we may just be cockroaches at the base of a very large garbage mountain
This happened last Thursday, but I've been meaning to comment about it...
Geoffrey Hellman of the University of Minnesota came to give a talk for the phil dept titled "Structuralism and the Open-Endedness of Mathematics", and it gave me a jolt. The coolest thing for me was that he gave a historical context to an idea I've been propounding maniacally for a while, originating in the question "What are axioms?" One view, is that axioms are assertions about the world, and therefore true or false. This is the view adopted in much misguided philosophy. But an alternative conception of axioms is that they are a set of conditions that define the objects they are about. For example it's not simply a true fact about groups that they have an associative operation with identity and inverses...having these properties is what it means to be a group. Likewise it's a silly question to ask whether the axiom of excluded middle is true -- accepting the excluded middle defines a particular kind of truth (classical), and rejecting it defines another (intuitionistic).
Hellman calls these respectively the "Fregean sense" and the "Hilbertian sense" of axioms. This was a bit surprising because for some reason -- probably from reading too much Girard -- I was operating on the assumption that Hilbert used the "Fregean sense". (In retrospect it makes sense though, given the whole formalization of mathematics project. I think I was confusing criticism of Hilbert's approach to logic, which is an ugly hack, with criticism of his approach to philosophy.) Hellman also claims that Dedekind promoted axioms-as-defining-conditions, and that Russell criticized him for this somewhere...
"Structuralism" can be summarized, Hellman says, as the project of trying to convert as many axioms as possible from the Fregean sense to the Hilbertian sense. Then I'm a structuralist! But he also says "as many as possible" -- we can't convert them all! This is where he gets weird. At some level he's probably right -- we have to be able to bootstrap our axioms to get something useful. But his assumption (which he waved hands a bit about when questioned) was that mathematical objects have to really "exist" at some point or else we fall into "deductivism" and "mathematics is just symbol manipulation". (Oh, if there were one word that I could banish from the Earth it would be you, "exist" -- the bane of my existence!) His talk actually examined different frameworks for the philosophy of mathematics -- set theory, category theory, "sui generis", and "modal" -- and at the end he showed how you can use a modal system to prove the existence of many things. This was kind of cool, but I had the foreboding that he was developing this theory out of misdirected discomfort. Maybe I'm totally wrong.
In the question session following the talk, for a while Dana Scott lay dormant (well not literally dormant, but quiet), then he spoke up, slowly but at length, you could tell carefully choosing his words, and producing two memorable quotes, which I'll reproduce as aphorisms:
1. "Mathematics is a method rather than a religion."
2. "We may just be cockroaches at the base of a very large garbage mountain."
Geoffrey Hellman of the University of Minnesota came to give a talk for the phil dept titled "Structuralism and the Open-Endedness of Mathematics", and it gave me a jolt. The coolest thing for me was that he gave a historical context to an idea I've been propounding maniacally for a while, originating in the question "What are axioms?" One view, is that axioms are assertions about the world, and therefore true or false. This is the view adopted in much misguided philosophy. But an alternative conception of axioms is that they are a set of conditions that define the objects they are about. For example it's not simply a true fact about groups that they have an associative operation with identity and inverses...having these properties is what it means to be a group. Likewise it's a silly question to ask whether the axiom of excluded middle is true -- accepting the excluded middle defines a particular kind of truth (classical), and rejecting it defines another (intuitionistic).
Hellman calls these respectively the "Fregean sense" and the "Hilbertian sense" of axioms. This was a bit surprising because for some reason -- probably from reading too much Girard -- I was operating on the assumption that Hilbert used the "Fregean sense". (In retrospect it makes sense though, given the whole formalization of mathematics project. I think I was confusing criticism of Hilbert's approach to logic, which is an ugly hack, with criticism of his approach to philosophy.) Hellman also claims that Dedekind promoted axioms-as-defining-conditions, and that Russell criticized him for this somewhere...
"Structuralism" can be summarized, Hellman says, as the project of trying to convert as many axioms as possible from the Fregean sense to the Hilbertian sense. Then I'm a structuralist! But he also says "as many as possible" -- we can't convert them all! This is where he gets weird. At some level he's probably right -- we have to be able to bootstrap our axioms to get something useful. But his assumption (which he waved hands a bit about when questioned) was that mathematical objects have to really "exist" at some point or else we fall into "deductivism" and "mathematics is just symbol manipulation". (Oh, if there were one word that I could banish from the Earth it would be you, "exist" -- the bane of my existence!) His talk actually examined different frameworks for the philosophy of mathematics -- set theory, category theory, "sui generis", and "modal" -- and at the end he showed how you can use a modal system to prove the existence of many things. This was kind of cool, but I had the foreboding that he was developing this theory out of misdirected discomfort. Maybe I'm totally wrong.
In the question session following the talk, for a while Dana Scott lay dormant (well not literally dormant, but quiet), then he spoke up, slowly but at length, you could tell carefully choosing his words, and producing two memorable quotes, which I'll reproduce as aphorisms:
1. "Mathematics is a method rather than a religion."
2. "We may just be cockroaches at the base of a very large garbage mountain."