An account of laws

According to the Lewisian best system analysis of laws, a proposition p is a fundamental law if and only if it is an axiom in the best system. There is room for variation in the concept of a best system, but a standard version in deterministic settings is that the best system comprises only truths and optimizes the brevity of its axioms and the informativeness of its theorems about the world. The biggest problem for me with the best system account is that the fact that something is an axiom in the best system simply does not make it be explanatory.

I think this is a better account. A proposition p is a fundamental law of nature provided that:

1. p is an axiom in the best system, and
2. God wills p, as such.

(I am not sure if the will in (2) should be taken to be antecedent or consequent. If miracles are counterinstances to laws, it must be antecedent. But a lot of people think that's a bad account of miracles, and that allows it to be consequent.)

The "as such" in (2) rules out a case where God instead of willing p, wills something that entails p.

This account solves the explanatory problem with Lewis's account by making the fundamental laws be explanatory. They are not explanatory directly because they are axioms in the best system, but rather because God wills them.

Interestingly, I think (1) may imply (2) in the actual world, by divine omnirationality. For that p has the kind of simplicity and fecundity that axioms in the best system are going to have gives God a reason to will p. And since p in fact holds, presumably God willed p. The only exception is going to be if p reports the sort of thing that God has reason to distance himself from. Suppose, for instance, that everyone who is tempted a certain way sins. Then that universal generalization might be a best system axiom, but God has reason not to will it. But in fact it does not seem that any axioms in our world's best system are going to be like that—such regularities don't seem to be far-reaching enough. All the candidates we hear about from physics are propositions that God does not seem to have reason to distance his will from.

If this is right, then in the actual world, all the axioms in the best system are fundamental laws, and Lewis is contingently right. Moreover, this line of thought shows that the fact that p is an axiom in the best system makes it likely that God wills it. Consequently, as long as we know that God exists, we get to keep the epistemological benefits of Lewis's system.

Of chocolates and laws

According to David Lewis, the fundamental laws of nature are those propositions that collectively optimize a balance of the twin desiderata of informativeness and simplicity. (You can get maximal informativeness by listing all the facts about the world, at the expense of maximal complexity of description, and you can get maximal simplicity by saying nothing, at the expense of minimal informativeness.) And laws are what follows from the fundamental laws.

But laws are explanatory. While Lewis's laws aren't.

Imagine a finite world w₁ which contains a powerful contingent magical being who creates a very large number N of golden boxes, and places a chocolate in each one, to fulfill some aesthetic goal—chocolate goes well with gold. No other explanation of the chocolate content of the boxes exists at w₁.

We can ask at w₁ why the third golden box contains a chocolate. And the answer is that the magical being put a chocolate in every golden box.

Now imagine a world w₂ just like w₁ but where the magical being doesn't exist and the boxes and chocolates come into existence ex nihilo. Nothing gets added to w₂ that wasn't there in w₁. Plainly at w₂ there just is no explanation of why the third golden box contains a chocolate.

If the number of boxes N is large enough, the proposition that every box contains a chocolate will be informative enough that it'll be included in the system that maximizes informativeness and simplicity (as N increases, the informativeness of the proposition increases but its simplicity stays constant).

But then if Lewis's account of laws is correct, then at w₂ it will be a law that all golden boxes contain chocolates. But laws are explanatory. So at w₂ we'll be able to explain why the third golden box contains a chocolate. But we said that at w₂ there is no explanation of this!

So the Lewisian account of laws is wrong.

Options: (1) Deny that laws are explanatory. (2) Abandon the Lewisian account of laws. (3) Deny that it is possible to have uncaused chocolates.

I think that (3) by itself won't do, because we can run the argument on a counterpossible. And (1) is unattractive. That leaves (2).

In summary, I think "Lewisian laws" aren't explanatory, and hence aren't laws.

Principle of Double Effect as an action-requiring principle

Normally the Principle of Double Effect (PDE) is taken to be a permissive principle: it gives permissibility conditions for actions that are foreseen to have an evil effect. Roughly, the conditions are: the action is not in itself wrong; the evil is not intended as an end or as a means; and the evil is proportionate to an intended good.

Suppose that we adopt the thesis that refraining from an action is itself a kind of action, and that refraining can have effects just as any other action can. Call this the Refraining Parity Thesis (RPT).

