It's not at all hard for a mathematician to come up with arbitrarily large numbers of statements that have about the same confidence as 2+2=4. There's lots of ways. Perhaps the most obvious is "n+2 = (n+1)+1" for arbitrary large n a whole number. It's rather silly to talk about how many lifetimes it would take to say these statements because there they are in 2 seconds.

I suppose the anticipated response would be to question whether these are independent statements. Why would they not be? If we are anticipating that 2+2 may not be 4 I don't see how we can for certainty say that any similar statement in arithmetic would imply any other. But perhaps it would be clearer if I changed the formula for the statements to be this: "2+2 is not equal to n", for arbitrarily large n, a whole number greater than 4. Of course this is no real difference except it now looks a lot like an argument for saying that a 1 in a million probability is sensible in cases where you have 1 million easily enumerated cases.

For example say I claim that the chance of winning a lottery by guessing a 6 digit number is 1 in a million. By the logic of the article this is a preposterous, egotistical notion unless I can come up with a million or so other statements of similar confidence. Easy enough. "the winning number is n" for each number 1 through 1 million. I think this has been used as an example in another article somewhere. These 1 million statements have a correspondence with similar statements like "2+2 is not 5" etc. What is 2+2? is it 4? is it 5? is it 6? etc If the lottery example counts as "independent" statements then so does the 2+2 series. And if they do not then are we saying it's egotistical to demand you know what the probability of the lottery is?

Incidentally the lottery example isn't a set of independent statements in the probability sense. Knowing if one statement is true or false gives me information about all the others. eg if you tell me the winning number is 1 then I know it's not 2. So what is the meaning of the word "independent" when asking for independent statements in this article? It seems to be some vague sense of not having much to do with each other somehow. Is it ever possible to have a large number of statements like this?

In the previous essay in this series evidence acceptable for thinking 2+2=3 was discussed. One example was that the person might be hypnotized. To me that seems like the only realistic explanation but certainly it's a likely one. That's great but if you've been hypnotized to think 2+2=3 like that isn't it suddenly much more likely that you might have been hypnotized to thing any number of other similar confidence level statements are true? So doesn't this challenge the real independence of all those supposedly independent statements of similar confidence you might have made?

It seems like this word "independent" is a problem within the article.

It's not at all hard for a mathematician to come up with arbitrarily large numbers of statements that have about the same confidence as 2+2=4. There's lots of ways. Perhaps the most obvious is "n+2 = (n+1)+1" for arbitrary large n a whole number. It's rather silly to talk about how many lifetimes it would take to say these statements because there they are in 2 seconds.

I suppose the anticipated response would be to question whether these are independent statements. Why would they not be? If we are anticipating that 2+2 may not be 4 I don't see how we can for certainty say that any similar statement in arithmetic would imply any other. But perhaps it would be clearer if I changed the formula for the statements to be this: "2+2 is not equal to n", for arbitrarily large n, a whole number greater than 4. Of course this is no real difference except it now looks a lot like an argument for saying that a 1 in a million probability is sensible in cases where you have 1 million easily enumerated cases.

For example say I claim that the chance of winning a lottery by guessing a 6 digit number is 1 in a million. By the logic of the article this is a preposterous, egotistical notion unless I can come up with a million or so other statements of similar confidence. Easy enough. "the winning number is n" for each number 1 through 1 million. I think this has been used as an example in another article somewhere. These 1 million statements have a correspondence with similar statements like "2+2 is not 5" etc. What is 2+2? is it 4? is it 5? is it 6? etc If the lottery example counts as "independent" statements then so does the 2+2 series. And if they do not then are we saying it's egotistical to demand you know what the probability of the lottery is?

Incidentally the lottery example isn't a set of independent statements in the probability sense. Knowing if one statement is true or false gives me information about all the others. eg if you tell me the winning number is 1 then I know it's not 2. So what is the meaning of the word "independent" when asking for independent statements in this article? It seems to be some vague sense of not having much to do with each other somehow. Is it ever possible to have a large number of statements like this?

In the previous essay in this series evidence acceptable for thinking 2+2=3 was discussed. One example was that the person might be hypnotized. To me that seems like the only realistic explanation but certainly it's a likely one. That's great but if you've been hypnotized to think 2+2=3 like that isn't it suddenly much more likely that you might have been hypnotized to thing any number of other similar confidence level statements are true? So doesn't this challenge the real independence of all those supposedly independent statements of similar confidence you might have made?

It seems like this word "independent" is a problem within the article.