's essence is His existence. This implies that the very first principle one must accept before being able to understand anything about God, in any way, is that He has real existence. So if the atheist has any bit of intent whatsoever, to understand God, this prejudice must first be dismissed. Otherwise it's a waist of time.

Reality is the realm of nature, and recall we put that to the question.

Your claim, "reality is the realm of nature" is fundamentally false. By saying "the realm of nature" you imply the possibility of other realms not contained within the realm of nature. And as a "realms" these must be real. So even the statement itself, as written, implies its own falsity. It's like saying "there is only one multiplicity". The statement is self-defeating.

Consider, that "the artificial" is often contrasted with "the natural". We cannot say that the artificial is not real. So many will class artificial as part of the natural. But by doing this we lose the meaning of "natural", which is defined as "not artificial". The intent of the person who redefines "natural" in this way, may be to include the artificial into the realm of the natural, to argue that only the natural is real, but what's the point? That statement is self-defeating as shown, and to class the artificial as natural, is to ignore the substantial difference between the two.

As to hearts, I have to own up to my ideas about "purpose" being pretty clearly not as clear as I thought they were, or would have liked them to be.

This is why it is a very good thread which you have started. If you learn something new then the thread is good, right? The issue here, I think, is the presuppositions which we commonly take for granted. These are what are commonly known as bedrock or hinge propositions. Since they are taken for granted they are not subjected to our doubt. Since we do not doubt them or subject them to any form of methodological skepticism, then we do not develop an adequate understanding of their meaning. So the use of many words, such as "purpose", just floats freely, being a facilitator of mundane communication, a word whose meaning is taken for granted allowing for fluid conversation. Because of this, the word's meaning gets shaped to the circumstances of conversation, and what comes out on top is the most common usage. If someone asks what is the meaning of "purpose", we have all sorts of examples in common usage to refer to. But since its such a commonly used word, we can restrict the meaning we express, to these common examples, and having not applied a methodic analysis like the skeptic does, the true deeper meaning escapes us.

However, I think I can distinguish between purpose and function.

This is a good start. Let's look at the difference between "purpose" and "function". At first glance, we can say that the two might commonly be interchangeable, "a thing's purpose is the thing's function". But invert that and say "a thing's function is the thing's purpose", and that's not necessarily the case. This implies, right off the bat, that "function" has an even broader meaning than "purpose". Not all functions are purposes.

Further, we can see that "function" is most often an activity, whereas "purpose" is more often the goal of the activity, the end, or objective. This opens an even bigger rift between the two. What is exposed here is that "purpose" is something we attribute to an activity, the property of an activity, which puts it into a specific relation with an end, a goal. This makes the activity a means to an end. "Function" in its common usage does not necessarily imply such a relation of means to an end, because the function may be the activity itself, regardless of the purpose of the activity. So we might say, of a thing, that the thing has a function, and this function is the activity of the thing, without even indicating the purpose of that activity, or whether it even has a purpose.

So for example, if I am involved in a cooperative effort, I have a function, which is to bring the others coffee. That can be referred to as my function, what I am doing, bringing the coffee, and this can be said without any reference to the purpose, why I am bringing the coffee. In the heart example, the function of the heart can be stated as "to beat". The beating is the function of the heart, and this may be stated with a complete disassociation from the purpose of the heart. The thing has a function, an activity, and this is completely irrelevant to whether there is a purpose, goal, or end to that activity.

You can see how this has become a very convenient way to separate "function" from "purpose" thereby ignoring the question of "purpose". This is the way language evolves according to social circumstances to avoid areas of doubt, and facilitate mundane communication. We can talk about all sorts of things, and the function of each thing, with complete disregard as to whether that function has a purpose or not. That helps us to avoid having to think about whether or not natural activities have a purpose, thus keeping us away from the volatile "God" question.
• Truth in mathematics
Ontology is choosing between languages. It consist in no more than stipulating the domain, the nouns of the language.

Oh my God! Save this poor lost soul.
• Purpose: what is it, where does it come from?
I'll try one more time: is God constrained in any way? Is He real? My point being that in belief in an idea, you can have what you want. But not in any reality.

God has to be real, because that is stipulated in the conception of "God", as an essential aspect of "God". If God was not real, then the conception would be contradictory, and there would be no God to talk about, just self-contradictory nonsense. So, if we are talking about God here, we are by definition talking about something real. You can dismiss talk about God as self-contradictory nonsense if you like, but please don't ask me if God is real, because it just indicates that you are totally ignorant.

As for your other question, I have no knowledge as to whether God is constrained or not. Some say that God is not constrained in any way, but I think that's just conjecture.

Take a look at this problem tim. I said to you that the purpose of an animal's heart is to circulate blood, and you said that's not the sense of "purpose" I am talking about. Now you clarify the sense of "purpose you are talking about, with the following.

On a good day, if I do something, it is for a reason. If my effort is successful, it might be said I had achieved my purpose in doing it. In this sense purpose like a work order or chore or task, a thing to be done.