Given RPT, it turns out that many of the actions that the PDE is used to justify are actually actions that the PDE actually requires. Suppose, for instance, that dropping a bomb on the enemy headquarters will cause a handful of civilian deaths but the deaths of the military leadership will lead to an early end to the war, saving thousands of civilian lives. Given RPT, consider the action of refraining from bombing. Refraining from bombing (considered maybe as an action of a military commander) has an intended good effect, the saving of the lives of the civilians near the headquarters, and an unintended evil effect, the deaths of thousands of civilians later. It is very plausible that the evil is disproportionate to the good, and hence PDE does not allow one to refrain from bombing.

There will, however, be cases where PDE allows but does not require an action. Suppose that I could save your life by jumping on a grenade. The PDE allows me to jump. The intended effect is saving your life by the absorption of kinetic energy, the foreseen evil is my death, and my death is not disproportionate to saving your life. But refraining from jumping is also permissible. The intended effect is saving my life, the foreseen evil is your death, and your death is not disproportionate to saving my life.

I don't know if RPT is true. I am inclined to think that refraining is not on par with positive action.

Paul Symington on Sophie's choice

Paul Symington has an intriguing post on Sophie's choice, where he defends the thesis that if virtue ethics is right she should have refused to cooperate. I disagree, but I think the post is really worth thinking about.

Animal consciousness

Sometimes I come up with an argument such that I can't tell for sure if it's more a joke or a really interesting argument. The following is a case in point:

1. (Premise) If some non-human earthly animals are conscious, all normal mammals are conscious.
2. (Premise) There have ever been several orders of magnitude more non-human mammals than humans.
3. (Premise, plausibly a consequence of 2) If all normal mammals are conscious, I should very strongly expect not to experience reality as a human.
4. (Premise) I experience reality as a human.
5. So, probably, not all normal mammals are conscious. (By 3 and 4)
6. So, probably, no non-human earthly animals are conscious.

As for (1), if some non-human earthly animals are conscious, a line must be drawn as to where consciousness is found. There are two main plausible places to draw such a line: (a) humans versus other animals, and (b) animals with sophisticated brains versus other animals. If we draw the line in the second place, all normal mammals will be conscious. As for (2), I don't have data as to how many mammals there are on earth. I saw an unreferenced "400 million" online, and a referenced somewhat smaller estimate for the number of birds (and I could run the argument with birds, too, I think). There are apparently roughly as many rats and mice in the world as humans. And there have been non-human animals for millions of years before there were humans.

I think the difficult philosophical question is whether (3) is true and what sense can be made of it.

I am more inclined to see this argument as a joke, or maybe as a challenge to figure out how anthropic arguments work.

Inductive scepticism and multiverses, theistic and secular

You have a minor ailment. You wonder to yourself if it's going to clear up by itself within a week. Suddenly you receive utterly conclusive evidence that an omniscient and perfectly honest being is speaking with you. Now consider the following case:

1. The being informs you that there is a twin-earth, where everything has hitherto been exactly like on earth. In particular, twin-earth contains a duplicate of you, whose life has hitherto been exactly like yours. In particular, your twin has the same minor ailment you do. Moreover, the being informs you that exactly one of you and your twin will have their ailment clear up within a week.

The following is clear: at the end of this, you should not be at all confident that your ailment will clear up within a week. I claim that this is still true in each of the following variant case:

2. Just like (1), except that the being also gives you some additional information afterwards. He tells you that on both earth and twin-earth, superb medical studies of the ailment have been done, and they have concluded that in the vast majority of cases like yours, the ailment clears up within a week.

The additional information about the medical studies is trumped by the information that of you and your twin exactly one will have the ailment clear up. You still should not be confident that your ailment will clear up in a week. Next suppose:

3. Just like (2), except that instead of the being telling you about the studies, you knew ahead of the communication from the being about these studies in your world, and after the being told you about the twin, you inferred that there must be such studies on twin-earth, too, since twin-earth has up to now been just like earth.

If in case (2) you shouldn't be confident that your ailment will clear up in a week, likewise you shouldn't be confident in case (3). Next consider:

4. Just like (3), except now the being informs you that there is an infinite number of such twin-earths, and if we let E be the set whose elements are earth and these twin-earths, then on infinitely many members of E, you or your twin (as the case might be) recovers within a week, and on an equal infinity of members of E, you or your twin does not recover.

