Antipresentism

Presentists think that the past and future are unreal but the present is real. I was going to do a tongue-in-cheek post about an opposed view where we have the past and future but no present. But as I thought about it, the position grew a little on me philosophically, at some expense of the tongueincheekness. Still, please take all I say below in good fun. If you get a plausible philosophical view out of it, that's great, but it's really just an exercise in philosophical imagination.

One way to think about antipresentism is to imagine the eternalist's four-dimensional universe, but then to remove one slice from it. Thus, we might have 1:59 pm and 2:01 pm, but no 2:00 pm. Put that way, the view isn't particularly attractive. Still, I do wonder why it would be more unattractive to remove just one time slice than to remove everything but that one time slice as the presentist does. It would, of course, be weird for the antipresentist to say that events first exist in the future, then pop out of existence just as one would have thought that they would come to be present, and then pop back into existence in the past. But perhaps no weirder than events coming out of nothing and going back into nothing, as on presentism. This way to think about antipresentism makes it a species of the A-theory.

But the antipresentisms I want to think about are ones that might be compatible with the B-theory. Start with the famous puzzles of Zeno and Augustine about the now. Augustine worried about the infinite thinness of the now. Zeno on the other hand worried about the fact that there are no processes in the now; there is no change in the now since within a single moment all is still.

One way of taking these ideas seriously is to see the present as an imaginary dividing line between the past and the future. There is in fact no dividing line: there is just the past and the future. (I think Joseph Diekemper's work inspired this thought.)

We might, for instance, instead of thinking of times as instants think of the basic entities as temporally extended events or time intervals, not made out of instantaneous events or moments. An event or interval might be past, or it might be future, or—like the writing of this post—it might be both past and future. (Thus, "past" and "future" is taken weakly: "at least partly past" and "at least partly future".) Some events or time intervals have the special property of being both past and future. We can stipulate that those events or time intervals are present. But they aren't real because they are present. They're just lucky enough to have two holds on reality: they are past and they are present. (In this framework, the presentist's claim that only present events are real sounds very strange. For why should reality require both pastness and futurity—why wouldn't one be enough?) There are no events or time intervals that are solely present.

There is a natural weakly-earlier-than relation e on events. If we had instants of time, we would say that EeF if and only if some time at which E happens is earlier than some time at which F happens. But that's just to aid intuition. Because there are noever instantaneous events, every event is weakly earlier than itself: e is reflexive. It is not transitive, however. The antipresentist theory I am sketching takes e to be primitive. There is also a symmetric temporal overlap relation o that can be defined in terms of e: EoF if and only if EeF and FeE.

If we like, we can now introduce abstract times. Maybe we can say that an abstract time is a maximally pairwise overlapping set of time intervals (or of events, if we prefer). We can say that t₁ is earlier than t₂ provided that some element of t₁ is strictly earlier than some element of t₂ (where E is strictly earlier than F provided EeF but not FeE). I haven't checked what formal properties this satisfies—I need to get ready for class now (!).

How to make an infinite fair lottery out of infinitely many coin flips

This is a technical post arising from a question Rob Koons asked me.

An infinite sequence of fair and independent coin flips determines a sequence of zeroes and ones (e.g., zero = tails, one = heads). Let Ω be the set of all infinite sequences of zero/one sequences, equipped with the probability measure P corresponding to the fair and independent coin flips.

Notice an invariance property capturing at least part of the independence and fairness assumption. If ρ_n is the operation of flipping the nth element in the sequence, and ρ_nA for a subset A of Ω is the set obtained by applying ρ_n to every sequence in A, then P(ρ_nA)=P(A) whenever A is measurable. Moreover, intuition extends this idea beyond the measurable sets: A and ρ_nA are always going to be probabilistically on par.

Let Ω₀ be the subset of Ω consisting of those sequences that have only finitely many ones in them. There is a natural one-to-one correspondence between Ω₀ and the natural numbers N. Suppose a=(a₀,a₁,...,a_k,0,0,0,...) is a member of Ω₀. Then let N(a) be the natural number whose binary digits are a_k...a₁a₀. Conversely, given a natural number n with binary digits a_k...a₁a₀, let n* be the sequence (a₀,a₁,...,a_k,0,0,0,...) in Ω₀. Thus, we can interpret the members of Ω₀ as binary numbers written least significant digit first.

For any members a and b of Ω, write a#b for the sequence whose nth element is the sum modulo 2 (xor) of the nth elements of a and b. For a subset B of Ω, let a#B = { a#b : b∈B }. We can think of a#B as a twist of B by a. If a is in Ω₀, I will call it a finite twist. Any finite twist can be written as a finite sequence of flips ρ_n, where the positions n correspond to the non-zero digits in the sequence we twist by. Thus, if A is measurable, a finite twist of it will have the same probability as A does, and even if A is not measurable, a finite twist will be intuitively equivalent to A.

Say that a~b if and only if a and b differ in only finitely many places. Thus, a~b if and only if a#b is a member of Ω₀. This is an equivalence relation. By the Axiom of Choice, there is a set A₀ such that for every b in Ω, there is a unique a in A₀ with a~b. (Thus, A₀ contains exactly one member of each equivalence class.) For any natural number n other than 0, let A_n=n*#A₀ and it's easy to check that this equation holds for n=0 as well.

