The More The Merrier Paradox

TheMadFool

As all of you must know by now one fundamental requirement in scientific observational data collection is repeatability. In essence the repeatability principle is that a single observation is dubious enough to require further corroboration as a confirmatory process.

The repeatability principle isn't restricted to highfalutin scientific enterprises. A less rigorous version of it exists in the lives of the ordinary person and is applied generously with great skill by all and sundry. In most cases, this "useful" heuristic remains hidden - an unspoken rule, that's second nature to us. On occasion though it needs to be hauled up from the depths of our subconscious, the abode of our habits, and attended to carefully. These occasions are, for the most part, novel experiences e.g. sights we've never seen, sounds we've never heard. It is then, during these times, we ask our companion, if one is available, "did you see that?" or "did you hear that?", etc.

However, if I'm correct, there seems to be serious flaw with the repeatability principle, a flaw that I'll outline in the paragraphs to follow.

Imagine, for the sake of simplicity and ease of calculation, that there's a world with only 3 people, X, Y, and Z. In this world an opportunity for an observation arises and X makes the observation, call it O. Now O is something none of them have ever seen. Naturally, like any one of us, when X observes O for the very first time, X is in doubt - is O real/not? A single observation just doesn't suffice on that score. Ergo, as the principle of repeatability recommends we need to elicit the aid of Y and Z. Three people observing the same thing, here O, should do the job, furnish the necessary justification to conclude that O is real, right?

Since this entails some calculations we have to start off by assigning a probability value for O being real. X doesn't know whether O is real/not and that in probabilistic terms means that, insofar as X is concerned, O is as likely to be real as O is likely to not be real. In other words, the probability that O is real = the probability that O is not real. The only probability value for O being real that satisfies this condition is 50%. This value of 50% is good for any single observation and so if Y alone were to observe O, the probability of O being real would be 50% and if Z alone were to observe O the probability of O being real would also be 50%.

Now we can get to the actual calculations:

A. The probability that O is real from a single observation (X's) = 50%

B. The probability that O is not real from a single observation (X's) = 50%

1. If all 3, X, Y, and Z observe O then the probability of O not being real (call this NR) = 50% * 50 * 50% = 12.5%

That's reassuring! The probability that O is not real has plummeted from a dismal 50% to a (phew!) 12.5%

Unfortunately for us, take a look at the probability that O is real given all 3, X, Y, and Z observe O below:

2. If all 3, X, Y, and Z observe O then the probability of O being real (call this R) = 50% * 50% * 50% = 12.5%

The numbers clearly show that NR = R. The probability of O being real given all 3, X, Y, and Z observe O = the probability of O not being real given all 3, X, Y, and Z observe O.

Conclusion:

Although the numbers show that the probability of an observation (O) not being real drops as the number of observers increase, the probability of that observation (O) being real also falls and falls to the same value (NR = R). The bottom line - the principle of repeatability fails to perform the intended task of increasing the chances that an observation is real/not. In fact, the chances of an observation being real = the chances of an observation not being real no matter how many observers there are.

Srap Tasmaner

X doesn't know whether O is real/not and that in probabilistic terms means that, insofar as X is concerned, O is as likely to be real as O is likely to not be real. — TheMadFool

This is the same thing you said before and it's plainly false, so I'll ask again why you think it's true.

TheMadFool

This is the same thing you said before and it's plainly false, so I'll ask again why you think it's true. — Srap Tasmaner

Is there something you don't know? Say there is and let's call it proposition S. What value of probability will you assign to the truth value of S? Less than 50% or more than 50% or 50%? You can't say less than 50% or more than 50% because that means you know something about the truth value of S which contradicts your initial position that you don't know anything about the truth value of S. To avoid contradicting yourself you must assign a value of 50% to the truth/falsehood of S.

Srap Tasmaner

↪TheMadFool

I place 98 red marbles and 2 blue marbles in an urn; then I blindfold you and have you select a marble from the urn.

Do you, before removing your blindfold, know what color marble you have selected? No. Do you know what color you've likely selected? Obviously.

Philosophim

Its a fun idea, but I see a few issues here.