Multiplying the twin-earths infinitely in this way should still not allow you to be confident that you'll be one of the lucky ones who recovers within a week. Finally, consider:

5. Just like (4), except that the twin-earths have not been exactly alike, but extremely similar, with any differences being far several orders of magnitude below the ability of our scientific experiments to discriminate between.

This should not make a difference. You still should not be confident of recovering within a week.

Now, if we are justifiably sure that there are infinitely many universes, then we should expect that there are universes that contain near twins like in (5), and we should expect that infinitely many of them your near twin recovers and on an infinity of them your near twin does not recover, and you should not be confident that you'll recover in a week. And if you aren't sure about there being infinitely many universes, but you simply assign a high probability to that hypothesis, then you should significantly lower your confidence that you will recover, below the confidence given by our best medical studies.

This applies to David Lewis's plurality of worlds. It seems to apply to scientific multiverse theories. And it applies to theistic versions of Lewis's theory, like those by Donald Turner or Klaas Kraay, on which only universes worthy of being created exist, since there plausibly is a just as big infinity of worlds worthy of creation where you don't recover in a week as ones where you do. (This is clear if our universe is far enough above the cut-off line that your failing to recover in a week will not push it below the cut-off line. If our universe is too close to the cut-off, then to ensure worthiness, some kind of a compensating good would also have to be present.)

Thus, infinite multiverse theories should sap our confidence in scientific predictions. This is particularly problematic for scientific multiverse theories.

I think the controversial move will be the transition from (3) to (4) actually.

Philosophy of cosmology blog

The Templeton-funded Philosophy of Cosmology project at Rutgers has a promising blog.

Sketches towards a theory of quantifiers and quantifier variance

Quantifier variance theorists think that there can be multiple kinds of quantifiers. Thus, there could be quantifiers that range over only fundamental entities, but there could also be quantifiers that range over arbitrary mereological sums. I will call all the quantifiers with a particular range a "quantifier family". A given quantifier family will include its own "for all x (∀x)" and "for some x (∃x)", but may also include "for most x", "for at least two x", and similar quantifiers. I will, however, not include any negative quantifiers like "for no x" or "for at most seven x", or partially negative ones like "for exactly one x". I will also include "singular quantifiers", which we express in English with "for x=Jones". In fact, I will be working with a language that has no names in it as names are normally thought of in logic. Instead of names, there will be a full complement of singular quantifiers, one for each name-as-ordinarily-thought-of; I am happy to identify names with singular quantifiers for the purposes of logic.

Say that quantifier-candidates are operators that take a variable and a formula and return a formula in which that variable is not open. Consider a set F of quantifier-candidates with a partial ordering ≤, where I will read "Q≤R" as "R is at least as strong as Q", and with a symmetric "duality" relation D on F. There is also a subset N of elements of F which will be called "singular". Then F will be a quantifier family provided that

1. There is a unique maximally strong operator ∀
2. There is an operator ∃ dual to ∀
3. If Q is dual to R then it can be proved that QxP iff ~Rx~P
4. ∃ is minimally strong
5. If R in F is at least as strong as Q in F, then from RxP one can prove QxP
6. From P one can prove QxP for any Q
7. From ∀xP one can prove P (note: open formulae can stand in provability relations)
8. If Q is singular, then Q is self-dual and Qx(A&B) can be proved from QxA and QxB

We may need to add some more stuff. But this will do for now.

One can set up a model-theory as well. A domain-model for a quantifier family will include a set O of "objects", and a set S of sets of subsets of O, such that if E is a member of S, then E is an upper set, i.e., if a subset A of O is in E, then so is any superset of A as long as it is still a subset of O. A member of S will be called an "evaluator". To get a model from a domain-model, we add the usual set of relations. An interpretation I in a given model for a language with a quantifier family will then involve an ordinary interpretation of the language's predicates, plus an assignment of quantifiers to members of S subject to the constraints that (a) if Q≤R, then I(R) is a subset of I(Q), (b) I(∀) is the evaluator {O}, (c) if Q is dual to R, then A is a member of I(Q) if and only if the complement O−A is not a member of I(R), (d) if Q is singular, then I(Q) is a filter-base. We can then define truth under I using the basic idea that QxP is true if and only if the set of objects o such that o satisfies P when put in for x is a member of I(Q) (this should be all done more carefully, of course).

