Glad you mentioned the LEM, because after posting I thought of a similar issue.

Many decades ago Michael Dummett noticed an uncanny similarity between lots of standard philosophical debates. He recognized the same moves as in the realism/anti-realism debate and suggested that many people were in effect having that same debate but within a restricted domain. Point being that the realists will assume that the LEM applies and the (local) anti-realists have often stumbled, because they don't recognize that they need to deny this. They feel boxed into true-or-false for propositions they really ought to say 'neither' for if they're to be consistent.

Note that the realists were just doing their thing -- applying the LEM is just part of their story, but it also functioned as an I WIN card without them intending or recognizing it.

I think philosophers (and maybe even ordinary folks) tend to be more sophisticated about that now than they were fifty years ago, but I think there's evidence around us of similar issues.

Infinite divisibility is the problem, which was Zeno's target all along (although in his case he wanted to argue that all is one, whereas I'm suggesting that there must be some fundamental unit of space/time (or at least movement) that cannot be halved). — Michael

Are you saying that Zeno's argument is sound, and that it shows that if space-time is continuous, then motion is impossible?

What about other variants, like the "starting and finishing" one I proposed?

Convergent series are necessarily inapplicable to supertasks. — Michael

Oh I think that's probably true, even though I'm feeling a bit uncertain about how supertasks should be analyzed.

(One reason I've been going through this is to get clearer about what your thinking is. Sometimes your objection is that a given task can't be finished, sometimes that it can't be started, sometimes, as here, that if it could be completed then something else you don't like could also be possible.)

Zeus, it should have been clear, is here as a stand-in for the power of mathematics itself, which isn't bound by many of the usual considerations. He is by stipulation magic. Thus if you encountered this on a test in a math class

17. If Zeus takes 1 second to say "1", one half a second to say "2", one quarter to say "3", and so on, how many seconds does it take Zeus to say all the natural numbers?

you'd answer "2 seconds", and you'd be right. I'm not arguing for informal pedagogy as serious philosophy, but I am interested in how the Zeus story does make perfect and uncontroversial sense in the right context.

*

Getting back to Zeno ... What turns out to be wrong with this family of arguments? It's not just about movement, for instance, but about there being any sort of change at all, about anyone, as I said before, ever doing anything. So what's going wrong here?

You've switched back to talking about movement, where there is a strong intuition that each step in the task of moving from A to B can be subdivided into just as many steps as the original task. (I.e., a lot.)

I was talking about Zeus reciting the natural numbers (with geometrically increasing speed).

If you now want to say that each step can be divided into, let's say, "starting" and "finishing", then we'd be back in your regress of being unable to start. Do we like that argument? Now the problem is that every task is an infinite task, and we needn't worry about whether any of them can be finished because none of them can even be started.

I'm still leaving aside movement. My point for the moment is that if anyone can do anything, then Zeus can recite all the natural numbers in 2 seconds.

Oh I see -- my spec is "move each time by half the distance remaining to be covered" and that works recursively. Your way makes it impossible to start. I'm content for the moment with defining task as something

1. I know how to start,
2. I know what the next step is, and
3. I know when I'm done.

Just off-the-shelf recursion.

Edit: nevermind. I see your argument. Hang on.

Can you specify the task of movement recursively? “Move to the first half-way point”? It’s a lot like counting the rational numbers between 0 and 1 in order — Michael

Those are not the same.

Moving by half the remaining distance can be specified recursively; doing the rationals between 0 and 1 in order cannot be if by "in order" you mean "smallest to largest". There's no smallest rational > 0 to be the first.

Counting up from 1 is a task but counting down to 1 isn't? Why is that? — Michael

Just in the sense that I don't know how to specify that task recursively. Is there a way? If not, is there some other way?