It's easy to see that the A_n are disjoint and their union is all of Ω. They are disjoint because if a is in n*#A₀ and m*#A₀, then a=n*#b and a=m*#c for b and c in A₀. It follows that b~c. But A₀ contains only one member from each equivalence class, so b=c, and so n*#b=m*#b, from which it obviously follows that n*=m* and so n=m. Their union is all of Ω, because if b is in Ω, and a is the unique member of A₀ such that a~b, then N(a#b)*#a=(a#b)#a=b (by obvious properties of addition modulo 2), and so b is a member of A_N(a#b).

But all the A_n are going to be intuitively probabilistically on par: they are each a finite twist of A₀.

Our lottery is now obvious. Given a random sequence of coin flips, we take its representation a in Ω and choose the unique number n such that a is in A_n.

This is really the Vitali-set construction applied directly to sequences of coin flips. Note that along the way we basically showed that Ω has nonmeasurable subsets. For the sets A_n cannot be measurable with respect to P, since they would all have equal probability, and so by countable additivity they would have to have probability zero, which would violate the total probability axiom.

The construction in this post is more complicated than the one here, I guess, but it has the advantage that it always works, while that construction only worked with probability 1.

A divine command and an open future

I'm piling on to the argument here.

Suppose God creates Adam and Eve, and gives them eternal life. He then commands them that:

1. They freely pray for at least a minute on each of the infinitely many Sabbaths starting with day t₇ (the day after their creation).

This seems a reasonable command. But it is unreasonable to command something that the agent cannot ever make true. And on open future views, it is impossible for (1) ever to be true. For at any time, (1) depends on future free choices. So on open future views, the command (1) is unreasonable. And that's a problem for open future views.

Yet another infinite population problem

There are infinitely many people in existence, unable to communicate with one another. An angel makes it known to all that if, and only if, infinitely many of them make some minor sacrifice, he will give them all a great benefit far outweighing the sacrifice. (Maybe the minor sacrifice is the payment of a dollar and the great benefit is eternal bliss for all of them.) You are one of the people.

It seems you can reason: We are making our decisions independently. Either infinitely many people other than me make the sacrifice or not. If they do, then there is no gain for anyone to my making it—we get the benefit anyway, and I unnecessarily make the sacrifice. If they don't, then there is no gain for anyone to my making it—we don't get the benefit even if I do, so why should I make the sacrifice?

If consequentialism is right, this reasoning seems exactly right. Yet one better hope that it's not the case that everyone reasons like this.

The case reminds me of both the Newcomb paradox—though without the need for prediction—and the Prisoner's Dilemma. Like in the case of the Prisoner's Dilemma, it sounds like the problem is with selfishness and freeriding. But perhaps unlike in the case of the Prisoner's Dilemma, the problem really isn't about selfishness.

For suppose that the infinitely many people each occupy a different room of Hilbert's Hotel (numbered 1,2,3,...). Instead of being asked to make a sacrifice oneself, however, one is asked to agree to the imposition of a small inconvenience on the person in the next room. It seems quite unselfish to reason: My decision doesn't affect anyone else's (I so suppose—so the inconveniences are only imposed after all the decisions have been made). Either infinitely many people other than me will agree or not. If so, then we get the benefit, and it is pointless to impose the inconvenience on my neighbor. If not, then we don't get the benefit, and it is pointless to add to this loss the inconvenience to my neighbor.

Perhaps, though, the right way to think is this: If I agree—either in the original or the modified case—then my action partly constitutes the a good collective (though not joint) action. If I don't agree, then my action runs a risk of partly constituting a bad collective (though not joint) action. And I have good reason to be on the side of the angels. But the paradoxicality doesn't evaporate.

I suspect this case, or one very close to it, is in the literature.

Aristotelian propositions, promises and an open future

Aristotelian propositions are "tensed propositions" that are supposed to be able to change their truth value. If I say "It is sunny", this is supposed to express an Aristotelian proposition p such that p is true today, but p was false on cloudy days.

Now, a necessary condition for me to have fulfilled a promise is that

1. the proposition that was the object of the promise is true.

Suppose yesterday—i.e., on Sunday—I promised:

2. Tomorrow, I will do a blog post on Aristotelian propositions.

And I do make such a post today, i.e., on Monday, but I won't make another one on Tuesday. If the propositions expressed by tensed sentences are Aristotelian, then I have not fulfilled my promise. For the tensed proposition expressed by (2) is not true.

So tensed sentences don't express Aristotelian propositions, it seems. Rather, the proposition that yesterday I expressed with (2) is different from the proposition that would have been expressed with (2) today. The proposition that I expressed with (3) yesterday is "tenseless".

The advocate of Aristotelian propositions does have a way out. She can modify the condition (1) for promise fulfillment to:

3. the proposition that was the object of the promise was true at the time of the promise.

Now, there is no difficulty. The Aristotelian proposition that would have been expressed by (2) was true yesterday (since today I do make such a blog post) but isn't true today (since tomorrow I won't—I hope!).

