Indexing

Indexing is mind-numbing. I've got over 700 proposed headings for the index of my One Body book. I need to search for key words for each heading, press ctrl-alt-I to invoke my index-marker entry macro, and then continue. After many hours, I've got about 50 headings done (not counting various "see"-type headings). But it's going faster now as I am going from most-prevalent to least-prevalent. Right now I'm indexing "male" and "female". The worst was indexing "love, romantic".

Anyway, indexing is why I haven't been posting much.

Sacrificing the fine-tuning argument to the argument from evil

The argument from evil is no stronger an argument than the fine-tuning argument. Moreover, the two are nicely paired up. Just as the fine-tuning argument seems to be seriously weakened by supposing a multiverse (since if there are infinitely many worlds, it's less surprising that some support life), so too the argument from evil is seriously weakened by supposing a multiverse of all creation-worthy worlds (since then there will presumably exist infinitely many worlds with lots of evils, as long as they are creation-worthy).

So here is a dialectical move a theist can make. Just sacrifice the fine-tuning argument to the argument from evil. Let the two cancel out! That still leaves the theist with a number of powerful arguments such as the cosmological argument, the argument from religious experience, the argument from moral epistemology, the argument from plausible miracle reports, the argument from consciousness and the argument from nomic regularity. The atheist, however, is left with little ammunition, besides some minor arguments concerning the exact formulation of divine attributes, which minor arguments can balanced off with less weighty arguments for theism, like the ontological argument or the argument from the experience of our lives as planned by another.

And so the balance of evidence, even if one does not take particular theistic arguments as apodeictic (I think one should do that in the case of the cosmological argument), strongly favors theism.

Good reasons, naturalism and evolution

Start with these three facts:

1. I have good (prima facie) reason to promote my own survival.
2. I believe (1).
3. I know (1).

Now, there is an excellent evolutionary explanation of fact (2). Beings that act on what they take to be reasons are unlikely to survive long enough to reproduce unless they take promoting their survival to be a good prima facie reason. However, this evolutionary explanation does not appear to have much connection with the normative fact (1). Thus, if this evolutionary explanation of fact (2) is the whole relevant story, then it is a mere coincidence that my belief that survival is reason-giving happens to be true. But then my belief isn't knowledge—it is at best a case of justified true belief, contrary to (3).

This means that the evolutionary explanation of fact (2) isn't the whole of the relevant story about why I have the belief that promoting my survival is worthwhile. We need, thus, an explanation of fact (2) that connects that belief with the normative fact (1). I doubt that naturalism can provide such a connection. Theism, on the other hand, can. For God could have deliberately created us in such a way that evolutionary processes would lead us to true belief in (1). On this theistic evolutionary story, it is no coincidence that (a) we believe proposition (1) and (b) proposition (1) is true. Thus, this story is to be preferred to a naturalistic one.

But what if the naturalist denies that (1) is true? Then she either does not believe that she has reason to promote her survival or she does. If she does believe it, then she contradicts herself—she believes something that she takes not to be true! But if she does not believe it, then for what reason does she act in her daily life as if she had reason to promote her own survival?

What I like about this argument is that it takes what seems an obvious strength of naturalistic evolution, namely its ability to handily explain facts such as (2), and turns it into a liability.

Defending what you don't really believe?

Here's a fascinating study. By changing what was in front of the subjects on the questionnaire they were filling out, the subjects were tricked into believing that they believed the opposite of what they had just affirmed. What is fascinating is that a slight majority not only would read out loud and affirm that opposite (say that something is permissible, which they first said was not), but would go on to defend that in argument.

I wonder what they were asserting when they seemed to affirm the opposite to their initial claim. It's tempting to say that they were simply misspeaking and hence we should not attribute to them the assertion of something that they didn't believe. But then they defended what they literally said, which suggests that this is what they were asserting.

I guess I am inclined to think they weren't asserting contrary to their beliefs, but they were arguing contrary to their beliefs. Maybe this is another way of seeing that arguments come apart from why one believes what one does.

Priors don't wash out

When I was a grad student, I was taught that in Bayesian epistemology the prior probabilities wash out as evidence comes in.

But that's false or at least deeply misleading.

