I'll repeat what I said to andrewk above:

There are some who claim that a supertask is possible; that if we continually half the time it takes to perform the subsequent step then, according to the sum of a geometric series, an infinite sequence of events can be completed in a finite amount of time.

Examples such as Thomson's Lamp show that this entails a contradiction and so that supertasks are not possible. Continually halfing the time it takes to perform the subsequent step does not just contradict the physical laws of our world but is a metaphysical impossibility.

With these paradoxes we shouldn't be looking for some answer that is consistent with the premises but should accept that they prove that the premises are flawed.

You seem to take issue with that first paragraph, but your reasoning against it doesn't make any sense. Unless the universe ceases to exist then 60 seconds is going to pass. The passage of time does not depend on the counter.

The counter stops after 60 seconds.

No mathematical thought experiment can determine the nature of reality. — fishfry

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.

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.

If time is infinitely divisible, the counter would go up to infinity. — Lionino

The counter resets to 0 after 9. It will only ever show the digits 0-9.

I see that 30 and 15 and 7.5 sums up to 52.5 seconds. I also see that as it progresses the sum approaches 60. But I do not see how it could ever get to 60. — Metaphysician Undercover

Because 60 seconds will pass. I don't understand the problem you're having. The passage of time does not depend on what the counter is doing.

So to make this simpler; I am watching a stopwatch whilst the counter is counting according to the prescribed rules. When the stopwatch reaches 60 I look at the counter. What digit does it show?

Except there have been plausible solutions given to Thomson's Lamp. — Lionino

I wonder if there's such a solution to my variation.

If we agree that time is infinitely divisible, it seems to follow that an infinite task may be completed in a finite amount of time — Lionino

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

There are some number systems that define division by zero as ${a\over0} = \infty$.

Clearly, what is implied by "and so on", contradicts "for 60 seconds". — Metaphysician Undercover

No it doesn't.

The "and so on" refers to repeating this formula:

Step 1 occurs after 30 seconds, step 2 occurs after a further 15 seconds, step 3 occurs after a further 7.5 seconds, and so on.

As per the sum of a geometric series this supertask takes 60 seconds.

we postulated the existence of a finite-sized mechanism that can switch state in an infinitesimally small time, which contradicts the laws of our world. — andrewk

That's precisely the argument being made.

There are some who claim that a supertask is possible; that if we continually half the time it takes to perform the subsequent step then, according to the sum of a geometric series, an infinite sequence of events can be completed in a finite amount of time.

Examples such as Thomson's Lamp show that this entails a contradiction and so that supertasks are not possible. Continually halfing the time it takes to perform the subsequent step does not just contradict the physical laws of our world but is a metaphysical impossibility.

With these paradoxes we shouldn't be looking for some answer that is consistent with the premises but should accept that they prove that the premises are flawed.

We don't directly see cows – according to the naive and indirect realist's meaning of "directly see"¹ – but we do indirectly see cows.

Given that the adverb "directly" modifies the verb "see", the phrases "I directly see a cow" and "I see a cow" do not mean the same thing. The phrase "I indirectly see a cow" entails "I see a cow" and so the phrases "I do not directly see a cow" and "I see a cow" are not contradictory.

_{¹ A directly sees B iff B is a constituent of A's visual experience.}

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

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

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. — fishfry

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.

Given the defined behaviour of the lamp, will the lamp be on or off after 60 seconds? If the answer is undefined, but if the lamp must be either on or off, then the behaviour is metaphysically impossible.

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

Yes, in other words rejecting iii), namely the idea that one can finish counting an infinite sequence. — sime

True, but that's only part of the issue.

If after 30 seconds he's flipped the switch once and if after a further 15 seconds he's flipped the switch a second time and if after a further 7.5 seconds he's flipped the switch a third time, and so on, then it would suggest that a supertask can be completed in 60 seconds.

So if a supertask can't been completed in 60 seconds then the time between each flip cannot continually decrease. At some point no further division is metaphysically possible.

For example, Thompson's proposed solution to his Lamp paradox is to accept (i) and (ii) but to reject (iii). — sime

I didn't think he proposed a solution. Rather, it was an example to show that supertasks are impossible.

It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off. This is a contradiction.

By such a method, one can count from negative infinity to zero. — noAxioms

Given that each division is some ${1\over{n}}m$ then such a movement is akin to counting all the real numbers from 0 to 1 in ascending order. Such a count cannot start because there is no first real number to count after 0.

