I concede (a little)!
In a previous Overcoming Bias post we learned that people sometimes believe the conjunction of events R and Q is more probable than event Q alone. Thus people can believe simple and strictly illogical things, and so I shouldn't throw around the word "unthinkable."
If I stretch my imagination, I can just maybe imagine this sort of logical blunder with small integers.
I draw the line at P AND ~P, though: just unthinkable.
This sentence of Eliezer's is where the action is:
I'm suspicious of claims that supposedly do not require justification and yet seem to be uniquely preferred within a rather large space of possibilities.
"There are no married bachelors" gets us to nod our heads because we uniquely prefer English syntax and semantics. We pick it out of the rather large space of possible languages because it's what everyone else is doing.
If Eliezer went around earnestly saying, "there are some married bachelors," I would guess he had entangled himself with an environment where people go around saying such things, with a different possible language.
Eliezer insists that he could be entangled with evidence such that he believes "there are some married bachelors" is true in English as we know it. I don't think he could; that proposition is unthinkable in good English.
Integers are slippery in a way that apples and poodles are not. If you say something unconventional about integers, you cease to talk about them. --- Does anyone disagree with that?
(1) Peter de Blanc asks what happens when I cannot follow a proof properly. I count that as a failure of rationality rather than an instance of being mislead by evidence. That is not, I think, what Eliezer intends when he says "convinced."
(2) If I observe some trick and say, "wow, two and two makes three," then I am dropping the integer system and adopting some other. My "wow" is the same one that we all said when we learned that Euclidean geometry doesn't hold in our universe.
Let me take another crack at this...
I do not believe any situation could ever convince Eliezer that 2+2=3.
If he proclaims "two and two makes three," then he must be talking about something other than the integers. You cannot be mistaken about the integers, you can only misunderstand them. It's like saying "some women are bachelors." You are not mistaken about the world, you've merely lost your grasp of the terminology.
I am confused by this discussion. Are we talking about integers or things?
Analytic truths may or may not correspond to our situations. When they don't correspond, I guess that's what you all are calling "false." So, if we're engineers working on building a GPS system, I might say to you, "Careful now, Euclidean geometry is false."
Similarly, quantum physicists on the job might say, "Watch out now, two and two isn't necessarily four."
I'm thinking of this excellent blog post I came across last week:
...Consider a basket with 2 apples in it. Now toss in 2 more apples. Examine the basket, and you will find (surprise!) 4 apples. However, you cannot prove a priori that there will be 4 apples in the basket. It is an empirical question, albeit a trivial one, whether baskets of apples (which are physical things) behave in the same manner as the non-negative integers under addition (which is an abstract logical construct).
This distinction might seem hopelessly pedantic at first, but you can easily go astray by ignoring it. For example, many people naively expect photons to behave in the same manner as integers under addition, but they don’t. “Number of photons” is not a conserved quantity in the way that “number of apples” is; photons can be created/destroyed, one photon can be split into two, et cetera....