How is this a different sense of "purpose" from when I said the purpose of the heart is to circulate blood? To circulate blood is "a thing to be done", by the heart, it is "the reason" for the heart. If the heart's effort is successful, it achieves its purpose. It's the very same sense of "purpose".
• Truth in mathematics
Classical Euclidean geometry is arguable not "real" mathematics. As Kant pointed out, it is incredibly married to sensory input, to the point that it is not pure reason.

That's the point, mathematics is always "married" to something, be it the world of sensory input, or the alternative, Platonic universe.

The fact that Euclidean geometry has too much meaning and does not fit the formalist narrative, points out a problem with Euclidean geometry and not with the formalist ontology. If it is not possible to interpret it as meaningless string manipulation, then it is not real mathematics.

You're jumping to a conclusion. How does the imaginary Platonic universe provide a less problematic grounding for meaning than the sensed universe?

You yourself described how the formalist approach does not get rid of correspondence, it only replaces the objects of the sensible universe, with the objects of the Platonic universe, as that which the mathematics corresponds with. But there is a huge problem here, the objects of the Platonic universe are simply whatever strikes the fancy of the mathematician. So for example, the mathematician might think, "I wish I could count to infinity". Then, one could create an axiom which states that the natural numbers are countable, and stipulate a 'set of the natural numbers', and voila, the mathematician has counted to infinity, within that imaginary universe of Platonic objects.

In its anti-realist take, mathematics is indeed "about nothing". In its realist take, mathematics is about an abstract, Platonic universe that is completely divorced from the physical universe. In both cases, any downstream application of mathematics is completely irrelevant to mathematics itself. That is a feature and not a bug.

What you describe is what formalism would be like if it could achieve its goal in an absolute sense. It would be "about nothing" but as I indicated, at the same time it would be "about everything", providing infinite applicability, and at the same time, as you say, no applicability. But of course, no mathematician would seek this, because it would be incomprehensible nonsense.

So formalist mathematics is always tainted, and rules of application always inhere within the axioms, whether the influences are the sensible universe, the Platonic universe, or both. Your claim, "any downstream application of mathematics is completely irrelevant to mathematics itself" is simply false, because that's what the axioms of mathematics do, lay the grounds for application.
• Truth in mathematics
Model theory makes anti-realist views unsustainable. Model theory makes mathematics decisively correspondentist. Because of model theory, mathematical realism and more specifically, Platonism, are unavoidable. Mathematics is about abstract Platonic worlds and is not just string manipulation.

How would you classify model-dependent realism? Clearly this is not "correspondentist". You can argue that it is a form of "realism" as the title suggests, but where does the correspondence lie. I suggest that you consider that correspondence inheres within the formal system itself. When the math is applied correctly there is correspondence between the symbols, and what you call the "Platonic universe".

But "correspondence" in the common sense, means to correspond with the empirical world of observable sense objects. Your thread does not seem to make a distinction between these two very different senses of "correspondence". The common sense is correspondence with an assumed observable, sensible world, and the sense you mention here is correspondence with an assumed Platonic universe.

In the op. you do not address the nature of axioms. In reality, the axioms dictate the applicability of mathematics, and it is how mathematics is applied which determines whether it is formalist or realist. In other words, the meaning of the symbols is dependent on the context of the application, and the applicability, therefore context, is dependent on the axioms. And, I would argue that in general, the axioms are intentionally extremely vague and ambiguous in this respect, for the very purpose of allowing the mathematics the widest possible context of applicability. You will see however, axioms like those of set theory, which are explicitly realist. Such restrictions limit the applicability of the mathematics, (which is evident from the recent paradox threads of @keystone), by doing things like limiting "infinite" to fit it into the confines of "an object".

Applied mathematics is actually not mathematics.

If you think clearly about this idea, you will see that the inverse is actually the case. To analyze, let's separate form from content, and assume that the formalist's goal is to remove all vestiges of content from the formal system. How can this be accomplished? If the axioms have no designated relationship with anything outside the logical system, then the system my be infinitely useful, but at the same time infinitely useless, because it is robbed of all meaning. So the formalist allows a little bit of meaning, content, to inhere within the form of the structure.

For example,

Mathematics proper seeks to establish the correspondence between an abstraction and a Platonic universe -- when interpreted according to realism -- or between an abstraction and another abstraction -- when interpreted according to anti-realism. Mathematics proper is never about the physical universe.

In this example, you have an assumed "Platonic universe". This assumed universe provides the content. So consider the following two possible sets of rules for the processes of counting. The first set of rules would be to produce a bijection between the symbols and the physical objects to be counted. You want to count chairs, you biject "1" and a chair, "2" and a chair, etc.. In that case, the formal structure, and the set of rules for application are completely distinct from the content, the content being the physical objects counted, which is dependent on the application The second possible set of rules for the process of counting would be to produce a bijection between the symbols and a "Platonic universe" of "numbers". In this case, the content, being the numbers as objects, is built right into the formal structure. The limits of applicability are built into the structure, instead of being defined by a further set of rules.

