From where I sit, these threads have raised the level of discussion on the forum. Specific problems are way more interesting than a battle of isms.

Appreciate it.

The layperson will not understand it if you tell them there are different sizes of infinity but we know it's true in math, but we don't take that as evidence against Cantor's work on infinity. — MindForged

Ah, no, I definitely wasn't saying that the recherche nature of things like the Liar or the Russell set is some kind of evidence they should be shrugged off.

We broadly agree, I think, that there is something reasonable and something unreasonable at work in producing the paradoxes. If forced to choose, my allegiance is with the LNC rather than naive set theory, but whatever. I do wholeheartedly approve of dialetheist tinkering. It's healthy.

But I am suspicious of a kind of magician's (or conman's) patter you see around these things. "I can have a set of numbers, a set of things that are red, a set of bald men. Perfectly natural, right? A set of cars. A set of cars that are blue. A set of all sets that don't have themselves as members. Most natural thing in the world..." I just want to pause over the "Wait, what?" reaction here, while you're always emphasizing how naturally these things arise. (I do understand that it's the principles not the example that's supposed to be natural.)

I guess emphasizing the weirdness of the counterexamples is holding onto the possibility that the counterexample itself is where the sleight-of-hand takes place, rather than in the principles it exploits. I certainly feel that way about the Liar. (Likewise Gettier, which is a whole lot like a magic act; Fitch's; the Slingshot.) Russell strikes me as something a little different, that absolutely unrestricted comprehension is bizarre and not what anyone needed or wanted and it's no surprise that if you explicitly let in anything at all, you'll get some pathological cases.

Have you read any Frank Ramsey? He might well say it's the other way round. — Pseudonym

"Truth and Probability" changed my life, but I'm no Ramsey scholar. What did you have in mind? What should I reread?

you might have to join the dots — Pseudonym

Here's the idea: science is part of philosophy in just the way I was talking about earlier, as an ideal to strive toward and as a tool we actually use. As a goal and a norm. (One of the many excellencies of Lewis's game theory scaffolding is that it clarifies how there can be a norm that is in some sense external to you, something you are beholden to, but at the same time you're responsible for it, helped make it.) There's a natural, even evolutionary process here of local competition enabling global cooperation. That's a bit of a fairy tale, sure, but that fairy tale is part of the system, as norm and goal.

((Going to bring this "speaking freely" to end soon.))

We engage in this activity because we're trying to assert power — Pseudonym

1. Cooperation is baked into language -- that much you should have learned from Wittgenstein. Vervet monkeys don't do their "predator" calls if there's no monkey near enough to hear them.

2. Thus if you want a purely competitive encounter with another human being, words are not the best tool for the job. You cite evidence of people straining against that limitation. That's interesting. Truly. It's a question how far you can get and what techniques you'll use to impose your will on others by imposing your will on words. As you say, that's rhetoric. But Humpty-Dumpty always falls.

3. From competitive use of language comes argument; from argument comes logic. Logic gives us both new ways to compete and new ways to cooperate, and it cannot do otherwise.

4. That's how philosophy becomes the incubator of science. Compete how you will and you are still also cooperating. (As you acknowledge -- you can gain something from my attempts to master you.) The war of all against all is, here, in this context, only a myth.

((Pardon the style -- in a weird mood today.))

The Russell set is not what anybody had in mind when they first had the idea of sorting the world into classes. The Liar is not what anybody had in mind when they first had the idea of communicating a fact about the world to another person. What you're both missing is how perverse the paradoxical cases are. As I've said elsewhere, the reaction of the average layperson will almost certainly be, "Oh, it's a trick." In my part of the world it might be worse: "I always figured math was bullshit -- guess I was right."

Both of these cases reveal the dangers of unrestrained generalization. We find an approach that works for some purpose, see that generalizing it allows us to use the same technique for many purposes, invent mathematics and conquer the world with Francis Bacon proudly leading the way.

But do paradoxes show that abstract thought is fundamentally flawed?

There are two cardinal sins from which all others spring: impatience and laziness. Because of impatience we were driven out of Paradise, because of laziness we cannot return. Perhaps, however, there is only one cardinal sin: impatience. Because of impatience we were driven out, because of impatience we cannot return. — Kafka

We rush ahead with our limited understanding of classes, or of truth, and prematurely declare a principle because the first phases of generalizing worked so well. The paradoxes are a warning not to be less ambitious but to be more careful. They are creatures of the work we've done so far -- this is why they have the peculiar form they do. If they go in a box with "incompleteness", "uncertainty", "undecidability" and all that jazz, the label on that box isn't "Not such a big shot now, are you human?" It's just "Hey, you're not done."