(The interpretation of a name is always an ultrafilter. If we wanted to, we could restrict names to being interpreted as principal ultrafilters, in which case names would correspond to objects, but I think things are more interesting like it is.)

Ideally, we'd want to make sure we have soundness and completeness at this point. I'm basically just making this up as I go along, so there may be a literature on this, and if there is, there will presumably be results about soundness and completeness in it. And maybe we need more rules of inference and maybe I screwed up above. This is just a blog post. Moreover, we might want some further restrictions about how particular quantifiers, like "for most", are interpreted (the above just constrains it to having an evaluator between the evaluators for ∀ and ∃). The point of the above is more to give an example of what a formal characterization of a quantifier family might look like than to give the correct one.

But now it is time for some metaphysics. The notion of a quantifier family is a purely formal one. Moreover, the model-theoretic notion of interpretation that I used above won't be helpful for quantifier variance discussions because it talks of "sets of objects", whereas what "metaphysically counts as an object" varies between quantifier families.

It is easy to come up with quantifier families and perverse interpretations such that under such an interpretation, we would not want to count the members of the quantifier family "quantifiers". Nor would it be a quantifier variance thesis to say that there are many such families and interpretations, since that there are such is not controversial.

I think a Thomist can give an answer: a quantifier family in the formal sense is a bona fide quantifier family provided that the family is analogous to some privileged family of quantifiers, say quantifiers over substances. In other words, the different kinds of existence are defined by analogy to existence proper. This won't satisfy typical quantifier variance folk, as I think they don't want a privileged family of quantifiers. But that's the best I can do right now.

Conscience, authority and moral intuition

A former student of mine wrote to me with a query on about how institutional Church authority could co-exist with the authority of individual conscience. She argued that ultimately my conscience will decide whether the authority is to be trusted, and quoted Anscombe as saying that one cannot help but be one's own pilot.

This made me think a bit more about conscience and authority. I had recently been reading about the Charles Bonnet and Musical Ear syndromes. In these, visual or hearing loss, respectively, apparently causes the brain to confabulate visual or auditory data, respectively, to fill in the sensorily deprived blanks. In Charles Bonnet Syndrome, the sufferers see things like colored patterns, faces, cartoons, etc. In Musical Ear Syndrome, they are apt to hear music. The significant thing about both syndromes is that the sufferers are quite sane and fully realize that the incorrect sensory data they are receiving is mere hallucination (that the hallucinations are limited to a single faculty must help there). They may, however, be distressed due to worries that they are insane, particularly if they are misdiagnosed by a psychiatrist, as in a case I recall hearing of.

A reasonable sufferer from one of these two syndromes will accept the testimony of reliable others that what she visually or auditorily perceives isn't there. In so doing, she is genuinely being her own pilot. Indeed, if she were to uncritically accept the visual or auditory data, she wouldn't be being her own responsible pilot: she would be replacing considered judgment with the flow of experience. Likewise, my colorblind son defers to the color judgments of others; an object may look light green to him, but when others testify that it is light pink, he accepts their judgment, and in so doing exercises his epistemic autonomy.

I think something similar can and does happen in moral matters. We have moral intuitions. These moral intuitions can be more or less reliable. But of course raw moral intuitions do not have a final say. Even apart from authority, moral intuitions need to be harmonized. And it may turn out that the best moral theory fitting the bulk of one's moral intuitions can go against some of one's moral intuitions, and then a judgment must be made.

Moreover, there is nothing contrary to being one's own pilot in making a reasonable judgment that a family of one's moral intuitions, or even all of one's moral intuitions, are less reliable than the testimony of an individual or institution one has reason to trust. That is just much an exercise of one's epistemic autonomy as it would be to accept the moral intuitions over that testimony.

I think that sometimes we confuse conscience with moral intuitions. The deliverance of conscience is an all-things-considered judgment of what is morally to be done. It may take moral intuitions into account, but it may also take other relevant data into account as well. The deliverance of moral intuition is not, as such, the deliverance of conscience, though of course in the absence of evidence against the moral intuition, conscience is apt to reasonably accept the content of the moral intuition as true.