But note that the advocates of an open future cannot go for (3). For on their view, the proposition that was the object of the promise wasn't true when I made the promise. Thus, there is a tension between holding that tensed sentences express Aristotelian propositions and accepting an Open Future. But a lot of Open Futurists do just that.

This is not an insoluble difficulty. One can, for instance, suppose an operator B_y that acts on an Aristotelian proposition and "shifts it backward by y. Thus, B_{1 day} applied to the Aristotelian proposition that tomorrow I will do a blog post on Aristotelian propositions is the Aristotelian proposition that today I do such a blog post. Then we replace condition (1) with:

4. I fulfill at t₂ a promise I made at t₁ only if B_y(p) is true at t₂, where y=t₂−t₁ and p is the object of the promise I made at t₁.

Still, it's weird, isn't it, that I fulfill a promise by bringing about something other than what I promised?

Propositions that never become true but are probable

According to open future views, the proposition that in 2015 a fair and indeterministic coin lands heads has some probability but is not true. However, that proposition is apt to become true in 2015. So the probability of the proposition isn't the same as the probability of the proposition being true, since it's certainly not true now, but might well become true in 2015.

So far so good (or bad). Suppose God promises you that from 2015 onward, every year, a fair and indeterministic coin will be tossed. Now let Q be the proposition that there are infinitely many years after 2014 during each of which a fair and indeterministic coin lands heads [I screwed up in the original formulation of Q and wrote "that every year from 2015 onward, ad infinitum, a fair and indeterministic coin lands heads"; Alan Rhoda's response targets my screwup; see my response to him below]. Now note that on open future views Q can never possibly become true. For on any date, the proposition requires for its truth that there will be infinitely many fair and indeterministic heads results still past that date, and on open future views a proposition that requires an undetermined future event won't be true.

So, open future views have to say that it's impossible for Q to ever be true. But a proposition such that it's impossible for it ever to be true should get probability zero. But the probability that of the infinitely many coin tosses, infinitely many will be heads is 1 according to classical probability theory. So open future views should be rejected.

Here's another argument in the same vein. Suppose I know I will have an eternal afterlife, and I promise you that I will freely pray for you every day, ad infinitum, starting November 1, 2014. On open future views, the object of my promise is a proposition that can never be true. But it's clearly a bad thing to promise something that can never be true. Yet what I promised wasn't a bad thing to promise. So open future views are false.

One might even have the direct intuition that one could keep the promise. That intuition is incompatible with open future views.

Yet yet another probability paradox

Start with a set M of countably infinitely many people, and a set D of countably infinitely many fair dice. Suppose that there are no natural orderings on the set D, and that each person in M has exactly one of the dice in D assigned to her. (Or if you prefer, these are sets of unique names of people and coins respectively.) You are a person in M, and you know what all the members of D are but have no information whatsoever on which member of D is yours. Now all the dice are simultaneously and independently tossed. Obviously, your probability that your die showed sixes is 1/6.

Then the set of all the dice that landed sixes is revealed to you. Call the revealed set D₆.

Suppose—this will be no surprise, as it had probability one—that the set of six-landing dice is infinite and the set of non-six-landing dice is infinite as well. Before it was revealed to you which dice landed sixes, your probability that your die yielded a six was 1/6. Did that probability change after you learned which set was the set of dice that landed sixes?

There are three options:

1. No, it didn't change at all—it stayed at 1/6.
2. Yes, it changed to an undefined value.
3. Yes, it changed to some other defined value.

To choose between the options, observe first that your current probability that your die landed six must now be exactly the same as the probability that your die is a member of D₆. But the fact that D₆ is in fact the set of the six-showing dice carries no information as whether your die is in D₆. Since all the dice are independent and fair, learning which dice landed sixes is completely irrelevant to finding your die. So whatever probability you assign to your die being among the members of D₆ after the revelation must be the same as the probability you assigned to it before the revelation.

So, if we choose option (1), then already before you found out that the double-six rollers were the members of D₆, you would have already assigned probability 1/6 to your die being in D₆. But there was no natural ordering on the set D of dice, so the set D₆ will be epistemically on par with its complement W−W₆. Both are simply countably infinite sets with countably infinite complements, and we can easily define an isomorphism of D onto itself that swaps the two sets. So if prior to learning the dice results you assigned 1/6 to your die being in D, you should have equally assigned 1/6 to your die being in D−D₆. But that's incoherent, since it's a given that the die is in D or D−D₆ but 1/6+1/6=1/3<1. So it seems that (1) is not an option.

That leaves (2) and (3). But those options are very strange. They imply that in such infinite die rolling scenarios, more data can always destroy your reasonable initial probability assignments.

Now, you might think that the above scenario only works when you don't know which die is yours, and that's kind of a strange scenario. But one can modify the scenario to work even when you do know which die is yours, but there is some other unique feature you don't know about your die, say, which of infinitely many (metaphysically) possible exotic particles is hidden inside the die, which of infinitely many angels has your die as a personal favorite, or what an independent sequence of rolls of the die yielded. Then the set D will be set of these unique features, and D₆ will be the set of these features among the dice that landed six.

