The Second Ace

Check out this post at Choice and Inference if you liked my earlier posts on probability puzles (here, here, and here).  The puzzle is given as a possible counterexample to reflection for credences, but is worked out nicely in the comments over there.  I was too late to the party to add what I learned from jd2718, but the solution to the second ace puzzle is pretty much the same as the solution to the dice puzzle that had initially fooled me when jd2718 presented it to me in comments.

Blogging about philosophy is hard.

I’ve got a few drafts of longer posts, but find myself always wanting to mull things over more and revise.  I suppose that, in thinking ahead to my dissertation, it’s right to be in a stage of thinking through a large project and that blogging lends itself to short bursts of ideas or to defending worked out positions.  Also, this still feels like launching words into the void; it’s hard to write anything if you don’t have a sure feeling of audience.

Sharing cool videos, on the other hand, is easy.

… but then neither is property.

Writing at National Review Online’s “The Corner” conservative commentator Stephen Spruiell claims:

Health care is not a right, at least not according to the conception of rights upon which this country was founded. Your rights include life, liberty and the pursuit of happiness. You may not be unjustly deprived of these things. Your rights do not include things that I or anyone else must be forced to provide for you, such as a home, a car, a job, or health care.

I would be surprised to find the founders univocal on the nature of rights, but I understand that the narrow conception of “negative rights” was historically important and represents a distinct concept of “right” from the one employed in claiming that health care or education is a right.  What I don’t think that Spruiell will like to admit is that “property rights” may well be argued to fall under the broader conception of “right” that includes positive rights.  Consider what Thomas Paine said in “Agrarian Justice”:

There could be no such thing as landed property originally. Man did not make the earth, and, though he had a natural right to occupy it, he had no right to locate as his property in perpetuity any part of it; neither did the Creator of the earth open a land-office, from whence the first title-deeds should issue. Whence then, arose the idea of landed property? I answer as before, that when cultivation began the idea of landed property began with it, from the impossibility of separating the improvement made by cultivation from the earth itself, upon which that improvement was made.

The value of the improvement so far exceeded the value of the natural earth, at that time, as to absorb it; till, in the end, the common right of all became confounded into the cultivated right of the individual. But there are, nevertheless, distinct species of rights, and will continue to be, so long as the earth endures.

Individual property rights are here recognized as “of a distinct species” from the “common right of all” to benefit from natural resources.  Paine is commenting directly on the Lockean standard and fairly clearly rejects it.  Continuing, Paine writes:

Cultivation is at least one of the greatest natural improvements ever made by human invention. It has given to created earth a tenfold value. But the landed monopoly that began with it has produced the greatest evil. It has dispossessed more than half the inhabitants of every nation of their natural inheritance, without providing for them, as ought to have been done, an indemnification for that loss, and has thereby created a species of poverty and wretchedness that did not exist before.

In advocating the case of the persons thus dispossessed, it is a right, and not a charity, that I am pleading for. But it is that kind of right which, being neglected at first, could not be brought forward afterwards till heaven had opened the way by a revolution in the system of government. Let us then do honor to revolutions by justice, and give currency to their principles by blessings.

The establishment by governments and quasi-governments of a regime of individual ownership has costs and benefits.  On the one hand, it creates the incentive to make the land more productive.  On the other hand, it creates an elite class of property owners and deprives others of opportunity to use resources that are naturally held in common.  As a remedy, Paine proposes a tax on property owners to redistribute goods to those that are deprived by the system of private ownership.  Indeed, if we follow Paine, it would seem that the redistributive scheme is justified by a right more basic than the system of private ownership.  The system of private ownership has a purely utilitarian justification: viz., it creates incentive to increase land productivity.  The redistributive correction of the negative and unintended consequences, on the other hand, is justified by appeal to a fundamental natural right of common propoerty, characterized thusly by Paine:

It is a position not to be controverted that the earth, in its natural, cultivated state was, and ever would have continued to be, the common property of the human race. In that state every man would have been born to property. He would have been a joint life proprietor with rest in the property of the soil, and in all its natural productions, vegetable and animal.

It may be controverted that others of the founders did not share Paine’s outlook, one that blurs the distinction between landlord and government in ways that would discomfort contemporary American libertarians (and those conservatives that pretend to be libertarians whenever it suits them).  I can’t say that I have a terrific grasp of every one of the founders’ views on “rights” and I won’t pretend (as Spruiell does) that I do.  Nevertheless, I find much to recommend in Paine’s outlook.  On this view, individual property rights are (again) “of a distinct species” from what we might call natural rights.  They are justified by their positive consequences, but to the extent that they also have negative effects by limiting every non-owner’s natural right to freely use the earth’s resources to sustain his or her life and to pursue happiness, I am with Paine that government is not only justified but obligated to provide compensation.  Taxation supported public services like health, education, welfare, and infrastructure fulfill this obligation.