Ah, but a man's reach should exceed his grasp,
Or what's a heaven for? — Browning

You have not given any argument as to why it must be denied — Jeremiah

Let's talk about the Barber.

Suppose we have a town with three men: a barber (B), a philosopher (P) who doesn't shave himself, and a mathematician (M) who does.

Now define a set R as all and only men who shave all and only men who don't shave themselves.

1. M is never a member of R because he shaves a man who shaves himself.
2. P can't be a member either because he doesn't shave himself, so he'd have to shave himself to be a member, but he doesn't.
3. What about B? He would have to shave P and not M. No problem. If he shaves himself, he'd be out, like M, but if he doesn't, he'd be out like P. So B can't be a member no matter what he does.

So R = { }. No one shaves all and only men who do not shave themselves, therefore the barber does not shave all and only men who do not shave themselves. The three cases are exhaustive, in fact: no one can be a member of R whether they shave themselves or not.

Your presentation is to start by defining R as {B}, and then saying

The paradox is whether he belongs to the set of men who get shaved. — Jeremiah

But we've already seen that B cannot be a member of R, so the premise is just false.

Now what about Russell? In the analysis given above, R was not the Russell set, but the set of all Russell sets, and it has been shown to be empty. It does not contain any set that contains all and only sets that do not contain themselves, because there can be no such set.

Therefore if you present the paradox by beginning, "Let S be the set of all sets that do not contain themselves as members," then I will deny the premise. No set can be formed in this way, which is exactly Russell's point.

I can't ever know if there's some definition about which we disagree even if I ask, because to explain it just requires further definitions about which I will not know if we agree. — Pseudonym

Let's call this the Skeptical Argument. (See what I did there?)

The Skeptical Argument appears to demonstrate that agreement in definitions is impossible. (It might not be; the SA might only show that such agreement cannot be achieved in this way.) Or, rather, that whether we agree cannot be known, or cannot be reliably known.

Then if agreement in definitions is necessary for communication of the sort contemplated, we cannot reliably know that we are communicating. (Or whatever formula goes there, again with the proviso that there may be some other way of reliably knowing this, not pictured here.) Call this the Extended Skeptical Argument.

Suppose this is true. Then there's this question:

(1) Why would we think we're communicating when we're not?

This question has a mate. Suppose the ESA is wrong, that we do communicate and the SA shows that agreement in definitions is not necessary to communicate.

(2) Why would we think we need agreement in definitions to communicate?

That's the first round of preliminaries.

Second round is a little historical context. Folks who've run into this before:

• Grice admits ("Meaning Revisited") that his scheme might lead to an infinite regress, and that maybe we never "quite" mean anything "in the technical sense", but we get close and "deem" that success.
• Lewis ( Convention ) compared the communication-oriented view of language (Grice, late LW, et al.), which he tidies up with game theory, and the formal view of the structure of a possible language (Frege, Tarski, et al.), but then reaches the weird conclusion that no one ever quite speaks a language in the formal sense.

The issue here (Grice says this in so many words) is the role of the ideal. Some versions of pragmatism find a place here, in the endlessly approached but never quite reached truth. How do we come to think there's an ideal, if it's something we never reach, never experience? How and why do we use it to regulate our non-ideal communication and reasoning? I'll throw in here Dummett eventually defining assertion as saying something that aims at the truth. It's aspirational. (Going to leave aside any reference to Davidson for now.)

Preliminaries out of the way, there are a few things we can note.

The "deeming" Grice mentions is not "declare victory and depart the field", but must be more like this: declare victory but remain on the field, and if there are signs we haven't actually won, make renewed efforts to win. Rinse and repeat.

Lewis thinks we get as far as figuring out that a member of some equivalence class of possible languages is being spoken, and that this might be good enough. There's a process of narrowing down the range of possibilities.

Now look at our contemplated method of discussion: work out a tentative agreement on terms, and if trouble arises, work out a new tentative agreement. We can deem ourselves to be in agreement, but we remember that this was "only" a matter of deeming, so there's no tragedy if it turns out we weren't. It seems to me that the process of deeming is itself governed by rules and norms, while functioning as the source of the rules and norms that govern communication.