Yet another probability paradox

You know for sure that infinitely many people, including yourself, each are independently tossing fair coins. You don't see your coin's result. But then you learn for sure something amazing: only finitely many of the coins came up heads. This is extremely unlikely—indeed, by the Law of Large Numbers it has zero probability—but it seems nonetheless possible. What probability should you now assign to your coin being heads?

Intuition: Very small, maybe zero, maybe infinitesimal.

Here's an argument, however, that you should stick to your guns and continue to assign 1/2. Let F be the proposition that only finitely many of the coins landed heads. Let G be the proposition that of the coins other than yours, only finitely many of the coins landed heads. Learning G does not affect your probability that your coin landed heads. The coins are all independent, so no information about the other tosses tells you about yours. But, now, necessarily (given the setup that you toss only one coin) F is true if and only if G is true. For your coin won't make the difference between infinitely and finitely many heads. So learning F does not affect your probability that your coin landed heads.

To make sticking to your guns even more amazing, note that this works for any infinity of people, even a very high uncountable infinity. Wow!

Scoring rules and epistemic rationality

Scoring rules measures the inaccuracy of one's credences. Roughly, when p is true, and one assigns credence r to p, then a scoring rule measures the distance between r and 1, while when p is false, the scoring rule measures the distance between r and 0. The smaller the score, the better.

Some scoring rules are better than others. Let's suppose some scoring rules are right. Then this thesis seems to be implicit in some applications of scoring rules (e.g., here):

1. If S is the right scoring rule, then a credence-assignment policy is epistemically rational only if following the policy minimizes expected total or average S-scores.

(And there will be a debate about whether we should have "total" or "average"—see link.)

But (1) is false. Here's a simple counterexample that works for most reasonable scoring rules. Consider a situation like this: A fair coin is flipped. If you assign credence 0.51 to heads, a mindreader who knows your credence assignments will immediately reveal to you how the coin landed. Otherwise, you will never have any information on how the coin landed.

Obviously, the epistemically rational thing to do is to assign 0.5 to heads. But this leads to higher expected total and average scores on most reasonable scoring rules. For if you assign 0.51, then once the mindreader tells you how the coin landed, you will update your credence to be very close to 0 or 1, and your score will be very low. And the only cost of this scenario is the slight inoptimality from briefly having score 0.51 instead of the optimal score of 0.5. So the epistemically rational policy for dealing with situations like this, namely assigning 0.5, does less well in expected scores than the epistemically irrational policy of assigning 0.51.

The case may seem farfetched. But there are real-life cases that may be similar. It may be that for psychological reasons when you are a bit more sure, or a bit less sure (depending on your character and the thesis), of a thesis than rationality calls for, you will be better able to investigate whether the thesis is true. Thus it may be better for your long term epistemic score that you do what is epistemically irrational.

Hair

For a while, I've thought that:

1. Hair is not alive.
2. Every part of me is alive.
3. So, hair is not a part of me.

This goes against the wisdom embodied in court precedent which has, I understand, held that cutting someone's hair without consent is battery rather than, say, theft.

Interestingly, in L'usage de la Raison, Mersenne talks of the human as a microcosm and mentions that humans, like the universe, have non-living parts, and gives hair as an example. So Mersenne denies (2). And on further reflection, I don't think I really had much reason to accept (2). Indeed, there seem to be other clear counterexamples to (2), such as the electrons in my heart (they are parts of my heart, and parthood seems transitive, at least in this case). Maybe one could argue that while the electrons are at least a part of a living part of me, hair isn't a part of a living part of me. But that would beg the question. For if my hair is a part of me, it's also a part of my head, and my head is surely a living part of me.

So I don't see much ground for denying that hair is a part of me. It's just one of my many nonliving parts.

Of course, speaking fundamentally, there is no such thing as hair (just as there are no hearts, chairs, stones, etc.). There is only I, who am hirsute.

Limiting frequencies and probabilities

You are one of infinitely many blindfolded people arranged in a line with a beginning and no end. Some people have a red hat and others have a white hat. The process by which hat colors were assigned took no account of the order of people. You don't know where you were in the line. Suppose you learn the exact sequence of hat colors, say, RWRRRRWRWRWWWWRWWWR.... But you still don't know your position. What should your probability be that your hat is red?

A natural way to answer this is to compute the limiting frequency of reds. Let R(n) be the number of red hats among the first n people, and then see if R(n)/n converges to some number. If so, then that number, call it r, seems to be a reasonable value for the probability. Call the assignment of r to the probability when the limit r exists the frequency rule.

Here's a curious and simple thing I hadn't noticed before. If you think the frequency rule is always the right rule, then for all integers n, you are committed to being almost certain that your position is greater than n. Here's why. Suppose that the sequence that comes up is n white hats followed by just red hats. The limiting frequency of R(n)/n is 1. So by the frequency rule, you're committed to assigning probability 1 to having a red hat. But since you have a red hat if and only if your position is greater than n, you are committed to assigning probability 1 to your position being greater than n. And since there is no connection between the hat color arrangement and the order of people on the line, if you have this commitment after learning the sequence of hat colors, you also had it before. The argument applies for all n, so for all n you must have been almost certain that your position in the sequence is greater than n.

