Why do people keep saying this?

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Kurushimi
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Why do people keep saying this?

Postby Kurushimi » Sat Dec 12, 2009 6:05 pm UTC

I've heard this time and time again "It's impossible to prove a negative." My understanding of logic makes me think of it like this. If we assume that A is true, and we reach a contradiction, we have just proved A is false. That is, we have proved not A. Isn't that a negative?

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Re: Why do people keep saying this?

Postby achan1058 » Sat Dec 12, 2009 6:23 pm UTC

I am assuming you are talking about something that appeared in the Serious Business/Science forum recently? While it is possible to prove a negative in mathematics, how can you do it with science, when your "proof" needs to observe every possible instance?

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Re: Why do people keep saying this?

Postby sje46 » Sat Dec 12, 2009 6:42 pm UTC

IF I told you there is a hippo in my room, you would probably think I'm lying. Because how would a hippo be in some New Hampshirite kid's room, and why is he just calmy sitting on the computer? It seems pretty illogical, but it's still possible.

So if you took an airplane over to where I live, found my dorm, entered my room, and failed to see a hippo, many things may have happened. Perhaps you got the address wrong. Perhaps the hippo is invisible. Perhaps you have selective blindness, and the only thing you can't see is the hippo. Maybe the hippo is three molecules tall. Maybe the light is reflecting off everything in the room to your eye, except the hippo, whose reflected light beams always just barely miss your pupils. Maybe you actually just dreamed about visiting me. Maybe you went into the room of another kid called Sean with the name sje46 on the xkcd fora from the state of New Hampshire with the same post count and sig as me, but this one doesn't have a hippo in his room, and I live in a room with the same number down the hall (because the 6 turned upside down to a 9), with a hippo.

All of these are valid possibilities, although unlikely.
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Re: Why do people keep saying this?

Postby Ended » Sat Dec 12, 2009 6:58 pm UTC

sje46 wrote:IF I told you there is a hippo in my room, you would probably think I'm lying. Because how would a hippo be in some New Hampshirite kid's room, and why is he just calmy sitting on the computer? It seems pretty illogical, but it's still possible.

So if you took an airplane over to where I live, found my dorm, entered my room, and failed to see a hippo, many things may have happened. Perhaps you got the address wrong. Perhaps the hippo is invisible. Perhaps you have selective blindness, and the only thing you can't see is the hippo. Maybe the hippo is three molecules tall. Maybe the light is reflecting off everything in the room to your eye, except the hippo, whose reflected light beams always just barely miss your pupils. Maybe you actually just dreamed about visiting me. Maybe you went into the room of another kid called Sean with the name sje46 on the xkcd fora from the state of New Hampshire with the same post count and sig as me, but this one doesn't have a hippo in his room, and I live in a room with the same number down the hall (because the 6 turned upside down to a 9), with a hippo.

All of these are valid possibilities, although unlikely.

I don't think that's a great example, since it would seem to suggest that you can't prove anything - e.g., I can't prove that there is a hippo in your room, because maybe I'm just seeing things, etc. Of course, 'proof' is really the wrong word when dealing with observations in the physical world, but I don't think that's what "you can't prove a negative" refers to.

The kind of example I'm thinking of is the difference between the two statements:

1) There are no pink unicorns in the universe
2) There is a pink unicorn in the universe

Clearly 1) is impossible to 'prove' (you can't search the whole universe) but 2) is trivial to 'prove' (just find a single pink unicorn).
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Re: Why do people keep saying this?

Postby sje46 » Sat Dec 12, 2009 7:05 pm UTC

I don't think that's a great example, since it would seem to suggest that you can't prove anything - e.g., I can't prove that there is a hippo in your room, because maybe I'm just seeing things, etc. Of course, 'proof' is really the wrong word when dealing with observations in the physical world, but I don't think that's what "you can't prove a negative" refers to.
Nah, man, it's a wonderful example, because it provesshows that you really can't prove anything based off a posteri data. You can only prove or disprove a priori, logical type thingies.
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Re: Why do people keep saying this?

Postby Ended » Sat Dec 12, 2009 7:13 pm UTC

True. Also hippos are pretty cool.
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Re: Why do people keep saying this?

Postby t0rajir0u » Sat Dec 12, 2009 7:38 pm UTC

Kurushimi wrote:I've heard this time and time again "It's impossible to prove a negative."

Could you provide some context? This is a very ambiguous phrase.

What it sounds like to me is that somebody's been arguing about the semantics of the word "proof." I think we can all agree that the word "proof" means something different in mathematics than it does in science, so was this in a mathematical or a scientific context?

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Re: Why do people keep saying this?

Postby mouseposture » Sat Dec 12, 2009 11:09 pm UTC

What it sounds like to me is someone's parroting a theory of inductive reasoning associated with Karl Popper, the Vienna Circle, and logical positivism. It ceased being respectable among philosophers in 1951 (Quine's "Two dogmas of empiricism") and among historians of science in 1962 (Kuhn's _The Structure of Scientific Revolutions_) The latter really should be required reading for any graduate student in the physical sciences. (I feel strongly about that, but my opinion seems not to be widely shared.)
Anyway, if I'm right about the sort of proof being discussed here, it's *inductive* reasoning, so this should really be in Science not Math.

