I agree that it can take a long time to prove simple things. But my claim is that one has to be very stupid to think 1+1=3, not so with the falsity of the Orthogonality thesis.
I agree that it can take a long time to prove simple things. But my claim is that one has to be very stupid to think 1+1=3
Or one might be working from different axioms. I don’t know what axioms, and I’d look at you funny until you explained, but I can’t rule it out. It’s possible (though implausible given its length) that Principia Mathematica wasn’t thorough enough, that it snuck in a hidden axiom that—if challenged—would reveal an equally-coherent alternate counting system in which 1+1=3.
I brought up Euclid’s postulates as an example of a time this actually happened. It seems obvious that “two lines that are parallel to the same line are also parallel to each other,” but in fact it only holds in Euclidean geometry. To quote the Wikipedia article on the topic,
Many other statements equivalent to the parallel postulate have been suggested, some of them appearing at first to be unrelated to parallelism, and some seeming so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid’s other postulates.
“So self-evident that they were unconsciously assumed.” But it turned out, you can’t prove the parallel postulate (or any equivalent postulate) from first principles, and there were a number of equally-coherent geometries waiting to be discovered once we started questioning it.
My advice is to be equally skeptical of claims of absolute morality. I agree you can derive human morality if you assume that sentience is good, happiness is good, and so on. And maybe you can derive those from each other, or from some other axioms, but at some point your moral system does have axioms. An intelligent being that didn’t start from these axioms could likely derive a coherent moral system that went against most of what humans consider good.
Summary: you’re speculating, based on your experience as an intelligent human, that an intelligent non-human would deduce a human-like moral system. I’m speculating that it might not. The problem is, neither of us can exactly test this at the moment. The only human-level intelligences we could ask are also human, meaning they have human values and biases baked in.
We all accept similar axioms, but does that really mean those axioms are the only option?
I agree that it can take a long time to prove simple things. But my claim is that one has to be very stupid to think 1+1=3, not so with the falsity of the Orthogonality thesis.
Or one might be working from different axioms. I don’t know what axioms, and I’d look at you funny until you explained, but I can’t rule it out. It’s possible (though implausible given its length) that Principia Mathematica wasn’t thorough enough, that it snuck in a hidden axiom that—if challenged—would reveal an equally-coherent alternate counting system in which 1+1=3.
I brought up Euclid’s postulates as an example of a time this actually happened. It seems obvious that “two lines that are parallel to the same line are also parallel to each other,” but in fact it only holds in Euclidean geometry. To quote the Wikipedia article on the topic,
“So self-evident that they were unconsciously assumed.” But it turned out, you can’t prove the parallel postulate (or any equivalent postulate) from first principles, and there were a number of equally-coherent geometries waiting to be discovered once we started questioning it.
My advice is to be equally skeptical of claims of absolute morality. I agree you can derive human morality if you assume that sentience is good, happiness is good, and so on. And maybe you can derive those from each other, or from some other axioms, but at some point your moral system does have axioms. An intelligent being that didn’t start from these axioms could likely derive a coherent moral system that went against most of what humans consider good.
Summary: you’re speculating, based on your experience as an intelligent human, that an intelligent non-human would deduce a human-like moral system. I’m speculating that it might not. The problem is, neither of us can exactly test this at the moment. The only human-level intelligences we could ask are also human, meaning they have human values and biases baked in.
We all accept similar axioms, but does that really mean those axioms are the only option?