And this in turn leads to the paradoxes of nonconglomerability. For instance, suppose that I flip a fair coin. If it's heads, I let N be your position number. If it's tails, I choose a number N at random such that P(N=n)=2⁻ⁿ. In either case, I reveal to you the value of N, but not how the flip went. For any number n, the probability that N=n is zero given heads (since you're almost certain that your position is greater than n), and the probability that N=n is greater than zero given tails, so by Bayes' Theorem you will be almost certain that the coin landed tails. So I can make you be sure that a coin landed tails, and thereby exploit you in paradoxical ways.

So the frequency rule isn't as innocent as it seems. It commits one to something like an infinite fair lottery.

Too late!

Let's say that something very good will happen to you if and only if the universe is in state S at midnight today. You labor mightily up until midnight to make the universe be in S. But then, surely, you stop and relax. There is no point to anything you may do after midnight with respect to the universe being in S at midnight, except for prayer or research on time machines or some other method of affecting the past. It's too late for anything else!

This line of thought immediately implies two-boxing in the Newcomb's Paradox. For suppose that the predictor will decide on the contents of the boxes on the basis of her predictions tonight at midnight about your actions tomorrow at noon when you will be shown the two boxes. Her predictions are based on the state of the universe at midnight. Let S be the state of the universe being such as to make her predict that you will engage in one-boxing. Then until midnight you will labor mightily to make the universe be in S. You will read the works of epistemic decision theorists, and shut out from your mind the two-boxers' responses. But then midnight strikes. And then, surely, you stop and relax. There is no point to anything you may do after midnight with respect to whether the universe was in S at midnight or not, except for prayer or research on time machines or some other method of affecting the past, and in the Newcomb paradox one normally stipulates that such techniques are not relevant. In particular, with respect to the universe being in S at midnight tonight, it makes no sense to choose a single box tomorrow at noon. So you might as well choose two. Though, if you're lucky, by midnight tonight you will have got yourself into such a firm spirit of one-boxing that by noon tomorrow you will be blind to this thought and will choose only one box.

Continuous Sleeping Beauty

A coin is tossed without the result being shown to you. If it's heads, you are put in a sensory deprivation chamber for 61 minutes. If it's tails, you are put in it for 121 minutes. Data from your past sensory deprivation chamber visits shows that after about a minute, you will lose all track of how long you've been in the chamber. So now you find yourself in the chamber, and realize that you've lost track of how long you've been there. What should your credence be that the coin landed heads?

Why is this a Sleeping Beauty case? Well, take the following discretized version. If it's heads, you get woken up 1,001,000 times and if's tails, you get woken up 2,001,000 times. There is no memory wiping, but empirical data from past experiments shows that you completely stop keeping track of wake-up counts after you've been woken up a thousand times. So now you've been woken up, and you know you've stopped counting. What should your credence be? This is clearly a version of Sleeping Beauty, except that instead of memory-wiping we have a cessation of keeping count, which plays the same role of being a non-rational process disturbing normal rational processes.

Oddly, though, in the sensory deprivation chamber case, I have the intuition that you should go for 1/2, even though in the original Sleeping Beauty case I've argued for 1/3. I don't have much intuition about my discretized version of the sensory deprivation chamber case.

P.s. I was thinking of blogging another Sleeping Beauty case, but it looks like LessWrong has beaten me to essentially it. (There may be a published version somewhere, too.)

Clumps and continuity

Our backyard had been free of black cats for as long as we've lived in this house, well over 400 days, except that over the last two nights, a black cat has visited our yard, meowing at the doors and windows. It's reasonable to think that it will visit again tonight. Yet 99.5% of evenings have been free of black cats. So how can it be inductively reasonable to think a black cat will visit tonight?

Presumably, it is because the data from the last two days is more relevant than the data from the earlier days, even though there are two orders of magnitude more black-cat-free days. But why is that data more relevant?

Granted, yesterday and the day before are more temporally similar to today than the other days. But why should temporal similarity override other kinds of similarity? No doubt there are many features (say, temperature, lunar phase, etc.) in respect of which today is more like some other day in the past 400 than like yesterday or the day before—after all, the earlier 398 days have a wide diversity of properties. But temporal similarity seems particularly important.

Maybe it is because we expect clumping, both in time and in space. Two black-cat evenings suggest the beginning of a clump.

I am curious: Is our expectation of clumping a priori justified or only a posteriori? Clumping seems to be a kind of
continuity. Is an expectation of continuity a priori justified or only a posteriori?

Not a finetuning argument

In The Impiety... (1624), as part of the 6th argument for the existence of God, Mersenne writes:

The proportion found between all the bodies of the world also shows that there is a God who has made all the universe in weight, in number and in measure: for the earth has no other ratio with the sun than 1:140, with the moon than 40:1, ... (pp. 98-99)

(I don't know off hand what the ratios are exactly meant to be; if they are ratios of volume, the moon is within 25% of the truth but the sun is off several orders of magnitude; if they are ratios of diameter, the sun is within an order of magnitude of the truth but the moon is an order of magnitude off.)

