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

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

There is a difference between saying that 1/2 + 1/4 + 1/8 + 1/16 + ... = 1 and saying that one can write out every 1/2n in order. The latter is not just a physical impossibility but a metaphysical impossibility.

Some say that the latter is not a metaphysical impossibility because it is metaphysically possible for the speed with which we write each subsequent 1/2n to increase to infinity, and so that this infinite sequence of events (writing out every 1/2n) can complete (and in a finite amount of time). Examples such as Thomson's lamp show that such supertasks entail a contradiction and so that we must reject the premise that it is metaphysically possible for the speed with which we write each subsequent 1/2n to increase to infinity.

So if you wish to define a final state, you can make it anything you like. I choose pumpkin. — fishfry

If you want to say that supertasks are possible but then have to make up some nonsense final state like "pumpkin" then I think this proves that your claim that supertasks are possible is nonsense and I have every reason to reject it.

Take the scenario here:

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

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

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

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

The conclusion, then, is that an infinite sequence of events cannot be completed, and the fact that we can sum the geometric series is a red herring. To resolve the fact that we can sum the geometric series with the fact that an infinite sequence of events cannot be completed we must accept that it is metaphysically impossible for an infinite sequence of events to follow a geometric series: we must accept that it is metaphysically impossible for time to be infinitely divisible.

The description of the Thomson lamp only actually specifies what the lamp is doing at each finite stage before 2 minutes. It says nothing about what happens at 2 minutes, especially given the lack of a converging limit. — Lionino

That's precisely the point. The lamp turning on and off and the square changing colours are each examples of an infinite sequence of events. If you claim that it is possible for an infinite sequence of events to complete then you should be able to determine the completed state of the lamp/square. If you cannot determine the completed state of the lamp/square then I will reject your claim that it is possible for an infinite sequence of events to complete.

Of course the solution doesn't work when you change the mechanism to be exactly like Thompson's lamp without the limit.

Likewise, Earman and Norton's solution doesn't work if you remove the limit (falling ball).

My example keeps the falling ball so I haven't "removed the limit".

Your "solution" doesn't work, as shown by this alternative:

The ball bounces at a rate such that it first strikes the panel after 30 seconds, then again after a further 15 seconds, then again after a further 7.5 seconds, and so on.

Each time the ball strikes the panel the colour of the panel changes, rotating through white, red, and blue.

What colour is the panel when the ball comes to a rest?

by whatever mechanism, the plate knows at what part of the parabola the ball is at, — Lionino

This is just a meaningless hand-wavy rationalisation and is inconsistent with the specific timing intervals:

Red after 30 seconds, blue after another 15 seconds, white after another 7.5 seconds, etc.

Each bounce of the ball is the timing interval, e.g. when it first hits the plate it turns red, when it hits the plate a second time it turns blue, when it hits the plate a third time it turns white, etc.

The simplest answer is that supertasks are illogical. It is metaphysically impossible for an infinite sequence of events to be completed in a finite amount of time.

Let's move away from numbers as that is clearly causing some confusion.

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

What colour is the square when this supertask completes (after 60 seconds)?

What contradiction? — Lionino

That the counter doesn't show 0 and doesn't show 1 and doesn't show 2 and doesn't show 3 and doesn't show 4 and doesn't show 5 and doesn't show 6 and doesn't show 7 and doesn't show 8 and doesn't show 9 even though it must show exactly one of them.

we already have the possibility of infinity as an assumption — Lionino

And that assumption entails a contradiction, proving the assumption false.

Now, you introduce another premise, "Unless the universe ceases to exist then 60 seconds is going to pass". This premise contradicts what is implied by the others which describe the supertask. — Metaphysician Undercover

No it doesn't.

But the counter only shows the standard 0-9 digits. At no point does it switch from showing some natural number to simply showing the ∞ symbol.

To repeat what I said earlier: 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.

Supertasks are illogical. Time cannot be infinitely divisible.

But then I am interested in a counter that would indeed count to infinity — Lionino

Assume the counter counts to infinity. After 30 (or 60) seconds, what is the first digit of the number it shows?

But does that imply necessarily that time and or space in our universe must be discrete and not continuous? — flannel jesus

If continuous space and/or time entail that supertasks are possible and if supertasks are not possible then space and/or time are not continuous.

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).