# Subjectivity vs. Objectivity, A Mathematical Analysis

To my knowledge, objectivity is more the merrier and subjectivity a loner. That is to say, the probability of something being objective is thought to increase with the number of observers. The whole point of corroboration and multiple witnesses is basically that.

Mathematically ...

Imagine 3 people, call them Tom, Dick, and Harry. The starting point here is that all 3, alone, are uncertain that an event E they're observing is real/not. So the individual probability that any one of them is right (E is real) is 50%

1. The probability that all 3 are wrong (E is not real) = AW = (50%)(50%)(50%) = 12.5%
2. The probability that at least 1 of them is right (E is real) = SR = 100% - 12.5% = 87.5%

87.5% > 12.5%. So, si, more the merrier. As the number of witnesses reporting the same observation increases, the probability that the observation is real (objective) also goes up.

However ...

1. The probability that all 3 are right (E is real) = AR = (50%)(50%)(50%) = 12.5%
2. The probability that at least one of them is wrong (E is not real) = SW = 100% - 12.5% = 87.5%

Comparing the probabilities we find:

1. AW = AR = 12.5% (The probability that all are wrong = The probability that all are right)
2. SW = SR = 87.5% (The probability that at least one is right = The probability that at least one is wrong)

The bottom line is that, based on the above calculations, numbers seem not to confer an advantage towards making the objective-subjective distinction (see points 5 and 6). This can be made clearer if we break down the 87.5% probability (SR and SW) into its components as below.

Someone is right, SR = 87.5% = 37.5% (exactly 1 is right) + 37.5% (exactly 2 are right) + 12.5% (all 3 are right).

Someone is wrong, SW = 87.5% = 37.5% (exactly 1 is wrong) + 37.5% (exactly 2 are wrong) + 12.5% (all 3 are wrong)

Matched, numerically, 1 to 1, 2 to 2, 3 to 3, the probability of being right = the probability of being wrong.

In short, true, the probability of being right increases with numbers, but this is canceled by an equal increase in the probability of being wrong.

Objectivity as a function of numbers is a myth.

What am I getting wrong?

Muchas gracias and have an awesome day!

• Mathematically, the calculations are incorrect. The probability that all 3 are right/wrong is not the product of individual probabilities, and cannot be computed from individual probabilities at all. It depends on statistical dependencies between their observations, and it would be strange if those were independent considering that they are observing "the same" event (that is the only case when the probability is the product). Those dependencies cannot even be estimated without many more details than the scenario provides. Commented Oct 12, 2023 at 0:35
• You are essentially sustaining your point upon an ad populum fallacy. According to your logic, the existence of God can be proven because most people are believers. In any case, complete objectivity (what science targets) is a philosophical problem because it implies the complete denial of the subject. See solipsism and George Berkeley. Commented Oct 12, 2023 at 6:17
• There's no simple inverse relation btw subjectivity & objectivity as 'merrier' vs 'loner'. Objectivity is nothing but reified (inter)subjectivity. Per David Lewis's principal principle one's belief about the objective chance of an outcome event (say, there'll be water out of the box) determines one's rational credence in the said outcome, thus any probability number you use in your extremely simplified analysis is closely related (if not typically determined) to some objectivity. Vasubandu mentioned mass hallucination of hungry ghosts seeing water as fire which is objective for them... Commented Oct 12, 2023 at 6:45
• This seems like a combinatorics problem: How many 3 ( or n) letter words can be formed using only two letters (T or F)(allowing repeated letters). The general case is 2^n possible n letter words. No matter how many letters are in the word, there is only 1 word that is all Ts and 1word that is Fs. All the words in between contain at least one T and one F. Commented Oct 12, 2023 at 18:33
• @Conifold, superb observation. Commented Oct 14, 2023 at 0:57

I wouldn't say objectivity and consensus are interchangeable ideas, but anyway.

the individual probability that any one of them is right (E is real) is 50%

That starting point is a huge problem.

50% is a coin flip - flipping 10 coins doesn't help you establish whether something (that isn't about the coin) is true any more than flipping 1 coin does.

Maths only works out in favour of consensus with a > 50% probability. With even just 60% probability, the probability of all 3 being right is 0.22, and all 3 being wrong is 0.06. If they agree (and they can only be right or wrong), those are the only relevant probabilities and that gets you to a 77% chance of being right, which is greater than the 60% from 1 alone. If they don't all agree, you should still improve the probability by going with the majority (but not in the case of a n-vs-n-1 majority, e.g. 2v1, as that just ends up canceling out down to the probability of a single person being right) - an agreement among 3/4 of 60%-ers gets you to 69% chance of being right.

That's why expert consensus means something, but random-anybody consensus doesn't. Not that we can really be sure that experts indeed have a > 50% probability of being right, but it seems like a reasonable position, at least for fields with a solid methodology and a history of demonstrated reliability* (e.g. science, not astrology).

That said, to multiply probabilities, we need to assume relative independence. If people believe things for similar bad reasons (which appears to often be the case), adding more people may not have much of an effect on the probability.