Mersenne's argument is full of such numerical (claimed) facts (the sun goes around the earth in 365.241 days, the moon traverses the Zodiac in 27 days, etc., etc.) and claims that God is needed to explain these facts. Now, I'm right now teaching on the fine-tuning argument, so I am sensitized to seeing such numbers in an argument for the existence of God. But it's striking that nowhere can I see Mersenne saying why these numbers are at all better than others, especially since surely some tuning facts seem very close at hand--surely, for instance, if the sun were much bigger or much smaller than it is, it would be too hot or too cold for life.

Mersenne explicitly insists that the numbers aren't explained by the essential natures of the objects, just before the above quote:

For the sun wouldn't be any the less the sun if it were closer or further from the earth, just as the stars could still be stars if they absented themselves from us by more than 14,000 earth radii.

Mersenne's argument seems to be a pure application of the idea that all contingent facts need explanation, and the arbitrariness of the numbers in the numerical statements seems to be cited precisely in order to show the contingency of the numerical statements. The argument suggests a strikingly strong commitment to a Principle of Sufficient Reason for contingent facts: all he needs to argue for a cosmic cause is to argue that there are contingent cosmic facts. Mersenne is confident that God has "many reasons" (as he says in the case of one of the numerical claims) for making the numbers be what they are, but these are reasons "which we aren't going to know except in Paradise" (101-102).

Mersenne's argument isn't a design argument--it doesn't advert to value-laden features that a God would have good reason to actualize. I think it's a kind of cosmological argument, but an eccentric one. Rather than arguing from generic features like motion or causation as Aquinas did, it focuses on very particular features.

The focus on these very particular features seems to have two benefits. The first is that it makes any appeal to necessity as the explanation implausible. Maybe it's necessary that there is motion, but it is incredible that it be necessary that the ratio of the diameter of the earth to that of the moon have to be 3.665:1 (to use modern numbers). So we get contingency very easily. The second feature is one I didn't notice right away. The astronomical features cited by Mersenne are ones that would reasonably be thought to be permanent features. They are thus prime candidates to be dismissed by it is so, as it has always been so. Mersenne's focus on the seeming arbitrariness of these features makes it very clear that would be no explanation. Thus Mersenne's cosmological argument works whether or not the past is finite. It is not disturbed by an infinite regress but does not need one either.

Of course, we no longer think that these particular features are permanent in the same way--the earth and sun changed in size in the formation of the solar system. But impermanent features are no better explained by an infinite regress than permanent ones--the permanence of the features in Mersenne's argument is only heuristic (and I don't see him explicitly drawing the reader's attention to the permanence). Plus we could run the argument on the basis of the apparently permanent but seemingly arbitrary elements in the laws of nature, such as precise values of constants.

The downside of Mersenne's argument, however, is that unless it is explained why the features are desirable, it is difficult to show that the cause of these features of the universe must be intelligent.

I am not a panentheist

In the past, while reading about panentheism I've had a hard time seeing how it's not simply a version of orthodox theism. That we are in God, that all our existence is a participation in God and that God is present in everything by his omnipresence and universal sustenance and causal concommittance are all perfectly orthodox ideas, ones that, for instance, any Thomist will embrace. Thus I thought that I probably am a panentheist if panentheism is understood as "All things are in God and God is in all things" (Matthew Fox). Of course, many panentheists would add to the above various unorthodox doctrines, but these did not seem to me to be a part of panentheism. However, I am now thinking, after a careful look at the rather confusing panentheism article in SEP that a core doctrine of panentheism is divine dependence on the the non-divine. And that's not compatible with orthodox theism, I think, and certainly not with what I think. So I guess I'm not a panentheist after all.

Another argument from Mersenne

In The Impiety of Deists, etc., Mersenne also gives this theistic argument:

And if there is no God, no independent being, it would be impossible that one exist, and thus our imagination would exceed all the beings of the word: and the being of our thoughts and our fantasies [phantasies] would infinitely exceed all real beings, and what would be imaginary would surpass the true, which cannot be. (p. 75)

There are a couple of interesting things. First, often Leibniz gets credited with noticing that if God possibly exists, then God actually exists. But here we see Mersenne claiming the contrapositive, almost two decades before Descartes' Meditations (in an objection to Descartes' Meditations, Mersenne also makes the point in the Leibniz form).

Second, we get an interesting argument:

1. If God doesn't exist, our imagination exceeds reality.
2. Our imagination does not exceed reality.
3. So, God exists.

We can also replace "exceed(s)" with "infinitely exceed(s)", which makes (2) even more plausible and (1) is still true. There are obvious connections between this and Descartes' infinity argument in the Meditations.

When thinking about this argument, I was initially puzzled why Mersenne starts the argument by arguing that if there is no God then the existence of God is impossible. After all, (1)-(3) doesn't seem to require the impossibility of God, just the non-actuality of God. My tentative interpretation is that Mersenne has in mind a fairly strong notion of "exceeds". Possibility has a certain foot in reality, and so for imagination to fully exceed reality, one would have to not only imagine something greater than what actually exists, but greater than what is possible. Now, God is greater than all non-divine possibilities. So if God is impossible, then the content of our thoughts outruns not just actuality but possibility, and that's what makes that content strongly outrun reality.

