I’m pretty sympathetic to this line of thought but haven’t made big updates based on these arguments (aside from preferring EDT/conditionals over CDT/counterfactuals for reasons similar to the OP). Some of my reasons:
On the other hand, the idea that the mathematical facts live even partially outside the universe is ontologically and epistemically questionable. How would we access these mathematical facts, if our behaviors are determined by physics? Why even assume they exist, when all we see is in the universe, not anything outside of it?
This argument (and the analogous one for moral non-realism) isn’t very convincing to me, because it doesn’t seem highly problematic that we can access mathematical facts that “live partially outside the universe” via “reasoning” or “logical correlation”, where the computations in our minds are entangled in some way with computations or math that we’re not physically connected to. Maybe the easiest way to see this is with the example of using one algorithm running on one computer to predict the output of a different algorithm running on a physically disconnected computer, or even predict a computation that exists in a different (real or hypothetical) universe.
One could still argue for non-realism/formalism by appealing to ontological minimality, i.e., let’s not assume the existence of mathematical structures or facts unless there are good reasons to, but I feel like the arguments in favor of some types of mathematical realism/platonism (e.g., universe and multiverse views of set theory) are actually fairly strong (and most working mathematicians and philosophers of math are realists probably for good reasons). For example one line of argument is that when mathematicians reason about math outside of a formal system, e.g. large cardinals, their reasoning still seem to be coherent and about something real.
Another reason I’m not ready to be super-convinced in this direction is I think philosophy is often very hard and slow, therefore as you say “It is somewhat questionable to infer from lack of success to define, say, optimal decision theories, that no such decision theory exists.”
Another more pragmatic reason is related to Ontological Crisis in Humans, namely if we currently have some entities in our ontologies that our values are expressed in terms of, and we’re not sure whether we’ll eventually keep them when we’re philosophically mature, and we don’t know how to translate these values to a new ontology that lack these entities, it seems better to keep them for now rather than to remove them (and only add them back later when we find good reasons to), because removing them and their associated values might constitute a form of value drift that we should want to prevent. See also Beware Selective Nihilism where I warned about something similar.
One could still argue for non-realism/formalism by appealing to ontological minimality, i.e., let’s not assume the existence of mathematical structures or facts unless there are good reasons to, but I feel like the arguments in favor of some types of mathematical realism/platonism (e.g., universe and multiverse views of set theory) are actually fairly strong
What are they, then? (I mean, I am familiar with the standard ones, and don’t find them convincing).
(and most working mathematicians and philosophers of math are realists probably for good reasons). For example one line of argument is that when mathematicians reason about math outside of a formal system, e.g. large cardinals, their reasoning still seem to be coherent and about something real.
What does “seems” mean here? Literally their subjective feeling about what they are doing?
And what does coherence have to do with reality? Surely, you can have coherent fictions.
it doesn’t seem highly problematic that we can access mathematical facts that “live partially outside the universe” via “reasoning” or “logical correlation”, where the computations in our minds are entangled in some way with computations or math that we’re not physically connected to.
While this is one way to think about, it seems first of all that it is limited to “small” mathematical facts that are computable in physics (not stuff like the continuum hypothesis). With respect to the entanglement, while it’s possible to have a Bayes net where the mathematical fact “causes” both computers to output the answers, there’s an alternative approach where the computers are two material devices that output the same answer because of physical symmetry. Two processes having symmetrical outputs doesn’t in general indicate they’re “caused by the same thing”.
arguments in favor of some types of mathematical realism/platonism (e.g., universe and multiverse views of set theory)
Not familiar with these arguments. I think a formalist approach would be, the consistency of ZFC already implies a bunch of “small” mathematical facts (e.g. ZFC can’t prove any false Π1 arithmetic statements). I think it’s pretty hard to find a useful a formal system that is strictly finitist, however my intuition is that set theory goes too far. (This is part of why I have been recently thinking about “reverse mathematics”, relatively weak second-order arithmetic theories like WKL0)
Another reason I’m not ready to be super-convinced in this direction is I think philosophy is often very hard and slow, therefore as you say “It is somewhat questionable to infer from lack of success to define, say, optimal decision theories, that no such decision theory exists.”