Rusell… Really.

In wondering why Russell has shown well in the Leiter poll, Weatherson actually winds up making a pretty decent case for Berty, or so it seems to me.  Fitelson and Chalmers speak up on Russell’s behalf in the comments.

Plantinga v. Dennett

I wish I’d been there…  Philosophy title fight between theist Plantinga and atheist Dennett.

I’ve seen Plantinga give a talk presenting some of his arguments.  They depend essentially on sophistical fabrication of prior probabilities.  The link I’ve given takes Dennett to task for rudeness, and I would not be surprised if this is true.  I don’t see what the “new atheists” think they’re accomplishing in mocking bare theism.  To be sure, any degree of literalism about Christian mythology is absurd (talking snakes, human sacrifice, drinking blood…) , but I think that bare theism–basically deism–can be respectably held.  Anyhow, I empathize with Dennett’s impulse to rude response to some extent because in a way mockery is about all that you can do when the “argument” is just the other person insisting on a bunch of a priori probability judgments, but I also agree with this reviewer that the rudeness is likely counterproductive.  Naturalists can and should try to do better.

The reviewer to whom I linked wrote:

Post-script: It has been about ten minutes since the session ended. I spoke to Peter Van Inwagen about the talk and he said it was an expected performance and that while it was a clash of worldviews, it was an interesting clash in two styles of doing philosophy. Initially, I thought to myself, “Yeah, Plantinga thinks philosophy is about arguments; Dennett thinks it is about stories.” But on further reflection I realized that Van Inwagen had a point. Dennett believes that science can tell us many things about metaphysics and epistemology, that we work from science to these positions. Plantinga thinks of these matters rather differently.

This seems to me to be correct, and there’s much to be said in Dennett’s favor on exactly this point.  Plantinga’s “thinking philosophy is about arguments”, as I’ve said, means making lots of appeals to claims about prior probabilities.  Establishing an objective framework for assigning prior probabilities seems to me to be a hopeless project philosophically; though how knowledge is established and advanced nevertheless is a demand that is fair to press.  Dennett’s “telling stories” reminds me of WVO Quine’s insistence that we inherit and rely on the “lore of our fathers” by working within the scientific community (and mothers, I might add) and Isaac Newton’s “standing on the shoulder’s of giants”.  Plantinga is a tall man, but not so tall.

Ulysses contracts and the voter paradox.

Lately I’ve been reading a lot and working things over, so not much to say on the philosophy of mathematics front. . . maybe not for a little while.  I’ve got some ideas brewing on the following topics: (1) category theory vs set theory foundations, (2) whether the application of SU(n) in physics and the discoveries derived from that application are “more surprising” than widely discussed examples in the history & philosophy of science (e.g. the discovery of Neptune), and (3) whether a wannabe-nominalist who accepts naturalized epistemology can answer Burgess and Rosen’s anti-nominalist arguments.  Even granting that a blog is a space for working through ideas in progress, I’m not sure I’m ready to say a whole lot about these things just yet, at least not until I work through at least some of the stack of books and articles I’m accumulating.

So, dear mathematically inclined readers (all three or four of you!), you’ll have to settle for something non-mathematical but hopefully interesting.  I had a very nice, intellectually stimulating evening last night in two parts.  It was a nice break from mathematical/logical slogging that lead to some free thinking about more open-ended questions.  First, Jason Hanna from U Colorado gave a colloquium titled “Ulysses Contracts and the Moral Relevance of Actual Consent”.  Second, Beth and I had dinner with a new professor in the political science department and his wife that lead to an interesting discussion of the voter paradox.  I will explain for you what these contracts are and what this paradox is, then relate the two issues and share some thoughts on them.

Ulysses contracts: Ulysses tells his crew to tie him to the mast and plug their ears.  Sail past the sirens so he can hear them sing.  Do not, under any circumstances, untie him.  Even if he begs, he insists that they not let him down.  A Ulysses “contract” (the word is a bit misleading since they need not be technically binding) is when one requests at timeA not to have one’s requests at a later timeB honored.

The philosophical issue that arises is how to deal with cases where we’re instructed by someone not to obey their future instructions.  These are not just philosopher’s fantasies.  Examples may arise in applied medical ethics.  For example, an expectant mother may tell her doctor not to administer an epidural even if she begs for it.  Suppose she does indeed beg for the epidural.  What is the ethical thing for the doctor to do?  How can the doctor best respect the woman’s autonomy?