How this works out, I don't know, but this is how I see the issues here. There seems to be a pattern of relying on an ideal in a particular way; when an argument shows us the ideal and shows us we can't reach it, or can't reliably know we've reached it, we're prone to some variety of skepticism. But the ideal is here as an element of the systems (of communication, of reasoning, of knowledge acquisition) we use. It's something we use, and thus at once what we hold up as the external goal to aim for and a tool we made and use ourselves.

Agreed, except that the existence of such a set is a presupposition, and it is that presupposition that must be denied. (In this way it's analogous to being asked if the present king of France is bald, or if you've stopped beating your children.}

What's curious is that if you consider these two commands

(a) Shave all and only those who do not shave themselves; and

(b) Form a set by selecting as members all and only sets that do not contain themselves as members,

then many people will conclude it is impossible to obey (a), but are confused by (b) and think it should be possible.

It just means this is a task the barber cannot carry out. It is an invalid way of specifying a task.

For the record, the barber can't exist, the Russell set we (most of us) don't allow to exist, but the word "heterological" does certainly exist and can go to hell.

Ronald Bogue — StreetlightX

Took a class from him in college. I remember once, hanging around his office talking, I mentioned I had just started reading Deleuze's book on Foucault and he said immediatley -- you know, sort of involuntarily -- "That's a beautiful book!" Then he looked embarrassed. I think he wanted to be careful not to push me in any particular direction, you know?

Maybe restricted quantification is not ad hoc at all though. Maybe "Some Gs are F" is a better paradigm than "Some xs are F".

learn a truth — fishfry

The axiom schema of specification blocks Russell. Would I be right in thinking that one reason to be cool with that approach (the truth learned) is that we don't need unrestricted quantification?

I think that following a rule is nothing other than thinking that you are following a rule. It is to hold a principle in one's mind and adhere to it. — Metaphysician Undercover

Are you talking about consciously thinking about the rule?

When you first learned how to play chess, you had to do that for a while, but no one who's played for a while ever thinks about the rules while they play, do they?

It's funny how often we (forum denizens) end up having the same discussion spread across several different threads ...

What occurs to me is that in a given discussion, there's no immediate need to pursue definitions ad infinitum. You define until you reach agreement. If what you agreed on later raises issues, you define again.

A few different ways to look at this:
(a) there must be common ground to have a discussion at all;
(b) to explain your position to someone, you must put it in terms they understand;
(c) to convince someone of the <correctness, usefulness, whateverness> of your position, you must give them reasons and reasoning they'll accept.

We all know this stuff, and it gets mentioned now & then, but (as should be clear from other posts, other threads) I'm not sure principles such as these get their due. What justifies such principles? Does their justification have philosophical import? Are they "innocent" -- transparent? inert? -- or do they actually affect the philosophy we do?

(Quine wrote pretty regularly about issues surrounding definition. Interesting stuff.)

Here's a quickish first response, but figuring this out was the whole point of the thread! (It's not all perfectly clear to me -- just pursuing a hunch, as I've said.)

For an argument that there is such a layer, here's an intro to implicature.

A: Should we go for a hike later today?
B: It's supposed to rain.

The plain language of B's response is a non sequitur, but we know, or rather we presume, that it isn't. How does that work?

If A assumes that B is following the principle of cooperation, then A can work out that B probably means that if it's going to rain then we shouldn't go hiking and it's going to rain therefore no, we shouldn't go.

There's nothing here about emotional reaction. There's also nothing here about interpretation: "It's supposed to rain" means exactly what it appears to. It does not, for instance, mean "No" in this particular case, even though B means "No" by saying "It's supposed to rain."

There's some logic here, but "It's supposed to rain" does not on its own logically entail that we shouldn't go hiking, and that's why there's room to cancel the implicature: "It's supposed to rain -- but let's go anyway." But if B does not want his response to be taken as "No", then he has to cancel the implicature. If he just says "It's supposed to rain", it will be taken as "No". Logic alone does not get you here: you have first to assume that B's response, which appears to be a non sequitur, is not.

So this is one of those layers. There are principles, rules, conventions that we rely on in using language that are neither just logical, dealing only with the plain language of what is said, nor extra-linguistic like emotional reactions.

