Of course, mathematics is full of “negatives” that can be proved in such a way, from the very easy (“No prime number ends in the digit 0”) to the very difficult (“No positive integers satisfy a[sup]n[/sup] + b[sup]n[/sup] = c[sup]n[/sup] where n is 3 or greater”).
If i had a box the size of a standard penguin and observe none inside, is it still possible that the box contains a penguin?
I call your Wittgenstein and raise you a Munroe.
Quite.
Of course the rest of the point is that mathematics is full of reliable generating rules where given inputs always produce given outputs and where logical deductions are fully reliable.
The material world, and especially the poorly understood world of human beliefs and behaviors has essentially no reliable generating rules and nothing where logical syllogisms apply with that desirable crystalline clarity. Witness the (admittedly silly) side kerfuffle just now about what is and is not a penguin. We would not have nearly the same opportunity for gamesmanship around what is or is not an integer.
Q. How does a mathematician capture all of the penguins in the world?
A. Build a fence around himself and define his side of the fence as “outside”.
Don’t get me started on auks. Found one in the oatmeal bag, I did.
This immediately reminded me of the “Black Swan Theory”
"Juvenal’s phrase was a common expression in 16th century London as a statement of impossibility. The London expression derives from the Old World presumption that all swans must be white because all historical records of swans reported that they had white feathers.[4] In that context, a black swan was impossible or at least nonexistent.
However, in 1697, Dutch explorers led by Willem de Vlamingh became the first Europeans to see black swans, in Western Australia.[5] The term subsequently metamorphosed to connote the idea that a perceived impossibility might later be disproven. Taleb notes that in the 19th century, John Stuart Mill used the black swan logical fallacy as a new term to identify falsification.[6]"
Also, for proving a negative, doesn’t there need to be a (possibly unstated) positive statement to contradict?
As in
I see no standard issue elephants in the living room
(if there was a standard issue elephant in the living room, I’d see it)
Therefore, there is no standard issue elephant in the living room
The opening post and most of the responses are off the mark. First off, it is NOT a rule of logic that “you can’t prove a negative.”
Second, “proving a negative” is not remotely the same as proving a statement is true, which contains what people think of as negative values in it.
In other words,even if there WERE a logical rule stating that you can’t prove a negative, the fact that you were talking about negative subjects, wouldn’t make all statements about those negative statements invalid.
This is a confusion of what IS true, which is that you can’t prove the NON-existence of most things, without first accepting parameters which would logically make them impossible.
Thus you can’t prove the NON-EXISTENCE of a god which is defined as Supernatural. This isn’t because the god does exist, it is because the parameters of the god exclude any and all logical tools.
Lots of people are extremely sloppy about real logic, and try to take shortcuts. Hence they get partway through being told that they can’t disprove the existence of certain things, they latch on to “can’t,” and a couple of other words, and run off joyously celebrating their delusion that “nothing negative can be proven.”
Again. There are TWO SEPARATE THINGS being bantered about here, and they aren't related, as many people seem to think.I don’t think anyone here has said it was.
:smack: I should have known better than to challenge a guy named Riemann. Smarty pants.
Of course I’m not taking you up on that. You strike me as just the kind of person who would hide their pet penguin in the bathroom, just to win a bet.
OK. That was* really* funny.
If I bring my stuffed Opus, can I win?
The point is that for a positive assertion, you never have a need to get to the silliness. “I have a penguin. And here it is.” The end.
I disagree. I think we all have a perfectly good grasp of what “penguin” reasonably meant in the context of the OP, and that (unless otherwise stated) it excluded stuffed penguins, penguins made of lego, penguins on TV, penguins that reside in books or in our imagination, penguins that are invisible, penguins that can teleport from one place in my living room to another without me noticing.
If we need to hire a team of lawyers and (God forbid) philosophers to pin down the definition to arcane precision in the case of the negative hypothesis, why not in the case of the positive hypothesis?
As I said above, if we’re going to do the silliness: the penguin that you claim to see might be an optical illusion; you might be drunk; it might be a meerkat in fancy dress. I can list 5,000 equally ridiculous and unreasonable exceptions like this, and demand that you address every single one before I accept that what you saw is actually a penguin.
