These are some thoughts in response to the excellent set of comments by Justin Clarke-Doane (a PhD student at NYU doing fascinating work on disagreement in mathematics--he has a pertinent paper on his homepage for those interested) in the thread on my paper on "Moral Skepticism and Moral Disagreement in Nietzsche." Clarke-Doane raised so many interesting issues that they deserve their own posts, on which I hope he and others can then comment. (I'll refer to him hereafter as JCD.)
In my paper, I had argued from the fact of "persistent and apparently intractable disaggrement on foundational questions" in moral theory to the conclusion that we ought to be skeptical about the existence of moral facts or properties. JCD claims that we find the same kind of disagreement in mathematics. This might, of course, lead us to skepticism about mathematical facts or it might lead us to worry about this strategy of argument. JCD doesn't take a position on that issue here. He wants, in his first set of comments, to make the case that the situation, when it comes to disagreement, is the same in math as I claim it is in ethics. (His second set of comments raise a different set of issues, to which I'll turn in a second post.)
As an example of "persistent" disagreement in mathematics, JCD notes that, "There has been disagreement over the axioms for set-theory since their formulation."
As evidence that this disagreement is "apparently intractable," JCD notes that "there is not merely disaggrement over the truth-values of mathematical sentences, but also disagreement over what would count as evidence for those sentences' truth or falsity." JCD gives the examples of disagreement over the Choice and Replacement axioms.
As evidence that this disagreement has a "foundational character," JCD notes that disagreement over the axioms for set-theory is "epistemically foundational" for mathematics, since these are "the only serious candidates which might be thought to justify axioms which imply them." Such disagreement is also "metaphysically foundational" since "the axioms of our 'explanatorily' fundamental theory, set theory" are the metaphysical foundation of mathematics.
JCD has the significant advantage here of knowing a lot more mathematics than I do, though I hope to hear from readers also conversant in the underlying mathematical debates. I don't want to take issue with the question of whether the putative disagreements in questions are "foundational" (other readers are welcome to do so); I do want to pose some questions about whether they are "persistent" and "apparently intractable."
Set theory dates from the late 19th-century, so has been subject to about 140 years of development and dispute by mathematicians. How does the state of disagreement today compare to 100 years ago? To 50 years ago? To 25 years ago? Reading the SEP entry on set theory , one is left with the impression of a progressive discipline with gradual agreement on many basic ideas. Why is this? Could any entry on the foundations of morality read like the SEP entry on set theory?
The latter, purely sociological, observations bear on the question of intractability. It appears that many foundational issues in set theory have been resolved since the 1870s. (Is there any foundational issue about the criteria of right action that has been resolved since the 1870s? Or since the 1670s?) Evidence of 'intractability' has partly to do with persistence, but partly to do with the terms in which disagreement is carried out. Foundational debates in ethics devolve into clashing intuitions and accusations of moral corruption and obtuseness rather quickly! What are the terms on which apparently 'intractable' debates about set theory are carried out?
JCD makes the interesting sociological point that "the correlation between relevant mathematicians' views and thsoe of their mentor is impossible to miss." How long does that pattern last? Over multiple generations? Or do we find that the student of X often comes to reject the views of his teacher's teacher?
I don't presuppose answers to these questions. But their answers might well suggest which horn of JCD's dilemma we should embrace (i.e., mathematical skepticism or skepticiam about the argument from disagreement).
I should note that I don't quite understand JCD's reference, near the end of his first set of remarks, to the fact that fictionalists in philosophy of mathematics deny simple arithmetical truths. The argument from moral disagreement appeals to first-order disagreement in ethical theory about the criteria of right action, the nature of moral goodnes, and the relative priority of rightnes and goodness in the evaluatoin of actions and persons, among other considerations. It does not depend on claims about the metaphysics, epistemology, or semantics of these judgments. JCD is obviously skeptical about drawing the line between the meta- and first-order disagreements in mathematics. I would like some further explanation of why, and whether there is any reason for that skepticism to carry over to the meta-ethical case.