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Re: Why do people keep saying this?

Postby greengiant » Sat Dec 12, 2009 11:12 pm UTC

The kind of example I'm thinking of is the difference between the two statements:

1) There are no pink unicorns in the universe
2) There is a pink unicorn in the universe


Surely the first one's easier to prove.

Here's my proof. According to my research http://dictionary.cambridge.org/define.asp?key=86577&dict=CALD&topic=legendary-and-mythological-characters a unicorn (this should be 'an unicorn' I guess but it sounds funny) is by definition white. So, assuming we accept that being white and being pink are mutually exclusive states, it follows that there are no pink unicorns.

Or even more directly, that definition says unicorns are imaginary. Therefore there are no unicorns in the universe. Therefore, no pink unicorns.

For my next trick I will attempt to prove that all bachelors are unmarried men.

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Re: Why do people keep saying this?

Postby Ended » Sat Dec 12, 2009 11:20 pm UTC

Spoilsport :) Fine, substitute 'extraterrestrial life' for 'pink unicorns'.
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Re: Why do people keep saying this?

Postby skeptical scientist » Sat Dec 12, 2009 11:40 pm UTC

It is of course possible to prove a negative. However, there are certain types of claims which are extraordinarily difficult or impossible to disprove, such as "there is a teapot in a synchronous orbit around neptune" or "there's an invisible incorporeal unicorn living in my backyard." The statement "it is impossible to prove a negative" is made to state that the burden of proof is on the person making such claims, not on the person trying to rebut them, and in the absence of evidence for or against, we our default state should be disbelieving a claim that some object or phenomenon exists (e.g. spontaneous human combustion, bigfoot, egg-laying mammals outside Australia/New Guinea/captivity, etc.) Of course we can can and should modify our beliefs when evidence is found.
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Re: Why do people keep saying this?

Postby greengiant » Sat Dec 12, 2009 11:48 pm UTC

Sorry, just couldn't resist.

More seriously, I'm not sure that there is a fundamental difference between proving

1) There are no pink unicorns in the universe, and
2) There is a pink unicorn in the universe

Assuming we accept that we're capable of recognising a pink unicorn, etc. (i.e. ignore Skepticism for the time being), the only difference between proving the two is a practical one. Sure, I might not have the necessary sensory equipment to determine that there are no pink unicorns in the universe, but I can't see any theoretical reason why it couldn't be checked. As long as we agree that I can tell that there is no pink unicorn in this room (which seems pretty reasonable if I'm able to tell when there is one) then it's just a case or covering more ground in my search.

Suppose some alien race has filled the universe with gazillions of pink-unicorn-detectors that scan for pink unicorns then report back their findings. Assuming their sensors are good enough to recognise a pink unicorn and they place them densely enough, I think they could 'prove' number 1 just as convincingly as anyone might 'prove' number 2.

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Re: Why do people keep saying this?

Postby Token » Sat Dec 12, 2009 11:57 pm UTC

skeptical scientist wrote:It is of course possible to prove a negative. However, there are certain types of claims which are extraordinarily difficult or impossible to disprove, such as "there is a teapot in a synchronous orbit around neptune" or "there's an invisible incorporeal unicorn living in my backyard." The statement "it is impossible to prove a negative" is made to state that the burden of proof is on the person making such claims, not on the person trying to rebut them, and in the absence of evidence for or against, we our default state should be disbelieving a claim that some object or phenomenon exists (e.g. spontaneous human combustion, bigfoot, egg-laying mammals outside Australia/New Guinea/captivity, etc.) Of course we can can and should modify our beliefs when evidence is found.

Putting that recent read-through of Less Wrong to good use? :P
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Re: Why do people keep saying this?

Postby skeptical scientist » Sun Dec 13, 2009 12:16 am UTC

Well, apart from my sig, I'm not sure what, if any, part of that post came from/was influenced by Less Wrong.
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Re: Why do people keep saying this?

Postby Kurushimi » Sun Dec 13, 2009 1:06 am UTC

achan1058 wrote:I am assuming you are talking about something that appeared in the Serious Business/Science forum recently? While it is possible to prove a negative in mathematics, how can you do it with science, when your "proof" needs to observe every possible instance?


I actually think this explained it pretty nicely to me. I wasn't thinking of it from an observational perspective. In that case, it would be physically impossible to prove some negatives. (Obviously it wouldn't be impossible to prove something like, "There is no hippo in my room". You can inspect your room yourself and be sure of that).

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Re: Why do people keep saying this?

