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I have moose on my feet!
Mar. 15th, 2006 09:23 pmJoel's Christmas/birthday gift from Andrea, Matt, and Owen arrived yesterday - and with it Andrea tucked in an early birthday present for me: socks with moose on them! I suppose she had to retaliate for the "Amsterdam" socks I sent her for Christmas. So now I have moose on my feet. :) I love getting neat socks. For a few years I never had to buy myself any new socks because the ones I bought in early high school were still lasting, and then my Grandma Friedemann would use pretty much any minor holiday as an excuse to send a package of candy and socks. I'm really going to miss those socks now that she's dead. :(
On a completely unrelated topic, constructive set theory is *so* *weird*. Today's seminar was the last on the topic, and I can confess to be more baffled by it than I was the first week of class, when I kicked things off by giving the first presentation.
I find it funny that intuitionistic logic is so unintuitive. I'm sure that part of this is because I've been trained from the start as a classical logician - every proposition is either true or false, every natural number is either even or odd, there are just as many even numbers as there are odd numbers, and there are a heck of a lot more real numbers. Countable infinities and uncountable infinities? Not a problem. Inaccessible cardinals? Sure thing. Classical mathematics just makes sense - and I still think this way even though I'm not nearly as strong a Platonist as I used to be. Right now I'm definitely much more of a mathematician than a philosopher - I don't care whether the things that we call numbers actually *exist*, I just care whether I can prove things from certain other things. Whether or not this reflects any real "deep structure" of the world - I just don't care.
So, given that, I think that's part of the reason why I don't find the constructive/intuitionistic viewpoint all that appealing. It's not because I think there's some Platonic universe out there were all these statements actually are true. However, given that we have these very nice sets of axioms that work very well in allowing us to prove very nice results about mathematics and logic and set theory, why not also go with the assumption that there is a mapping from propositions to the truth values? Why not do that, and accept the fact that sometimes, we may not be able to determine what truth value a certain proposition has? If we can accept Choice, then a truth value for every proposition seems hardly worth a second thought. Why should that be so frightening?
Because it seems like to the intuitionist or the constructivist, it is frightening. You give an argument of the form "suppose, for all x phi(x). Oh, but look - that gets us a contradiction. So it must be the case that exists (x) not phi(x)." Great! We know something we didn't know before. But the intuitionist is unhappy. He wrings his hands and cries, "But, but, but, I need a witness! Which object is it that doesn't have the property phi? Tell me!" What is so much better about actually being able to name the object that you've just proved something about? What's so much better about having a method for *constructing* the object? What do you get, if you drop 'p or not p', and go the intuitionistic/constructivist route?
You get really bizarre things. You get numbers which are not provably equal to any real number. You get Dedekind reals which are not Cauchy reals. You get the fact that the power set of 1 is so huge, you can't even count it - and yet, you have the set of functions from 1 into 2, all two of them, though somehow you can't use this to construct the power set. You get really weird axioms like Subset Collection. When you switch from talking about truth to talking about provability, suddenly everything becomes a whole lot weirder, and after 6 weeks or so of different presentations on the topic, I'm far less convinced I understand what you gain than I was when we started.
On a completely unrelated topic, constructive set theory is *so* *weird*. Today's seminar was the last on the topic, and I can confess to be more baffled by it than I was the first week of class, when I kicked things off by giving the first presentation.
I find it funny that intuitionistic logic is so unintuitive. I'm sure that part of this is because I've been trained from the start as a classical logician - every proposition is either true or false, every natural number is either even or odd, there are just as many even numbers as there are odd numbers, and there are a heck of a lot more real numbers. Countable infinities and uncountable infinities? Not a problem. Inaccessible cardinals? Sure thing. Classical mathematics just makes sense - and I still think this way even though I'm not nearly as strong a Platonist as I used to be. Right now I'm definitely much more of a mathematician than a philosopher - I don't care whether the things that we call numbers actually *exist*, I just care whether I can prove things from certain other things. Whether or not this reflects any real "deep structure" of the world - I just don't care.
So, given that, I think that's part of the reason why I don't find the constructive/intuitionistic viewpoint all that appealing. It's not because I think there's some Platonic universe out there were all these statements actually are true. However, given that we have these very nice sets of axioms that work very well in allowing us to prove very nice results about mathematics and logic and set theory, why not also go with the assumption that there is a mapping from propositions to the truth values? Why not do that, and accept the fact that sometimes, we may not be able to determine what truth value a certain proposition has? If we can accept Choice, then a truth value for every proposition seems hardly worth a second thought. Why should that be so frightening?
Because it seems like to the intuitionist or the constructivist, it is frightening. You give an argument of the form "suppose, for all x phi(x). Oh, but look - that gets us a contradiction. So it must be the case that exists (x) not phi(x)." Great! We know something we didn't know before. But the intuitionist is unhappy. He wrings his hands and cries, "But, but, but, I need a witness! Which object is it that doesn't have the property phi? Tell me!" What is so much better about actually being able to name the object that you've just proved something about? What's so much better about having a method for *constructing* the object? What do you get, if you drop 'p or not p', and go the intuitionistic/constructivist route?
You get really bizarre things. You get numbers which are not provably equal to any real number. You get Dedekind reals which are not Cauchy reals. You get the fact that the power set of 1 is so huge, you can't even count it - and yet, you have the set of functions from 1 into 2, all two of them, though somehow you can't use this to construct the power set. You get really weird axioms like Subset Collection. When you switch from talking about truth to talking about provability, suddenly everything becomes a whole lot weirder, and after 6 weeks or so of different presentations on the topic, I'm far less convinced I understand what you gain than I was when we started.
