Re: Philosophy as interminable debate, another way to put the relationship between math and philosophy:
Philosophy as weakly verifiable argumentation
Math is solving problems by looking at the consequences of a small number of axiomatic reasoning steps. For something to be math, we have to be able to ultimately cash out any proof as a series of these reasoning steps. Once something is cashed out in this way, it takes a small constant amount of time to verify any reasoning step, so we can verify given polynomial time.
Philosophy is solving problems where we haven’t figured out a set of axiomatic reasoning steps. Any non-axiomatic reasoning step we propose could end up having arguments that we hadn’t thought of that would lead us to reject that step. And those arguments themselves might be undermined by other arguments, and so on. Each round of debate lets us add another level of counter-arguments. Philosophers can make progress when they have some good predictor of whether arguments are good or not, but they don’t have access to certain knowledge of arguments being good.
Another difference between mathematics and philosophy is that in mathematics we have a well defined set of objects and a well-defined problem we are asking about. Whereas in philosophy we are trying to ask questions about things that exist in the real world and/or we are asking questions that we haven’t crisply defined yet.
When we come up with a set of axioms and a description of a problem, we can move that problem from the realm of philosophy to the realm of mathematics. When we come up with some method we trust of verifying arguments (ie. replicating scientific experiments), we can move problems out of philosophy to other sciences.
It could be the case that philosophy grounds out in some reasonable set of axioms which we don’t have access to now for computational reasons—in which case it could all end up in the realm of mathematics. It could be the case that, for all practical purposes, we will never reach this state, so it will remain in the “potentially unbounded DEBATE round case”. I’m not sure what it would look like if it could never ground out—one model could be that we have a black box function that performs a probabilistic evaluation of argument strength given counter-arguments, and we go through some process to get the consequences of that, but it never looks like “here is a set of axioms”.
Re: Philosophy as interminable debate, another way to put the relationship between math and philosophy:
Philosophy as weakly verifiable argumentation
Math is solving problems by looking at the consequences of a small number of axiomatic reasoning steps. For something to be math, we have to be able to ultimately cash out any proof as a series of these reasoning steps. Once something is cashed out in this way, it takes a small constant amount of time to verify any reasoning step, so we can verify given polynomial time.
Philosophy is solving problems where we haven’t figured out a set of axiomatic reasoning steps. Any non-axiomatic reasoning step we propose could end up having arguments that we hadn’t thought of that would lead us to reject that step. And those arguments themselves might be undermined by other arguments, and so on. Each round of debate lets us add another level of counter-arguments. Philosophers can make progress when they have some good predictor of whether arguments are good or not, but they don’t have access to certain knowledge of arguments being good.
Another difference between mathematics and philosophy is that in mathematics we have a well defined set of objects and a well-defined problem we are asking about. Whereas in philosophy we are trying to ask questions about things that exist in the real world and/or we are asking questions that we haven’t crisply defined yet.
When we come up with a set of axioms and a description of a problem, we can move that problem from the realm of philosophy to the realm of mathematics. When we come up with some method we trust of verifying arguments (ie. replicating scientific experiments), we can move problems out of philosophy to other sciences.
It could be the case that philosophy grounds out in some reasonable set of axioms which we don’t have access to now for computational reasons—in which case it could all end up in the realm of mathematics. It could be the case that, for all practical purposes, we will never reach this state, so it will remain in the “potentially unbounded DEBATE round case”. I’m not sure what it would look like if it could never ground out—one model could be that we have a black box function that performs a probabilistic evaluation of argument strength given counter-arguments, and we go through some process to get the consequences of that, but it never looks like “here is a set of axioms”.