First, repeatability in science is not about confirmation of results, it is to check if the results are irrefutable. In putting an experiment before different people, we're trying to introduce a new element to see if the observation can be refuted. If everyone is just interested in confirming what is before their eyes, repeatability has no merit.

The second is how you assess probability. Probability is based on knowables. When you talk about the probability of a jack being pulled from a regular deck of 52 cards, we know that there are four jacks. Probability is making a prediction based off of the limitations of what we do, and do not know.

You cannot assign the probability of something being real, without first constructing some limitations. What does it mean to be real? What are the circumstances in which you observed something, and it was not real? Is there chance involved based on these limitations? Once defined, we can say, "Its either real or not", but that still does not determine the likelihood of it being real or not.

It would be like saying that I'm either alive or dead, therefore its a 50% chance that I'm either alive or dead. It doesn't work that way.

TheMadFool

I place 98 red marbles and 2 blue marbles in an urn; then I blindfold you and have you select a marble from the urn.

Do you, before removing your blindfold, know what color marble you have selected? No. Do you know what color you've likely selected? Obviously. — Srap Tasmaner

Well, I know it's likely to be red. Probability that it's a red marble = 98/100.

Consider now the situation that there are 50 red and 50 blue marbles and I make a blind-folded selection It's equally likely that the marble is red or blue, I can't say that it's likely that I've chosen a red or a blue marble.

Srap Tasmaner

↪TheMadFool

Well, yeah. The point is not knowing whether P and not knowing anything at all about the likelihood of P are really obviously not the same thing.

TheMadFool

First, repeatability in science is not about confirmation of results, it is to check if the results are irrefutable. — Philosophim

That's the same thing. If refuted then disconfirmed. If confirmed then not refuted.

You cannot assign the probability of something being real, without first constructing some limitations. — Philosophim

There are two options: an observation is real or not real. Ergo, the probability it's real = 1/2 = 50% and the probability that it's not real = 1/2 = 50%

TheMadFool

Well, yeah. The point is not knowing whether P and not knowing anything at all about the likelihood of P are really obviously not the same thing — Srap Tasmaner

And...

Anthony

In essence the repeatability principle is that a single observation is dubious enough to require further corroboration as a confirmatory process. — TheMadFool

Then science doesn't study the actual...which is limited to one place and time ad infinitum. And how modern science (cognitive science) handles incomplete information is wrong. The only true objectivity includes the unconscious/unknown. Which introduces a paradox, inasmuch as saying anything about what is unknowable is fraudulent. What modern science is. Models are the new God...clap! That is to say, if you reject God but accept computer models...you are incontrovertibly nuts. Not that I think in terms of "God"...but rather in terms of incomplete information...which is eternal. Each new moment in time is nothing like the previous, which is clearer when considering space...obviously one space is capable of executing events another space does not. But what with spacetime...it starts to become lucid the difference between what scientists are interested in and the actual.

Understanding is pretty much always in retrospect. The importance of prediction is lost on me.

Srap Tasmaner

↪TheMadFool

So stop saying they are.

TheMadFool

↪Srap Tasmaner

The point is not knowing whether P and not knowing anything at all about the likelihood of P are really obviously not the same thing — Srap Tasmaner

I'll change your "P" to R for convenience and use the notation P(R) = probability of R

1. Not knowing whether P. There are two possibilities in the sample space: R or ~R. P(R) = 1/2 = P(~R) = 1/2 = 50%

2. Not knowing anything at all about the likelihood of P. P(R) = P(~R). Since P(R) + P(~R) = 1, it follows that P(R) = P(~R) = 1/2 = 50%

Srap Tasmaner

↪TheMadFool

Alright, then for both my urn and your urn, we'll say the sample space is { the marble I picked was red, the marble I picked was blue }, so in both cases the chances are 1 in 2 that you picked a red marble. Sound good?

TheMadFool

Alright, then for both my urn and your urn, we'll say the sample space is { the marble I picked was red, the marble I picked was blue }, so in both cases the chances are 1 in 2 that you picked a red marble. Sound good? — Srap Tasmaner

You need to work on it a bit more.