In different cases our intuitions may be different.  We may think that Ullyses crew definitely should not let him down, but that the doctor definitely should administer the epidural.  The philosopher’s task is to see if there’s a principled way to account for all of the judgments that seem intuitive.  Here’s one suggestion: Honor the most recent request unless it’s obviously deficient.  Ullyses’ request to be let down is paradigmatically deficient because he’s under the spell of the siren song.  But the mother’s request is not obviously deficient.  It is made under the duress of pain, to be sure, but there’s a sense in which that makes it a more informed choice than her earlier decision and hence closer to the optimum.  The matter is debatable, but that implies that it’s not obvious.  This was not the account that our speaker gave.  He was pressed by Greg Landini and Richard Fumerton to come up with a “pure case”.  It seemed like all of the cases where we have the intuition to honor the earlier choice were cases where the later choice was made under the influence of addiction, delusion, mental illness, etc.  Is there any case where we are inclined to honor the earlier choice even though the latter choice is closer to optimally rational?

Which brings me to:

The voter paradox: The expected utility of voting is really, really low. After all, one vote is not going to have any impact on the election. So it is irrational to vote.

Of course, this isn’t a paradox if you accept the conclusion, but I think that a lot of people will be disinclined toward accepting that conclusion and it is at least worthwhile to explore the account of rationality that leads to the conclusion and to think about alternatives.  Before we do that, however, I want to return to the Ulysses contracts and offer what I think may be a pure case.  The relevance of the voter paradox will become clear as we proceed, so keep it in mind.

Suppose I tell my wife the following.  We’re going to the gym tomorrow before class.  I know that I’ll want to sleep in, but no matter what I say don’t listen.  Drag me out of bed if you have to.  Even if I tell you that I take everything I’m saying now back, I want you to bang on pots and pans until I get up…  Ok, you get the picture.  It’s at least plausible that my wife should honor my initial decision even though I take it back in the morning.  In a discussion of weakness of the will, Donald Davidson argues (citation not handy) that it can be irrational to brush one’s teeth if one believes one will in the future brush one’s teeth regularly.  That case is similar to this one.

But wait!  When in the morning langor rises, reaching to shut off the alarm isn’t it the case that I’m making a deficient decision, one clouded by grogginess.  So is this really a pure case?  I say that it is.  In fact, in the morning I’m more informed about how good it will feel to sleep in.  Furthermore, I’m not under the influence of addiction or mental illness or anything like that.  In particular, I reason that skipping the gym just this once maximizes my expected utility.  After all, one day skipped is not going to have any impact on my overall health. My decision to sleep in is more rational than my decision the night before to tell my wife to drag me out of bed whatever it takes!  (By the same type of reasoning that led to the voter paradox, that is).

In my gym case, the intuition to honor the earlier decision is, I maintain, at least respectable.  But it’s not a case where the later decision is over-ruled because it’s deficient in some obvious way.  I claim, in fact, that the later decision is more rational than the earlier decision by the standard of expected utility.  Yet, the voter paradox seems to point to some problems with that standard and I think so does the gym case.  After all, if every rational voter reasons that it’s irrational for them to vote then only irrational people will vote, which can’t be good.  Similarly, if I reason every morning that I should sleep in rather than go to the gym then I’ll never go to them gym, and this will have predictably rotund consequences.  Just as the voter paradox relies on the supposition that other people will not reason as you do, the gym case relies on the supposition that my future selves will not reason as I do now.

So, here is what I think is suggested.  Rationality is not a simple matter of maximizing expected utility with each individual decision taken in isolation.  If it were, the person who tells him or herself every day that tomorrow is the day that they’ll start eating healthy and working out would be rational, even if tomorrow never comes.  We need to make, as it were, Ulysses contracts with ourselves in order to consistently make decisions that have good results in aggregate, even if in each of the moments that we make them they have low expected utility.  It really is (sort of) like this for me, with a internal voice of conscience speaking: “Self, you swore to yourself that you’d go to the gym and I’m holding you to it even if you don’t like it”.  I don’t think that the voter paradox is really that different.  To be sure, it’s members of a group (i.e., citizens of a democracy) making a sort of Ulysses contract with one another to act as an aggregate in a way that everyone agrees is good in aggregate but everyone recognizes does not maximize expected utility individually.  The voice of conscience thus speaks on behalf of the shared interests of the group rather than the interests of the trans-temporal self, but I don’t see that this is a difference that makes the solution good in the gym case but bad in the voter case.

