I misinterpreted this: “we can never know all the truths of euclidean geometry, but we can still specify euclidean geometry via a set of axioms. Not so for arithmetic.”
Eliezer: There are an infinite number of truths of euclidean geometry. How could our finite brains know them all?
This was not meant to be a profound observation; it was meant to correct Silas, who seemed to think that I was reading some deep significance into our inability to know all the truths of arithmetic. My point was that there are lots of things we can’t know all the truths about, and this was therefore not the feature of arithmetic I was pointing to.
A decision procedure is a finite specification of all truths of euclidean geometry; I can use that finite fact anywhere I could use any truth of geometry. I suppose there is a difference, but even so, it’s the wrong thing to say in a Godelian discussion.
it was meant to correct Silas, who seemed to think that I was reading some deep significance into our inability to know all the truths of arithmetic. My point was that there are lots of things we can’t know all the truths about, and this was therefore not the feature of arithmetic I was pointing to.
Yes, it was. When I and several others pointed out that arithmetic isn’t actually complex, you responded by saying that it is infinitely complex, because it can’t be finitely described, because to do so … you’d have to know all the truths.
Am I misreading that response? If so, how do you reconcile arithmetic’s infinite complexity with the fact that scientists in fact use it to compress discriptions of the world? An infinitely complex entity can’t help to compress your descriptions.
Decidability of Euclidean geometry#Some_decidable_theories).
I don’t know where Landsburg gets the claim that we can know all the truths of arithmetic.
Richard Kennaway:
I don’t know where Landsburg gets the claim that we can know all the truths of arithmetic.
I don’t know where you got the idea that I’d ever make such a silly claim.
I misinterpreted this: “we can never know all the truths of euclidean geometry, but we can still specify euclidean geometry via a set of axioms. Not so for arithmetic.”
Richard: Gotcha. Sorry if it was unclear which part the “not so” referred to.
Note that Landsburg is thus also incorrect in saying “we can never know all the truths of euclidean geometry”.
Eliezer: There are an infinite number of truths of euclidean geometry. How could our finite brains know them all?
This was not meant to be a profound observation; it was meant to correct Silas, who seemed to think that I was reading some deep significance into our inability to know all the truths of arithmetic. My point was that there are lots of things we can’t know all the truths about, and this was therefore not the feature of arithmetic I was pointing to.
A decision procedure is a finite specification of all truths of euclidean geometry; I can use that finite fact anywhere I could use any truth of geometry. I suppose there is a difference, but even so, it’s the wrong thing to say in a Godelian discussion.
Yes, it was. When I and several others pointed out that arithmetic isn’t actually complex, you responded by saying that it is infinitely complex, because it can’t be finitely described, because to do so … you’d have to know all the truths.
Am I misreading that response? If so, how do you reconcile arithmetic’s infinite complexity with the fact that scientists in fact use it to compress discriptions of the world? An infinitely complex entity can’t help to compress your descriptions.