I don't know. It seems a truism that if one has completed a series of consecutive tasks then some task was the final task. I know it's not much of an argument, but I honestly can't make sense of it being any other way. — Michael

And my argument is that you're generalizing something that happens to be true of finite tasks. In effect your claim is that "infinite task" is a contradiction. My claim is that this is false, because (1) is a side-effect, and we needn't consider it part of the very definition of "task".

This is Thomson's lamp paradox, except the question in this case will be "was the last number odd or even?" — Michael

I suggested there are two criteria for "having finished a task":

(1) Having performed the last step;
(2) Having performed all of the steps, in some specified order.

For finite tasks, these are the same: "last" can just be defined as "all but one have already been performed."

But what about infinite tasks?

I see your argument as something like this:
1. If you have recited all the members of a set, there is some member of the set that is the last one you recited.
2. Zeus has recited the natural numbers in order. At step n, he recited the natural number "n".
3. By (1) and (2), there is a natural number z that was the last one Zeus recited.
4. By (2) and (3), z is the largest natural number.
5. Since there is no largest natural number, (2) is false.

I'm questioning step (5). We have the option of discarding premise (1) instead of (2).

Look at how criterion (1) works with finite tasks. Each time you perform a step, the number of steps remaining to be performed is one smaller. You're done when that number is 0. But this is just not true for infinite tasks. The number of natural numbers remaining to be recited is the same after reciting any finite number.

In fact, it looks to me like (1) is derivative of (2). We need a closer look at what it means to specify a task.

Suppose I give you a jar of marbles and tell you to count them. I come back half an hour later to find you haven't even started. Your explanation is that I didn't tell you what order to count them in. Fine. I know order doesn't matter, but evidently you don't, so I instruct you to pick one, take it out of the jar, add 1 to your running total, then pick any remaining marble as the next one. Go on until there are no marbles left.

Is it reasonable now to say you cannot count the marbles because I didn't tell you which one is the last one? No, of course not, because my recursive specification is enough. Here's how to start; here's how to continue; here's how to know when you're done.

It might be useful to consider a similar scenario. Zeus counts backwards to 1, getting slower as he counts. It took him 1 second to count from 2 to 1, half a second to count from 3 to 2, and so on. — Michael

I think we could play around with "first" as I have been with "last", but for many cases recursive specifications are exactly what we want, so I can just as well say that what you describe here is not a task at all.

I think these calculus solutions are just a bewitchment — Michael

I think if there's an intuition pump in the room, it's not calculus but Thompson's lamp.

I think it could be that some tasks we specify by specifying the last step -- maybe that's all we care about and are indifferent about what steps are or aren't taken. Really that seems more like a direction just to bring about a certain state of affairs.

But some tasks we naturally specify using recursion, and the infinite tasks we're talking about are clearly that kind. (Counting all the marbles is not the same as making the jar empty; the jar being empty is just how you know you're done.)

So is there an argument for (1), or an argument that it is not just a special case of (2)?

If propositions are understood as something like 'bearers of reference which are truth apt', then this is precisely what is in question here. — StreetlightX

But why? Why not reach for "This vocabulary allows me to say things I couldn't say before -- some things true and some things false"?

Just over to the side of this, there's what Ornette Coleman said: "It's when I realized I could make a mistake that I knew I was onto something."

It shows that no matter how fast you go you can never finish. — Michael

Suppose Zeus is reciting all of the natural numbers infinitely quickly.

What does it mean to say that he never finishes?

Does he ever recite the largest integer? No. There isn't one. That's a point for not finishing.

But is there any integer he never gets to? Nope. That's a point for finishing.

Suppose we do it differently: Give Zeus one second to recite "1", half a second to recite "2", a quarter to recite "3", and so on. Is there any natural number he hasn't recited after 2 seconds have passed?

Our views are very close.

I have almost irresistible impulses toward naturalism and nominalism. Almost started a thread yesterday on abstract objects as imaginary, supported by certain sorts of speech acts. I cannot make the details work though ... As it turns out, Darth started one, so I'll probably chime in there.

I often wish I'd never even heard of ontology.