So, what’s the upshot of all of this?  Well, maybe my pure case is not so pure afterall, if voter paradox style reasoning is in fact deficient reasoning.  My discussion has suggested that it is deficient for the following reason.  The locus of rationality ought not to be thought of as the present self.  Assessments of rationality may equally well be made from the standpoint of the transtemporal self (which argues: sure that makes sense now, but what if you always thought like that) or from the standpoint of shared interests and intentions (which argue: sure that makes sense for each of us individually, but what if we all thought like that).  From these standpoints, the gym case and the voter paradox reasoning are obviously deficient since such reasoning will leave me fat and governed by morons, maybe even rotted of tooth. It is the deficiency of these decisions (not to go to the gym, brush my teeth, vote, etc.) that warrants the voice of conscience to enforce (e.g.,through guilt) the Ulysses contract.

Your thoughts?  Are there reasons to answer the gym case or the voter paradox differently than I have?  Are there relevant disanalogies?  Have I left the self too splintered into too many “voices”?  See my post on Lackey’s counterexamples to knowledge as the norm of assertion for more splintering along these lines (though not exactly the same); I worried then as I do now that the kinds of analyses I’m suggesting will leave us all with philosophical mutliple-personalities.

Two More Probability Puzzles.

Both of these are from Ernest Adams’ “A Primer on Probability Logic”, which I’m finding to be a nice, introductory level book with lots of examples I’ll be using in my Principles of Reasoning course next semester to compare deductive and probabilistic reasoning.  Since the first has a pretty straightforward solution and the second lends itself to a somewhat more complex but well understood analysis, I’m not sure about labeling these “paradoxes”, but perhaps the difference between a paradox and a puzzle is purely psychological; i.e., a puzzle is a paradox only when the wrong answer is gripping, which is a subjective matter.  So, different individuals will find these more or less paradoxical.

The first, “Simpson’s Paradox”, is given as a homework problem.  Quoting Adams (p. 62):

During the 1985 baseball season Steve Sax of the Los Angeles Dodgers outhit Ron Oester of the Cincinnati Reds both playing on grass (.257 for Sax and .225 for Oester) and playing on artificial turf (.333 for Sax and .323 for Oester), yet Sax’s overall batting average was lower than Oester’s (.279 for Sax and .295 for Oester).

What gives?  How could Sax have done better both on grass and on turf but not better overall?  Well reflect that both players hit much better on artificial turf than on natural grass.  With that in mind, what stands out as different between Riverfront Stadium where Oester played half of his games and had roughly half of his at-bats vs Dodger Stadium where Sax played his home games?  It didn’t take me long to find Simpson’s “paradox” not particularly paradoxical. However, in Adams’ framework basic, static “probability logic” is an extension of the standard propositional calculus.  In place of truth-tables with0/1’s for each primitive proposition (i.e., a “world for each combination of truth assignments to the primitive propositions) Adams allows for probability distributions subject to the constraints of the Kolmogorov axioms (i.e., a world for each probability distribution).  Working out what the primitive propositions should be for the Sax/Oester scenario and an assignment of probabilities that gives the corrrect answer is a nice exercise and nontrivial.
(Question: Is “trvial/nontrivial” also a psychological matter?)

The second paradox/puzzle is also given by Adams as an exersize (p. 69).  Again, quoting Adams:

If each team in the World Series has a 50-50 chance of winning any given game, does it follow that the team that wins the final game of the Series has a 50% chance of winning that game?  If not, why doesn’t this follow as a particular instance of the universal statement “each team in the World Series has a 50-50 chance of winning any given game”?  And if it doesn’t folow, can you restate the universal statement so as to make it clear exactly what follows from it?

It took me a bit to see what the questions were driving at, so let me explain what I think the problem is through a slight restatement.  Suppose the Cubs win the first 3 games of the series.  There’s a 1/8 prior chance of that.  Suppose further that all three games were very close, so that we have no reason to revise our belief that the teams are very evenly matched, just like we have no reason to revise our robust probability of 1/2 when three straight heads are flipped.  Aside from personal joy unprecedented except by my wedding day, some things follow from this.  There are four games left, any of which will be the last if the Cubs win it.  The chance that they win one of the remaining four is thus one minus the chance that they lose them all: viz., 1 – 1/16.  So it would seem that the chance that they win the final game, having won the first 3, is 15/16, but doesn’t this contradict the universal statement that they have a 50-50 (i.e., 1/2) chance of winning any given game, which was never contradicted and was in fact employed in the reasoning that lead to the value 15/16?

Adams asserts in a footnote that the puzzle, which I find a bit more paradoxical than Simpson’s Paradox, relates to the problem of “referential opacity”, discussed in detail by Quine and raised prior to that in Frege’s discussions of the informativity of identity statements and of apparent violations of Leibniz’s Law.  The trouble, briefly, is that we cannot, it seems, infer (even leaving aside the complications of the fictional setting) as follows:

(1a) Lois Lane believes Superman is a hero

(2a) Superman = Clark Kent

(3a) Thus, Lois Lane believes that Clark Kent is a hero.