But I've been arguing that the above reasoning is fallacious. Yes, each division must be passed, and each division is preceded by other divisions (infinitely many), and yes, from that it can be shown that there is no first division. All that is true even in a physical journey (at least if distance is continuous).

But it doesn't follow that the journey thus cannot start, since clearly it can. — noAxioms

It does follow that the journey cannot start. Therefore given that the journey can start then the premise that there is no first division is false. It's a proof by contradiction.

As such there is some first division and so movement is discrete.

The paradox does not require the physical possibility of such a counter. It simply asks us to consider the outcome if we assume the metaphysical possibility of the counter. If the outcome is paradoxical then the counter is metaphysically impossible, and so we must ask which of the premises is necessarily false. I would suggest that the premise that is necessarily false is that time is infinitely divisible.

It is metaphysically necessary that there is a limit to how fast something can change (even for some proposed deity that is capable of counting at superhuman speeds).

I don't understand what you are saying.

The example is simply: after 30 seconds a single-digit counter increments to 1, after a further 15 seconds it increments to 2, after a further 7.5 seconds it increments to 3, and so on for 60 seconds, resetting to 0 at every tenth increment.

What digit does the counter show after 60 seconds?

Your suggestion that the above entails that 60 seconds won't pass makes no sense.

Naive realists. That's why they are naive realists. See What’s so naïve about naïve realism?:

The second formulation is the constitutive claim, which says that it introspectively seems to one that the perceived mind-independent objects (and their features) are constituents of the experiential state. Nudds, for instance, argues that ‘visual experiences seem to have the NR [Naïve Realist] property’ (2009, p. 335), which he defines as ‘the property of having some mind-independent object or feature as a constituent’ (2009, p. 334), and, more explicitly, that ‘our experience […] seems to have mind-independent objects and features as constituents’ (2013, p. 271). Martin claims that ‘when one introspects one’s veridical perception one recognises that this is a situation in which some mind-independent object is present and is a constituent of the experiential episode’ (2004, p. 65).

...

... Intentionalism typically characterizes the connection between perception (taken as a representative state) and the perceived mind-independent objects as a merely causal one. But if the connection is merely causal, then it seems natural to take the suitable mind-independent objects to be distinct from the experience itself and, therefore, not literally constituents of it.

Note the distinction between the constitutive claim of naive realism and the merely causal claim of intentionalism.

Indirect realism is the rejection of naive realism (and is compatible with intentionalism; see Semantic Direct Realism).

60 seconds will pass in the universe. The counter is just one thing that exists in the universe and it changes according to the prescribed rules.

So given the prescribed rules, when the universe is 60 seconds older, what digit will the counter show?

Not in the distal world; in the world. — Pierre-Normand

Well, yes. Phenomenal character exists in the brain, the brain exists in the world, and so phenomenal character exists in the world. But it is still the case that phenomenal character exists in the brain, not outside the brain, and so the naive realist's claim that distal objects and their properties are constituents of phenomenal character is disproven by the fact that distal objects and their properties do not exist in the brain.

For example, one group defines "direct perception" as "ABC". They claim that "ABC" is true and so call themselves "direct realists". Another group defines "direct perception" as "XYZ". They claim that "XYZ" is false and so call themselves "indirect realists".

It is possible both that "ABC" is true and that "XYZ" is false and so that both the group that call themselves "direct realists" and the group that calls themselves "indirect realists" are correct.

I suppose it's also why people have invited you to reconsider the kind of things that can count as direct realism! — fdrake

You can call anything you like "direct realism", but it is not a given that you are saying anything that contradicts indirect realism. Each group just means different things by the word "direct".

To suggest that if your direct realism is true then my indirect realism is false is to equivocate.

That's right. The phenomenal character of experience is something that is constructed and not merely received. The perceiving agent must for instance shift their attention to different aspect of it in order to assess the phenomenal character of their experience. But this is not a matter of closing your eyes and inspecting the content of your visual experience since when you close your eyes, this content vanishes. You must keep your eyes open and while you attend to different aspects of your visual experience, eye saccades, accommodation by the lens, and head movements may be a requirement for those aspects to come into focus. This is an activity that takes place in the world. — Pierre-Normand

The phenomenal character doesn't take place in the distal world. The phenomenal character takes place in the brain, albeit is (in the veridical case) causally determined by the body's interaction with the distal world. Indirect realists accept this.

