This essay offers a new interpretation of Nietzsche's argument for moral

skepticism (i.e., the metaphysical thesis that there do not exist any objective

moral properties or facts), an argument that should be of independent

philosophical interest as well. On this account, Nietzsche offers a version of

the argument from moral disagreement, but, unlike familiar varieties, it does

not purport to exploit anthropological reports about the moral views of exotic

cultures, or even garden-variety conflicting moral intuitions about concrete

cases. Nietzsche, instead, calls attention to the single most important and

embarrassing fact about the history of moral theorizing by philosophers over two

millennia: namely, that no rational consensus has been secured on any

substantive, foundational proposition about morality. Persistent and apparently

intractable disagreement on foundational questions, of course, distinguishes

moral theory from inquiry in the sciences and mathematics (perhaps in kind,

certainly in degree). According to Nietzsche, the best explanation for this

disagreement is that, even though moral skepticism is true, philosophers can

still construct valid dialectical justifications for moral propositions because

the premises of different justifications will answer to the psychological needs

of at least some philosophers and thus be deemed true by some of them. The essay

concludes by considering various attempts to defuse this abductive argument for

skepticism based on moral disagreement and by addressing the question whether

the argument "proves too much," that is, whether it might entail an implausible

skepticism about a wide range of topics about which there is philosophical

disagreement.

## Thursday, December 11, 2008

### "Moral Skepticism and Moral Disagreement in Nietzsche"

I've posted a revised version of the paper I gave at the annual NYU "History of Modern Philosophy" conference in November (which generated an excellent and very helpful discussion). I hope the paper may interest moral philosophers generally, as well as Nietzsche scholars. Here is the abstract for the paper:

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## 8 comments:

Hi, Brian. I'm excited to read this provocative paper, which I've heard so much about. As you know, I doubt that "Persistent and apparently intractable disagreement on foundational questions…distinguishes moral theory from inquiry in...mathematics (perhaps in kind, certainly in degree)." As a result, I also wonder whether Nietzsche’s argument doesn’t “prove too much” (though I certainly believe that mathematical “skepticism” [error-theory] is worth taking seriously). But, as you also know, I am presently writing a paper on relevant matters, so let me here merely gesture at some of the reasons for doubt that show up in my paper.

“Persistent and apparently intractable disagreement on foundational questions” has been present in mathematics at least since its unified axiomatization in set-theory.

Persistence: There has been disagreement over the axioms for set-theory since their formulation. For example, there remains disagreement over Union, Infinity, Power-Set, Replacement, Foundation, Choice, (unrestricted) Determinacy, the existence of an inaccessible, Martin’s Axiom, and the existence of omega-many Woodin cardinals.

Apparent Intractability: There appears to be no method by which to decide disputes over foundational questions in mathematics, since there is not merely disagreement over the truth-values of mathematical sentences, but also disagreement over what would count as evidence for those sentences’ truth or falsity. For example, some regard such consequences of Choice as “there are non-measurable sets of reals” or “a sphere can be ‘cut up’ into a finite number of pieces and reassembled using only rigid motions into a sphere twice as large” as evidence against Choice, while others regard such consequences as evidence in favor of Choice. Similarly, some regard the consequence of Replacement, that there is a least ordinal greater than any f(n), where f(0) = Alephsub0 and f(n + 1) = Alephsubf(n) for any natural number, n, as evidence in favor of Replacement, while others regard it as evidence against Replacement.

Foundational Character: Disagreement in ethics might be thought to concern both “epistemically” and “metaphysically” foundational propositions, and in both senses of “foundational”, disagreement in mathematics concerns foundational propositions as well. There is disagreement over epistemically foundational propositions in mathematics since there is disagreement over the only serious candidates for these – namely, axioms, and theorems which might be thought to justify axioms which imply them. What I have said in the prior paragraphs should suffice to establish this. There is disagreement over metaphysically foundational propositions since there is disagreement over the only serious candidates for these as well – namely, the axioms of our “explanatorily” fundamental theory, set-theory (on one view, the axioms of set-theory are both epistemically and metaphysically foundational).

It remains open whether disagreement in mathematics has the apparently “pervasive” character of disagreement in ethics. Let me briefly motivate the thought that it does.

According to one sense of “pervasive”, disagreement with respect to propositions from some domain, D, is pervasive if there are many propositions from D which are such that some pair of people disagrees with respect to those propositions. According to another sense of “pervasive”, disagreement with respect to propositions from D is pervasive if there are many pairs of people that disagree with respect to some proposition from D. It might be supposed that moral disagreement is pervasive in both such senses.

Both senses of “pervasive” need to be clarified. With respect to the first, it is unobvious how to understand the relevant sense of “many”. Any proposition entails its double negation, so there is disagreement over at least (and at most) a countable infinity of propositions whenever there is disagreement over any proposition. This suggests that the relevant sense of “many” should be thought to concern kinds of propositions. I shall assume that we can make clear sense of the claim that disagreement from some domain is pervasive on this understanding of “many”.