If this is right, then we can expand the argument as follows:

4. If God doesn't exist, it is impossible for God to exist. (Premise)
5. God is greater than all possibilities and actualities other than God. (Premise)
6. We can think of God. (Premise)
7. We cannot think of anything that exceeds all actualities and possibilities.
8. God doesn't exist. (Supposition for reductio)
9. God is not a possibility or actuality. (4 and 8)
10. We can think of something that exceeds all actualities and possibilities. (5, 6 and 9)
11. Contradiction! (7 and 10)
12. So, God exists. (By reductio)

Finally, it is rather interesting how Mersenne argues for the thesis if God doesn't exist, he can't exist. In the context of another argument, he says:

He isn't a being, as we supposed, he can't exist: since who would make him, and who would give him being [qui luy donneroit estre]? (p. 119)

My first thought on this was that Mersenne subscribes to the causal theory of possibility that I've defended. My second thought, however, was that his argument may be broader. The "who would give him being?" rhetorical question may work on any view on which possibility is grounded in actuality given the plausibility that God's possibility couldn't be grounded in anything other than himself, or else he wouldn't truly be an independent being (and notice the focus on independence in the first Mersenne quote).

By the way, while I am relying on my own translations in the above (partly for fun), professional translations can be found here.

Two applications of 'ought implies can'?

First application:
1. God ought to exist.
2. So, God can exist. (Ought implies can)
3. So, God exists. (By Ontological Argument)

Second:
4. It's an evil for a person to die.
5. If naturalism is true, people can't live forever.
6. Evils ought not be there.
7. So, people ought not die.
8. So, people can live forever. (Ought implies can)
9. So, naturalism is not true.

In the second argument, for 5 to be true, the "can" must be stronger than metaphysical possibility. Thus that argument requires a stronger ought implies can principle.

All that said, I am sceptical of 1 and 7. An impersonal "ought" not addressed to any person is weird.

The core of the second argument, however, can perhaps be rescued:
10. It's an evil for a person to die.
11. What is normal for a kind of being is not an evil.
12. If naturalism is true, it's normal for people to die.
13. So, if naturalism is true, it's not an evil for a person to die.
14. So, naturalism is not true.

Mersenne's supreme bad vs. supreme good argument

In his 1624 The Impiety of Deists, Atheists and Libertines of This Time... (dedicated to Cardinal Richelieu, by the way), Mersenne gives this fascinating little argument:

Nobody fails to acknowledge that if there is a supremely good being [un estre souverainement bon], it merits the name of God, since we don't mean anything by that name other than that which has all [the] sorts of perfections, and which lacks nothing. Now I will show that this supreme good exists. If it didn't exist, its privation would exist, which would be a supreme bad [mal], and consequently the supreme non-being, since the bad and the non-being are the same thing: but it doesn't in the least seem that the privation exists more than its actuality, which must necessary precede it. Thus one must confess that there is a supreme goodness, and then that there cannot be a supreme badness. So we have a supreme being, since we deny a supreme non-being, it being necessary that the one or the other exist....

There is actually more than one argument here. There is an interesting and deeply metaphysical argument based on evil as the privation of a good. But there is also the kernel of a rather interesting and simple argument:

1. It would be supremely bad if God doesn't exist.
2. The world doesn't exemplify a supreme bad.
3. So, God exists.

My son suggests using the goodness in the world to argue for (2). That would be an interesting hybrid design argument.

Desire as belief

David Lewis argues against an anti-Humean toy theory on which the value of a proposition A is the agent's credence in the proposition A^v which says that A is valuable. Thus: V(A)=C(A^v) where C is credence. His argument depends crucially on the independence assumption that the value of A doesn't depend on whether A is true:

1. C(A^v|A)=C(A^v).

But independence understood this way is plainly false. Suppose my initial valuation of my daughter making dessert (D) is V(D)=C(D^v)=1/2. But suppose I completely trust her judgments of value, she had informed me that she made dessert if and only if making dessert was valuable, and this was all in the background knowledge. Then obviously my valuation of D had better change upon learning whether D is true. If D is true, then her making dessert was valuable; if not, it wasn't. Thus: C(D^v|D)=1 and C(D^v|~D)=0, contrary to (1).

Divine hiddenness and a fallacy of practical reasoning

There is an old Soviet joke. A visitor arrives in the Soviet Union and by the airport he sees two workers with shovels. The first digs a hole. Then the second covers up the hole. He asks the workers what they are doing. They say: "The worker who puts the trees in the holes didn't show up."

The joke illustrates this fallacy of practical reasoning:

1. I have good (very good, excellent, etc.) reason to make p hold.
2. A necessary condition for p is q.
3. Thus, I have good (very good, excellent, etc.) reason to make q hold.

There is good reason to plant a tree. Digging a hole and filling in a hole are necessary conditions for planting a tree. But that only gives one reason to dig the hole when one expects a tree to be put in, and it only gives one reason to fill in the hole when the tree has been inserted.

One's reason to make p hold transfers to a similar weight reason to make the necessary condition q hold only when it is sufficiently likely that the other conditions needed for p will come to be in place.

I will call inferences like (3) instances of the Necessary Condition Fallacy.

Now consider this familiar line of thought.