Call this “argument.a”.  The term “believes” creates a “referentially opaque” context.  Logicians have marked those contexts in which the simple substitution of coreferring terms does not preserve truth by the term “intensional” (with an “s”–not “intentional” with a “t”), to be contrasted with “extensional” contexts, in which substitution of coreferring term preserves truth-values. Propositional attitudes, like “believes”, “knows”, “desires”, etc., are paradigmatically intensional.  Modal operators, like “possibly” and “necessarily”, are also taken by logicians to create intensional contexts.

Suppose, to my delight, that the Cubs in fact win the Series in game 4.  We now have the antinomy between the following, “argument.b” and “argument.c”.  Recall that we are given in my restatement of the problem that the probability of the Cubs winning any game is 1/2 and that they have won the first three:

(1b) The probability of the Cubs winning game 4 is 1/2 (by instantiation of the given).

(2b) Game 4 = the last game of the series.

(3b) The probability of the Cubs winning the last game is 1/2.

_______________________________________________

(1c) The probability the Cubs winning any given game is 1/2.

(2c) The probability that they win one of the remaining four is one minus the chance that they lose them all: i.e.,  1 – 1/16.

(3c) So, the probability of the Cubs winning the last game is 15/16.

What gives?  There are a number of ways to approach paradoxes of referential opacity.  One approach, championed by Bertrand Russell, is to pay very close attention to “definite descriptions”.  Those are terms formed by using the word “the”: viz., in our case “the last game”.  Russell maintained that although these terms behave grammatically like genuine proper names, that the correct analysis of their truth conditions will show that they have a disguised quantificational structure.  (See Russell’s treatment of the truth conditions for the sentence “The present King of France is bald.” for details).  The standard descriptivist approach to paradoxes of referential opacity would restate (2b) in a way that shows this quantificational structure, which will transform it from a simple identity statement to one with an existential quantifier out front and thereby block the inference to (3b).

I don’t find that to be the most insightful analysis of this puzzle, however, even if it is logically adequate.  For one, there’s a question whether the Russellian approach generalizes to cover all opacity puzzles, since some do not involve definite descriptions.  Russellians have claimed that ordinary proper names actually function as disguised definite descriptions, a claim that many have found gives an implausible account of the semantics of ordinary names.  But we are digressing already.

I think that the proper analysis of this puzzle is in fact rather simpler.  The puzzle arises from static logical reasoning.  That is, in argument.b the information given in premise (2b) should allow a rational person to revise the belief expressed by (1b).  On a good dynamical account of reasoing (1b) is abandoned upon learning that (2b) is true, provided all the background information that was given in the set-up of the puzzle.  After all, if the Cubs have won the first three games and game 4 is the last game of the series then the probability that they win game 4 is not at all 1/2; it’s, in fact, 1!  Oh, joyous day!!!!

There is, however, quite a bit more that could be said here about the relationship between descriptions and singular terms.  What seems to be distinctive about descriptions, in contrast to singular terms, is that a when description is used to refer it does so in virtue of some piece of information about the referent.  This is very important, given the requirement of total evidence, in probability contexts.  If an entity is referred to using a description then our beliefs about that entity ought to be updated in light of that information.  The same does not hold when a proper name is used to refer; we may have information associated with the name, but the name itself does not communicate any information in the way that a description does.

Quine on Naturalized Semantics and Ontological Relativity

My thinking in philosophy has for a fairly long time now been animated by thinking about WVO Quine’s writings.  His views have, of course, been tremendously influential over the past 50+ years within philosophical circles.  My readers with more background in mathematics and less in philosophy and philosophical logic, however, may not be familiar with Quine’s central theses and arguments.  Since I’ll be working through responses to my comps questions in future posts, and will sometimes need to mention Quinean ideas without fully explicating them, it will be useful to have up some basic exegetical posts.  This is one of them, a response to a final exam question in Greg Landini’s Spring 08 Quine course.  The intro is a bit cheezy, but it gets the point across how Quine’s challenge to the “myth of the museum” attacks presuppositions of both platonism and the mentalism shared alike by enlightenment rationalists and empiricists.  In place of the myth Quine proposes a naturalized semantics motivated by Dewey, Wittgenstein, and behaviorist psychology.  Ok, here we go.