Postby Twistar » Sun Dec 13, 2009 6:13 am UTC

mouseposture wrote:What it sounds like to me is someone's parroting a theory of inductive reasoning associated with Karl Popper, the Vienna Circle, and logical positivism. It ceased being respectable among philosophers in 1951 (Quine's "Two dogmas of empiricism") and among historians of science in 1962 (Kuhn's _The Structure of Scientific Revolutions_) The latter really should be required reading for any graduate student in the physical sciences. (I feel strongly about that, but my opinion seems not to be widely shared.)
Anyway, if I'm right about the sort of proof being discussed here, it's *inductive* reasoning, so this should really be in Science not Math.

Just had a test over these guys. By the way I'm going to talk in a this-should-probably-be-in-the-science-forum way right now. Yeah, on this note here, The point is that science is illogical, maybe alogical is a better wordl. Every hypothesis has test implications right? We run experiments to see if those test implications come to bear. So lets say I have some hypothesis:
All swans are white
The test implication would be I see whiteness when I look at a swan.
So I go look at 10 million swans and they are all white.
My hypothesis must be correct.
No, I have just commited the fallacy of affirming the consequent. This is the problem of induction. This says that we CANNOT CONFIRM ANY HYPOTHESIS on empirical grounds such as these. The logic does not support it. If I go to Australia I will find black swans.
So now we get into Falsificationism by Popper. Popper would say, oh you found a black swan now you have to trash your theory and get a new one.
Wrong again. Scientists don't do this and it isn't clear that science would work if they did. Quine pointed out that instead of throwing away the hypothesis we can say we made some wrong assumption but all swans are still white. Maybe we assumed that the swans weren't painted. What this says is that WE CANNOT FALSIFY HYPOTHESES on these empirical grounds.
So now when we run a test we don't know what to do if it went positive or negative. At least using logic.
And this is all entirely accepted stuff in the philosophy of science. Logic doesn't tell scientists what to do. We cannot compare our theories directly to nature. This is where Kuhn takes off and I believe it is safe to say this stuff is where we get the beginings of post-modernism?

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Re: Why do people keep saying this?

Postby Jorb » Sun Dec 13, 2009 9:17 am UTC

greengiant wrote:this should be 'an unicorn' I guess but it sounds funny

Nope, a unicorn, an unseen hippo, a hippo, an hour (couldn't think of a relevant word). See the pattern? Vowel sound gets an an.
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Re: Why do people keep saying this?

Postby the tree » Sun Dec 13, 2009 11:01 am UTC

There are all types of sort-of ways around the problem of induction. I was happy throwing Occam's Razor at it until recently, now I'm not happy with Occam's Razor at all (learning philosophy of science screws you up a bit), but that's where modern philosophers step in like Popper with falsification and Kuhn with protecting paradigms (then urm.. Lakatos? with something that was better than Kuhn iirc but a little confusing).

When it comes to not being able to prove a negative though, a real bitch of a problem is the Raven's Paraox - that since any statement has a contrapositive - everything is a negative.

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Re: Why do people keep saying this?

Postby mouseposture » Sun Dec 13, 2009 12:23 pm UTC

kurushimi wrote: (Obviously it wouldn't be impossible to prove something like, "There is no hippo in my room". You can inspect your room yourself and be sure of that).


Nope. You can't even prove that. If you claim to have searched the room exhaustively, I can propose various counterhypotheses, such as a cloak of invisibility for the hippo, or the hippo is teleported around the room so it's always behind you, or you're a brain in a vat ... and so on. Of course, you rightly reject all these as absurd, but the point is you're rejecting them on the basis of a mere heuristic (Occam's razor).

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Re: Why do people keep saying this?

Postby Kurushimi » Sun Dec 13, 2009 1:04 pm UTC

mouseposture wrote:
kurushimi wrote: (Obviously it wouldn't be impossible to prove something like, "There is no hippo in my room". You can inspect your room yourself and be sure of that).


Nope. You can't even prove that. If you claim to have searched the room exhaustively, I can propose various counterhypotheses, such as a cloak of invisibility for the hippo, or the hippo is teleported around the room so it's always behind you, or you're a brain in a vat ... and so on. Of course, you rightly reject all these as absurd, but the point is you're rejecting them on the basis of a mere heuristic (Occam's razor).


Hippo's don't have cloaks of invisibility. And hippo's cannot teleport. Besides, if you want to put it like that, like someone said before you can't prove anything. You can't even prove you're on a computer right now. As far as you know, you're delusional. Or if I asked you to prove there is a hippo in your room and you turned around and saw one, I could say it could be a hologram. It could be a foam construction of a hippo. It could be one of various alien species that bear striking similarities to hippos. In fact, I don't think Science actually proves anything (except for obvious mathematical relationships between their constructs). They're just sure beyond all reasonable doubt. I think that's why theories need refinement as years go on and every now and then they get overturned.

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Re: Why do people keep saying this?

Postby Diadem » Sun Dec 13, 2009 2:54 pm UTC

Seems like people are using two different definitions of 'proof' without distuingishing clearly between them.

In mathematics proof means absolute proof.

In science proof means 'certain beyond reasonable doubt'.