It is quite possible for one to reasonably come to the conclusion that one's moral intuitions are less reliable than the teaching of an authority. In such a case, when there is a conflict between one's moral intuition and a teaching of the authority, one's considered moral judgment will at least typically go with the teaching. (I say "at least typically" to leave open the possibility that, say, a particularly strong moral intuition might be judged more likely to be accurate than a teaching that the authority gives quite low weight to.) In so doing, one may very well be a responsible pilot of one's self, if the reasons for accepting the authority as reliable were very good ones.

And one is not going against conscience then. On the contrary, in such a case, it would go against conscience to follow the moral intuitions, because one's considered judgment is that the authority is more reliable than the intuitions.

Our moral intuitions while being a genuine source of moral knowledge are often distorted by the desire to find excuses for our own faults or, more excusably, those of friends. Moral intuitions should not be glorified with the name "conscience". Like a Charles Bonnet Syndrome patient, one can be reasonable in judging that one ought to submit to the judgment of another, and then the other's judgment is the deliverance of one's conscience.

At the same time, I should note that normally our moral intuitions will play a significant role in figuring out that a putative authority should be listened to. When the putative authority's teachings harmonize particularly with those moral intuitions that we take to be more reliable, that will count in favor of the claim to authority, and when they disagree, that will count against the claim to authority. Here I think there is a useful rule of thumb: moral intuitions that something is permissible are less to be trusted than moral intuitions that something is impermissible. An action is impermissible provided there is a conclusive moral reason not to do it. An action is permissible provided that there is no conclusive moral reason not to do it. Generally, perceptions of absence are less to be trusted than perceptions of presence. Moreover, the space of reasons is large, and to judge that none of the infinitely many considerations in that space gives conclusive reason not to do A is fraught witih difficulty. (Of course, judgments about permissibility are very often right, but perhaps only because of the base rate: most actions people perform are right.)

Do riches lead to vice?

This is a fascinating piece on social class and vice. Apparently, either being of a higher socioeconomic class, or seeing oneself as of a higher socioeconomic class, leads to vicious behavior. The article itself says in the abstract: "Mediator and moderator data demonstrated that upper-class individuals’ unethical tendencies are accounted for, in part, by their more favorable attitudes toward greed." So the love of money is a root of at least some other evils, science says.

Consolidating evidence

Here's something that surprised me when I noticed it, though it probably shouldn't have surprised me. The following can happen: My evidence favors p. Your evidence disfavors p. I know you are rational and competent. After talking with you, and consolidating evidence, I rationally increase my evidence for p.

Here's a case. Suppose we have a coin which is either biased 3:1 in favor of heads or biased 3:1 in favor of tails. We don't know which. I have observed a few coin tosses, and they included four tails and seven heads. My evidence supports the hypothesis that the coin is biased in favor of heads. You have observed a few coin tosses, and they were four tails and two heads. Your evidence supports the hypothesis that the coin is biased in favor of tails. Intuitively, I should lower my credence in the heads-bias hypothesis when I learn of your evidence.

But imagine further that the four tail tosses you observed are the same four tail tosses that I observed, but the two heads tosses you observed were not among the seven heads tosses I observed. Then consolidating our evidence, we get four tails and nine heads, which supports the heads-bias hypothesis.

This is humdrum: When we consolidate evidence, we need to watch out for double counting in either direction. The above case makes this striking, because when we eliminate double counting, we get confirmation in the opposite direction to what we would initially have expected.

There is a very practical moral of the above story. It is important not only to remember one's credence in the propositions one believes and cares about, but also the evidence that gave rise to this credence. For if one does not remember this evidence, it will be difficult to avoid double counting (or subtler failures of independence).

By the way, I think it is helpful to think of the disagreement literature as well as discussions about the nature of arguments and other social epistemology stuff as interpersonal data consolidation problems. Getting clear on what we are aiming at should help. You have data, I have data, we all have data. What we are aiming for are methods (algorithms, virtues, etc.) that help us consolidate data across persons to get a better picture of reality than we are likely to have individually.