I would like to point out that my particular penguin was neither teleporting, imaginary, fictional, on TV, or anything of the sort. It was a perfectly average tuxedoed seafowl, which was simply positioning itself, quietly and without Riemann’s notice, under the table. This was happening while **Riemann **was looking behind the sofa. To be clear, the penguin did not teleport from behind the sofa. It was hiding, for instance, behind a curtain. The purpose of the sofa in this context was simply to obscure Riemann’s line of sight, keeping him from noticing the movement of the bird, as it relocated to a place where **Riemann **had already checked earlier. Surely, nothing about this is beyond the capacity of a motivated member of the species honkus magnificentus. Everyone on board with that? Great.
Let’s continue the hypothetical from this point, and say that **Riemann **now concludes his search by also checking behind the aforementioned curtain, where the penguin no longer is. We can now picture the following exchange:
Riemann: I have performed an exhaustive search, to my satisfaction, as far as my logistical capacity allows. Ain’t no penguin in my living room, bro.
Me: Dude, there’s one right there. Under the table.
**Riemann **is free, at this point, to suggest that the penguin I am observing is an optical illusion, or a dressed up suricata suricatta. But if so, I’m not the one adding on unreasonable complications.
And assuming a well-defined and commonly understood penguin, there does seem to me to be a difference between Riemann’s proof that there is no penguin, and my following proof that there actually is one. All I need is one penguin, and, as **Smeghead **put it, the end. For Riemann, however, a limited search, although exhaustive, doesn’t seem to be enough. He needs an infinity of non-penguin.
It should also be pointed out that this principle is closely related to Occam’s Razor (the real Occam’s Razor, not “the simplest explanation is usually the right one”), and general skepticism.
We assume the non-existence of any entity until we have good reason (usually empirical evidence) that it exists. Otherwise we’d have our hands full trying to prove the non-existence of an infinity of possible entities.
Which might be Schrodinger’s penguin, proving and disproving the positive or negative at the same time, except, there is no room left in the box, for the ambiguity has put on weight.
Not Kreskin-[Criswell](That’s the negative statement. You can’t prove it didn’t happen because you can’t come up with positive evidence to show it didn’t happen. Any evidence you can give is to say, "It didn’t happen in examples A, B, C, D . . . " But the retort is that “you didn’t show it couldn’t have happened somewhere else.” There is an infinite number of events and you’ll never be able to disprove all of them.).
edited to add: But of course I cannot prove that Kreskin never said it. 
But any such “prove-ables” require mathematical theories that assert properties of the items under consideration. I’m not aware of any theory that states anything about intermediate digits of prime numbers… the whole point of the assertion.
So I guess the corollary is that you cannot prove a negative if the only proof is by inspection and the set is too large (too complex?) for inspection. Which I assume is the essential meaning of the saying “you can’t prove a negative”.
Of course, it is possible that the penguin just happens to be moving to a different spot every time Riemann looks where it used to be. But there are only two chances of that happening repeatedly within a closed room: fat and slim. Yet because of this minuscule possibility of always and forever looking in the place where it happens to be not, you impose upon Riemann a literally infinite burden of proof. Your own words: “infinity of non-penguin”.
Well, we could just as easily apply this brazen burden of proof to your own observational skills.
When you look at your “one penguin”, you might be mistaking a penguin toy or robot. It might be a penguin picture or mobile, moving in the wind, that gives an illusion of living penguin. You might be having penguin-based delusions. You might be looking at a penguin hologram. There might be a strange system of mirrors that makes you think a penguin is in this room, when in fact it is in the next room. Or you might have just checked the two rooms, found a penguin in one and not the other, and then later mis-remembered which room had the penguin in it when you last looked. “There is a penguin in this room” is a statement based on personal observation, and human observation can be entirely mistaken. Regardless of whatever level of confidence you personally have in your ability to truthfully assert the existence of one penguin, based on your purported observation of such, there is no reason for anyone else to believe you… if we were to apply the same burden of proof to your observation as you would want to apply to Riemann.
The infinite regress cuts both ways.
There is, quite literally, an infinite number of scenarios we could create that might have inclined you to think you saw a penguin when, in fact, there was no penguin present. Especially if there were some tricksome person around who very much wanted to deceive you. The list I gave above was a pathetically small and finite subset of the infinite reasons that your penguin confirmation skills might have failed.
This is exactly the same point made earlier:
If we’re going to impose an infinite burden of proof in one direction, then that opens up an infinite burden of proof the other way, as well. In case you’re not aware: there are ways to search rooms where people are very careful about their line of sight not being blocked by the sofa. These ways are not perfectly secure, no. But then again, nothing is perfectly secure, not even the physical act of observing a penguin, because the observation itself might be false.