The location of the museum had been in dispute for centuries. Some held that meanings inhabited a mysterious third realm. Some thought it was in our minds. Some thought that it was in the world we inhabit. There was little doubt that there was a museum, however, wherein meanings were kept for consultation on the occasion of any philosophical dispute. So goes the myth Quine says was busted by Dewey and Wittgenstein. In place of mythology Quine promises to develop the ideas of Dewey and Wittgenstein regarding alternatives to the myth of the museum along thoroughly modern and scientific lines. To understand a word will not be to be in some relation with its meaning, whatever that may be, but rather to have a kind of know-how.

Quine indicates two aspects to knowing a word: (a) the phonetic part: knowing its sound, and (b) the semantic part: knowing how to use it. For a simple example, one may know how to use the word to refer to an object in one’s environment. To refer is to draw another’s attention to the object, so learning how to do this includes learning how to bring about desired behavior in one’s fellows. What one does, in general, in learning how to use language is learn about the behavior of one’s fellows. Quine demands that in naturalized semantics one must recognize that “even in the complex and obscure parts of language learning, the learner has no data to work with but the overt behavior of other speakers.” Quine argues that one consequence of this view of meaning is ontological relativity: the doctrine that there is no language transcendent fact of the matter regarding “what there is”.

Relativism, or opposition to it, was one of the reasons for the introduction of the myth in the first place. Meno presented Socrates with the following dilemma: If we know the meaning of “virtue” then we don’t need to search for a definition, but if we don’t know the meaning then we won’t recognize a good definition if we find it. Either way, it’s pointless to go looking for a definition. The puzzle presented to Socrates is now known as the “paradox of analysis”, which states that philosophical analysis is pointless because it is either trivial or false. Plato presents Socrates as solving the problem through the theory of the forms and anamnesis (i.e., recollection). We recognize an analysis of a term such as “virtue” as adequate by recalling our former acquaintance with a certain museum item: viz., the form of virtue. The analysis is non-trivial in light of the difficulty of recollection. The account has left many a bit uneasy, including Socrates himself. After giving his answer to the dilemma, Socrates confesses to Meno that the idea of a third realm of forms that we inhabited in a past life seems a bit dubious. The alternative, Socrates fears, is that talk of virtue, justice, etc. is just a bunch of squawk. The pernicious Callicles and Thrasymachus would jump on such a result.

Quine is less concerned about pernicious relativism and more concerned about giving a scientifically plausible and naturalistic semantics. Platonism is out. Mentalism is also out. The mentalist places the museum in our minds rather than a mysterious third realm. She gains some ground on Socrates epistemologically, it would seem. Some have learned from Descartes to insist that nothing could be plainer than one’s own mind. The terminology has varied regarding the exhibits in the mental museum, but Hume’s “ideas” and “impressions” are typical. Impressions are sense experiences. An idea is thought of as a “dim copy” of an impression. An expression like “blue” is understood by reference to an exemplar idea bearing the label “blue” in one’s mental museum. The problems with mentalism are numerous. Wittgenstein’s Philosophical Investigations offers trenchant criticisms of the view that concept words refer to private mental exemplars, and there are well-known metaphysical difficulties with mind-body dualism (though it is perhaps debatable that the mentalist is committed to dualism). Quine has been criticized for dogmatism in his embrace of naturalistic alternatives and for dismissing mentalism too quickly without much argument. I think that a fairer assessment of Quine’s work is to view him as more interested in the positive project of developing the implications of alternatives to mentalism, such as behaviorism, than in beating what he plausibly took to be a dead horse.

Quine’s case for ontological relativity builds from his theses of the inscrutability of reference and the indeterminacy of translation. Indeed, the inscrutability of reference would show that the physical world, no more than the platonic or mental realms, cannot serve as a museum of fixed meanings. One way to present the case is consider the maneuvers available to an ethno-linguist clinging to a seemingly strange hypothesis: that the native’s ‘gavagai’ uttered regularly in the presence of rabbits actually means ‘undetached rabbit parts’. As Quine notes, it’s a rather unnatural hypothesis and one that actual practicing ethno-linguists would never and should never make; that is beside the point though. How could we ever disprove the hypothesis? Quine insists that based on the behavior of the natives alone, we could not. We might be able to show that the hypothesis is inconsistent with an established translation manual specifying, for instance, the individuative apparatus of the language, but our ethno-linguist may simply cling to his hypothesis by abandoning the translation manual, which functions evidentially as an auxiliary hypothesis, and retranslating the relevant native utterances in a way that is consistent with the clung-to hypothesis. The opportunities for revision of auxiliary hypotheses seem endless and the suggestion that behavioral data alone can uniquely determine one correct translation seems hopeless.