Suppose Sam and Jennifer start with significantly different priors, but have the same relevant conditional probabilities and the same evidence. Then their posterior probabilities will always be significantly different. For instance, suppose Sam and Jennifer start with with priors of 0.1 and 0.9 respectively for some proposition p. They then get a ton of evidence, so that Sam's posterior probability is 0.99. But Jennifer's posterior probability will be way higher, about 0.99988. Suppose Sam's is 0.999. Then Jennifer's will be 0.999988. And so on. Jennifer's probabilities will always be way higher.

If the difference between 0.999 and 0.999988 seems small, that's because we're using the wrong scale. Notice that Sam assigns 81X higher probability to not-p than Jennifer does.

And in fact, if with Turing we measure probabilities with log-odds (log-odds(A) = log (P(A)/(1−P(A))), then no matter how much Sam and Jennifer collect the same evidence, Jennifer's log-odds for p minus Sam's will always equal about 4.39.

A potential solution to the Benacerraf problem

The Benacerraf problem is that there seem to be too many answers to the question of what the numbers are—too many constructions will do the trick—and no reason to say that one of them is the correct answer. In an earlier post, I suggested a way of biting the bullet. Here I want to suggest a very different solution.

It seems that often purely mathematical facts are part of the explanation of physical phenomenon. For instance, let's suppose that the fact that some real-valued f solves some differential equation Df=0 (where D is some complicated differential operator) partly explains some physical phenomenon F. Then it is reasonable to say that the values of f are really the real numbers.

This strategy may well fail. Here's how that could be. It could be that f is, say, a function that assigns a temperature to each position in spacetime, and temperatures aren't numbers, but numbers-with-a-unit. In that case, the fact that Df=0 might be analogous to a purely mathematical fact, because maybe temperatures (whose Platonic ontology could be that of determinates of a determinable, or which might get some non-Platonic ontology) satisfy the same axioms as the real numbers do. And then strictly speaking the "0" in Df=0 might have some unit attached to it. But the purely mathematical fact doesn't do any explaining—however, because of the axiomatic correspondence, it gives us reason to believe the corresponding fact about temperature-valued functions of position. If it turns out that the ontology of physics works in this kind of a way, then that would be a reason to adopt some structuralist type of solution to the Benacerraf problem.

But it could also turn out that the ontology of physics actually deals in numbers, not just numbers-with-a-unit. For instance, it could be that what it is to have a certain temperature (temperature is just an arbitrary example here; in fact, temperature surely reduce to more fundamental quantities) is to stand in a certain relation to a number. That number would then be the temperature, and what the values of those numbers are would determine what the objectively natural units of temperature are. In that case, we would have a solution to the Benacerraf problem.

So it could well be the case that the solution the Benacerraf problem depends on what is to be said about determinables like mass-energy and charge.

More amusement with infinitesimal probabilities

Suppose we uniformly pick a random number from the interval [0,1] ([a,b] is the closed interval from a to b, both inclusive). On standard probabilistic measures, the probability of picking any particular number is zero. Some people don't like that, instead insisting that the probability should be infinitesimal.

So, here's one plausible desideratum for uniform probabilities on [0,1]:

1. For every x in [0,1], P({x})>0.

Here are two more very plausible desiderata:

2. P is a finitely additive probability measure that makes all intervals measurable.
3. If I and J are closed intervals and J has twice the length of I, then P(J)=2P(I).

It turns out that 1-3 are incompatible. The argument is super-simple. Let J=[0,1]. By two applications of (3), we have P([0,1/2])=P([1/2,1])=(1/2)P([0,1]). But by finite additivity we have P([0,1])=P([0,1/2])+P([1/2,1])−P({1/2}) (we need to subtract the midpoint as it's in both of the sub-intervals. Thus P([0,1])=(1/2)P([0,1])+(1/2)P([0,1])−P({1/2}). Hence P([0,1])=P([0,1])−P({1/2}). And the only way this can be is if P({1/2})=0, contrary to (1).

I don't think this little fact is a very big deal, in that perhaps we can still have (3) for half-open intervals (ones that contain one but not the other endpoint). But it does show that we're not saving all intuitions about the measure structure of [0,1] when we add infinitesimals—we violate the scalability intuition in (3).

Sceptical scenarios and theism

There are many large-scale sceptical scenarios: brains in vats, evil demons, anti-inductive worlds, evolutionary scenarios that lead astray, mathematical faculties out of touch with Platonic reality, Boltzmann brains, the five minute hypothesis, etc. I'll just call these "sceptical scenarios". The crucial feature of a sceptical scenario is that some doxastic faculty of ours is completely out of whack with reality in such a way that we have no way of correcting for the error by using this and other faculties.