In the mathematical sense, you can not proof that there isn't a hippo in your room. But you can't proof there is one either. You can't proof anything. Which is why we don't use that meaning of proof in science :)

In the scientific sense, you can definitely proof that there is not a hippo in your room. You can also proof that there is. The latter however is easier than the former, because spotting an hippo makes the latter immidiately obvious. But to make sure there wasn't a hippo you'd have to convince yourself yourself your search was exhaustive enough, and you took into account all reasonable explanations for not seeing one. In this case there aren't too many. "Maybe it's behind the couch? Nope. Under the bed? Nope. In the fridge maybe? Hmm, I don't think a hippo would fit into my fridge, but let's check. Nope, no hippo. None in my closet either. I think that pretty much covers it. Yep, there's definitely not a hippo in my room". That was easy, but still harder than walking into your room and immidiately thinking "WTF is this hippo doing here!".

And the broader the claim, the harder it becomes to proof the negative. Stricly speaking it is never 'impossible'. You could theoretically proof that there is no teapot in orbit around Saturn, by doing an exhaustive enough search. But that'd be very hard.
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Re: Why do people keep saying this?

Postby skeptical scientist » Sun Dec 13, 2009 5:36 pm UTC

Twistar wrote:Every hypothesis has test implications right? We run experiments to see if those test implications come to bear. So lets say I have some hypothesis:
All swans are white
The test implication would be I see whiteness when I look at a swan.
So I go look at 10 million swans and they are all white.
My hypothesis must be correct.
No, I have just commited the fallacy of affirming the consequent. This is the problem of induction. This says that we CANNOT CONFIRM ANY HYPOTHESIS on empirical grounds such as these. The logic does not support it. If I go to Australia I will find black swans.

Nonsense, it just means the standard of proof is different in science than in pure logic. If I examine 10 million swans and they are all white, it doesn't prove that all swans are white, but it does make it exceedingly likely from my point of view. I haven't committed any logical fallacies, because I accept the possibility that my observations may fail to generalize completely, but nevertheless my observations are already sufficiently general that generalizing further is a reasonable guess. And the process of science is still quite logical, even if the conclusions are not true logical implications, because this process gives a very accurate heuristic for separating truth from fiction, and works better than any other heuristic that's been proposed which works on similar information.

Popper would say, oh you found a black swan now you have to trash your theory and get a new one. Wrong again. Scientists don't do this and it isn't clear that science would work if they did. Quine pointed out that instead of throwing away the hypothesis we can say we made some wrong assumption but all swans are still white. Maybe we assumed that the swans weren't painted. What this says is that WE CANNOT FALSIFY HYPOTHESES on these empirical grounds.

No you do have to trash your theory. This nonsense about maybe the swan was painted is just that: nonsense, and this is what scientists do. What you may be confused by is that if there was a lot of data supporting the theory before it was falsified, the best thing to do is not throw it out and start from scratch, but rather throw it out and replace it with something that gives similar predictions but takes into account the newly discovered discrepancies. For example, when it became clear that Newton's theory of gravitation was not entirely correct, it still made good predictions over a wide range of circumstances, so the correct thing to do was not to throw it out and start from nothing, but rather to look for a theory which was approximated by Newton's theory over the ranges where Newton's theory gave good predictions. For your swan example, I will indeed throw out my original hypothesis, but I will make a new hypothesis, perhaps proposing that all North American swans are white. Again this may turn out to be false, but hopefully it is closer to the truth than my original hypothesis.

Scientific theories may not always be true, but they have this wonderful property of converging towards the truth as more experiments are done and more observations are made, and the results used to refine the theories. This is what separates science from other proposed methods of discovering truth.
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Re: Why do people keep saying this?

Postby Twistar » Sun Dec 13, 2009 6:45 pm UTC

the tree wrote:There are all types of sort-of ways around the problem of induction. I was happy throwing Occam's Razor at it until recently, now I'm not happy with Occam's Razor at all (learning philosophy of science screws you up a bit), but that's where modern philosophers step in like Popper with falsification and Kuhn with protecting paradigms (then urm.. Lakatos? with something that was better than Kuhn iirc but a little confusing).

When it comes to not being able to prove a negative though, a real bitch of a problem is the Raven's Paraox - that since any statement has a contrapositive - everything is a negative.


Yes, there are ways around the "irrationaly of science" as it comes from Kuhn. As you said Popper gets around the problem of induction but he runs into the problem of rejecting auxiliary assumptions over the hyptothesis. Lakatos proposes progressive problem shifts (we look at an old theory and new theory, if the new theory explains everything the old one does, predicts something new, and has some corroborated evidence we accept the new theory.) This makes a lot of sense, but note that it isn't really based on 'logic.' It is actually a lot like Occam's Razor if I think about it. And Occam's Razor isn't logical, it is just an outlook that humans seem to have a preference for. There is no REASON the simplest solution is right, there's not just no reason its wrong and its simpler.