I think I'm cool with most of this. (But what's that intuition worth?)

We've talked elsewhere about the special use of introspection in linguistics. (The stuff about speaking on behalf of your speech community.) Something I'd like to know more about is the idea of theory in linguistics. It's my (limited) understanding that in trying to model a language formally, whether any speaker is or even could be conscious of the theory is not an issue. I think there are constraints on what could be computable, and thus conceivably instantiated in a human brain. And I guess there are also learnability constraints. But we know -- by introspection, no less! -- that people do not and need not consciously work through their knowledge of a language in order to speak it. I think there is some residual uncertainty, the usual anxiety of modeling, about whether it would be meaningful much less correct to say that your complete theoretical description must in fact be instantiated in the brains of speakers. I don't know where people come down on that.

Anyway, that would be a principled way of leaning away from what's agreeable to your intuition and toward whatever has the most predictive power, because the theory's structure might not look much like what you think the structure of your language is, if you think about it, or much like the sorts of things you think about when you do consciously intervene in the production of speech.

And in a broad sense I think the stuff we've learned about cognitive biases the last several decades -- with, you know, research, not anecdotes! -- is all to the good. It's notable that awareness of such biases can lead to corrective conscious intervention. ("Man this rookie shortstop is hitting like a god! -- Okay, okay, dial down the excitement a bit, that's only 50 PAs ...")

Not sure where that leaves me. There's tons of stuff that goes on "below" -- it's always "below", isn't it? -- the level of consciousness, and I'm cool with that. How we do intervene consciously in those unconscious, automatic processes is pretty interesting, especially since there's a whole lot of reasoning I'm interested in that does seem to take place in the exception room instead of the business-as-usual room. The use of intuition there does require considerable care -- not least because it might represent the automatic department trying to assert control and get consciousness off its turf!

I was wondering whether it makes sense, yes.

I'm not sure your code tests that exactly. I mean, you specify that it runs infinitely fast, but the question you ask is just whether the last integer is even. There's no last integer.

Does this show that no matter how fast you go, counting integers takes an infinite amount of time? If so, that's interesting. It does quash the assumption that doing anything faster makes it take less time -- not true if the task is infinite.

Is this our conclusion?

At the end of what? When it reaches the largest integer?

If it took me 1 second for each hop, it would take me countably many seconds to do all the rationals, the same number of seconds it would take to hop to all the natural numbers in order. The only point here is that you cannot hop to all the reals in countably many seconds.

A supertask has countably many steps completed in a finite amount of time. Seems like we could get go "infinitely fast" and get a finite amount of time, instead of taking an infinite amount of time at a finite speed. I'm not clear whether the definition of "supertask" precludes going infinitely fast in this sense. Maybe "infinitely fast" doesn't even make sense the way "infinite amount of time" does. Maybe that's part of the point?

This is getting confusing, so big thanks to @Jeremiah!

What you're pointing out now, I think, is that the rationals (or, I guess $\{ \frac{1}{2^n}\}$) are not well-ordered under $\lt$, and that's true.

The reason we care is because we're talking about movement, and movement looks like a matter of going from one place to the next, where "next" is already defined in a particular way.

Which gives rise to another paradox? Given an infinite amount of time I could hop to all the rationals between my starting point and any destination (inclusive), but I cannot do them in order from closest to where I start to farthest (i.e., at my destination). So whatever that is, it doesn't look much like movement in the usual sense.

Getting back to our finite world, any finite subset of $\{ \frac{1}{2^n}\}$ is well-ordered under $\lt$, so that's what we're looking for I guess.

I am unclear on whether progress is being made, which is pretty freaking ironic.

Except to keep this analogous to movement the counting has to be ordered. We don't jump to the half-way point and then back to some earlier point. — Michael

I'd have to brush up on this to answer properly, but my instinct is that that's an interpretation problem, essentially a matter of labeling. There's the standard interpretation, associated with the number line, of what order numbers are in, but they don't have to be. That may not look like much of an answer.