4. If God exists, then for each sufficiently epistemically rational person x, God has an overriding reason to bring it about that x enters into a love relationship with him.
5. A necessary condition for a sufficiently epistemically rational x's entering into a love relationship with God is that x will believe that God exists.
6. A necessary condition for a sufficiently epistemically rational x's coming to believe that God exists is x's having evidence of God's existence.
7. So, a necessary condition for a sufficiently epistemically rational x's entering into a love relationship with God is that x have evidence of God's existence. (5 and 6)
8. So, if God exists, for any sufficiently epistemically rational human x, God has an overriding reason to bring it about that x has evidence of God's existence. (4 and 7)
9. But what God has an overriding reason to do always happens.
10. So, if God exists, every sufficiently epistemically rational person has evidence of God's existence.
11. But not every sufficiently epistemically rational person has evidence of God's existence.
12. So God doesn't exist.

But the derivation of (8) is a clear instance of the Necessary Condition Fallacy.

So the question now is whether there is a way of deriving (8) without making use of this fallacy. If it were the case that

13. every sufficiently epistemically rational creature would be very likely to enter into a love relationship with God upon receiving evidence that God exists,

then (8) would have some plausibility. (I say "some", because I am not sure the overridingness transfers from (4) to (8) given merely a high probability of success in producing a love relationship.) But (13) is not particularly plausible, especialy given that it seems likely that there are people who rationally believe in God but don't love him. (One thinks here of the line from the Letter of James about demons who know that God exists and tremble—but surely don't love.)

So, at least, further argument is needed for (8).

Lying and politics

Suppose lying were not wrong when great goods—goods at least of the order of magnitude of a human life—are at stake. Then many politicians running for high office would have—at least by their own lights—carte blanche on lying in order to get elected. For many politicians take their policies to be significantly better than those of their opponents' and once a policy concerns a large enough polity, the "order of magnitude of a human life" bar is surely very easy to meet (for instance, a policy that increases employment is likely to decrease suicide rates).

The conclusion that politicians have such carte blanche, however, is unacceptable.

A Kierkegaardian response to the problem of hiddenness

Kierkegaard somewhere says that the best argument for the Gospel is something like this: Everyone agrees that the moral teaching of the Gospel—love of neighbor and so on—is morally good. Thus, one should wholeheartedly strive to live it. And one will find oneself unable to do so, which will lead to one's throwing oneself on the grace of God.

Abstracting from the details, Kierkegaard is committed to something like this thesis:

1. A sufficiently wholehearted attempt to live the morally upright life puts one in a position to have an epistemically justified faith.

By contraposition, if one never attains the epistemic conditions for an epistemically justified faith, one did not make a sufficiently wholehearted attempt to live the morally upright life.

But if one fails to make that wholehearted effort, then one cannot reasonably complain about hiddenness. This story may seem offensive to non-Christians, but I suspect that Kierkegaard also thinks very few Christians make that wholehearted attempt.

Is divine hiddenness an evil?

There is a discussion in the literature on whether the argument from divine hiddenness is a special case of the argument from evil. Here is an interesting difference. While many evils, like suffering and death, are evils no matter whether there is a God, divine hiddenness—the state of affairs of some epistemically virtuous agents not believing in God—can only be an evil if God exists. Indeed, if there is no God, then non-belief in God is intrinsically good.

This has an interesting implication. Say that an evil is "unjustified" provided that God does not (or would not?) have good moral reason to allow it. Then divine hiddenness is not an unjustified evil. For:

1. Either God exists or God does not exist.
2. If God exists, there are no unjustified evils.
3. If God does not exist, divine hiddenness is not an evil.
4. So, either way, divine hiddenness is not an unjustified evil.

This means that the hiddenness argument cannot be a special case of the argument from apparently unjustified evils. It's tempting to run the argument in counterfactual mode and make it a close parallel to the argument from apparently unjustified evil by contending:

5. If there were a God, then hiddenness would be an unjustified evil.

But (5) is implausible in light of the fact that

6. If there were a God, there would be no unjustified evils.

In fact, the following argument is valid and only the first premise is controversial:

7. Divine hiddenness is an evil.
8. If God doesn't exist, divine hiddenness is not an evil.
9. So, God exists.

Hiddenness is not the only case where this logical issue comes up. It comes up in the case of any state of affairs that has the property that necessarily it is an evil only if God exists. In all such cases, the state of affairs simply cannot be an unjustified evil, and one has an argument from the state of affairs being an evil to God existing. Blasphemy against God is another such state of affairs.

(And if it could be shown that, necessarily, there is only evil if God exists, then this would mean that all arguments from evil are in this boat.)

None of this means that there is no argument from hiddenness for atheism that's worth discussing. It just means that one shouldn't see it as just a special case of the problem of evil. A much more cogent formulation of the hiddenness argument is this simple one (inspired by Trent Dougherty):

10. P(this pattern of hiddenness | God) << P(this pattern of hiddenness | no God).
11. So, probably, no God.

Here, "this pattern of hiddenness" includes all the relevant features of this world that might be taken to justify hiddenness if God existed.

That said, one needs to be cautious about (10). For a part of this pattern of hiddenness is that some people have thought long and hard about whether God exists. And it's not clear that they would have been likely to think long and hard about whether God exists if there were no God.