Srap Tasmaner

↪TheMadFool

I admit I haven't quite figured out the best way to describe or explain the fallacy here, but that it is a fallacy should be clear. Your conclusions make it clear something has gone wrong, don't they?

TheMadFool

I admit I haven't quite figured out the best way to describe or explain the fallacy here, but that it is a fallacy should be clear. Your conclusions make it clear something has gone wrong, don't they? — Srap Tasmaner

Well, either the entire community of mathematicians and scientists is correct or I'm correct. It seems I have a snowball's chance in hell. Could you give it some thought and get back to me if you're so disposed?

Srap Tasmaner

↪TheMadFool

I certainly will.

TheMadFool

I certainly will. — Srap Tasmaner

:ok: :up:

Mww

X makes the observation.....
....A single observation just doesn't suffice.... — TheMadFool

we need to elicit the aid of Y and Z....
......."did you see that?" or "did you hear that?", etc. — TheMadFool

Hmmmm.....what are Y and Z going to do, tell X he didn’t make an observation? Better not, lest the gedankenexperiment immediately contract itself.

A single observation does suffice, at least for the determination that perception has been met with something. Doesn’t matter that the observation doesn’t relate to extant knowledge, it’s still an observation of something real, otherwise it couldn’t have been an observation. It’s just an O, but it must be a real O.

assigning a probability value for O being real. — TheMadFool

How could it not be real? There are no observations of the not-real. Even a mirage is real, albeit mistakenly judged as a false reality. Besides, even Y and Z, all else being equal, would observe a mirage just as mistakenly as X.

if I'm correct, there seems to be serious flaw with the repeatability principle — TheMadFool

For a mere observation, I would agree; the repeatability principle is irrelevant. For assigning a name to the observed, given lack of extant knowledge of it by X, Y, and Z, that would require some kind of three-way agreement. Or, they could all just call it what they want, and since there’s only three of them, probably wouldn’t hurt much. But let any one of the three, in turn, tell a forth, and the forth guy is gonna have some trouble.

Interesting, even without all the probability stuff.

Philosophim

↪TheMadFool

First, repeatability in science is not about confirmation of results, it is to check if the results are irrefutable.
— Philosophim

That's the same thing. If refuted then disconfirmed. If confirmed then not refuted. — TheMadFool

No, they really aren't. Let me give you an example. Lets say that some one sees the Loch Ness monster in the lake at a distance. Its really just a man in a submarine having some fun. But the first person invites another to stand where they are, and they too are convinced its the Loch Ness monster. Tons of people are invited, and everyone confirms it must be the Loch Ness monster, because that's what they want to see. Such repeatability is confirmation, but useless. Confirmation when someone is trying to refute a claim, like going down to the water for a closer inspection, is when it is useful.

There are two options: an observation is real or not real. Ergo, the probability it's real = 1/2 = 50% and the probability that it's not real = 1/2 = 50% — TheMadFool

This is also incorrect. You ignored the point about probability being based off of knowables. I'll give another example if it wasn't clear. There is possibility, and probability. It is possible I win the lottery. Either I win it, or I don't. That does not mean I have a 50% chance to win or lose the lottery. We know this, because there are very known instances in which I would win, and many known instances in which I would lose.

The same with something being real. It is possible it is real, or possible that it is not real. That does not mean it has a 50% probability of being real, or not real. To determine a probability, we would need to know instances in which the observation would not be real, and compare it to all the known instances in which it could be real. If we have no knowledge of any of this, we cannot use probability.

TheMadFool

No, they really aren't. Let me give you an example. Lets say that some one sees the Loch Ness monster in the lake at a distance. Its really just a man in a submarine having some fun. But the first person invites another to stand where they are, and they too are convinced its the Loch Ness monster. Tons of people are invited, and everyone confirms it must be the Loch Ness monster, because that's what they want to see. Such repeatability is confirmation, but useless. Confirmation when someone is trying to refute a claim, like going down to the water for a closer inspection, is when it is useful. — Philosophim

Confirmation isn't useless. It is the lifeblood of the scientific method. I'm sure you already know this but just in case you don't allow me to explain. A scientific theory T becomes the basis of some predictions P i.e. T implies P. If P is observed then theory T is confirmed but notice that the refutation of T is still about P, specifically the failure to observe P. P decides both confirmation and refutation.