There are a couple of ways this shows up in doing philosophy. One is that philosophy, being an almost exclusively verbal activity, depends on such conventions of language use. We try to bring everything out into the open and rely on logic and plain language as much as possible, but there may be limits to that or there may be things we miss. (Will come back to this.) Another issue is that to evaluate a position, we often turn to verbal evidence of one sort or another -- "If you're right then it would make sense to say ..." Gotta be careful there, because our intuitions about "what it makes sense to say" are not based only in logic and plain language but also in these other conventions about how we use language.

Example of the latter. Austin claims that in normal circumstances it makes no sense to say either "He sat in the chair voluntarily" or "He sat in the chair involuntarily" -- only in special circumstances would we reach for those adverbs. He's sidling up to the issue of free will, but note this is also an implicit rebuke of the way philosophers use the law of the excluded middle. You might feel compelled to say that one of these sentences must be true and the other false. What's uncomfortable here is that either way you'll be opting for a sentence whose plain language you're cool with, but whose ordinary use carries baggage you don't want. You'll have to reject something.

Which gets us back to our other issue, I think. How do you justify your choice of what to reject? What kind of justification do you have to give? A question like "Have you stopped beating your significant other?" isn't a loaded question in the sense that it only pretends to offer a choice but only one answer is okay; it carries a presupposition. If you can figure out the presupposition -- easy to do in this case, but not always -- you can deny that, but that means refusing to answer simply "yes" or "no".

I think there's some other stuff going on in philosophical discussion too. I should answer your questions about my position, but it's easy to come up with a question a philosopher won't want to answer (because of what it presupposes, for instance). But I'm also supposed to convince you, by meeting some unspecified standard of yours, not mine, and that standard might not be simply logical but include, say, answering your questions. Much of this sort of discussion has to flicker between figuring out what the words themselves mean and entail, etc., and what you mean by saying them, so we bounce between layers of conventions a lot. Getting clearer about that was my goal in this thread.

Yes, it's not just a matter of emotional reactions, but I think that emotional reactions are of relevance to the topic. The emotional reactions are emotional reactions to something, so that something is also of relevance. What has been said is of relevance, as is how it has been or might be interpreted, and as is how it should be interpreted. — Sapientia

Here's how I understand your view of (philosophical) discussion:
1. There's the plain language of what people say. Logic has a place here, and truth.
2. There's how people react to what people say, and that might be colored by emotional reactions, misinterpretation, etc., none of which has anything to do with logic and truth.

The very model of philosophy for you is saying the emperor has no clothes, speaking the unfiltered truth and if people find that impolitic or impolite or if they misinterpret it, that's on them. Have I gotten this wrong?

In my view, this view is cripplingly simplistic. There are layers between (1) and (2), and everything I've written relies on that fact.

A now seems to have a choice between admitting that he might be anti-Semitic or admitting that he cannot claim to know he's not. He appears to be playing a losing game. — Srap Tasmaner

If losing is judged based on something as superficial as emotional reaction, then yes, he appears to be playing a losing game. He would be at risk of sending the wrong message.

But if admitting that he might be antisemitic is like admitting that he might be a Martian, then, in that sense, it's trivial. — Sapientia

"Sending the wrong message" is not just a matter of emotional reactions.

But that is not the only issue here. What is the status of Philip Roth's belief that he is not anti-Semitic? Is it right to call this a belief at all, something that could be veridical or not? Is it something he learned by gathering evidence and comparing hypotheses? What if his saying this is a direct report to be treated the way we would treat "I'm in pain." We're rightly not sure whether to call this knowledge at all, but people do resort to this sort of language because what else is there?

I would guess that a survey would show that the vast majority of people on this site, if asked, "Have you stopped beating your significant other?" would like to answer "C. I have not because I never started, and I would have to have started to stop." "No" doesn't logically entail that you are still beating your SO, but the implication is not just a matter of emotional reaction. (Don't forget that "No" could also be selected because it is true full-stop.)

The natural next step around here is usually to say something like, "The question 'Do I know that's a tree?' is wrongheaded." The problem with that is that unless you can reframe the issues in a really compelling way, you'll be taken simply as dodging the question because your position is untenable. As if you're trying to change the rules of chess because you're about to be checkmated.

What interests me is that the two choices presented don't come out of the same box at all.

Since a philosophical discussion is a conversation like any other, there are of course norms in play that aren't specifically philosophical. What I'm interested in are those cases where one can be mistaken for the other.