Well that is why I have spent 60 pages trying to explain that much of the dispute between indirect and non-naive direct realists is a confusion borne from each group using the words "direct" and "see" to mean different things.

The relevant consideration is the epistemological problem of perception. Do we have direct knowledge of distal objects and their properties or only direct knowledge of the phenomenal character of experience?

The dispute between naive and indirect realists concerns the phenomenal character of experience. You can use the word "experience" to refer to something else if you like but in doing so you're no longer addressing indirect realism.

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.

And note that this only considers progressive interpretations of these paradoxes (i.e. how they can complete). Regressive interpretations (i.e. how they can start) must also be considered. I don't think mathematical limits are relevant to these at all.

I think the scientific evidence strongly suggests that experience is either reducible to brain states or supervenes on brain states.

I think the scientific evidence strongly suggests that distal objects are not constituents of brain states and are not constituents of any phenomena that supervene on brain states.

Therefore, I think the scientific evidence strongly suggests that distal objects are not constituents of experience.

I think we have direct knowledge of the constituents of experience and indirect knowledge of anything that is causally responsible for experience and causally covariant with its constituents.

I think the constituents of experience are mental phenomena (e.g. smells, tastes, and colours) and that distal objects are causally responsible for experience and their properties causally covariant with its constituents.

Therefore, I think we have direct knowledge of mental phenomena and indirect knowledge of distal objects and their properties.

This is all I understand indirect realism to be. Whether or not to describe this as "experiencing mental phenomena" is an irrelevant grammatical choice with no philosophical or physiological implications (e.g. it no more implies an homunculus than "I feel pain" does).

Instead, I say that our perception of real objects is direct (in a non-naive sense) because perceptions are mental representations. — Luke

The indirect realist opposes the naive realist position, saying that we do not directly perceive a real object but that we directly perceive only a mental representation of the real object. — Luke

What is the physical/physiological difference between us seeing a mental representation and a mental representation existing in our heads?

This is where I think you're getting so confused by grammar.

If mental representations exist and if distal objects are not constituents of these mental representations and if our knowledge of distal objects is mediated by knowledge of these mental representations then indirect realism is true, because that's all that indirect realism means.

Similarly with the Thomson's Lamp case. When we ask "is the lamp on or off at one minute" we are asking for something that the set-up doesn't give us enough information to answer. The setup tells us whether the lamp is on or off at every instant in [0,60) and tells us nothing about whether it is on or off at 60 or later. We cannot infer whether it would be on or off at 60 because we know nothing about the physics of the world in question, which must be enormously different from that of our own, in order to allow complete switching of a finite-sized lamp in infinitesimally small time periods. I expect we could invent some physical rules to support either an on or an off assumption. — andrewk

I don't think the physics is relevant. The question can be asked of any universe with any physical laws. The thought experiment is entirely metaphysical.

Repeating my specific example:

After 30 seconds a single-digit counter increments to 1, after a further 15 seconds it increments to 2, after a further 7.5 seconds it increments to 3, and so on, resetting to 0 at every tenth increment.

What digit does the counter show after 60 seconds?

The issue we have is that if there is no smallest unit of time then the counter is metaphysically possible, but this entails a paradox as the answer to what the counter shows after 60 seconds is undefined yet the counter will show something after 60 seconds. Assuming that paradoxes are metaphysically impossible then the counter is metaphysically impossible, and that suggests that it's metaphysically impossible for time to be infinitely divisible.

We could replace the counter with some supernatural deity capable of keeping such a count if it makes things easier to consider (similar in kind to Benardete's Paradox of the Gods).

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

I suggested that movement was discrete, not that space was discrete. In other words, at a sufficiently small scale, when an object (esp. particle) moves from A to B it does so without passing any half-way point. Your use of the phrase "quantum jump" is fitting.

Experience exists within the brain, distal objects exist outside the brain, therefore distal objects do not exist within experience.

Experience is causally determined by our interaction with distal objects, and its qualities causally covariant with the distal object's properties, but there's nothing more to the connection than that. It's naive, and inconsistent with the scientific evidence, to suggest otherwise.

How could we have reliable knowledge of objects if they were not experienced by us? — Janus

Are you referencing the problem of induction? There is no clear answer to that. According to the scientific method a statistical significance of five sigma is accepted as the criterion for reliable knowledge of unobservable entities like the Higgs boson. In everyday life we're not so strict. I did not directly experience Joe Biden's inauguration but I think I have reliable knowledge that he was inaugurated.