The interesting one I think is Laudan. He starts off by saying we got the aim of science wrong. Up through at least Kuhn people had been taking the aim or goal of science to be to find Truth. This is why we got that problem that science is illogical. Humans aren't in a position to directly compare their theories to nature therefore any science based on finding the truth will inevitabely fail. And also, I am using very strong definitions of prove, and truth. Basically mathematical definitions. For example, to say a hypothesis is probable because its test implications have been bourne out many times really doesn't mean anything. No matter how many times we run the test our hypothesis is either right or wrong, and as I explained before logic cannot tell us which.
So what Laudan does is he says that the goal of science is rather problem solving efficacy. Then he outlines various types of problems including empirical and conceptual problems. Then we can compare scientific theories TO EACHOTHER to see which solves the most problems (and has the least problems). However, Laudan says that he doesn't know how the cost-benefit analyses would look. It certainly wouldn't be strictly 'logical' or mathematical because we can't ascribe mathematical values to particular problems. But the process of comparing two theories in this way does seem to have produced a quite good science.


skeptical scientist wrote:
Twistar wrote:Every hypothesis has test implications right? We run experiments to see if those test implications come to bear. So lets say I have some hypothesis:
All swans are white
The test implication would be I see whiteness when I look at a swan.
So I go look at 10 million swans and they are all white.
My hypothesis must be correct.
No, I have just commited the fallacy of affirming the consequent. This is the problem of induction. This says that we CANNOT CONFIRM ANY HYPOTHESIS on empirical grounds such as these. The logic does not support it. If I go to Australia I will find black swans.

Nonsense, it just means the standard of proof is different in science than in pure logic. If I examine 10 million swans and they are all white, it doesn't prove that all swans are white, but it does make it exceedingly likely from my point of view. I haven't committed any logical fallacies, because I accept the possibility that my observations may fail to generalize completely, but nevertheless my observations are already sufficiently general that generalizing further is a reasonable guess. And the process of science is still quite logical, even if the conclusions are not true logical implications, because this process gives a very accurate heuristic for separating truth from fiction, and works better than any other heuristic that's been proposed which works on similar information.

Popper would say, oh you found a black swan now you have to trash your theory and get a new one. Wrong again. Scientists don't do this and it isn't clear that science would work if they did. Quine pointed out that instead of throwing away the hypothesis we can say we made some wrong assumption but all swans are still white. Maybe we assumed that the swans weren't painted. What this says is that WE CANNOT FALSIFY HYPOTHESES on these empirical grounds.

No you do have to trash your theory. This nonsense about maybe the swan was painted is just that: nonsense, and this is what scientists do. What you may be confused by is that if there was a lot of data supporting the theory before it was falsified, the best thing to do is not throw it out and start from scratch, but rather throw it out and replace it with something that gives similar predictions but takes into account the newly discovered discrepancies. For example, when it became clear that Newton's theory of gravitation was not entirely correct, it still made good predictions over a wide range of circumstances, so the correct thing to do was not to throw it out and start from nothing, but rather to look for a theory which was approximated by Newton's theory over the ranges where Newton's theory gave good predictions. For your swan example, I will indeed throw out my original hypothesis, but I will make a new hypothesis, perhaps proposing that all North American swans are white. Again this may turn out to be false, but hopefully it is closer to the truth than my original hypothesis.

Scientific theories may not always be true, but they have this wonderful property of converging towards the truth as more experiments are done and more observations are made, and the results used to refine the theories. This is what separates science from other proposed methods of discovering truth.


the Newton-Einstein switch seems to be riddled with good examples about this. (sorry if I get the facts a bit off, the point is clear still) Scientists were looking at Uranus and noticed it wasn't moving like Newton's theory predicted it was. STOP. by falsificationism that means we need to throw out Newton's theory of gravity because it one of its test implications was not borne out.
If H then I
Not I
Therefore Not H
H being Newton's gravity, I being Uranus' orbit.
But scientists did not reject Newton's gravity. In fact, they kept at it and discovered the planet Neptune. So this is a wonderful example of when scientists did not act as falsificationists and science prospered for it. So what did they do? Quine points this out:
If (H and A1 and A2...) Then I
Not I
Therefore Not H or Not A1 or Not A2...
in the case of Uranus scientists rejected the auxiliary assumption that there are no planets beyond Uranus. But what this reveals is that no logical process tells us if we should accept or reject a hypothesis based on its test implications. However, other rational processes can tell us what to do. This is important. These processes include Occam's Razor, and theory comparison and problem a lot of other things.

But I guess the real point of what I'm saying is that science simply cannot discover Truth. The logic isn't behind it. Once again, when I say truth i think I basically mean mathematical truth. I think Popper said it best when he talks about corrobaration. We can't conclusively confirm our hypothesis, but we can corroborate it. That basically means that we can observe that a given hypothesis is the best one out there. That doesn't mean it is any closer to the Truth than other theories, we can't know that, it just means that it is going to be the one we use.


edit: I want to address this again
skeptical scientist wrote:Nonsense, it just means the standard of proof is different in science than in pure logic.