Either I win it, or I don't. That does not mean I have a 50% chance to win or lose the lottery. We know this, because there are very known instances in which I would win, and many known instances in which I would lose. — Philosophim

Of course the probability of you winning/losing isn't 50%. Nothing I said would've led anyone to make that egregious mistake.

It is possible it is real, or possible that it is not real. That does not mean it has a 50% probability of being real, or not real. — Philosophim

Suppose that you see something extraordinary, say E. you can't believe your eyes, You don't know if what you saw was real or not. What numerical value would you assign to the probability that E is real?

TheMadFool

Better not, lest the gedankenexperiment immediately contract itself. — Mww

The gedanken experiment isn't so unrealistic. I have, on many occasions, seen people ask others to confirm their perceptions, especially when it's something out of the ordinary.

A single observation does suffice, at least for the determination that perception has been met with something — Mww

The qualifier weakens the statement to the point of being self-negating.

Even a mirage is real — Mww

So, if I see a mirage of Winston Churchill then it's actually Churchill?

For a mere observation, I would agree; the repeatability principle is irrelevant. For assigning a name to the observed, given lack of extant knowledge of it by X, Y, and Z, that would require some kind of three-way agreement. Or, they could all just call it what they want, and since there’s only three of them, probably wouldn’t hurt much. But let any one of the three, in turn, tell a forth, and the forth guy is gonna have some trouble. — Mww

Irrelevant. The thought experiment has real-life instantiations.

Philosophim

↪TheMadFool

Confirmation isn't useless. It is the lifeblood of the scientific method. — TheMadFool

I'll link you an article from an a person who has a Phd in Astrophysics. Its an easy read though, no worry. https://www.forbes.com/sites/startswithabang/2017/11/22/scientific-proof-is-a-myth/#14b8fc7a2fb1
Perhaps this will better explain why I mean by science being something which has not been refuted, and not science is something which is done through confirmation.

Suppose that you see something extraordinary, say E. you can't believe your eyes, You don't know if what you saw was real or not. What numerical value would you assign to E being real? — TheMadFool

There is no numerical value to assign with that limited information. We have nothing that entails a pattern. It is like a face down deck of cards which could contain anything on each of its faces. The only way to know is to flip the cards one by one until we see the pictures.

TheMadFool

I'll link you an article from an a person who has a Phd in Astrophysics. — Philosophim

:ok: :up:

There is no numerical value to assign with that limited information. — Philosophim

Probability is all about working with limited information. It's not that hard. There are 3 possibilities:

1. Less than 50%
2. 50%
3. More than 50%

What's your choice and why?

Philosophim

↪TheMadFool

Probability is all about working with limited information. — TheMadFool

Yes, you are correct. But so are possibilities. While they are both inductions, they are different kinds of inductions with different rules. Probabilities are ratios of all knowable outcomes involved, while possibilities are one or more outcomes that can happen, but have no indication of their likelihood. Lets break it down to see what I mean.

To have a probability of your observation being real, or not real, we need to define a situation in which its not real. Lets say you've been up for 48 hours and have lived on coffee and sugar bombs. In that situation, you find yourself occasionally looking at something, and not being sure if its real. It turns out that occasionally the thing you doubt isn't actually real. You pass your hand over it, and it vanishes. Turns out its your mind playing tricks. You start counting the times you pass your hand on something and it vanishes, versus the time it does not vanish. After counting 20 times, you find 2 instances in which it was your mind playing tricks. So we could say at that point that there is a 10% chance that when you question whether what you are seeing is real, that it is not real.

Without examining actual instances of the outcomes, you can't use probability. Its just like the lottery odds example. You can win the lottery, or lose the lottery. Those are possibilities. Probabilities can only enter by examining the combination of outcomes, and counting up the ratio of times when a win versus a loss happens.