Regularity, predictability, and common sense seem like sufficient criterion for most cases. But it's not infallible, hence the warrant for healthy scepticism.

If movement is continuous then an object in motion passes through every ${1\over{n}}m$ marker in sequential order, but there is no first ${1\over{n}}m$ marker, so this is a contradiction.

I'm interested in your take on the nonexistent 'barrier' thing described at the lower half of my prior post in this topic. It also is a variation on something somebody else authored, but I cannot remember what it was originally called. — noAxioms

Bernadete's Paradox of the Gods:

A man walks a mile from a point α. But there is an infinity of gods each of whom, unknown to the others, intends to obstruct him. One of them will raise a barrier to stop his further advance if he reaches the half-mile point, a second if he reaches the quarter-mile point, a third if he goes one-eighth of a mile, and so on ad infinitum. So he cannot even get started, because however short a distance he travels he will already have been stopped by a barrier. But in that case no barrier will rise, so that there is nothing to stop him setting off. He has been forced to stay where he is by the mere unfulfilled intentions of the gods.

It's the same principle as Zeno's dichotomy, albeit Zeno uses distance markers rather than barriers. Given that each division must be passed before any subsequent division, and given that there is no first division, the sequence of events cannot start.

I think it's the crux of Zeno's paradox that the mathematics of an infinite series fails to address. The solution, similar to my proposed solution above, is that movement is not infinitely divisible (either because space is discrete or because movement within continuous space is discrete).

Thanks, although it's actually a variation of Thomson's lamp.

I think there’s a simpler way to phrase this problem.

After 30 seconds a single-digit counter increments to 1, after a further 15 seconds it increments to 2, after a further 7.5 seconds it increments to 3, and so on, resetting to 0 at every tenth increment.

What digit does the counter show after 60 seconds?

If there is no answer then perhaps it suggests a metaphysically necessary smallest period of time.

I'll offer you the same answer as given to Frank, above. Blind, illiterate mutes can herd cows. You account seems a bit ableist...

We do not simply passively "experience" cows. we feed them, move them into yards, slaughter them and eat them.

All this by way of pointing out that the "constituents of our experience" are not one way, from world to mind; we also change what is in the world, and this is part of our experience of the world. While you read this, you are already formulating your reply.

And you do not feed, herd, slaughter and eat sense impressions. — Banno

The debate between naive and indirect realists does not concern whether or not we can feed or slaughter cows. It concerns whether or not our perception of cows counts as "direct perception" according to some relevant meaning of "direct perception".

To put it simply, "we feed cows, therefore direct realism is true" is a very obvious non sequitur.

Specifically, the debate between naive and indirect realists concerns the relationship between the phenomenal character of experience and the mind-independent nature of distal objects (and its epistemological implications).

Either we see distal objects, or we see mental phenomena. It cannot be both. — creativesoul

Yes it can. I feel pain and I feel the fire. I see cows and I see colours.

I don’t see how this relates to whether we perceive objects directly or indirectly or, in particular, how it relates to the supposed perception of representations or perceptual intermediaries. This is the philosophical substance of the dispute as I understand it. Direct realists claim we do not perceive any perceptual intermediary or representation, whereas indirect realists claim that we do.

Furthermore, I don’t see why a direct realist must hold the view that “distal objects and their properties are constituents of experience” in the physical sense that you suppose. A direct realist can have an unmediated perception without the perception needing to be the perceived object. Otherwise, it’s just a strawman of perception. — Luke

What does it mean to directly see something?

By "directly see X" naive and indirect realists mean that X is a constituent of experience, so when naive realists say that we directly see distal objects they are saying that distal objects are constituents of experience and when indirect realists say that we don't directly see distal objects they are saying that distal objects are not constituents of experience.

You're welcome to mean something else by "directly see X" but in doing so you're no longer addressing indirect realism.

I think there's this assumption you're making that there's some singular agreed upon meaning of "directly see X" that everyone is using and that each group just disagrees on which things satisfy this meaning, but that's a mistake. See Semantic Direct Realism.

What you're missing is the fact that light carries a great deal of information about distal objects, from which it follows that, contrary to your claims, we do have reliable knowledge of distal objects. Perhaps you're trading on the absurd demand for certainty. We have reliable, certain in the relative but not certain in the artificial "absolute" sense, knowledge of external objects. — Janus

I didn't say that we don't have reliable knowledge. I said that we don't have direct knowledge.