Well first off, I want propose right then and there that science is illogical. Anything other than Truth (with a capital T) is not Truth. We can't even say it is close to truth or not because humans don't have an advantaged enough position of observation in the universe to discern that. This is again why the aim of science cannot be to find truth because you will find that science is no better at find the Truth of how the universe really is than dogmatic religion because nothing humans do can find absolute Truth.

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Re: Why do people keep saying this?

Postby achan1058 » Sun Dec 13, 2009 8:28 pm UTC

Truth is not well defined. QED.

More seriously, the world is not black and white. Any attempt to view it as such will guarantee to drive you to insanity. Just because science isn't perfectly logical does not mean it is completely illogical. How about this, instead of a boolean value truth, we will define a real function T:statement -> [0,1] so that T(x) = 1 is x is completely true, and 0 if x is completely false. In that aspect, T(Newton's theory) is quite close to 1, even though T(Newton's theory) < T(Einstein's theory), according to current experiments.

For Uranus, you do not know which one is right or wrong, until in retrospect. You simply weight the following against each other:
P(Newton is wrong) vs P(there is another planet)
Ease(demonstrating that Newton is wrong and coming up with a better theory) vs Ease(finding a new planet)

Seems perfectly logical to me that they would try to find another planet first.

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Re: Why do people keep saying this?

Postby Twistar » Sun Dec 13, 2009 9:14 pm UTC

achan1058 wrote:Truth is not well defined. QED.

More seriously, the world is not black and white. Any attempt to view it as such will guarantee to drive you to insanity. Just because science isn't perfectly logical does not mean it is completely illogical. How about this, instead of a boolean value truth, we will define a real function T:statement -> [0,1] so that T(x) = 1 is x is completely true, and 0 if x is completely false. In that aspect, T(Newton's theory) is quite close to 1, even though T(Newton's theory) < T(Einstein's theory), according to current experiments.

For Uranus, you do not know which one is right or wrong, until in retrospect. You simply weight the following against each other:
P(Newton is wrong) vs P(there is another planet)
Ease(demonstrating that Newton is wrong and coming up with a better theory) vs Ease(finding a new planet)

Seems perfectly logical to me that they would try to find another planet first.


This is asking for a logic of induction which philosophers have not been able to develop. here,

"Seems perfectly logical to me that they would try to find another planet first."
say
"Seems perfectly rational to me that they would try to find another plate first."

and I agree

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Re: Why do people keep saying this?

Postby gmalivuk » Sun Dec 13, 2009 9:17 pm UTC

Twistar wrote:Anything other than Truth (with a capital T) is not Truth. We can't even say it is close to truth or not because humans don't have an advantaged enough position of observation in the universe to discern that.
achan1058 wrote:More seriously, the world is not black and white. Any attempt to view it as such will guarantee to drive you to insanity.

Right. If there is any such thing as Truth, we can be closer or farther from it.
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Re: Why do people keep saying this?

Postby Twistar » Sun Dec 13, 2009 9:30 pm UTC

gmalivuk wrote:
Twistar wrote:Anything other than Truth (with a capital T) is not Truth. We can't even say it is close to truth or not because humans don't have an advantaged enough position of observation in the universe to discern that.
achan1058 wrote:More seriously, the world is not black and white. Any attempt to view it as such will guarantee to drive you to insanity.

Right. If there is any such thing as Truth, we can be closer or farther from it.

But we can't know what we are closer or farther from since we don't know what the truth is.
Its like Einstein's watch in a glass box. We can look at it and propose mechanism that would re-create what we see in the box, but we don't know how the watch ACTUALLY works. So lets say we propose a mechanism that looks the same and functions the same (outwardly) as the watch in glass box. What we have done is come up with a useful explanation that allows us to recreate watch objects. We've solved the problem of "how can we make a watch." But we have not found the truth of how the watch in the glass box really works.

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Re: Why do people keep saying this?

Postby gmalivuk » Sun Dec 13, 2009 9:40 pm UTC

Perhaps not, but we're certainly closer to the truth in a practical sense than an explanation that doesn't match observations at all.
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Re: Why do people keep saying this?

Postby skeptical scientist » Sun Dec 13, 2009 11:34 pm UTC

Isaac Asimov once said, "When people thought the earth was flat, they were wrong. When people thought the earth was spherical, they were wrong. But if you think that thinking the earth is spherical is just as wrong as thinking the earth is flat, then your view is wronger than both of them put together." That's right, Twistar, Asimov called you wronger than a flat-Earther.

We may not know the whole truth, and we may not be in "an advantaged enough position of observation" to ever know the whole truth, but given the state of our knowledge today compared with the state of our knowledge 5 centuries ago, I don't understand how you or anyone can think we don't know enough to say that some things are truer than others.

Since for some reason we seem to be in the math forum rather than the science forum, I'll bring things back to mathematics. I like to think of science as a limiting process. We never know anything for sure, but we settle on the likeliest-looking explanation we currently have, and believe it. All the while we make new hypotheses, perform new experiments, and continue observing the universe, and eventually a new explanation may become the likeliest-looking explanation we currently have, and we start believing it, and stop believing the old one. Eventually the true explanation becomes the likeliest-looking explanation we currently have, and never stops being that explanation*. In the limit, we only believe true explanations, and that is the virtue of science.