Now, most people aren't in any sceptical scenario. I am not claiming I know this (though in fact I think I do know this), but only that it is true. What explains the striking fact that most people aren't in any sceptical scenario? For any particular sceptical scenario, the naturalist can try to explain why most people aren't in it. That explanation may or may not be very good. But the theist can explain all at once why most people aren't in any sceptical scenario: God is unlikely to create a world where most people are in a sceptical scenario. This is a significant explanatory advantage of theism.

Notice also what happens when a new sceptical scenario is invented, such as my scenario that all the apparently random processes in our world are probabilistically non-measurable, but look like they were measurable. The theist's explanation automatically extends to cover it. But the naturalist may well need to scramble to create a new explanation or posit yet another brute fact.

Vagueness and the foundations of mathematics

There are many set-theoretic constructions of the natural numbers. For instance, one might let 0 be the empty set ∅, 1 be {0}, 2 be {1,2}, and so on. Or one might let 0 be ∅, 1 be {∅}, 2 be {{∅}}, and so on. (The same point goes for the rationals, the reals, the complex numbers, and so on.) Famously, Benacerraf used this to argue that none of these constructions could be the natural numbers, since there is no reason to prefer one over another.

My graduate student John Giannini suggested to me that one might make a move of insisting that there really is a correct set of numbers, but we don't know what it is, a move analogous to epistemicism about vagueness. (Epistemicists say that there is a fact of the matter about exactly how much hair I need to lose to count as being bald, but we aren't in a position to know that fact.)

It then occurred to me that one might more strongly take the Benacerraf problem literally to be a case of vagueness. The suggestion is this. Provable intra-arithmetical claims like that 2+2=4 or that there are infinitely many primes are definitely true. Claims dependent on one particular construction of the naturals, however, are only vaguely true. Thus, it is vaguely true that 1={0}. Depending, though, on what sorts of naturalness constraints our usage might put on constructions, it could be that some conditional claims are definitely true, such as that if 3={0,1,2}, then 4={0,1,2,3}.

There are some choices about how to develop this further on the side of foundations of mathematics. For instance, one might wonder if some (all?) unprovable arithmetical claims might be vague. (If all, one might recover the Hilbert program, as regards the definite truths.) Likewise, extending this to set theory, one might wonder whether "set" and "member of" might not be vague in such a way that the Axiom of Choice, the Continuum Hypothesis and the like are all vague.

Vagueness, I think, comes from to our linguistic practices undeterdetermine the meanings of terms. Likewise, our arithmetical practices arguably undetermine the foundations.

The above account neatly fits with our intuition that intra-mathematical claims are much more "solid" than meta-mathematical claims. For the meta-mathematical claims are all vague.

The next step would be to consider what happens when plug the above into various accounts of vagueness. Epistemicism is one option: our arithmetical terminology does have reference to one particular choice of foundation, but we aren't in a position to see what it is. I find promising a theistic variant on epistemicism. Supervaluationism seems particularly neat here. There will be one precification which precisifies things consistently with one foundational story, and another with another. can also consider other options.

There might even be some elements of epistemicism and some of supervaluationism. For there might be facts beyond our ken that say that some foundational stories are false—the epistemicism part of the story—but these facts may be insufficient to determine one foundational story to be right.

That said, I think I still prefer a more ordinary structuralism, though this story has the advantage that it takes the logical form of mathematical claims at face value rather than as disguised conditionals.

Deflation of the foundations of probability

I don't really want to commit to the following, but it has some attraction.

Question 1: What is probability?

Answer: Any assignment of values that satisfies the Kolmogorov axioms or an appropriate analogue of them (say, a propositional one).

Question 2: Are probabilities to be interpreted along frequentist, propensity or epistemic/logical lines?

Answer: Frequency-based, propensity-based and epistemically-based assignments of weights are all probabilities when the assignments satisfy the axioms or an appropriate analogue of them. In particular, improved frequentist probabilities are genuine probabilities when they can be defined, but so are propensity-based objective probabilities if they satisfy the axioms, and likewise logical probabilities. Each of these may have a place in the world.

Question 3: But what about the big metaphysical and epistemological questions, say about the grounds of objective tendencies and epistemic probabilities?