With respect to the second sense of “pervasive”, what seems to be relevant to realism with respect to a domain, D, is not how many people disagree with respect to propositions from D per se, but rather what proportion of people who are aware of relevant issues disagree with respect to propositions from D. The fact that there is no disagreement over propositions from some domain that no one has ever considered is not evidence in favor of realism with respect to that domain. This suggests that, given the second sense of “pervasive”, “many” should be thought to concern proportions of people who are aware of relevant issues.

Given these clarifications, mathematical disagreement, like moral disagreement, seems to be pervasive in both of the relevant senses. To see that mathematical disagreement is pervasive in the first sense, note that there is disagreement over axioms of set-theory which have consequences for a wide variety of branches of mathematics. For example, Choice implies fundamental results in analysis (the Banach-Tarski thoerem or the Baire Category theorem for complete metric spaces), algebra (every field has an algebraic closure or the Nielsen-Schreier theorem), topology (every Tychonoff space as a Stone-Cech compactification or that a uniform space is compact just in case it is complete and totally bounded), and set-theory (the union of countably-many countable sets is countable or that every infinite game on a Borel subset of a Baire space is determined). Indeed, the Choice is actually equivalent to fundamental results in these areas – such as Zorn’s lemma in analysis, that every vector space has a basis in algebra, Tychonoff’s theorem in topology, and the well-ordering theorem in set-theory. But alternatives to the Axiom of Choice that critics of it advocate “disagree” with respect to such consequences. The (unrestricted) Axiom of Determinacy is one such alternative.

To see that mathematical disagreement is pervasive in the second sense, one need merely consult the relevant mathematical literature (one might also check the FOM archive). Of course, some views are more popular than others in mathematics. For example, ZFC and extensions thereof are more popular than weaker systems. But some views are surly more popular than others in ethics as well.

Let me conclude (wake up!) with two observations.

First, one might wonder whether certain mathematical propositions, as opposed to moral ones, are “off limits” from (reasonable) disagreement. This does not seem to be so. For instance, fictionalists in the philosophy of mathematics deny even such apparent trivialities as “2 + 2 = 4”. And mathematicians, such as Edward Nelson, reject the principle of (unrestricted) mathematical induction (Nelson himself even conjectures that Primitive Recursive Arithmetic is inconsistent). Of course, one might claim that such positions are still motivated by “philosophical” rather than “mathematical” considerations. But I can see no way to draw the needed divide (were the reasons to reject infinitesimals, to introduce imaginary numbers, to allow impredicative definitions, or to identify cardinals with initial ordinals, philosophical or mathematical?) Moreover, to whatever extent the said divide can be drawn, it seems that radical moral views, such as that it is not wrong to cause needless suffering, are likewise motivated by philosophical considerations (such as moral nihilism).

Second, one might wonder whether moral disagreements don’t seem to reflect social forces more directly than mathematical ones. But this does not seem to be so either. Of course, in the case of mathematics, the relevant social forces are not societies, but rather “schools of thought” and mathematical departments. But the correlation between relevant mathematicians’ views and those of their mentors is impossible to miss.

Assume that Nietzsche’s argument does generalize to mathematical realism. Does this mean that we should reject Nietzsche’s argument from disagreement against moral realism, or that we should rather endorse an analogous argument against mathematical realism? I won’t take a stand on that here.

Hi, again. I've now read your paper, which I found to be very enjoyable. Your attribution of the relevant argument from disagreement to Nietzsche is well-motivated, and your defense of that argument is subtle.

In addition to my above worry that Nietzsche’s argument applies with comparable force to mathematical realism, the following additional worries occurred to me.

You suggest that disagreement through the ages with respect to moral claims is peculiarly problematic for moral realism when its participants are philosophers in particular. Accordingly, you concentrate on establishing the existence of disagreement through the ages with respect to moral claims among philosophers. In contrast, I wonder whether such disagreement isn’t peculiarly unproblematic for moral realism when its participants are philosophers. You point out rightly that philosophers tend to be peculiarly reflective and clear. But philosophers have also disagreed through the ages about everything. In particular, they have disagreed through the ages about all of the following:

-first-order intuitively common-sense claims (such as that there are “tables and chairs”)

-first-order logical and mathematical claims (such as the law of non-contradiction or the

least upper-bound theorem)

-first-order empirical scientific claims (such as that spacetime is non-Euclidean)

-first-order intuitively metaphysical claims (such as that there are properties,

propositions, or possible worlds)

-second-order claims about what it is to make claims of any of the aforementioned sorts

(such as that to claim that spacetime is non-Euclidean is to make a claim about possible

observations, or that to assert the law of non-contradiction is to make a claim about

linguistic conventions, or that to claim that there are properties is to claim that some

things are propertied)

-epistemic claims about the justifiability of believing claims of the aforementioned sorts

(such as that we are justified in believing the law of non-contradiction but not in

believing that spacetime is non-Euclidean)

-intuitively meta-philosophical claims (such as that there are no non-trivial determinately

true philosophical propositions, that philosophical claims “aim” at truth, that philosophy

has a method, etc.)