And infinite tasks of any kind are still beyond the capabilities of finite beings. The difference between the rationals and the reals is that even if you had infinite time or could count infinitely fast, you still couldn't count the reals.

Of course if space is granular, then our task is finite, yes?

Just for clarity's sake: the problem you're pointing up is that the reals are uncountable. You could look at the rationals and say, there is no first one after zero, but this doesn't actually matter, because the rationals can be re-arranged into a list. You can just pick what to count as the first, the second, etc. You cannot do this with the reals.

Thanks Srap. — Metaphysician Undercover

You're welcome?!

This is not a circle, because it is derived from an infinite number of points equidistant from a central point, rather than a circular line. — Metaphysician Undercover

The whole post was good, really good, but this is my favorite part.

Carry on.

Ah. Cool.

Sorry, I don't know Mill, but the way "connotation" is used in ordinary language, it would be the sort of thing Frege calls "coloring", among other things. Aspects of meaning that don't affect reference, the truth value of assertions a word is used in, and so on. Connotations are at least usually public, rather than just private associations, and Frege definitely wants sense to be public, but may not affect how reference is determined. (Maybe there's a different connotation for "the President" and "the Prez", but not the kind of difference that would change the truth value of sentences I used those words in.)

The analogy he gives is this: when you look through a telescope, there's the object you're looking at (reference), the image of the object on the mirror, viewable by whoever uses the telescope (sense), and then there's the entirely private image formed on your retina.

?

And I agree. (Should have made that clear. The computability approach actually makes more sense.)

What's curious is that even in a high school science class there's likely a kid who'll argue that you can't subdivide matter infinitely -- everyone's heard something about particles, even if it's hard to understand. But the idea that space is granular just seems crazy!

When I was a kid, I was taught, like Jeremiah here, that limits and convergent series and calculus "solve" Zeno's paradox. Greeks just didn't have as much as math as we do. Of course they didn't teach me about computability when I was 17.

Here are two more versions.

1. Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct?

A) 25%
B) 50%
C) 60%
D) 25%

Only when you go to the answer sheet, you see (I'm putting the letters where there would be bubbles in labeled columns)

1. A. B. C

Wait, so is D part of the sample space or not, since it turns out I can't "choose" it?

2. Multiple Choice: If you choose an answer to this question at random, what is the chance you will be correct?

A) 25%
B) 50%
C) 60%
D)

Now what? There's "D", but can I answer "D"? If I assert that D is the answer, what would I be asserting?

Which brings me to this point: a question and answer pair should be recastible as an argument or at least an assertion. If it can't, then it's either inconsistent or a contradiction.

I get the point you're making about mathematics, I think. If I drag in some context, it's too show that mathematics doesn't stand naked on its own, but relies on a broader conception of rationality. That conception can be seen at work in the way we talk to each other, and in the way we make tests. It's those conventions your question violates, and that's why it's a conundrum.

I actually like the puzzle, else I wouldn't spend my time trying to figure out how it works and why it resists solution.

I mean, without even reading the questions.

See, there could be a point to that. Suppose it were done this way:

16. ...
17. Do not select an answer to this question.
18. ...

Then the instructor could assess a heavy penalty on anyone who bubbled in any answer to (17). It would be a check on students answering randomly.

I’m construing it as a true/false check on a send/receive transaction. — Fool

But we're not asked what the chance is that we've succeeded in answering; we're asked what the chance is that we've answered correctly.

Anyone else want to back the horse that the second 25% was put there by mistake? Or can we all at least agree the question was purposely designed with it? — Jeremiah

I see two possibilities:

(1) I assume you've made a mistake, because the question cannot be answered as posed (Principle of Charity and all that);
(2) You disabuse me of (1) by demonstrating the correct answer.

But not answered correctly. And the question is about answering correctly, so it's not like I'm forcing my own preconceptions on it.

That's the definition of a broken question.

