I don’t think you’re just rationalizing. I think this is exactly what the philosophy of mathematics needs in fact.
If we really understand the foundations of mathematics, Godel’s theorems should seem to us, if not irrelevant, then perfectly reasonable—perhaps even trivially obvious (or at least trivially obvious in hindsight, which is of course not the same thing), the way that a lot of very well-understood things seem to us.
In my mind I’ve gotten fairly close to this point, so maybe this will help: By being inside the system, you’re always going to get “paradoxes” of self-reference that aren’t really catastrophes.
For example, I cannot coherently and honestly assert this statement: “It is raining in Bangladesh but Patrick Julius does not believe that.” The statement could in fact be true. It has often been true many times in the past. But I can’t assert it, because I am part of it, and part of what it says is that I don’t believe it, and hence can’t assert it.
Likewise, Godel’s theorems are a way of making number theory talk about itself and say things like “Number theory can’t prove this statement”; well, of course it can’t, because you made the statement about number theory proving things.
There is a further subtlety here. As I discussed in “Syntacticism”, in Gödel’s theorems number theory is in fact talking about “number theory”, and we apply a metatheory to prove that “number theory is “number theory”″, and think we’ve proved that number theory is “number theory”. The answer I came to was to conclude that number theory isn’t talking about anything (ie. ascription of semantics to mathematics does not reflect any underlying reality), it’s just a set of symbols and rules for manipulating same, and that those symbols and rules together embody a Platonic object. Others may reach different conclusions.
I don’t think you’re just rationalizing. I think this is exactly what the philosophy of mathematics needs in fact.
If we really understand the foundations of mathematics, Godel’s theorems should seem to us, if not irrelevant, then perfectly reasonable—perhaps even trivially obvious (or at least trivially obvious in hindsight, which is of course not the same thing), the way that a lot of very well-understood things seem to us.
In my mind I’ve gotten fairly close to this point, so maybe this will help: By being inside the system, you’re always going to get “paradoxes” of self-reference that aren’t really catastrophes.
For example, I cannot coherently and honestly assert this statement: “It is raining in Bangladesh but Patrick Julius does not believe that.” The statement could in fact be true. It has often been true many times in the past. But I can’t assert it, because I am part of it, and part of what it says is that I don’t believe it, and hence can’t assert it.
Likewise, Godel’s theorems are a way of making number theory talk about itself and say things like “Number theory can’t prove this statement”; well, of course it can’t, because you made the statement about number theory proving things.
There is a further subtlety here. As I discussed in “Syntacticism”, in Gödel’s theorems number theory is in fact talking about “number theory”, and we apply a metatheory to prove that “number theory is “number theory”″, and think we’ve proved that number theory is “number theory”. The answer I came to was to conclude that number theory isn’t talking about anything (ie. ascription of semantics to mathematics does not reflect any underlying reality), it’s just a set of symbols and rules for manipulating same, and that those symbols and rules together embody a Platonic object. Others may reach different conclusions.