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*For this to work, we need to use Occam's razor or a similar principle. Working out why this is is a good exercise for the interested reader.
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Re: Why do people keep saying this?

Postby Yakk » Mon Dec 14, 2009 2:04 am UTC

There is a ridiculous information/probability theory result that describes a function that provides the best prediction of the future, given past knowledge.

This function is incomputable. A closely related decimal constant has the neat property that the number of bits you can calculate of this constant is a bounded by a (computable?) function of the number of bits it takes to describe the axioms in which you attempt to calculate it, or something equally strange.

I only read of it casually -- does anyone remember what it is I'm blathering about?

The really neat thing is that the information/probability theory weird function is effectively Occam's Razor.
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Re: Why do people keep saying this?

Postby skeptical scientist » Mon Dec 14, 2009 2:13 am UTC

I seldom have any idea what you're blathering about. :P

I find it highly amusing that you asked that question immediately below a signature mentioning Solomonoff induction, which is what you are referring to. (The related constant is referred to as Chaitin's constant or the halting probability, although it seems strange to refer to a real in binary form as a "decimal constant.") Solomonoff induction provides a good mathematical model of exactly what I was talking about in the last paragraph of my last post.
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Re: Why do people keep saying this?

Postby Goplat » Mon Dec 14, 2009 5:41 am UTC

Yakk wrote:There is a ridiculous information/probability theory result that describes a function that provides the best prediction of the future, given past knowledge.
No, there isn't. If you look into what this function actually is, you can easily see it does nothing of the sort. As described in the page that skeptical scientist linked to: "we can define the discrete universal a priori probability, m(x), to be the probability that the output of a universal prefix Turing machine U is x when provided with fair coin flips on the input tape"

But hold it. Which universal prefix Turing machine? There's an infinite number of them, after all. By choosing the machine appropriately, you can make Solomonoff induction give any hypothesis you want a prior as high or as low as you want. You can even make it so extreme that no amount of evidence will ever make more than a tiny dent in it.

The only time Solomonoff induction would actually be correct is if you were actually dealing with some known UPTM or equivalent "computer" with unknown, randomly chosen input. Real life is nothing of the sort.

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Re: Why do people keep saying this?

Postby skeptical scientist » Mon Dec 14, 2009 6:31 am UTC

Goplat, you seem to be missing the point. It's true that the priors depend on the choice of universal Turing machine, but that doesn't really matter. The essence of Solomonoff induction is that you learn as you go, and once you have learned enough, your guesses at the distribution get arbitrarily close to the actual distribution (or as close as one can get with a computable distribution). Your choice of universal machine doesn't really matter because that only affects the duration of the learning period, and not this property of convergence. So no, you don't actually need to have any prior knowledge about the source of the distribution you are trying to approximate.

(Of course, the measure given by Solomonoff induction is incomputable, so it's kind of moot in practice anyways. Solomonoff induction is more of a theoretical model of things like machine learning than an actual procedure one would imagine following to make inductive inferences.)
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Re: Why do people keep saying this?

Postby Goplat » Mon Dec 14, 2009 7:05 am UTC

skeptical scientist wrote:Goplat, you seem to be missing the point. It's true that the priors depend on the choice of universal Turing machine, but that doesn't really matter. The essence of Solomonoff induction is that you learn as you go, and once you have learned enough, your guesses at the distribution get arbitrarily close to the actual distribution (or as close as one can get with a computable distribution). Your choice of universal machine doesn't really matter because that only affects the duration of the learning period, and not this property of convergence.
If my choice gives the correct hypothesis a prior of 1/g64, it's going to take a very, very long time before that convergence happens. But if you're positing an infinite amount of "learning", there's no reason to even bother with trying to make the prior follow Occam's Razor, which the page you linked to did consider important.

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Re: Why do people keep saying this?

Postby skeptical scientist » Mon Dec 14, 2009 8:03 am UTC

g64 is a finite, not infinite, amount, so even then it still takes a finite amount of learning. But more importantly, the machine will only assign the correct hypothesis such a small prior if it takes log2 g64 bits to describe the correct hypothesis given our choice of universal machine. For example, if we're predicting the results of a sequence of coin flips, the correct hypothesis might be a seemingly random sequence of log2 g64 coin flips, that then repeats every log2 g64 flips. It's not like a human would ever discern the pattern anyways. In such a case it shouldn't bother us that Solomonoff induction takes an ungodly amount of time to converge on the correct hypothesis, since any human way of learning the correct hypothesis would take far longer.
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Re: Why do people keep saying this?