So until we can determine the times and situation in which you have observed something, not been sure it is real, and it has not been real, we can't use probabilities. Any guess or stab at a number is a misapplication of the number. Or like Abraham Lincoln once said, "94.5% of statistics on the internet are fake".

Dawnstorm

The problem is that you're defining O as constant, but then treat it as a variable during the calculations.

Basically, you're saying that O can either be real or not, and that's how you justify your 50 %, but then you treat O as a variable for the reliability of O(x), O(y), and O(z). Of course, both likelihoods are going to become ever smaller, because you're only considering the extremes. It's easier to show you what I mean, if we choose different values.

Let's say the likelihood that O is real is 90 % and the likelihood that O is not real is 10 %.

You're only considering the following cases:

O is real: 0.9 * 0.9 * 0.9

And O is not real: 0.1 * 0.1 * 0.1

There are a number of cases you're ignoring:

0.9 * 0.1 * 0.1
0.9 * 0.9 * 0.1
0.1 * 0.9 * 0.1

And so on.

It's understandable that you'd ignore those cases, since in all those cases people would be hallucinating things that are actually there. But your maths describes those cases and ignoring them gives you results you falsely interpret as a paradox, because you're not looking at the whole picture.

There's a theoretical problem here you have to solve. What do you do with cases in which all people see the same thing but only some of them see something real? Your math doesn't allow you to ignore those cases.

Deleted User

↪TheMadFool

The problem with this is you are drawing an objective probability from a subjective not knowing.

So, I don't know if X is true or false. And I have no criteria at all for deciding if it is more likely to be true or not. So FOR ME it might be useful to think that there is a 50% chance it is true or false, but it doesn't mean there is a 50% chance it is true or false. Perhaps there are experts who know things that swing it one way or another. That's one clue that my lack of knowledge does not split any issue into 50 percent packages. It just means that I have no reason to weight true more or less than false. I have no reason to.

That's very different from saying that given my state of unknowing we can draw the conclusion that X is 50% likely to be real or true.

TheMadFool

↪Philosophim

You're talking experimental probability and it's a fallback measure when theoretical probability can't be calculated.

In our case, theoretical probability can be calculated. There are [i[two possibilities[/i]: the observation O is real/not real. X, the first observer, doesn't know if O is real or not.

What's the probability that O is real?

X's options:

1. The probability that O is real is less than 50%
2. The probability that O is real is 50%
3. The probability that O is real is greater than 50%

lottery — Philosophim

If you're trying to compare the lottery to the situation X is in then fine. Like in a lottery you either win or you lose just like O can be real ir not. In a lottery the probability of winning is not the same as the probability of losing as so many who've seen their hard-earned cash greedily devoured by the lottery monster will attest to. Likewise, X isn't sure what value the probability that his observation O is real will assume. That's why I asked you to make a choice after giving you 3 option which are mutually exclusive and jointly exhaustive. Again...

X's options are:

1. The probability that O is real is less than 50%.

2. The probability that O is real is 50%

3. The probability thaf O is real is greater than 50%

Which among the 3 options I provided above is the correct numerical value for the probability of O being real?

Remember that choosing option 1 or 2 means that you know that O is unlikely/likely respectively but that contradicts X's initial epistemic state - X doesn't know whether O is real/not. For X, it's as likely as it's unlikely that O is real; in other words, the probability of O being real = 50% and that takes back to the More The Merrier Paradox.

↪Dawnstorm

You're right. Some people are more likely to hallucinate than others who, in turn, are more likely to observe the real. That means I have to calculate probabilities for each possible scenario.

However, in my defense, I'd like to point out that the variations are not so extreme as your numbers suggest. The Bell curve should be good enough to allay your concerns - most cluster around the mean.

What do you do with cases in which all people see the same thing but only some of them see something real? — Dawnstorm

You need to give me more to go on.

So FOR ME it might be useful to think that there is a 50% chance it is true or false, but it doesn't mean there is a 50% chance it is true or false. — Coben

There are two things to consider here. First the thing that's been observed, O, and X's perception of that thing. The issue concerns the reliability of our perceptions and not the things themselves that are being observed. The reliability of our perceptions can be framed in a probabilistic context i.e. we can assign probability values to them and do some number crunching after that.