Of course, there has been less disagreement among philosophers with respect to some such claims than there has been with respect to others. For example, there has been less disagreement among philosophers with respect to the law of non-contradiction than there has been with respect to the claim that there are possible worlds. But, first, the mere possibility that philosophers have held conflicting views with respect to a given claim in the absence of a cognitive shortcoming seems to me to be just as worrisome as the actuality of this. And I believe that we must grant the possibility that philosophers have held conflicting views with respect to practically any philosophical claim of interest in the absence of a cognitive shortcoming. Second, there has been less disagreement among philosophers with respect to some moral claims than there has been with respect to others as well. For example, there has been less disagreement among philosophers with respect to the claim that one ought not cause needless harm than there has been with respect to the categorical imperative. If the relative difference in actual disagreement among philosophers with respect to non-moral claims is supposed to correspond to a relative difference in the plausibility of realism with respect to those claims, then the relative difference in actual disagreement among philosophers with respect to moral claims should correspond to a relative difference in the plausibility of realism with respect to those claims too.

These considerations naturally suggest the following proposal: if any disagreement with respect to claims from some domain, D, is relevant to realism with respect to D, it is disagreement among non-philosophers. This proposal is most plausible with respect to scientific domains. It seems natural that if there is virtual unanimity among *physicists* with respect to the claim that spacetime is non-Euclidean, for example, then the mere fact that there is disagreement with respect to that claim among *philosophers* shouldn’t call realism with respect to spacetime-structure-discourse into question. Ditto perhaps for pure mathematics. The mere fact that the philosopher, Quine, endorses V = L, and so rejects the hypothesis of a measurable cardinal, shouldn’t trouble the realist about pure mathematics, until it is demonstrated that mathematicians themselves are divided on the latter hypothesis (as I believe that they are). Two problems with this proposal are that (a) it isn’t as plausible with respect to non-scientific discourses, and that (b) it is unclear how to draw the needed divide between philosophers and non-philosophers (certainly professional affiliation will not do).

Two final points, and then I (really!) have to go.

You suggest that disagreement over explanatorily fundamental moral claims (or justificatorily fundamental moral claims in the moral, rather than epistemic, sense of “justification”) is peculiarly problematic for realism with respect to moral discourse. Accordingly, you concentrate on establishing the historical prevalence of this sort of disagreement among philosophers in particular. But, setting the relevance of disagreement among philosophers aside, one might doubt whether such disagreement in a domain is ever relevant to realism with respect to that domain. There seems to be reasonable, and apparently intractable, disagreement over explanatorily fundamental propositions, rather than over "mid-level", relatively untheoretical, propositions, in practically all domains. Consider, for instance, explanatorily fundamental principles in physics. Such principles include the likes of the principle of special relativity. It is precisely such principles as these that seem to generate enduring, apparently intractable, disagreement -- perhaps because these propositions are the (logically) strongest, and so also the easiest to call into question (consider the extensive Lorentz-violation literature). Nevertheless, realism with respect to physical discourse does not seem to be thereby threatened.

Finally, toward the end of your paper you anticipate the objection to Nietzsche’s argument that it applies with comparable force to realism with respect to philosophical discourse generally. You respond by claiming that the strength of this reply needs to be measured on a case by case basis. Perhaps there are particular philosophical issues over which there is comparatively little disagreement. If so, then realism with respect to discourse about those issues is not threatened by considerations of disagreement. I wish merely to point out that such a reply seems open to the moral realist in response to Nietzsche’s argument as well. To repeat, you concentrate on establishing historically prevalent disagreement among philosophers over explanatorily fundamental moral claims. You even note that such disagreements often fail to translate into disagreements over what is right or wrong in concrete cases. It remains open for the moral realist to claim that Nietzsche’s argument has, at best, called into question realism with respect to explanatorily-fundamental-moral discourse – a discourse that is arguably at far remove from day to day moral discussion. Such a response raises the difficult question of how to draw the needed partition among moral sentences (or, intuitively, the question of how to specify the “reach” of a given argument from disagreement).

My thanks to Justin Clarke-Doane for two sets of excellent and penetrating comments. I will reply to those in the next week or two, and invite others to weigh in as well.