Answer: Those questions are intact. But they are not questions about the interpretation of probability as such. They are questions about the grounds of objective propensity or about the grounds of epistemic assignments. Thus, the former question belongs to the philosophy of science and the metaphysics of causation and the latter to epistemology.

Question 4: But surely one of the interpretations of probability is fundamental.

Answer: Maybe, but do we need to think so? Take the axioms of group theory. There are many kinds of structures that satisfy these axioms. Why think one kind of structure satisfying the axioms of group theory is fundamental?

Question 5: Still, couldn't there be connections, such as that logical probabilities ultimately derive from propensities via some version of the Principal Principle, or the other way around?

Answer: Maybe. But even if so, that doesn't affect the deflationary theory. There are plenty more structures that satisfy the probability calculus that do not derive from propensities.

Question 6: But shouldn't we think there is a focal Aristotelian sense of probability from which the others derive?

Answer: Maybe, but unlikely given the wide variety of things that instantiate the axioms. Maybe instead of an Aristotelian pros hen analogy, all we have is structural resemblance.

A new sceptical argument

Let's say that an infinite sequence of real-numbered observations is generated by independent runs of a random process. Suppose that we can represent the runs of the random process as independent and identically distributed random variables X₁,X₂,.... Recall that a random variable is a function f from some probability space Ω to the reals R with the measurability property that f⁻¹[B] is a measurable subset of Ω for every Borel-measurable subset B of the reals R (and it's enough to check this for B an interval, since the intervals generate the Borel sets).

It turns out that under these assumptions we can almost surely recover the distribution of the random variables X_i from the observed sequence. For almost surely the frequency of the observations fitting into any given interval with rational numbers as ends will converge to the probability that X_i is in that interval. And since there are only countably many such intervals, almost surely for every such interval I we can read off the probability P(X_i in I) from the observed frequencies. And then by the uniqueness condition in the Caratheodory extension theorem, we can recover the probability of X_i being in A for any Borel subset, not just a rational-ended interval.

So far this sounds like a kind of vindication of infinitary frequentism. It is a helpful, optimistic result.

But notice a crucial assumption the recovery of the distribution of the X_i made: that the X_i are measurable when considered as functions to the Borel-measure space R. But there are infinitely many other σ-algebras on R besides the Borel one. When recovering the distribution from the observations, what justifies the assumption that the X_i are measurable as a function to R considered as coming with the Borel σ-algebra?

We might have some hope that the recovery process will be likely to fail if the X_i aren't measurable in this way, so that if the recovery process succeeds, we have good reason to think that the X_i have this measurability condition. But I suspect that some analogue of these results will show that this won't work.

So I guess we just need to assume measurability with respect to Borel sets. But why? Because God loves the Borel sets? It's not so crazy. I love the Borel sets, and God made me in his image, after all. :-)

Not only am I myself, but I am my self

1. (Premise) I am experiencing writing this post.
2. (Premise) My self is experiencing writing this post.
3. (Premise) Only one entity is experiencing writing this post.
4. So, I am my self.

This is an oblique partial Olsonesque response to a paper by Himma. It's not fully a response.

An argument from evil against naturalism

Consider this valid argument:

1. (Premise) Moral outrage at an event is misplaced when no one is responsible for the event.
2. (Premise) Moral outrage at the suffering of animals before the advent of humankind is not misplaced.
3. (Premise) If naturalism is true, then no one is responsible for the suffering of animals before the advent of humankind.
4. So, naturalism is false.

I don't know if (2) is true, though. But this argument does put pressure on the naturalist running an argument from the suffering of animals against the existence of God. For that argument is persuasive in large part by creating moral outrage in the reader. But if naturalism is true, that outrage is misplaced.

What if theism is true? Is the outrage misplaced? That depends. If, say, the devil is behind that suffering, it's not misplaced.

Values, persons and wholes

Start with these premises:

1. Value personalism: Nothing is more valuable than a good person except perhaps for another good person.
2. If A has finite value and B has positive value, and the mereological sum A+B exists, then the mereological sum A+B has more value than A.

Thus, it is not possible to have a mereological sum of a finitely valuable good person, say Socrates, and one or more (the "or more" will use transitivity of "is more valuable than") positively valuable objects, say frogs or oak trees, unless the sum is a person. But whether things compose a whole shouldn't depend on their values. Thus it is not possible to have a mereological sum of a finite person and one or more things, unless the sum is a person.