Yeah that makes sense. I think maybe what I’ve become more reluctant to endorse over time, is a jump from “an intuition that something here works, plus alternative solutions failing” to “here, this thing I came up with or something a lot like it is going to work”. Like going from failure of CDT to success of EDT, or failure of CDT+EDT to TDT. There is not really any assurance that the new thing will work either.
we’re not sure whether we’ll eventually keep them when we’re philosophically mature, and we don’t know how to translate these values to a new ontology that lack these entities
I see this is a practical consideration in many value systems, although perhaps either (a) the pragmatic considerations go differently for different people, (b) different systems could be used for different pragmatic purposes. It at least presents a case for explaining the psychological phenomena of different ontologies/values even ones that might fail in physicalism.
I’m pretty sympathetic to this line of thought but haven’t made big updates based on these arguments (aside from preferring EDT/conditionals over CDT/counterfactuals for reasons similar to the OP). Some of my reasons:
This argument (and the analogous one for moral non-realism) isn’t very convincing to me, because it doesn’t seem highly problematic that we can access mathematical facts that “live partially outside the universe” via “reasoning” or “logical correlation”, where the computations in our minds are entangled in some way with computations or math that we’re not physically connected to. Maybe the easiest way to see this is with the example of using one algorithm running on one computer to predict the output of a different algorithm running on a physically disconnected computer, or even predict a computation that exists in a different (real or hypothetical) universe.
One could still argue for non-realism/formalism by appealing to ontological minimality, i.e., let’s not assume the existence of mathematical structures or facts unless there are good reasons to, but I feel like the arguments in favor of some types of mathematical realism/platonism (e.g., universe and multiverse views of set theory) are actually fairly strong (and most working mathematicians and philosophers of math are realists probably for good reasons). For example one line of argument is that when mathematicians reason about math outside of a formal system, e.g. large cardinals, their reasoning still seem to be coherent and about something real.
Another reason I’m not ready to be super-convinced in this direction is I think philosophy is often very hard and slow, therefore as you say “It is somewhat questionable to infer from lack of success to define, say, optimal decision theories, that no such decision theory exists.”
Another more pragmatic reason is related to Ontological Crisis in Humans, namely if we currently have some entities in our ontologies that our values are expressed in terms of, and we’re not sure whether we’ll eventually keep them when we’re philosophically mature, and we don’t know how to translate these values to a new ontology that lack these entities, it seems better to keep them for now rather than to remove them (and only add them back later when we find good reasons to), because removing them and their associated values might constitute a form of value drift that we should want to prevent. See also Beware Selective Nihilism where I warned about something similar.
What are they, then? (I mean, I am familiar with the standard ones, and don’t find them convincing).
What does “seems” mean here? Literally their subjective feeling about what they are doing?
And what does coherence have to do with reality? Surely, you can have coherent fictions.
While this is one way to think about, it seems first of all that it is limited to “small” mathematical facts that are computable in physics (not stuff like the continuum hypothesis). With respect to the entanglement, while it’s possible to have a Bayes net where the mathematical fact “causes” both computers to output the answers, there’s an alternative approach where the computers are two material devices that output the same answer because of physical symmetry. Two processes having symmetrical outputs doesn’t in general indicate they’re “caused by the same thing”.
Not familiar with these arguments. I think a formalist approach would be, the consistency of ZFC already implies a bunch of “small” mathematical facts (e.g. ZFC can’t prove any false Π1 arithmetic statements). I think it’s pretty hard to find a useful a formal system that is strictly finitist, however my intuition is that set theory goes too far. (This is part of why I have been recently thinking about “reverse mathematics”, relatively weak second-order arithmetic theories like WKL0)
Yeah that makes sense. I think maybe what I’ve become more reluctant to endorse over time, is a jump from “an intuition that something here works, plus alternative solutions failing” to “here, this thing I came up with or something a lot like it is going to work”. Like going from failure of CDT to success of EDT, or failure of CDT+EDT to TDT. There is not really any assurance that the new thing will work either.
I see this is a practical consideration in many value systems, although perhaps either (a) the pragmatic considerations go differently for different people, (b) different systems could be used for different pragmatic purposes. It at least presents a case for explaining the psychological phenomena of different ontologies/values even ones that might fail in physicalism.