Mr. C-D:

Although I have a background in higher math, it was too long ago and too shallow to even begin to follow the details of your arguments. But I at least think I sense - both in the content of your contentions and in the analogies with unquestionably man-made philosophical propositions - support for the "made" rather than "found" view in the philosophy of math, ie, for "embodied-mind" theories. Am I right, wrong, or merely hopelessly confused?

Thanks - Charles

Hi, Charles.

I didn't mean to suggest any view on the semantics of mathematical discourse in what I wrote above. In particular, I didn't mean to suggest an answer to the question of whether mathematical truths are "made" or more generally "mind-dependent".

I assume that the "unquestionably man-made philosophical propositions" to which you allude are moral propositions. One might take the comparisons that I drew between mathematical propositions and moral propositions to suggest that mathematical propositions are "made" or "mind-dependent". One might also take those comparisons to suggest that moral propositions are "discovered" (the scare quotes that I lack a clear understanding of the corresponding concept).

As a matter of autobiography, I am sympathetic to what one might call a "made view in the philosophy of math". In particular, I believe that mathematical truth may depend on our ideal mathematical theory in a way that physical truth does not. As a result, if there are undecidable sentences with respect to our ideal mathematical theory, then I believe that those sentences may lack a determinate truth-value. The Continuum Hypothesis is often regarded as an undecidable sentence with respect to our ideal mathematical theory, though recent work by Hugh Woodin suggests that it is not.

Justin -

Thanks for the reply, which shows me that I need to think some more about the philosophy side of the analogy (where I'm even weaker than on the math side).

When I think about the made/discovered question, I recall the philosophical riddle "if a tree falls in the woods and is unheard, does it make a sound?" So, re math I ask something like "if an apple falls off a tree to the ground and another apple falls, if unfound by a human have two apples fallen off the tree?".

I have a vague sense that the answer to the "physical" riddle is "yes" because there could be effects on entities that are mindless or at least lacking human mental capabilties. But I see the answer to the "mathematical" apples question as being "no" because in the absence of an intelligence that needs and therefore develops integer arithmetic, the question has no meaning since "two" has none. Ie, sound waves can exist without mind but counting can't.

If philosophical questions have answers that have meaning only to humans (eg, moral questions), presumably the same "reasoning" would apply. But I need to think about both the premise and the conclusion.

- Charles

Hi, Charles.

If you think that it is possible for trees to have unrecognized sonic properties, then I wonder why you don't think that it is possible for apples to have unrecognized numerical ones. Insofar as hearing is the recognition of sonic properties, not a sonic property itself, then I should think that counting is the recognition of numerical properties, not a numerical property itself.

One could grant the existence of recognition-independent numerical properties of concrete things without granting the existence of recognition-independent mathematical objects (such as numbers). Indeed, typical intuitively arithmetic truths about concrete things are formalizable as truths of first-order logic plus identity, and seem to be consistent with the view that there are no numbers. Perhaps your question, then, is better understood, not as the question of whether it is possible for the likes of apples to have recognition-independent numerical properties, but rather as the question of whether it is possible for mathematical objects to exist or to have whatever properties they do independent of recognition. A positive answer to the latter question, as opposed to the first, is not obviously recommended by the common-sense view that trees can have recognition-independent sonic properties.

Justin -

Re trees and apples, let's simplify to just apples. First, I'll describe more precisely what I have in mind and try to fit what I take to be your view into that description.

While still in the tree, an apple has two latent (in the sense of emerging only when recognized) properties: the sonic property that its falling will cause a physical disturbance of the environment that could be recognized (ie, heard) were a "hearer" present, and the numerical property of being recognized (ie, counted) were a "counter" present.

In my mind, the distinction (if any) turns on what is meant by the prospective "hearer" of the sonic property and the prospective "counter" of the numerical property. If one means by each a person, then there does appear to be no distinction. And even if one thinks of the "hearer" as a device that records some measurable effect of an apple's fall, then one could also imagine a device that "counts" apples and records the result. In a broad sense the former is a primitive sensory system and the latter is a primitive "mind". Alternatively, since the devices must have been designed by entities with "minds", one could consider the devices to be merely proxies for the real recognizers. Again, one can argue that there is no distinction.

But if one considers effects that are "heard" by non-sensory entities, it gets more complicated. For example, a falling apple causes air movements which could in turn cause a nearby leaf to move, and one could argue that the sonic property was "recognized". I see nothing analogous with respect to the "numerical property", recognition of which seems to require some form of "mind". Of course, one can also argue that in the former case the leaf is merely another step in a series ("if a leaf waves in the breeze and no one recognizes it, ...") which either is infinite or must terminate in some form of "mind".

In any event, I agree with your suggested question re math objects, which was really what I had in mind (though not necessarily very clearly). Injecting apples, trees, etc was intended only to make the issue more concrete - and fun.

Thanks for your replies. I hope Prof Leiter ultimately responds as promised to your comments. I look forward to reading your exchanges on the actual subject of his post.

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