So, if there are mereological sums, finite persons are not a part of them, unless the sums themselves are persons. But it would be weird if it were possible for finite persons to be proper parts of mereological sums that are persons but not of other mereological sums. So it is reasonable to conclude that finite persons can't be proper parts of mereological sums. But neither can God be a proper part of any mereological sum. So, no person can be a proper part of a mereological sum.

I think the best explanation of all these facts is that there can't be such a thing as a mereological sum.

Defending infinitary frequentism from some arguments

Frequentism defines probabilities in terms of long-term frequencies of outcomes. This doesn't work very well with finite frequencies—for one, it's going to conflict with physics in that finite sequence frequentism can only yield rational numbers as probabilities while quantum physics is quite happy to yield irrational numbers. As a result frequentism is often extended to suppose a hypothetical infinite sequence of data for defining frequencies. Alan Hajek has a paper that gives fifteen arguments against such a frequentism. Fourteen of them are strong arguments against standard hypothetical frequentism (I am unmoved by argument 15 involving infinitesimals, since I doubt that infinitesimal probabilities are much use to us).

But it turns out that one can formulate a frequentism that escapes or partly escapes some of Hajek's arguments.

A representative of these six arguments is the observation going back to De Finetti that probabilities defined via frequencies fail to satisfy the Kolmogorov Axioms (arguments 13 and 14). But my modified frequentist probabilities satisfy the Kolmogorov Axioms.

For simplicity, our data will be real valued, but the extension to Rⁿ is easy. Let s=(s_n) be our sequence of real numbers in R. For any subset A of R, let F_n(s,A) be the proportion of s₁,...,s_n that is in A. Let L(s,A) be the limit of F_n(s,A) if that limit exists, and otherwise L(s,A) is undefined.

Say that a (classical) probability measure m on the Borel subsets of R fits s provided that for all subintervals I of R, L(s,I) is defined and L(s,I)=m(I).

If there is a probability measure m that fits s, let a frequentist probability measure P_s defined by s be the completion of m (basically, the completion of a measure sets the measure of all subsets of null sets to be measurable and have measure zero).

Proposition:

1. If s defines a frequentist probability measure, it defines a unique frequentist probability measure.
2. Suppose that P is a probability measure and X=(X_n) is a sequence of independent identically distributed random variables. Let P₁ be the measure on R defined by P₁(A)=P(X₁ in A). Then with probability one, X defines a frequentist probability measure on R which coincides with P₁.

Because P_s is an ordinary Kolmogorovian probability measure, Hajek's arguments 13 and 14 do not apply. Argument 15 is anyway not very convincing, but is weakened since our version of frequentism handles the case of a dart thrown at [0,1] about as well as one can expect classical probabilities to handle it. (There is also a tension between arguments 13-14 and 15, in that probabilities involving infinitesimals are unlikely to be Kolmogorovian.) It is plausible that our frequentist probability measure will provide frequencies only when there is a probability, which makes argument 8 not apply, and non-uniqueness worries from argument 4 are ruled out by (1). I think the frequentist can bite the bullet on arguments 5 and 6, whether with standard frequentism or our modified version, given that the problem occurs only with probability zero.

Remark: The big philosophical problem with this is the reliance on intervals.

Quick sketch of proof:

Claim (1) is easy, because two Borel measures that agree on all intervals agree everywhere.

Claim (2) is proved by letting S be the collection of (a) all intervals with rational numbers as endpoints and (b) all singletons with non-zero P₁ measure, and using the Strong Law of Large Numbers to see that for each member A of S almost surely L(X,A) exists and equals P₁(A). But since S has countably members (obvious in the case of the intervals, but also easy in the case of the singletons), almost surely for every A in S we have L(X,A) existing. Moreover, almost surely, no singleton with null P₁ measure will be hit by infinitely many of the X_n, and hence L(X,A) will be defined and equal to zero for all such singletons.

Thus there is a set W of P-measure one such that on W, we have L(x,A) existing and equal to P₁(A) for every A that is either an interval with rational number endpoints or a singleton. Approximating any other interval A from below and above with monotone sequences of rational-number-ended intervals plus or minus one or two singletons, we can show that L(x,A) exists and equals P₁(A) for any other interval everywhere on W.