Your thoughts are in line with Philosophim's. You might want to read my reply to him/her as the case may be.

Dawnstorm

You're right. Some people are more likely to hallucinate than others who, in turn, are more likely to observe the real. That means I have to calculate probabilities for each possible scenario.

However, in my defense, I'd like to point out that the variations are not so extreme as your numbers suggest. The Bell curve should be good enough to allay your concerns - most cluster around the mean. — TheMadFool

The bell curve isn't very relevant to my point. It's about distributions. And you'd first have to clearly define the variable that's distributed (the mean of what?). None of that is very relevant to the point.

You're talking about a paradox that doesn't exist, because you <i>don't</i> have a clear grasp of your variables.

Let me try to explain it again: You're multiplying your 0.5 probability as if it were a independent random variables. Of course, they're going to get ever smaller, no matter if they're all real or all unreal. Mathematically, what you're calculating is equivalent to coin flips. Three times heads in a row is more unlikely than two times heads in row. Same goes for tails. That's the sort of maths your using.

What you're ignoring is the likelihood that O is real when X sees it, unreal when Y sees it, and real again when Z sees it, and so on. The more people you add, the greater number of possible events you ignore.

Your 12.5 % is the chance that O is real for all three people, and the chance that O is real for none of them. The other 75 % distribute over all the other permutations (like, for example, real for X, but unreal for Y and Z, or real for X and Y, but unreal for Z).

If that state of events doesn't make sense to you (i.e. you're not a relativist about what's real), you've used inappropriate maths.

TheMadFool

What you're ignoring is the likelihood that O is real when X sees it, unreal when Y sees it, and real again when Z sees it, and so on. The more people you add, the greater number of possible events you ignore. — Dawnstorm

I understand your point. There's more to it than the 12.5% + 12.5% = 25%. However, remember that I'm only concerned about the principle of repeatability which is basically the belief that the probability of an observation being real increases with the number of observers. While there might be a lot going on in between, I only have to consider the worst case scenaro (everyone [all 3, X, Y, and Z] observing something not real) and the best case scenario (everyone [all 3, X, Y, and Z] observing something real). Everything falling betwixt these two can be ignored. If this doesn't make sense to you then it's because you're not looking at it the right way. If I were to claim that my car is the best in the world, the proper way to test the truth of that claim would be to put my car through extreme conditions, in other word, the best and the worst environments should be chosen as a proving ground. This I've done.

The bell curve isn't very relevant to my point. — Dawnstorm

Au contraire, if you're going to raise an issue from variability in the observers then, the Bell curve is the right mathematical object to consult - extreme variations, variations that could upset our calculations are outliers, unlikely to be encountered in a random sample of observers.

Dawnstorm

However, remember that I'm only concerned about the principle of repeatability which is basically the belief that the probability of an observation being real increases with the number of observers. — TheMadFool

Then you should choose a mathematical model that's up to the task. An independent random variable with the sample space of {Real, Unreal] isn't it.

While there might be a lot going on in between, I only have to consider the worst case scenaro (everyone [all 3, X, Y, and Z] observing something not real) and the best case scenario (everyone [all 3, X, Y, and Z] observing something real). — TheMadFool

Not if you treat O as an independent random variable. If you do that the math forces you to consider those cases, lest the math be rendered useless.

You <i>can</i> ignore those cases of coure. Let me show you:

RRR - 12.5 %
RUU - 12.5 %
RUR - 12.5 %
RRU - 12.5 %
URR - 12.5 %
UUR - 12.5 %
URU - 12.5 %
UUU - 12.5 %

Turns into:

RRR - 12.5 %
RUU - 12.5 %
RUR - 12.5 %
RRU - 12.5 %
URR - 12.5 %
UUR - 12.5 %
URU - 12.5 %
UUU - 12.5 %

And your probability that O is real remains 50 %, because 12.5 % are 50 % of 25 %.

You're not going to get very far if you don't understand the maths. There is no paradox.

The More The Merrier Paradox

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