You could imagine a teacher doing that and still successfully grading the test. — Fool

By marking all answers as wrong?

The second 25% is obviously a typo that leaves no correct answer as an option.

To the student who just bubbles away, there's no difference between a broken question and a question he just didn't luckily answer correctly.

To the test designer, a duplicated answer is always a mistake that can only help such students.

If you choose an answer to this question at random — Jeremiah

Given a multiple choice test with a fixed format, say, every question having four possible answers, there are two ways to choose randomly:

(1) reading the questions and the answers first;
(2) not reading them and just bubbling in something on the answer sheet.

By design these are equivalent on multiple choice tests because answers are never deliberately duplicated. If use method (2), it won't even matter to you that this question directs you to choose randomly -- that's what you're doing already. People tend to use method (1) in part so they can throw out "(a) a fish" as a possible answer to "What is 6 x 7?" and then randomly select among the other three.

I claim all this is relevant because you rely on our expectations about how multiple choice tests work but then screw with them. It is literally a trick question, just not the sort that's typical on these tests.

Suppose this was a real test, and the duplication was a mistake. Then the instructor would discover that this question is broken. Admittedly it's broken in a really unusual way, but the result is that no student can give a correct answer. As far as that goes, it's no different from a typo in which the correct answer was supposed to be "(d) 33%" but was printed as "(d) 55%".

And again, all of this is lost on the student who just randomly bubbled in A or B or C or D without even reading the broken question. To him, the question is just

Blah blah blah
A something
B something
C something
D something

Do you have a proof that he is Doing It Wrong™?

On hand-waiving — csalisbury

Tell the guy looking at his hands not to worry, since you've relaxed the requirement for actually having hands in order to engage in hand-waving.

Have not read Ramachandran, but cognitive science is enough in the air it's not hard to have a sense of these sorts of things. I'm just never sure what the philosophical upshot is. That was the point of the "machine code" post after the one you quote: of course a person's brain is doing all sorts of stuff below the level of consciousness, but that doesn't necessarily mean that everything we think about people and how they reason is wrong. That looks like a category mistake.

Suppose I argue that you don't "really" hear Giancarlo Stanton's bat striking the ball, that there is a vibration in the air, and your brain processes that as an auditory signal in some complicated way that isn't simply veridical, and puts it together with a highly processed version of the visual sensations you're having, makes some adjustments for a direction for the "sound" to be perceived to have come from, and "arbitrarily" assigns it to the image your brain has created of the Stanton-object swinging. I've left out ~1200 pages of detail. I've probably also left out too many of the "justs" and "onlys" that this sort of account relies on, but I got an "arbitrarily" in there.

Should be clear I don't think this is anything like proof I don't hear bat striking (the crap out of) ball. It's just an account of how I do that -- not sure what a more neutral phrase there would be -- at another level that doesn't include me or bats or balls.

Here's another version. When it catches a "need an explanation" signal, your brain thinks, "I could show you the machine code for what actually happened but you wouldn't understand it, maybe not even with drugs. I could show it to you in Python -- shit, you never learned Python. What have we got? Turbo Pascal? Are you kidding? Okay, here's what happened in Turbo Pascal.”

Okay, but the vibe I'm getting here is that this "explanation" is essentially fictive, that the right word for all this sort of stuff is "rationalization". Is that your view?

If so, is it the connections made that are fictive, or what is connected, or both? For example, if you're nervous about your intentions, maybe your brain rummages around among your actual beliefs and preferences and so forth, finds some stuff that will pass for an explanation and serves that up as why you want to do what you want to do. What's fictive there is not the beliefs and such, and hey -- maybe not even the logical connections between everything, since after all this has to be convincing. What's fictive is that this is the process you went through in forming your present desire.

Feels like this is cognitive science now, rather than philosophy, so I'm getting confused ... Are we waiting on brain scans to find out if beliefs are fictive?

What happened to the connection between "I want to find my keys" and "I want to look for them in the kitchen"?