"Being correct" isn't always the same as "knowing"

If you gave me a math problem such as 5 + 5, and I say "10", you'd say I was right. Easy peasy. Then you ask me to solve 4567738 ÷ 652534, and I almost instantly say "7". You take the time to do the calculation and see that the answer is, indeed, 7. It's possible I might have quickly calculated the answer, but did I? I did get it correct, but how?

Maybe I say I had a secret way to do math like that quickly in my head. Maybe I say I thought that "7" is quite frequently an answer to math problems. Maybe I claim that "7" just "felt right". Or maybe I just guessed.

In this particular case, you could quickly test my real arithmetic ability by testing me further: What's 4321513 x 1518? What's 1354521 ÷ 5425? etc. And you'd see I rarely got the correct answer. So, even though I did get "7" earlier, you'd not credit me with actually knowing that answer: you'd just say "He didn't know that, it was merely luck he got the answer right." In short, you'd not come to me for help with arithmetic.

The method I used to get the correct answer the first time doesn't matter. Even though I had the correct answer, I had no sound reason to believe it was "7". I stumbled upon the correct answer by chance. Maybe there were a hundred people guessing, and I happened to be the first to yell "7!"

I had the correct answer, but I didn't know the answer.

Sometimes though, you can't test someone's knowledge quite so easily. Suppose we're in an ancient pyramid and we stumble across a problem like 5643.34 ը 3433 + 42 ⑃ 342.7, and I yell out "The answer is 6!" Would you believe me? You'd ask how I know, and I might say I say I have a secret way of figuring it out. Or I give some complicated explanation of how I deduced that ը obviously means blah blah blah and ⑃ must be an operator for blah blah blah.

"OK, people often have specialized knowledge," you think. "Maybe he's a archeo-mathematician?"

So you bring that problem to the math/archeology departments at the local university. No has ever seen those symbols before: they don't know what they are. You bring it to a larger university, same deal. You post it on the internet and in ancient-math interest forums: no dice. No one has a clue what that equation means. But I'm positive the answer is 6.

There's no way to test my knowledge, really, because we have no way of checking if I'm right. You ask me what 5644 ը 454 is, and I say "65!" Is that correct? You don't know. Then you ask me what 78643 ⑃ 983923 is, and I say "-2"! Is that correct? You can't say. You can test if my answers are consistent, but not if they match up with the ancient-pyramid math.

Now we discover more ancient math problems, and archaeological mathematicians quickly decipher what the ը and ⑃ symbols mean. It turns out "6" was correct for the first problem, but it was simply an accident. My ideas of ը and ⑃ were not well thought out, and most of the rest of my answers were wrong.

So, even though I had the "correct answer", I didn't know. No matter how sure I was, no matter that I even had the correct answer: I was "speculating" (ie guessing) based on insufficient evidence. You were not wrong to disbelieve me.

Why is this important? Well, because this kind of thing happens all the time in other contexts. Someone will end up with the correct answer, but not for the right reasons. Their theories are bunk, they just happen to land on a correct answer by chance. This is important, because that person isn't the only one yelling out answers: thousands of people claim that theirs is the right answer. None have good evidence. Some of them will be right just by chance, but most will be wrong, and you don't know which is which ahead of time. Just because you picked the "right" one without evidence, doesn't mean you chose wisely: you just got lucky. You were really just as unwise as those who picked the wrong answers.

Examples:

• In politics, as usual, many people have many theories about what will happen if specific actions are taken (or not taken). They are all just guessing/hoping based on their political biases. Some people will turn out to be correct just by chance. It does not mean they are correct in all their other theories.

• A friend tells you that oregano oil will help with your diabetes. Another friend says to try cayenne. Where are the medical studies which show either one works? There are none? Then they don't know. They both might have anecdotes they can point to, or say the ancient Chinese used one or the other, but they don't know. Later, some large scale tests are done and replicated (and replicated and replicated and replicated) and now there's lots of good evidence that oregano oil does help, but cayenne does nothing (or maybe causes ulcers). The first friend still didn't know, even though she had the correct answer. Before the medical studies, the two friends had the same level of knowledge, and you had equal reasons to believe (or disbelieve) both of them. They were both, for all intents and purposes, guessing.

• You see someone swerving on the road and say "drunk driver!" Then spend the rest of your day believing you saw a drunk driver. But in reality, there are many reasons why that person might have swerved. If you had a way to check, it might turn out that the driver was, indeed, drunk, in which case you'd be correct, but you didn't really know that. It was a guess… maybe an educated guess, but a guess nonetheless. You didn't know. If I could test your ability to distinguish drunk drivers from swerving-but-not-drunk drivers a few dozen times, and you were always right, then I'd know that you know, but short of that, it's a guess.

• Someone claims to be a psychic and predicts 20 different things will happen. One of those things does (seem to) happen, the other 19 don't. Did the person actually know, or was he/she guessing (or playing the odds)?