Postby Goplat » Mon Dec 14, 2009 8:40 am UTC

skeptical scientist wrote:g64 is a finite, not infinite, amount, so even then it still takes a finite amount of learning. But more importantly, the machine will only assign the correct hypothesis such a small prior if it takes log2 g64 bits to describe the correct hypothesis given our choice of universal machine. For example, if we're predicting the results of a sequence of coin flips, the correct hypothesis might be a seemingly random sequence of log2 g64 coin flips, that then repeats every log2 g64 flips. It's not like a human would ever discern the pattern anyways. In such a case it shouldn't bother us that Solomonoff induction takes an ungodly amount of time to converge on the correct hypothesis, since any human way of learning the correct hypothesis would take far longer.
And what if the correct hypothesis is the sequence consisting of nothing but "heads" each time (which looks very simple to a human), but in our favorite machine it takes log2 g64 bits to describe that? Suppose for instance that it works like
input starts with 0 -> TTTTTTTTTTTT...
input starts with 10 -> HTHTHTHTHTHT...
input starts with 110 -> HHTHHTHHTHHT...
input starts with 1110 -> HHHTHHHTHHHT...
and so on up to (log2 g64 - 1) 1s followed by 0. Other computable sequences can only be produced by inputs starting with log2 g64 1s. Solomonoff induction is not going to be making any correct predictions here for a while.

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Re: Why do people keep saying this?

Postby skeptical scientist » Mon Dec 14, 2009 3:42 pm UTC

Yes, okay, if you go to a lot of effort to design a UTM that is really bad for a given problem, then Solomonoff induction doesn't work so well. If you take a reasonable UTM, it doesn't matter what UTM you take, since the Kolmogorov complexities given by any reasonable UTMs will differ by a small constant.
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Re: Why do people keep saying this?

Postby Twistar » Mon Dec 14, 2009 5:23 pm UTC

skeptical scientist wrote:Isaac Asimov once said, "When people thought the earth was flat, they were wrong. When people thought the earth was spherical, they were wrong. But if you think that thinking the earth is spherical is just as wrong as thinking the earth is flat, then your view is wronger than both of them put together." That's right, Twistar, Asimov called you wronger than a flat-Earther.

We may not know the whole truth, and we may not be in "an advantaged enough position of observation" to ever know the whole truth, but given the state of our knowledge today compared with the state of our knowledge 5 centuries ago, I don't understand how you or anyone can think we don't know enough to say that some things are truer than others.

Since for some reason we seem to be in the math forum rather than the science forum, I'll bring things back to mathematics. I like to think of science as a limiting process. We never know anything for sure, but we settle on the likeliest-looking explanation we currently have, and believe it. All the while we make new hypotheses, perform new experiments, and continue observing the universe, and eventually a new explanation may become the likeliest-looking explanation we currently have, and we start believing it, and stop believing the old one. Eventually the true explanation becomes the likeliest-looking explanation we currently have, and never stops being that explanation*. In the limit, we only believe true explanations, and that is the virtue of science.

———
*For this to work, we need to use Occam's razor or a similar principle. Working out why this is is a good exercise for the interested reader.


See, I agree with the last paragraph. Just because science doesn't have to do with The Truth doesn't mean it is willy-nilly. I say science is alogical, but still rational. We can give explanations that seem to retain that science is logical like "it is logical to take the theory which explains the most stuff," but that really isn't logic and it is isn't necessary for science to be logical.

So as far as the flat earthers or spherical earthers being more wrong... Based on what I've said so far I can't really give a comment on who is "closer to the truth." what I can say for certain is that taking the earth to be spherical solves WAY more problems than a flat earth does, so science will adopt this view.

But the closer to the truth thing does bother me. Because I agree that if there is a way the universe is we can be farther or closer from it in our descriptions of the universe. The problem is we have no objective way of knowing where we stand on the spectrum. I mean by my reasoning thus far, we could develop a working theory of everything that will predict every event (even quantum randomness if you so desire - hidden variables) and it could still be "wrong." But at that point notions of rightness and wrongness seem to break down. So what my resolution to this problem is, is to divorce science from truth entirely and say science is that thing which seeks to solve (mostly empirical) problems about the world. If it happens to get towards the Truth of how things are.. sweet.

Well, anyhow, in my line in this discussion we seem to just be arguing over a little bit of semantics so I might leave it off here. The Solomonoff induction seems a lot more intesting but I really don't understand it to well!

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Re: Why do people keep saying this?

Postby Goplat » Mon Dec 14, 2009 5:57 pm UTC

skeptical scientist wrote:Yes, okay, if you go to a lot of effort to design a UTM that is really bad for a given problem, then Solomonoff induction doesn't work so well. If you take a reasonable UTM, it doesn't matter what UTM you take, since the Kolmogorov complexities given by any reasonable UTMs will differ by a small constant.
The page you linked to neglected to mention that anything had to be "reasonable". Please define "reasonable UTM" in a non-subjective way.

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Re: Why do people keep saying this?

Postby Token » Mon Dec 14, 2009 6:43 pm UTC

Goplat wrote:The page you linked to neglected to mention that anything had to be "reasonable". Please define "reasonable UTM" in a non-subjective way.
How come you get to insist on non-subjective definitions, but use subjective phrases like "a very, very, long time" at the same time?
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