The position that makes the most sense to me is that uncomputable mathematics exist in a platonic sense, we just happen to inhabit a part of the ultimate ensemble that appears computable, but (1) there’s a chance that our universe is actually uncomputable (or alternatively, we’re “simultaneously” in computable and uncomputable parts of the multiverse), and (2) we can use computable machinery to learn some facts about uncomputable math.
The alternative of saying that only computable math exists seems to imply that all of the work done on understanding uncomputable math (such as Turing degrees and the like) is merely a game of manipulating meaningless symbols. In other words, those mathematicians who think they are reasoning about uncomputable math are not actually doing anything meaningful. That is hard to swallow.
What do you mean by computable or uncomputable math? In one sense, all math is computable (you can enumerate all valid proofs in a formal system, even if it talks about something mind-bogglingly huge like Grothendieck universes). In another sense, all math is uncomputable (you can’t enumerate all true facts about the integers, never mind more complex stuff). Is there some well-defined third way of cashing out the concept?
I don’t have an answer for that, since I’m not proposing to make a distinction based on computable vs uncomputable math. But I think Juergen Schmidhuber has defended some version of “only computable math exists” on the everything-list, so you can try searching the archives there.
It seems like your conception is isomorphic to mine to the extent that the ultimate ensemble can be considered as a set of counterfactual (with respect to local physical law) universes. You’ll see below that I brought up Zeno machines, and noted that they seem to follow this paradigm. Does this seem accurate?
The alternative of saying that only computable math exists seems to imply that all of the work done on understanding uncomputable math (such as Turing degrees and the like) is merely a game of manipulating meaningless symbols. In other words, those mathematicians who think they are reasoning about uncomputable math are not actually doing anything meaningful. That is hard to swallow.
Even the most abstract symbol manipulation can turn out to be functionally equivalent to a more concrete mode of reasoning and hence be quite useful. It is still an exploration of some sort of algorithmic reasoning, and coming from the view that mathematics is something like the art of predicting the outcome of various manipulations of structured data, such abstract games are not totally invalid.
My intuition is currently along the lines of thinking that uncomputable mathematics is disguised computable math, since the manipulation and creation of the system itself is computable (again, given the assumption that the human mind is computable), and so it is perfectly valid but misinterpreted in some sense.
I feel that I should say that my primary motivation for this line of thought is a desire to think about how a machine might create mathematics from scratch. I’ve started reading Simon Colton’s work, it seems quite interesting. Perhaps you could offer some insight into this sort of issue?
The position that makes the most sense to me is that uncomputable mathematics exist in a platonic sense, we just happen to inhabit a part of the ultimate ensemble that appears computable, but (1) there’s a chance that our universe is actually uncomputable (or alternatively, we’re “simultaneously” in computable and uncomputable parts of the multiverse), and (2) we can use computable machinery to learn some facts about uncomputable math.
The alternative of saying that only computable math exists seems to imply that all of the work done on understanding uncomputable math (such as Turing degrees and the like) is merely a game of manipulating meaningless symbols. In other words, those mathematicians who think they are reasoning about uncomputable math are not actually doing anything meaningful. That is hard to swallow.
What do you mean by computable or uncomputable math? In one sense, all math is computable (you can enumerate all valid proofs in a formal system, even if it talks about something mind-bogglingly huge like Grothendieck universes). In another sense, all math is uncomputable (you can’t enumerate all true facts about the integers, never mind more complex stuff). Is there some well-defined third way of cashing out the concept?
I don’t have an answer for that, since I’m not proposing to make a distinction based on computable vs uncomputable math. But I think Juergen Schmidhuber has defended some version of “only computable math exists” on the everything-list, so you can try searching the archives there.
It seems like your conception is isomorphic to mine to the extent that the ultimate ensemble can be considered as a set of counterfactual (with respect to local physical law) universes. You’ll see below that I brought up Zeno machines, and noted that they seem to follow this paradigm. Does this seem accurate?
Even the most abstract symbol manipulation can turn out to be functionally equivalent to a more concrete mode of reasoning and hence be quite useful. It is still an exploration of some sort of algorithmic reasoning, and coming from the view that mathematics is something like the art of predicting the outcome of various manipulations of structured data, such abstract games are not totally invalid.
My intuition is currently along the lines of thinking that uncomputable mathematics is disguised computable math, since the manipulation and creation of the system itself is computable (again, given the assumption that the human mind is computable), and so it is perfectly valid but misinterpreted in some sense.
I feel that I should say that my primary motivation for this line of thought is a desire to think about how a machine might create mathematics from scratch. I’ve started reading Simon Colton’s work, it seems quite interesting. Perhaps you could offer some insight into this sort of issue?