I don’t consider these mistakes to be no problem at all. What I meant to say is that the existence of these noise errors doesn’t reduce the reasonabliness of me going around and using logical reasoning to draw deductions. Which also means that if reality seems to contradict my deductions, then either there is an error within my deductions that I can theoretically find, or there is an error within the line of thought that made me doubt my deductions, for example eyes being inadequate tools for counting pebbles. To put it more generally: If I don’t find errors within my deductions, then my perception of reality is not an appropriate measure for the truth of my deductions, unless said deductions deal in any way with the applicability of other deductions on reality, or reality in general, which mathematics does not.
It’s not as if errors in perceiving reality weren’t much more numerous and harder to detect than errors in anyone’s faculty of doing logical reasoning.
It’s not as if errors in perceiving reality weren’t much more numerous and harder to detect than errors in anyone’s faculty of doing logical reasoning.
And the probability of an error in a given logical argument gets smaller as the chain of deductions gets shorter and as the number of verifications of the argument gets larger.
Nonetheless, the probability of error should never reach zero, even if the argument is as short as the proof that SS0 + SS0 = SSSS0 in PA, and even if the proof has been verified by yourself and others billions of times.
ETA: Where ever I wrote “proof” in this comment, I meant “alleged proof”. (Erm … except for in this ETA.)
The probability that there is an error within the line of thought that lets me come to the conclusion that there is an error within any theorem of peano arithmetic is always higher than the probability that there actually is an error within any theorem of peano arithmetic, since probability theory is based on peano arithmetic and if SS0 + SS0 = SSSS0 were wrong, probability theory would be at least equally wrong.
the probability that there actually is an error within any theorem of peano arithmetic.
(Emphasis added.) Where ever I wrote “proof” in the grandparent comment, I should have written “alleged proof”.
We probably agree that the idea of “an error in a theorem of PA” isn’t meaningful. But the idea that everyone was making a mistake the whole time that they thought that SS0 + SS0 = SSSS0 was a theorem of PA, while, all along, SS0 + SS0 = SSS0 was a theorem of PA — that idea is meaningful. After all, people are all the time alleging that some statement is a theorem of PA when it really isn’t. That is to say, people make arithmetic mistakes all the time.
That is true. However, if your perception of reality leads you to the thought that there might be an error with SS0 + SS0 = SSSS0, and you can’t find that error, then it is irrational to assume that there actually is an error with SS0 + SS0 = SSSS0 rather than with your perception of rationality or the concept of applying SS0 + SS0 = SSSS0 to reality.
I think so, if I understand you. But I think that you’re referring to a more restricted class of “perceptions of reality” than Eliezer is.
In the kind of scenario that Eliezer is talking about, your perceptions of reality include seeming to find an error in the alleged proof that SS0 + SS0 = SSSS0 (and confirming your perception of an error sufficiently many times to outweigh all the times when you thought you’d confirmed that the alleged proof was valid). If that is the kind of “perception of reality” that we’re talking about, then you should conclude that there was an error in the alleged proof of SS0 + SS0 = SSSS0.
That is all good and valid, and of course I don’t believe in any results of deductions with errors in them just based on said deductions. But that has nothing to do with reality. Two pebbles plus two pebbles resulting in three pebbles is not what convinces me that SS0 + SS0 = SSS0; finding the error is, which is nothing that is perceived (i.e. it is purely abstract).
If we’re defining “situation” in a way similar to how it’s used in the top-level post (pebbles and stuff), then there simply can’t exist a situation that could convince me that SS0 + SS0 = SSSS0 is wrong in peano arithmetic. It might convince me to check peano arithmetic, of course, but that’s all.
I try to not argue about definition of words, but it just seems to me that as soon as you define words like “perception”, “situation”, “believe” etcetera in a way that would result in a situation capable of convincing me that SS0 + SS0 = SSS0 is true in peano arithmetic, we are not talking about reality anymore.
Okay, I just thought of a possible situation that would indeed “convince” me of 2 + 2 = 3: Disable the module of my brain responsible for logical reasoning, then show me some stage magic involving pebbles or earplugs, and then my poor rationalization module would probably end up with some explanation along the lines of 2 + 2 = 3.