This is roughly the debate over ordinary language philosophy, or one version of it, or one perception of one version of it, etc. It goes like this:

A: In normal circumstances, it makes no sense to say 'I know that's a tree'.
B: In normal circumstances, it may be inappropriate to say 'I know that's a tree', but it may nevertheless be true.

This spirals off into a debate about meaning, because both sides know what they're about.

But watch how which side is which can flip:

A: I'm not going to debate this. I am not anti-Semitic.
B: Then you admit you might be.
A: No I don't.
B: But it can only make sense to say "Roth isn't anti-Semitic" if it also makes sense to say "Roth is anti-Semitic."

A now seems to have a choice between admitting that he might be anti-Semitic or admitting that he cannot claim to know he's not. He appears to be playing a losing game.

Unless all communication on an internet forum is conducted using video, a significant amount of information is lost in transmission. — Galuchat

This only follows if you do not adapt your verbal style to the medium you are using. The studies you cite in the previous paragraph only show that there are verbal shortcuts available when you can rely on other channels carrying the rest of the signal. (Trivial example for clarity: I can't point at things here as I might IRL so I have to substitute a description.)

What's more, I'd expect that a certain amount (I don't know how much) of the information I'm not getting by having a purely verbal exchange with you is information I'm not interested in when I'm on this forum. Your ideas are all I'm interested in. I'm not trying to get to know you as a person.

Think of it as a barrier to understanding, if you like. Until I know what's wrong with Zeno's argument, I don't really understand the physics.

Whenever @StreetlightX talks about creative concept construction, I always think about this good proof vs. bad proof thing. Good proofs are the ones that show you why the thing is true, and are crucial to the pedagogy of mathematics.

My hunch had been that the logical character of philosophical debate would provide a sort of natural camouflage for moves that are successful or unsuccessful on another plane, so to speak, the rules of conversation rather than logic per se, and that this camouflage might be so good that people on both sides might think they're winning or losing for logical reasons when it's really something a little different. (A really good read here is Sellars' article about Strawson, "On Presupposing".)

(Complication for me is that I'd like to understand what connects the two domains, and I suspect the laws of argument emerge from the laws of arguing -- but that's not what I'm after here.)

Imagine my surprise when some contributors don't even bother with a pretense of logical argument, but go straight for the psychodrama.

the recent Abel Prize winner won the prize: not just what he proved, but the way he proved stuff and the exciting conjectures his methodological development allowed, — fdrake

Something we haven't talked about -- and I'd really like to hear your thoughts on -- comes out in the article @StreetlightX linked: one reason the proof was ignored is because it was a bad proof. That is, everyone expected a proof that would show what the connection to convex polygons is (if I'm remembering the issues correctly), but it turns out there's a purely statistical proof that leaves those connections, which have emerged in the years of attempted proofs, entirely unexplained. The reception seems now to be, well it's nice to know for certain that it's true, but that's not really what we wanted.

A proof being good or bad depends on what you wanted out of it. I could see this proof being eventually recognized as good if it leads to some deep insight about statistics. (There really shouldn't be proofs that are bad in an absolute sense.) Such a deep insight might even eventually link back to the algebraic geometry it passed by.

(Sorry if I'm garbling the math -- these aren't areas I know at all.)

?

Things I may not have articulated clearly in the OP, if anyone cares:

• Playing a "winning game" (or a "losing game") means employing a strategy guaranteed to win (or to lose), no matter what your opponent does, or perhaps only if your opponent continues as they have.
  
  
• I'm trying to tease out a difference between conflict "on the merits" and conflict on another level I'm not sure how to describe -- argumentation? dialogue? exchange? Philosophers will tend, I think, to assume or pretend there is no such difference, but I think there might be. Maybe not.

I'm not sure exactly what Roth meant. Part of it might have been the "proving a negative" thing. Part of it was probably also that defending someone accused of anti-Semitism would open you to at least a suspicion of being likewise anti-Semitic, and that makes your defense suspect and self-serving.

My best guess is that Roth saw something that's more sinister, in a way. Mounting any defense is agreeing to debate the claim that Philip Roth is, among other things, anti-Semitic, to look at evidence for and against the proposition, etc., and that's agreeing to treat the claim as something that might be true, might be false. But no! There's no reason to allow this claim into the category of "might be true". So that's one sense in which defending would be losing.

I'm totally not following this. Could you take another run at it?

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?