Let’s see. What else would I have to believe in order to accept a statement like “~(p&~p) is not a theorem in propositional logic?”
A statement of the form “X is a theorem in this particular formal mathematical system” means that I can use the operations allowed within that system to construct a “proof” of the sentence X. In theory, I can make a machine that takes a “proof” as input and returns “true” if the proof is indeed a correct proof and “false” if there is a step in the proof that is not allowed by the formal system. If the machine works as intended, then if the machine says that a proof is a correct one, it really is a correct proof and that statement really is a theorem in that system.
My brain is like such a machine. I can look at a mathematical proof in a formal system and check if the proof really is a correct proof within that system. In order to believe that 2+2=3, I would have to believe that the “theorem checker” module in my brain—that part responsible for deductive reasoning—is not operating properly. What I perceive as a correct step in a proof is, in actuality, an incorrect step, and my brain is hardwired to make that particular kind of mistake without realizing it. In other words, in order to believe that “2+2=4″ is false, I would have to believe that the proof of “2+2=4” in my head does not mean what I think it means.
I would have to believe that I am not capable of correct deductive reasoning. A person not capable of correct deductive reasoning is insane.
If “2+2=4” is false, then I am insane.
All of my beliefs have to come with the background assumption that I am not insane. If I am, in fact, insane, then all my internal models of the universe have to be replaced with the black hole of maximum entropy. I can no more trust my probability estimates than I could trust the probability estimates of a rock.
I choose to act as though I am sane because if I am sane, I gain maximum benefits from that belief, and if I am not sane, it doesn’t matter anyway. I can be convinced I am mistaken about something, but I cannot be convinced that I am insane.
Let’s see. What else would I have to believe in order to accept a statement like “~(p&~p) is not a theorem in propositional logic?”
A statement of the form “X is a theorem in this particular formal mathematical system” means that I can use the operations allowed within that system to construct a “proof” of the sentence X. In theory, I can make a machine that takes a “proof” as input and returns “true” if the proof is indeed a correct proof and “false” if there is a step in the proof that is not allowed by the formal system. If the machine works as intended, then if the machine says that a proof is a correct one, it really is a correct proof and that statement really is a theorem in that system.
My brain is like such a machine. I can look at a mathematical proof in a formal system and check if the proof really is a correct proof within that system. In order to believe that 2+2=3, I would have to believe that the “theorem checker” module in my brain—that part responsible for deductive reasoning—is not operating properly. What I perceive as a correct step in a proof is, in actuality, an incorrect step, and my brain is hardwired to make that particular kind of mistake without realizing it. In other words, in order to believe that “2+2=4″ is false, I would have to believe that the proof of “2+2=4” in my head does not mean what I think it means.
I would have to believe that I am not capable of correct deductive reasoning. A person not capable of correct deductive reasoning is insane.
If “2+2=4” is false, then I am insane.
All of my beliefs have to come with the background assumption that I am not insane. If I am, in fact, insane, then all my internal models of the universe have to be replaced with the black hole of maximum entropy. I can no more trust my probability estimates than I could trust the probability estimates of a rock.
I choose to act as though I am sane because if I am sane, I gain maximum benefits from that belief, and if I am not sane, it doesn’t matter anyway. I can be convinced I am mistaken about something, but I cannot be convinced that I am insane.