I fear that I might be currently trapped in this error: I’ve always resented Gödel’s Incompleteness Theorems. When I was about 17 I thought I’d disproved 1IT (turned out I’d just reconstructed the proof of 2IT and missed the detail that Con(T)≠ProvT(Con(T))). It took me about a year after that to realise that, no, I wasn’t going to disprove the ITs no matter how much I wanted to, and I accepted that trying to disprove them anyway would be a crackpot thing to do. Since then I’ve been trying to construct a philosophical framework of mathematics in which the ITs become irrelevant. Have I, in fact, taken the Crackpot Offer?
From your description it looks like you might have. You should retract failed conjectures, not rectify them. Another (less efficient) way to recover is to get expertise in the topic strong enough to sever incorrect intuitions (it doesn’t always work in itself, human “ability” for rationalization is strong too). I think if you know math (specifically logic, algebra and set theory) less than on graduate level, you should either drop what you’re doing, or get to that level.
Well, I’m studying for an undergraduate degree in mathematics at a good university; the “trying to construct...” is just one of several things I do in my copious free time. Also, I’m spending a much smaller proportion of my time on this project than I was spending on trying to disprove the ITs. So it looks to me as though I’m actually behaving rationally, but maybe that’s just how the algorithm looks from the inside.
I think that by “make the ITs become irrelevant” I mean that I’m trying to find a philosophy in which the things that make me want the ITs to be false are no longer represented, because if I have any assumption that implies “And therefore the ITs are false” then that assumption is wrong. But again, is that just me rationalising?
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 fear that I might be currently trapped in this error: I’ve always resented Gödel’s Incompleteness Theorems. When I was about 17 I thought I’d disproved 1IT (turned out I’d just reconstructed the proof of 2IT and missed the detail that Con(T)≠ProvT(Con(T))). It took me about a year after that to realise that, no, I wasn’t going to disprove the ITs no matter how much I wanted to, and I accepted that trying to disprove them anyway would be a crackpot thing to do. Since then I’ve been trying to construct a philosophical framework of mathematics in which the ITs become irrelevant. Have I, in fact, taken the Crackpot Offer?
From your description it looks like you might have. You should retract failed conjectures, not rectify them. Another (less efficient) way to recover is to get expertise in the topic strong enough to sever incorrect intuitions (it doesn’t always work in itself, human “ability” for rationalization is strong too). I think if you know math (specifically logic, algebra and set theory) less than on graduate level, you should either drop what you’re doing, or get to that level.
Well, I’m studying for an undergraduate degree in mathematics at a good university; the “trying to construct...” is just one of several things I do in my copious free time. Also, I’m spending a much smaller proportion of my time on this project than I was spending on trying to disprove the ITs. So it looks to me as though I’m actually behaving rationally, but maybe that’s just how the algorithm looks from the inside.
I think that by “make the ITs become irrelevant” I mean that I’m trying to find a philosophy in which the things that make me want the ITs to be false are no longer represented, because if I have any assumption that implies “And therefore the ITs are false” then that assumption is wrong. But again, is that just me rationalising?
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.