Moreover, I think that morally speaking it is very important when engaging in argumentation to remember what we are doing: the telos of arguing is to consolidate data across persons in order to get to truth and understanding. This telos is social, as befits social animals. It is not the telos of an argument that I convince you of the argument's conclusion. Rather it is that I convince you of the truth or show you how truth hangs together. If instead of convincing you of the argument's conclusion I convince you by modus tollens of the falsity of one of the premises, and in fact the conclusion is false and so is that premise, then the point of arguing has perhaps been fulfilled. And if in a case where the conclusion is false my argument convinces you of that conclusion, then the argument is a failure qua argument.

Reasons of trust

Suppose you promise me to do something and suppose I should trust you. Then I have a moral reason not to check whether you did what your promised. Of course, if I have a special responsibility for it, I may also have a moral reason to check. But generally speaking, I think we have an imperfect duty not to check up on people when we should trust them. Moreover, we should trust people unless we have good reason to the contrary. I would be wronging a colleague if, out of the blue, I were to start running his papers through TurnItIn.com to look for plagiarism. Such an action would be a failure to show required trust. It would thus be contrary to collegial love.

Natural love, thus, requires natural faith of us. But our supernatural love for Christ requires supernatural faith of us.

Infinite lotteries and infinitesimal probabilities

The argument in this post is based on a construction by Dubins (see Example 2.1 here) that I've switched into an infinitesimal case.

Suppose you can have an infinite lottery with ticket numbers 1,2,3,... and each ticket has infinitesimal probability (perhaps the same one for each). Then really weird stuff can happen. Say I toss a fair coin, but don't show you the result. Instead, you know for sure that I will do this:

• If the coin was tails, I run an infinite lottery with ticket numbers 1,2,3,... and with each ticket having infinitesimal probability
• If the coin was heads, I run an infinite lottery with the same ticket numbers, but now the probability of ticket n is 2⁻ⁿ.

And you know for sure that I will then announce the result of the lottery.

Here's the oddity. No matter what my announcement, you will end up all but certain—i.e., assigning a probability infinitesimally short of 1—that the coin was heads. Here's why. Suppose I announce ticket n. Now, P(n|heads)=2⁻ⁿ but P(n|tails) is infinitesimal. Plugging these facts into Bayes' theorem, and assuming that your prior probability for heads was 1/2 (actually, all that's needed is that it be neither zero nor infinitesimal), your posterior probability P(heads|n) ends up equal to 1−a where a is infinitesimal.

So I can rationally force you to be all but certain that it was heads, simply by telling you the result of my lottery experiment. And by reversing the arrangement, I could force you to be all but certain that it was tails. Thus there is something pathological about the infinite lottery with infinitesimal probabilities.

This is, to me, yet another of the somewhat unhappy results that show that probability theory has a quite limited sphere of epistemological application.

How likely am I to be misled?

Suppose a hypothesis H is true, and I assign a positive prior probability P(H)>0 to H. I now perform some experiment such that I have a correct estimate of how likely the various possible outcomes of the experiment are given H as well as how likely they are given not-H. (For simplicity, I will suppose all the possible outcomes of the experiment have non-zero probability.) It could turn out that the experiment supports not-H, even though H is true. That's the case of being misled by the outcome of the experiment.

How likely am I to be misled? Perhaps very likely. Suppose that I have a machine that picks a number from 1 to 100, and suppose that the only two relevant hypotheses are H, the hypothesis that the machine is biased in favor of the number 1, which it picks out with probability 2/100, while all the other numbers are picked out with probability 98/9900, and not-H, which says that all the numbers are equal. My experiment is that I will run the machine once. Suppose that H is in fact true. Then, by Bayes' theorem, if I get the number 1, my credence for H will go up, while if I get anything else, my credence for H will go down. Since my probability of getting something other than 1 is 98/100, very likely I will be misled. But only by a little—the amount of confirmation to the fairness hypothesis not-H that is given by a result other than 1 is very small.

Could one cook up cases where I am likely to be misled to a significant degree?

It turns out that one can prove that one cannot. For we can measure the degree to which I am misled by the Bayes' factor B=P(E|H)/P(E|~H), where E is the outcome of the experiment. When B<1, then I am misled by the result of the experiment into downgrading my confidence in the truth H, and the smaller B is, the more I am misled. But it turns out that there is a very elegant inequality saying how likely I am to be misled to any particular degree:

1. P(B≤y)≤y.

This inequality holds for any experiment where all the possible outcomes have non-zero probability that I could perform, as long as my estimates of the conditional probabilities of the outcomes are correct.