I don’t consider these mistakes to be no problem at all. What I meant to say is that the existence of these noise errors doesn’t reduce the reasonabliness of me going around and using logical reasoning to draw deductions. Which also means that if reality seems to contradict my deductions, then either there is an error within my deductions that I can theoretically find, or there is an error within the line of thought that made me doubt my deductions, for example eyes being inadequate tools for counting pebbles. To put it more generally: If I don’t find errors within my deductions, then my perception of reality is not an appropriate measure for the truth of my deductions, unless said deductions deal in any way with the applicability of other deductions on reality, or reality in general, which mathematics does not.
It’s not as if errors in perceiving reality weren’t much more numerous and harder to detect than errors in anyone’s faculty of doing logical reasoning.
And the probability of an error in a given logical argument gets smaller as the chain of deductions gets shorter and as the number of verifications of the argument gets larger.
Nonetheless, the probability of error should never reach zero, even if the argument is as short as the proof that SS0 + SS0 = SSSS0 in PA, and even if the proof has been verified by yourself and others billions of times.
ETA: Where ever I wrote “proof” in this comment, I meant “alleged proof”. (Erm … except for in this ETA.)
The probability that there is an error within the line of thought that lets me come to the conclusion that there is an error within any theorem of peano arithmetic is always higher than the probability that there actually is an error within any theorem of peano arithmetic, since probability theory is based on peano arithmetic and if SS0 + SS0 = SSSS0 were wrong, probability theory would be at least equally wrong.
(Emphasis added.) Where ever I wrote “proof” in the grandparent comment, I should have written “alleged proof”.
We probably agree that the idea of “an error in a theorem of PA” isn’t meaningful. But the idea that everyone was making a mistake the whole time that they thought that SS0 + SS0 = SSSS0 was a theorem of PA, while, all along, SS0 + SS0 = SSS0 was a theorem of PA — that idea is meaningful. After all, people are all the time alleging that some statement is a theorem of PA when it really isn’t. That is to say, people make arithmetic mistakes all the time.
That is true. However, if your perception of reality leads you to the thought that there might be an error with SS0 + SS0 = SSSS0, and you can’t find that error, then it is irrational to assume that there actually is an error with SS0 + SS0 = SSSS0 rather than with your perception of rationality or the concept of applying SS0 + SS0 = SSSS0 to reality.
Can we agree on that?
I think so, if I understand you. But I think that you’re referring to a more restricted class of “perceptions of reality” than Eliezer is.
In the kind of scenario that Eliezer is talking about, your perceptions of reality include seeming to find an error in the alleged proof that SS0 + SS0 = SSSS0 (and confirming your perception of an error sufficiently many times to outweigh all the times when you thought you’d confirmed that the alleged proof was valid). If that is the kind of “perception of reality” that we’re talking about, then you should conclude that there was an error in the alleged proof of SS0 + SS0 = SSSS0.
That is all good and valid, and of course I don’t believe in any results of deductions with errors in them just based on said deductions. But that has nothing to do with reality. Two pebbles plus two pebbles resulting in three pebbles is not what convinces me that SS0 + SS0 = SSS0; finding the error is, which is nothing that is perceived (i.e. it is purely abstract).
If we’re defining “situation” in a way similar to how it’s used in the top-level post (pebbles and stuff), then there simply can’t exist a situation that could convince me that SS0 + SS0 = SSSS0 is wrong in peano arithmetic. It might convince me to check peano arithmetic, of course, but that’s all.
I try to not argue about definition of words, but it just seems to me that as soon as you define words like “perception”, “situation”, “believe” etcetera in a way that would result in a situation capable of convincing me that SS0 + SS0 = SSS0 is true in peano arithmetic, we are not talking about reality anymore.
Okay, I just thought of a possible situation that would indeed “convince” me of 2 + 2 = 3: Disable the module of my brain responsible for logical reasoning, then show me some stage magic involving pebbles or earplugs, and then my poor rationalization module would probably end up with some explanation along the lines of 2 + 2 = 3.
But let’s not go there.