I was initially intrigued to read this because it seemed like you were going to make an interesting case somewhere along the lines of “mathematics involves hermeneutics” because ultimately mathematics is done by humans using a (formal) language that they must interpret before they can generate more mathematics that follows the rules of the language. It seems to me you never quite got there, a stumbled towards some other point that’s not totally clear to me. Forgive me if this is an uncharitable reading, but I read your point as being “look at all the cool stuff we can do because we use formal languages”.
Pointing out that formal languages let us do cool stuff is something I agree with, although it feels a bit obvious. I suspect I’m mostly reacting to having hoped you’d make a stronger case for applying hermeneutic methods and having an attitude of interpretation when dealing with formal systems, since this is a point often ignored when people learn about formal systems by remaking what I might call the “naive positivist” mistake of thinking formal systems describe reality precisely, or put in LW terms, confuse the map for the territory.
Additionally, I found your proof of “There exist a Turing Machine that recognize membership in the language.” somewhat inadequate given what you presented in the text. It’s only thanks to knowing some details about TMs already that this proof is very meaningful to me, and I think the uninitiated reader, whom you seem to be targeting, would have a hard time understanding this proof since you didn’t explain the constraints of TMs. For example, an easy objection to raise to your proof, lacking a clear definition of TMs, would be “but won’t it halt incorrectly if it sees a ‘2’ or a ‘3’ or a ‘c’ on the tape?”
I appreciate your clear writing style. If my comments seem harsh it’s because you got my hopes up and then delivered something less than what you initially lead me to expect.
A few comments.
I was initially intrigued to read this because it seemed like you were going to make an interesting case somewhere along the lines of “mathematics involves hermeneutics” because ultimately mathematics is done by humans using a (formal) language that they must interpret before they can generate more mathematics that follows the rules of the language. It seems to me you never quite got there, a stumbled towards some other point that’s not totally clear to me. Forgive me if this is an uncharitable reading, but I read your point as being “look at all the cool stuff we can do because we use formal languages”.
Pointing out that formal languages let us do cool stuff is something I agree with, although it feels a bit obvious. I suspect I’m mostly reacting to having hoped you’d make a stronger case for applying hermeneutic methods and having an attitude of interpretation when dealing with formal systems, since this is a point often ignored when people learn about formal systems by remaking what I might call the “naive positivist” mistake of thinking formal systems describe reality precisely, or put in LW terms, confuse the map for the territory.
Additionally, I found your proof of “There exist a Turing Machine that recognize membership in the language.” somewhat inadequate given what you presented in the text. It’s only thanks to knowing some details about TMs already that this proof is very meaningful to me, and I think the uninitiated reader, whom you seem to be targeting, would have a hard time understanding this proof since you didn’t explain the constraints of TMs. For example, an easy objection to raise to your proof, lacking a clear definition of TMs, would be “but won’t it halt incorrectly if it sees a ‘2’ or a ‘3’ or a ‘c’ on the tape?”
I appreciate your clear writing style. If my comments seem harsh it’s because you got my hopes up and then delivered something less than what you initially lead me to expect.