We can also state the result in terms of probabilities. Suppose I start with assigning probability 1/2 to H. I then perform an experiment and perform a Bayesian update. How likely is it that I end up with a posterior probability at or below p? It turns out that the probability that I will end up with a posterior probability at or below p is no more than p/(1−p). So, for instance, the probability that an experiment will lead me to assign probability 1/10 is at most 11%, no matter how carefully the experiment is set up to be unfavorable to H, as long as I have correct estimates of the likelihoods of the experiment's outcomes on H and on not-H.

This has an interesting corollary. We know that it need not be the case that a series of experiments will yield convergence of posterior probabilities to truth, even if we have correct likelihood estimates. For the experiments might get less and less discriminating, and if they do so fast enough, the posteriors for the true hypothesis H will not converge to 1. But could things be so bad that the posteriors would be sure to converge to 0 (i.e., would do so with probability one)? No, it cannot be the case that with probability one the posteriors will converge to 0, because we could aggregate a sequence of experiments into a single large experiment (whose outcome is an n-tuple of outcomes) and use inequality (1).

So even though Bayesian convergence doesn't always happen, even if one has correct likelihoods, nonetheless we can be confident that Bayesian mis-convergence won't happen if we have correct likelihoods.

Gentler structuralisms about mathematics

According to some standard structuralist accounts, a mathematical claim like that there are infinitely many primes, is equivalent to a claim like:

1. Necessarily, for any physical structure that satisfies the axioms A₁,...,A_n, the structure satisfies the claim that there are infinitely many primes.

There are two main motivations for structuralism. The first motivation is anti-Platonic animus. The second is worries about uniqueness: if there are abstract objects, there are many candidates for, say, the natural numbers, and it would be arbitrary if our mathematical language were to succeed in picking out on particular family of candidates.

The difficulty with this sort of structuralism is that while it may be fine for a good deal of "ordinary mathematics", such as real analysis, finite-dimensional geometry, dealing with prime numbers, etc., it is not clear that there are enough possible physical structures to model the axioms of such systems as transfinite arithmetic. And if there aren't, then antecedents in claims like (1) will be false, and hence the necessary conditional will hold trivially. One could bring in counterpossibles but that would be explaining the obscure with the obscurer.

I want to drop the requirement that the structures we're talking about are physical structures. Thus, instead of (1), we should say:

2. Necessarily, for any structure that satisfies the axioms A₁,...,A_n, the structure satisfies the claim that there are infinitely many primes.

If we do this, we no longer have a physicalist reduction. But that's fine if our motive for structuralism is worries about arbitrariness rather than worries about abstracta.

Next, restrict the theory to being about what modern mathematics typically means by its mathematical claims. If we do this, the claim becomes logically compatible with Platonism about numbers. Let us suppose that there really are numbers, and our ordinary language gets at them. Nonetheless, I submit, when a modern number theorist is saying that there are infinitely many primes, she is likely not making a claim specifically about them. Rather, she is making a claim about every system that satisfies the said axioms. If the natural numbers satisfy the axioms, then her claims have a bearing on the natural numbers, too.

Here is one reason to think that she's saying that. Mathematical practice is centered on getting what generality you can. What mathematician would want to limit a claim to being about the natural numbers, when she could, at no additional cost, be making a claim about every system that satisfies the Peano axioms?

Now, if we go for this gentler structuralism, and allow abstract entities, we can easily generate structures that satisfy all sorts of axioms. For instance, consider plural existential propositions. These are propositions of the form of the proposition that the Fs exist, where "the Fs" directly plurally refers to a particular plurality. We can define a membership relation: x is a member of p if and only if x is said by p to exist. Add an "empty proposition", which can be any other proposition (say, that cats hate dogs) and say that nothing is its member. Then plural existential propositions, plus the empty proposition, with this membership relations should satisfy the axioms of a plausible set theory with ur-elements. If all one wants is Peano axioms, we can take them to be satisfied by the sequence of propositions that there are no cats, that there is a unique cat, that there are distinct cats x and y and every cat is x or is y, that there are distinct cats x and y and z and every cat is x or is y or is z, and so on.

I am not completely convinced that this sociological thesis about modern mathematics is correct. Maybe I can retreat to the claim that this is what modern mathematics ought to claim.