If one wished, one could deal with the stubborn ethno-linguist as follows: simply come up with a translation manual for him. One could translate his ‘undetached rabbit parts’ as ‘rabbit’, supposing that he speaks a bizarre dialect of English.  One will have to undertake some considerable gerrymandering the terms of his apparatus of individuation for the hypothesized dialect: viz., the use of pronouns, plurals, identity, etc. It seems, however theoretically, possible to translate our ethno-linguist so that the composition of his translation of the natives (call this t1) with our translation of him (call this t2) yields the more natural translation of the natives call this (t3): i.e., t2(t1(‘gavagai’)) = t3(‘gavagai’) = ‘rabbit’. He no longer has a bizarre hypothesis, just bizarre linguistic dispositions. This reflection brings radical translation home. We will have heterophonically translated our neighbor. As Quine notes, our ordinary practice is to homophonically translate one another, but there are exceptions. Frequently, for instance, when I hear someone say that they are “anxious” about something they seem to anticipate without the least bit of anxiety, I translate them as meaning they are “eager”.

Nevertheless, heterophonic translation is the exception and homophonic translation is obviously a rule that we cannot get by without. Suppose we want to test t2, our proposed translation manual of the ethno-linguist. How will we test it? We can ask the ethno-linguist whether by ‘undettached rabbit parts’ he means rabbit or undetached rabbit parts. Of course, he’ll answer “I mean undettached rabbit parts, obviously”. This neither confirms nor disconfirms t2. In fact, an answer to the question whether the ethno-linguist has a bizarre hypothesis or speaks a bizarre language cannot be, Quine argues, empirically determined at all. So, if we are verificationists, we should reject the question as senseless. The point is not restricted to the term ‘rabbit’. Which of the numerous series satisfying the Peano axioms are the numbers? We could always interpret some person’s specification of which series they are talking about by heterophonically translating their utterances to have them come out referring to our preferred series; again, the question whether they have a different view in philosophy of mathematics or a different language comes out senseless if we have the strict verificationist/empiricist criteria of meaning.

We have considered the ethno-linguists’ radical translation of the native and our own radical translation of the ethno-linguist. This all must stop somewhere. Could one radically translate oneself? Does the question we have not been able to answer empirically, the question whether the hypothesis or the language is bizarre, have an answer for us “from the first-person”? Ask yourself: “Self, by ‘rabbit’ do you mean rabbit or undetached rabbit parts?” Answer yourself: “I mean rabbit, obviously.” The answer is no more informative when you ask yourself than it was when you asked the ethno-linguist. If one could put the question thusly, the matter might be resolved: “Self, by ‘rabbit’ do you mean this (mentally pointing to an idea) or that (mentally pointing to another idea)?” You could then confidently hold the former idea before your mind as what you mean. It would be the same if one could consult forms. However, if we are to reject platonism and mentalism for naturalism we do not have such recourse, and I am with Quine in finding neither the platonist nor the mentalist picture to be particularly accurate, illuminating, or useful.

As long as one is taking seriously the possibility of heterophonically translating oneself, it won’t help to make reference to the apparatus of individuation to convince oneself that one means rabbit by ‘rabbit’. Just as with the natives and the ethno-linguist, that will be just as up for grabs. These are just as immanent to our natural language as all the rest. The tempting conclusion is that one cannot tell even of oneself whether one is speaking of rabbits or undetached rabbit parts. Quine’s way of resisting this temptation is to point out that the whole business of translation presumes a language to be translated into. To pose the question—“Self, by ‘rabbit’ do you mean rabbit or undetached rabbit parts?”—one has to use, not merely mention, the term ‘rabbit’. In even posing this question to oneself, one acquiesces in the language one was brought up in, and that is where radical translation stops. In particular, recall that it was crucial to the argument for indeterminacy of translation that the individuative apparatus of the language be subject to translation. Acquiescence in this apparatus, which in its regimented is just first-order logic, provides a fixed frame of reference in which to draw distinctions.

So, Quine maintains that just as it is only relative to an inertial frame that a thing can be said to be in motion, it is only relative to a linguistic frame that we can say what a term refers to. Quine extends the analogy further, claiming that just as according to the relational theory of space it makes sense only to speak of relations between inertial frames and relations between things and not of absolute position or absolute motion, it also only makes sense to speak of relations of interpretation and reinterpretation between languages and not of absolute interpretation. When “interpretation” above is taken, as Quine means it, in the sense of Tarski the wide implications of Quine’s arguments for the theory of truth are laid bare. The correspondence theory can be understood as a theory of absolute interpretation. Truth in the sense of correspondence is truth in the sense of Tarski plus the specification of an intended interpretation. The facts correspond to the sentence when and only when the sentence is Tarski-true on the intended model. The upshot of Quine’s reflections on ontological relativity for the correspondence theory of truth is that there may be no way of specifying an intended interpretation for the language in which one finally acquiesces.

