One of the huge sources of map/territory relation issues is that we use systems that let us avoid the problem discussed here
Definitely worth spending time on.
vajrabum 21 minutes ago [-]
I would have thought that the proof shows this problem is unavoidable. How in your view do we avoid this problem?
jxmorris12 1 hours ago [-]
I'm currently a machine learning grad student taking a meta-complexity class and came across this blog post. I found the whole thing very interesting. In particular the idea that somethings are uncomputable seems fundamentally unaddressed in ML. We usually assume that (a) the entire universe is computable and (b) even stronger than that, the entire universe is _learnable_, so we can just approximate everything using almost any function as long as we use neural networks and backpropagation, and have enough data. Clearly there's more to the story here.
23 minutes ago [-]
Xcelerate 1 hours ago [-]
Another weird one related to Gödel’s theorems is Löb’s theorem: given a sound formal system F and a sentence s, if F proves that “if s is provable in F, then s,” then F also proves s. That is:
F ⊢ (Prov_F(“s”) → s) → s
Which is strange because you might think that proving “if s is provable, then s” would be possible regardless of whether s is actually provable. But Löb’s theorem shows that such self-referential statements can only be proven when s itself is already provable.
mcphage 2 hours ago [-]
I don't think you need anything fancy to tackle the "surprise examination" or "unexpected hanging" paradox. This is my take on it, at least:
> The teacher says one day he'll give a quiz and it will be a surprise. So the kids think "well, it can't be on the last day then—we'd know it was coming." And then they think "well, so it can't be on the day before the last day, either!—we'd know it was coming." And so on... and they convince themselves it can't happen at all.
> But then the teacher gives it the very next day, and they're completely surprised.
The students convince themselves that it can't happen at all... and that's well and good, but once they admit that as an option, they have to include that in their argument—and if they do so, their entire argument falls apart immediate.
Consider the first time through: "It can't be on the last day, because we'd know it was coming, and so couldn't be a surprise." Fine.
Now compare the second time through: "If we get to the last day, then either it will be on that day, or it won't happen at all. We don't know which, so if it did happen on that day, it would count as a surprise." Now you can't exclude any day, the whole structure of the argument fell apart.
Basically, they start with a bunch of premises, arrive at a contradiction, and conclude some new possibility. But if you stop there, you just end up with a contradiction and can't conclude anything.
So you need to restart your argument, with your new possibility as one of the premises. And now you don't get to a contradiction at all.
jerf 1 hours ago [-]
I can't help but think the "surprise examination paradox" rests too much in English equivocation for it to be a properly logical paradox. In particular, the fact that "surprise" changes over time, and the fact that if I've logically deduced that it is "impossible" for the test to occur on the last day then it is ipso facto a surprise if it happens then.
Sit down and make the argument really rigorous as to the definition of "surprise" and the fuzz disappears. You can get several different results from doing so, and that's really another way of saying the original problem is inadequately specified and not really a logical conundrum. As "logical conundrums" go, equivocation is endlessly fascinating to humans, it seems, but any conundrum that can be solved merely by being more careful, up to merely a normal level of mathematical rigor, isn't logically interesting.
ogogmad 1 minutes ago [-]
You did not understand the paradox. If you're not willing to engage in good faith, you should probably not bloviate. The word "surprise" means that the prisoner won't know his date of execution until he is told. You're pretty bad at these thought experiments if you need to be told this, and should sit them out.
astrobe_ 24 minutes ago [-]
It is like the infamous 0.999999... = 1. That one uses sloppy notation (what is "..."?) to make students think and talk about math.
mcphage 20 minutes ago [-]
I'm not sure the "..." is sloppy notation—it can be made rigid pretty easily. The surprise is that students' expectations that if two decimal expressions are distinct, that the real number they correspond to must be distinct also. (Even there, students have already gotten used to trailing zeros being irrelevant).
bongodongobob 37 minutes ago [-]
I agree. The premise itself is spurious. I've never liked this paradox because I don't think it makes sense from the get go.
griffzhowl 8 minutes ago [-]
But it's stipulated that the test will happen on one of the days - it's not a possibility that it won't happen at all, hence the paradox.
One resolution is that what the teacher stipulates is impossible. It should really be
"You'll have a test within the next x days but won't know which day it'll be on (unless it's the last day)"
robinhouston 48 minutes ago [-]
I would encourage anyone who's intrigued by this paradox to read Timothy Chow's comprehensive paper on the subject (https://arxiv.org/abs/math/9903160).
In particular, he discusses what he calls the meta-paradox:
> The meta-paradox consists of two seemingly incompatible facts. The first is that the surprise exam paradox seems easy to resolve. Those seeing it for the first time typically have the instinctive reaction that the flaw in the students’ reasoning is obvious. Furthermore, most readers who have tried to think it through have had little difficulty resolving it to their own satisfaction.
