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see, the halting problem is much more appropriate for this audience :) However, there is the calculus of constructions, which, as far as I can tell, tries to b

see, the halting problem is much more appropriate for this audience :) However, there is the calculus of constructions, which, as far as I can tell, tries to be provable in all admissible formulas. It's a totally functional language. So, is the universe totally functional, will it halt? I'm not sure that's the actual question, neither whether the universe is beyond our understanding, that's out of the question, only, does the universe follow rules or is it deterministic and not only in the quantum sense. > all computable mathematical structures (in Gödel's sense) exist - en.wikipedia/wiki/Mathematical_universe_hypothesis That's trivially true, because they exist in the heads of the mathematicians constructing them.

pron 9y ago
> However, there is the calculus of constructions, which, as far as I can tell, tries to be provable in all admissible formulas. It's a totally functional language. So, is the universe totally functional, will it halt? Total programming languages (or more precisely, their term language), is not Turing complete, and is not universal (although such languages, like Agda, are Turing complete, but not in their terms; i.e. their expressions are not Turing complete, but they allow simulating Turing machines using an interaction with the runtime using something called codata). If the universe is finite, then it, too, cannot fully simulate a Turing machine (and the formalism, or language, used to desrcibe it is completely irrelevant; total functional languages aren't just formalisms, and the universe is a true system). > That's trivially true, because they exist in the heads of the mathematicians constructing them. Well, it's not so simple. First, we have to philosophically determine what we mean by "exists". Platonists believe that there exists a world of abstract forms. If we restrict existence to physical existence, we still need to determine what it means for an abstract mathematical structure to exist in the mathematician's mind. Brouwer believed that for those structures to "exist", the mathematicians should be able, at least in principle, to construct them. So only constructive math "exists". On the other hand, Hilbert, also a finitist, said that math is made of two parts: "real" propositions (finite, about the physical universe), and ideal propositions (finite expressions but infinitary objects, those don't "exist" as anything other than the finite formulas that express them). But even in Hilbert's formulation, there are many inifinitary mathematical objects for which no finite description could exist (by a simple cardinality argument). Do those exist? For Hilbert, probably not, but for Platonists, they still do.

pron 9y ago

> However, there is the calculus of constructions, which, as far as I can tell, tries to be provable in all admissible formulas. It's a totally functional language. So, is the universe totally functional, will it halt? Total programming languages (or more precisely, their term language), is not Turing complete, and is not universal (although such languages, like Agda, are Turing complete, but not in their terms; i.e. their expressions are not Turing complete, but they allow simulating Turing machines using an interaction with the runtime using something called codata). If the universe is finite, then it, too, cannot fully simulate a Turing machine (and the formalism, or language, used to desrcibe it is completely irrelevant; total functional languages aren't just formalisms, and the universe is a true system). > That's trivially true, because they exist in the heads of the mathematicians constructing them. Well, it's not so simple. First, we have to philosophically determine what we mean by "exists". Platonists believe that there exists a world of abstract forms. If we restrict existence to physical existence, we still need to determine what it means for an abstract mathematical structure to exist in the mathematician's mind. Brouwer believed that for those structures to "exist", the mathematicians should be able, at least in principle, to construct them. So only constructive math "exists". On the other hand, Hilbert, also a finitist, said that math is made of two parts: "real" propositions (finite, about the physical universe), and ideal propositions (finite expressions but infinitary objects, those don't "exist" as anything other than the finite formulas that express them). But even in Hilbert's formulation, there are many inifinitary mathematical objects for which no finite description could exist (by a simple cardinality argument). Do those exist? For Hilbert, probably not, but for Platonists, they still do.