Clever.

Traditionally, the problem with nominalism in mathematics has been not so much that Occam’s razor has been dulled by Plato’s beard, but rather that it has managed to remove the beard only by severing the head at the neck.

–Geoffrey Hellman, “Structuralism Without Structures”

Logic and Mathematical Method

In the comments to my post Follow-up on “A Probability Puzzle” Jonathan of jd2718 posed another probability question that I answered incorrectly and somewhat pretentiously (note to self: beware the phrase “deep epistemological and logical question” unless you’re really sure you’re on the right track).  In my defense, I quickly saw the force behind the correct answer and began hedging my defense of the incorrect answer.  In my further defense, I was wrong in an interesting way that shed light on the relationship between the puzzle Jonathan posed and my own.

One thing that Jonathan impressed upon me was to “keep it simple stupid”, though he was polite enough not to put it that way.  List the outcomes, he kept insisting, and count.  The emphasis on basic methods of counting and charting relates nicely to some of Wittgenstein’s criticisms of logicism, which I have been reading about lately.  Especially in his later writings, but also to some extent in his early work, Wittgenstein attacked the logicist project as overly reductive.  He emphasized the dependence of mathematical understanding on basic rule-governed practices like counting, which rules are conceptually prior to and independent of logic (for detail see this reconstruction of Wittgenstein’s argument for this priority claim).

Wittgenstein also placed emphasis on the inventiveness involved in mathematical reasoning.  Neither rules nor invention, it was maintained, could be reduced to logic; hence, the Frege-Russell project could only give an at-best-incomplete account of the epistemology of mathematics.  (I should menition that the logicist line of defense would likely be a charge of psychologism, but it is not my purpose in this post to pursue that point).    Juliet Floyd, in her essay “Wittgenstein’s Philosophy of Logic and Mathematics”, summarizes the point about invention thusly:

Wittgenstein’s emphasis on the image of the mathematician as inventor or fashioner of models, pictures, and concepts was, in the main, directed at the philosophical talk of those, like Hardy and Russell, who insisted on speaking of mathematical reality in a freestanding way, picturing the logician or mathematician as a zoologist embarked on an expedition to new, hitherto unseen land, analogous to an empirical scientist…  For Wittgenstein, the mathematician is an inventor, not in the sense of making up truth willy-nilly as he or she goes along, as the pure conventionalist would suppose, but in the sense of engaging in the activities of fashioning proofs, diagrams, notations, routines, or algorithms that allow us to see and accept (understand, apply) results as answering to what does and does not make sense to us.

I think that this point about invention and its relationship to logicism, or indeed to any reductive/foundationalist mathematical epistemology, can be brought out by reflection on two proofs of the Pythagorean theorem.  First, consider this pictorially motivated proof by the (utterly brilliant) 12th century Indian mathematician Bhaskara:

The proof has the virtue of immediate surveyability and visual intuitiveness.  This is in contrast with Euclid’s proof, which while proceeding in simple and intuitive steps, takes a bit of effort to take in as a whole.  I’m struck by the inventiveness of Bhaskara’s proof.  It could not have been, in any straightforward way, been uncovered by just running through sequents with a computer program.  Given the axioms, Euclid’s proof, it seems to me, could have been discovered by a computer.  This is not to say that Euclid lacked inventiveness, of course, since stating his axioms and realizing that they could form a foundation for geometric study was an incredible insight.

I’ll let Floyd, on behalf of Wittgenstein, have the last word for now, but needless to say I think that there is much, much more to say about the relationship between “invention” and foundations in mathematics.  (When Floyd employs the term “metamathematics” you may think of set theory as well as Frege and Russell’s approach to foundations in higher-order logics):

The trouble with metamathematics, for Wittgenstein, is that it tends to mislead philosophers into thinking that the metamathematical language gives us a single way of surveying the core, or interpreting the meaning, of apparently fundamental mathematical and logical notions.   But ascent to the metalanguage is just another perspective on practices that gain their character within language from their working applications in human life.  Such ascent may change our perspective on our own language, but it grows from our current practices, and it is parasitic upon them: it cannot make them more epistemologicall certain.

An analogy: Bhaskara’s proof is to Euclid’s, perhaps, as a familiar arithmetic calculation is to a set-theoretical or logicist derivation.  It must be granted that if “surveying the core” or “interpreting the meaning” is our interest then the more familiar, the more picturable, and the more simply and totally grasped are to be prefered. . .  but that is only the last word for now.