> The second (astonishing) fact is that to date nearly a hundred papers on the paradox have been published, and still no consensus on its correct resolution has been reached. The paradox has even been called a “significant problem” for philosophy [30, chapter 7, section VII]. How can this be? Can such a ridiculous argument really be a major unsolved mystery? If not, why does paper after paper begin by brusquely dismissing all previous work and claiming that it alone presents the long-awaited simple solution that lays the paradox to rest once and for all?
> Some other paradoxes suffer from a similar meta-paradox, but the problem is especially acute in the case of the surprise examination paradox. For most other trivial-sounding paradoxes there is broad consensus on the proper resolution, whereas for the surprise exam paradox there is not even agreement on its proper formulation. Since one’s view of the meta-paradox influences the way one views the paradox itself, I must try to clear up the former before discussing the latter.
> In my view, most of the confusion has been caused by authors who have plunged into the process of “resolving” the paradox without first having a clear idea of what it means to “resolve” a paradox. The goal is poorly understood, so controversy over whether the goal has been attained is inevitable. Let me now suggest a way of thinking about the process of “resolving a paradox” that I believe dispels the meta-paradox.
mcphage 24 minutes ago [-]
That sounds interesting—thanks for sharing, I'll check it out.
3 hours ago [-]
Rendered at 18:05:22 GMT+0000 (Coordinated Universal Time) with Vercel.
Definitely worth spending time on.
F ⊢ (Prov_F(“s”) → s) → s
Which is strange because you might think that proving “if s is provable, then s” would be possible regardless of whether s is actually provable. But Löb’s theorem shows that such self-referential statements can only be proven when s itself is already provable.
> The teacher says one day he'll give a quiz and it will be a surprise. So the kids think "well, it can't be on the last day then—we'd know it was coming." And then they think "well, so it can't be on the day before the last day, either!—we'd know it was coming." And so on... and they convince themselves it can't happen at all.
> But then the teacher gives it the very next day, and they're completely surprised.
The students convince themselves that it can't happen at all... and that's well and good, but once they admit that as an option, they have to include that in their argument—and if they do so, their entire argument falls apart immediate.
Consider the first time through: "It can't be on the last day, because we'd know it was coming, and so couldn't be a surprise." Fine.
Now compare the second time through: "If we get to the last day, then either it will be on that day, or it won't happen at all. We don't know which, so if it did happen on that day, it would count as a surprise." Now you can't exclude any day, the whole structure of the argument fell apart.
Basically, they start with a bunch of premises, arrive at a contradiction, and conclude some new possibility. But if you stop there, you just end up with a contradiction and can't conclude anything.
So you need to restart your argument, with your new possibility as one of the premises. And now you don't get to a contradiction at all.
Sit down and make the argument really rigorous as to the definition of "surprise" and the fuzz disappears. You can get several different results from doing so, and that's really another way of saying the original problem is inadequately specified and not really a logical conundrum. As "logical conundrums" go, equivocation is endlessly fascinating to humans, it seems, but any conundrum that can be solved merely by being more careful, up to merely a normal level of mathematical rigor, isn't logically interesting.
One resolution is that what the teacher stipulates is impossible. It should really be
"You'll have a test within the next x days but won't know which day it'll be on (unless it's the last day)"
In particular, he discusses what he calls the meta-paradox:
> The meta-paradox consists of two seemingly incompatible facts. The first is that the surprise exam paradox seems easy to resolve. Those seeing it for the first time typically have the instinctive reaction that the flaw in the students’ reasoning is obvious. Furthermore, most readers who have tried to think it through have had little difficulty resolving it to their own satisfaction.
> The second (astonishing) fact is that to date nearly a hundred papers on the paradox have been published, and still no consensus on its correct resolution has been reached. The paradox has even been called a “significant problem” for philosophy [30, chapter 7, section VII]. How can this be? Can such a ridiculous argument really be a major unsolved mystery? If not, why does paper after paper begin by brusquely dismissing all previous work and claiming that it alone presents the long-awaited simple solution that lays the paradox to rest once and for all?
> Some other paradoxes suffer from a similar meta-paradox, but the problem is especially acute in the case of the surprise examination paradox. For most other trivial-sounding paradoxes there is broad consensus on the proper resolution, whereas for the surprise exam paradox there is not even agreement on its proper formulation. Since one’s view of the meta-paradox influences the way one views the paradox itself, I must try to clear up the former before discussing the latter.
> In my view, most of the confusion has been caused by authors who have plunged into the process of “resolving” the paradox without first having a clear idea of what it means to “resolve” a paradox. The goal is poorly understood, so controversy over whether the goal has been attained is inevitable. Let me now suggest a way of thinking about the process of “resolving a paradox” that I believe dispels the meta-paradox.