If you categorize the first as "formalist", then you have a separation between the formal structure and the content (physical things) which the structure is applied to. However, the structure is useless without rules of application, so we proceed toward axioms of geometry, and rules of categorizing, to provide rules of application. The rules of application are still a part of the formal system, and there is no proper "formalist" separation. If you categorize the second as "formalist", then the content inheres within the formal structure, and there is no proper separation, as required for a true formalism.

Either way, mathematics cannot escape the need for, or its dependence on, application. There is always some form of application built into the formal structure, as axioms. Either the rules for application are a distinct part of the structure, as in the first case, or the application itself is already built into the structure, as in the second case. In no way can mathematics completely escape application, without it becoming something other (a useless bunch of symbols) than mathematics. So the inverse of your statement is actually the truth. With absolutely no application, mathematics would be absolutely nothing. And in reality mathematics is nothing other than application, pure means without any defined end.
• The Riddle Of Everything Meaningful

Oh I see, "I exist now", is an eternal truth. So the "eternal truth" is a truth which obtains the highest degree of certainty. The other less certain truths are not eternal because we allow that they may fall out of the status of being true at some time, just like what happened to "Pluto is a planet". The truths with a really low level of certainty, which we employ commonly in our mundane thinking, like "it will not rain today", are only true for as long as they are useful.
• The Riddle Of Everything Meaningful
What if we could consider 'cogito ergo sum' as an eternal truth?

Alas, being aware that we exist or being aware of our consciousness could be an eternal truth.
I can't imagine a decrease in the level of meaningfulness in Cartesianism.

Wouldn't this mean that your existence is eternal?
• Purpose: what is it, where does it come from?
If you'd read the OP, you could not have failed to observe that this, your sense of purpose here, is not the topic, and so without relevance.

If you'd have read what you said to me, you would know that you asked me "in terms of purpose - of any kind -". I assume that the clearly stated "any kind", implies that any such restrictions are to be put aside.

Anyway, I just reread your op, and cannot understand your proposed restrictions at all. Can you explain clearly how you are proposing to restrict the meaning of "purpose" in this thread?

And in passing since you claimed earlier that there could be no propose before purpose, I assume you also would hold that there can be no hearts until there was a heart.

What are you talking about? Of course there can be no hearts until there is a heart. That's a self-evident truth. But that's not at all relevant to what I said. I said purpose is prior to a display of purpose. A heart is a display of purpose, so purpose (or intent if you prefer) must be prior to the heart. How do you get from this to the self-evident truth of "there can be no hearts until there was a heart"?

But let's try these: is God constrained in any way? Is He real? My point being that in belief in an idea, you can have what you want. But not in any reality.

How is this relevant? In reality, sometimes you get what you want, sometimes you do not. In what way do you believe that the constraints placed on human beings are related to the constraints placed on God, if there are any?
The point of my example with the ship was to counter your assertion of Newton forces not being necessary to move and free will being enough. I said you'd need help from Newton. Asking for a line to be thrown to you is you admitting the help from Newton was necessary. That's what the tether is: a way to do it by exerting an external force, since the free will couldn't do it itself.

It seems you misunderstood.
• Purpose: what is it, where does it come from?
I understand reality as being the world we all live in, and also a set of constraints which things not of or in reality are not subject to. I don't object to beliefs, except when, as concerning things not of or in reality, the believer tries to place them into reality.

I cannot quite apprehend what you mean by "a set of constraints which things not of or in reality are not subject to". I assume you are saying that there are things which are not part of reality, and those things are not subject to this particular set of constraints you are referring to. Are these non-real things subject to any kind of constraints, and what kind of existence do they have if they are non-real?

And as God is supposed to be unconstrained, he cannot be in reality nor rationally supposed to be there.

Your conditions for "reality" do not state that there cannot be an unconstrained real thing. You said that things not in reality are not constrained by a specific set of constraints, but you didn't say that things in reality are necessarily constrained. What exactly do you mean by "a set of constraints"? I understand "sets" to be things created by human beings. Are these constraints artificial as well, or is it just the classifying of them into a specific set which is artificial?

And in terms of purpose - of any kind - can you point to or articulate any that do not come into being through a man's or a woman's speech or writing?

Are you serious? Is it not the case that the purpose of an animal's heart is to circulate blood, and the purpose of sense organs is to sense, etc..?
• The Riddle Of Everything Meaningful
Aren't there things with a constant meaningful duration?

That would be eternal truth, if there is such a thing. Some would attribute this to God, others to mathematics, and some perhaps to physics. It seems like people generally have a desire to assume some kind of eternal meaning, as a sort of principle of balance, because life, while it seems to strive in that direction, fails in its capacity to give us this.
• Purpose: what is it, where does it come from?
Fair enough? And may we say as well, boot-strapped? By which I mean valued because they are valued, any other value being derivative and incidental.

Doesn't "valued because they are valued" imply infinite regress, or maybe a vicious circle, rather than bootstrapping?

In the op you say "Bottom line, purpose is boot-strapped", but how could this be possible? Isn't it true that boot-strapping is a purposeful process? This would imply that purpose is necessarily prior to, as the intentional cause of any boot-strapping activity. Then purpose itself cannot be boot-strapped.

#### Metaphysician Undercover

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