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"?

Yeah I get that. And my point was that this sort of post hoc fails if I don't act on my preference. You'd have to say my acquiring the preference is what I just do, and that belief figures in my post hoc justification for the new preference I have.

We'd have to work on that. Even if we just dropped all talk of belief or rationality or community norms, what would turn a desire to find my keys into a desire to look for them in the kitchen? Maybe there are substitutes for belief, but something has to get you from one to the other.

Edit: autocorrect

There's surely a difference of some kind. We can say there's A's and there's B's, or we can say there's two kinds of A's. I don't suppose it matters unless we want to say "All A's are F"; then we'd want to be sure we don't mean "All type 1 A's are F."

In recent posts here, I've been kicking the can of outward behavior into another zip code.

It's rather that Pat searches the kitchen, and justifies his behaviour post-hoc by claiming to believe that the keys are there.

The belief is irrelevant. Pat does what Pat does. — Banno

But that can't be right, because of the knife-wielding psycho in the kitchen. I can form a preference to look there even if it's overridden by my preference to go on breathing. Rationality does seem to have a foothold here: given some preferences and beliefs, you should also have this preference. Maybe you don't act on it for whatever reason, or for no reason. Different issue.

All of that assumes by "behavior" you mean outward, publicly observable actions. Are you throwing in what I think as behavior?

I still feel pretty good about the preference version, because I get to say "If you don't want to look for your keys in the kitchen, either you don't think they're there or you don't want to find them (or you don't reason like the rest of us)." That feels solid to me.

So it would be weird if I felt a deep sense of kinship with this man ...

Anywho, this sort of gamesmanship is practically built into the multiple choice test and it would be easy to (ahem) multiply examples.

None of these strategies look like a counter to the random chooser though. (The "all B" sequence messes with our faulty intuitions.)

All of which points to something weird in Jeremiah's puzzle.

My values are halves of the 1 in 3, and 1 in 4 totals, duh. The only reason to look at it by cases as I was -- and I should have done the other two -- is to see if there's any point in a test designer choosing one of these options, and I don't see any such reason, or any reason to follow some other strategy like randomly duplicating, etc.

Dude, I didn't even think of that!

Not sure if I did the simulation right, and I'm at work now. :-(

(I had right answer being chosen from {b, c} and two students: one chooses from {a, b, c, a} and one from {a, b, c}. I don't understand the result though, so must've muffed it.)

Also, I got something wrong about the duplicated correct answer: that will help the student who doesn't read the questions even more than it helps the one who does. So whether the duplication helps the reader or non-reader more flips depending on whether it's a right duplicated or a wrong. Cool.

In any case, it still seems that duplication can only help random choosing, even if it helps in a differentiated way.

What I wanted to get back to eventually was how this puzzle conflicts with our expectations about how tests work...

(Pointless anecdote: I had a professor in college who didn't give multiple choice questions because he said they helped poor students and hurt good ones.)

Why don't multiple choice questions ever have repeated answers? If you were taking a math test and got to a question that had a repeated answer, you'd walk up to the teacher's desk and show him, and he'd make an announcement like, "Sorry, folks, Question 17 has a typo. Answer E should be '33%'."

Presented with a multiple-choice question, there are several methods you can use to choose an answer. Best method is knowing which answer is correct -- a word which here means "will be graded by the test preparer as correct". If you don't know, or don't think you know, you can go with your gut or choose randomly, and you can also eliminate answers you know are wrong before doing either of those to improve your (subjective) chances. There are some other methods, but the main point is that random choice is a fall-back when no better option is available.

Multiple choice tests are designed to test knowledge or reasoning, not luck. Tests are often designed with answers that will appear tempting if your reasoning is faulty, but not to foil a test taker who is lucky. That test takers have the option of making random choices is interesting, but test design needn't take this into account, and it's not perfectly clear that it can.

You could in fact help the random chooser by repeating correct answers, and people do this sort of thing for comic effect. What are the three most important principles of retail? Location, location, location. There are only two rules for working here: 1 is "Do what I tell you" and 2 is "Do what I tell you".

So here's the answer to the question at the top of this post: multiple choice questions never (deliberately) have repeated answers because a duplicated incorrect answer will not foil the random chooser, while a duplicated correct answer will help him. And the test preparer has no reason to help the random chooser.

The question that remains is whether a duplicated incorrect answer also helps the random chooser by changing his chance of getting the right answer from, say, 1 in 4 to 1 in 3. That would seem to depend entirely on the test taker -- that is, on whether he reads the question at all or just bubbles in something on the answer sheet.

And that opens up the possibility of a gap -- noted by several people in this thread -- between the chance of my picking the answer that is correct, on the one hand, and the chance of the answer I pick being correct. In the usual case, with no duplicated answers, these are identical, by design. But if there are duplicated wrong answers, will the random chooser who reads the questions out-perform the random chooser who doesn't?

If I did the simulation right, I get 16.6162 for the test taker who reads the questions, and 12.4914 for the test taker who doesn't.

Bonus thought about constraints: there's prescription and proscription. Foucault talked about this with different styles of morality: you can have the "default" be that everything is allowed and proscribe specific behaviors (don't wed or kill kin, etc.); or you can prescribe The One True Way to live and count every deviation as wrong. Negative versus positive constraints -- don't do that vs do this.

The funny thing about words and concepts is that even though we seem bound to think of them (and teach them) prescriptively -- here is how we use this word, this is what this concept applies to -- we're always ready to rewrite the rules so long as the rule-breaker makes sense. And that's curious, at least because the prescriptive view suggests that being successful in this way isn't even possible. And indeed this is what we will do, add a click. (Metaphors, for instance, are just recently added literal uses.) We always come back to rules, even though the rules are ever changing.

Maybe it's simplest to say that our rules are always open-ended: so far we have found the following uses for this word or concept, there will probably be others. The rules would then in essence all be permissives -- you may at least use this word or this concept in the following way -- which is not how we generally think of them.