* You could also evaluate conclusions directly based on the methodology to reach them and its history of reliability. This may be better where possible. Although human knowledge is built on many lifetimes of knowledge gathering, so you can't evaluate much of that in your one life, and your own attempt to evaluate any given claim may be flawed if you lack sufficient knowledge of the subject.

• Perhaps I'm mistaken about subjectivity-objectivity. Commented Oct 14, 2023 at 1:37

Your numeric analysis is correct for just an average of opinions. What I think you are struggling with is that empiricism uses a consensus of experts as its final stage of a truth check. If this consensus were just raw opinion, it would be subject to your numeric concern.

Consensus of experts, however, involves DIALOG, and justification. This dialog includes discussion about what reasoning or evidence is relevant, and when one expert's opinion disagrees with others, they provide this justification to a community debate. Not all such debate operates perfectly, and expert opinion that sided one way is sometimes overturned to support what had been a minority view decades or centuries later, as more data or reasoning is brought forward.

Error correcting statistics are different from random statistics, which evolutionary thinkers have cited against the random statistics used by creationists to claim that building any useful enzymes can never be done by evolution. Error correction statistics, however, have to include the possibility of ERROR IN THE ERROR CORRECTION too, which is something very hard to quantify for the peer consultation process.

The numerics to show this will always take us in a good direction of better knowledge still do not work. The concept that science will incrementally approach "truth" is called verisimilitude, and Popper's (and subsequent efforts) to show that verisimilitude will always be positive was shown to be numerically in error.

However, our inability to demonstrate the validity of our dialog/mutual-questioning methodology is not fatal, as Godel has shown that there are math/logic truths that cannot be demonstrated within a math/logic system. Our pragmatic success of incrementally better understandings of our world are EMPIRICALLY rather than logically demonstrated. One can reasonably infer from the success of science that we have a useful methodology to arrive at increasing likely approximations to truth in our dialog/consensus approach within an expert community -- which yes, relies upon a "majority rules" of the judgement among those experts.

• Following dialog and justification, disagreement may (and often does) remain, and if consensus indeed doesn't improve probability, we're still at the original problem of why you should prefer consensus above a single expert opinion (... but consensus does improve probability, so that's not an issue). Or is the implication here that if a majority concludes or decides something, there is simply not enough people to oppose that conclusion or decision? Commented Oct 13, 2023 at 14:47
• @NotThatGuy Generally dialog identifies reasons to suspect each of several competing views, and establishes methodologies to investigate these suspicions. During this stage of dialog there is no consensus and a non expert can treat the subject as open between the actively competing views. When consensus has been achieved, such as with the ASME pressure vessel standards, or Anthropogenic Global Warming, non experts should treat the consensus as reality. These are indicated by formal professional society recommendations. Commented Oct 13, 2023 at 16:26
• Expert communities continue dialogue after a formal position is taken, and sometimes recommendations are withdrawn, or more often are modified. But the recommendations remain what non experts should follow. Commented Oct 13, 2023 at 16:28
• There will always remain logical possibilities of gross error for ANY empirical conclusions. The justification for following this process is the massive utility we have experienced based on it. Commented Oct 13, 2023 at 16:31
• I guess the question is: why should we care about consensus at all? Does it provide any benefit above just listening to one expert? The maths in the question seems to suggest not (although the maths is based on a flawed premise). Commented Oct 13, 2023 at 17:22

There are two points you could consider.

The first is that you have assumed that the probability of any individual being right is 50%, which is the same as guessing a coin toss. You can't predict a coin toss more accurately by have more people guessing what it will be, so you should not be surprised by the result of your calculations. Try repeating your calculation with a higher probability as a starting assumption, say 80%, and see where that gets you.

The second is that objectivity is not the same as popularity. That a large number of people believe something does not make it true.

• I guess I wasn't as clear-minded as I thought I was. With 80% probability as our square 1, sure, more the merrier. Commented Oct 15, 2023 at 18:48
• @AgentSmith indeed, and I imagine that if you start with 20%, say, you'll find it the fewer the merrier. I'm sure there's an important conclusion in that somewhere, but it's too late at night for me to think straight! Commented Oct 15, 2023 at 19:15
• 😁, right you are! Commented Oct 15, 2023 at 19:32

Linking objectivity to probability and to a consensus of what is objective seems to be wrong here I think. Because facts speak for themselves, such as, apples are fruit they do not require consensus nor probability so whether more or less people believe whether apples are fruit is irrelevant.

• The further you go in philosophy, the more you find that seemingly obvious and stable taxonomies admit of exponentially increasing fine-grained refinements, until whether the category of fruit has much, if any, nontrivial significance is a question that itself starts to evaporate (as to its own significance) in light of uncertainty over the definitions of words like "the" and "two." So in the end, it turns out to be epistemically possible that apples aren't fruit because nothing is really fruit: c.f. issues with words like "tomato" and "berries," which get categorized "wrongly" quite often. Commented Oct 12, 2023 at 0:21
• Indeed, a point well made. Thank you for your response. Commented Oct 12, 2023 at 0:22