Friday, May 31, 2013

A Cosmological Argument based on the Empire State Building


  1. Necessarily, every exact duplicate of the Empire State Building has a cause.
  2. Necessarily, if an exact duplicate of the Empire State Building never changes, then neither it nor any of its parts cause its existence.
  3. Possibly, the only contingent beings ever are an unchanging duplicate of the Empire State Building and parts thereof.
Let w be a world where the only contingent beings ever are the unchanging duplicate of the Empire State Building and its parts. By (1), it has a cause. By (2), this cause cannot be the Empire State Building or a part thereof. Since those are all the contingent beings, the cause must be a necessary being, or include a necessary being as a part. So, possibly, there is a necessary being. By S5:
  1. There is a necessary being.

Premise (1) is a version of the Causal Principle specialized to the sorts of entities that we are most confident of there being causes of. One might wonder about why one needs the "never changes" in (2). But there is reason for it. Some objects can perhaps be caused by their parts. Imagine a bunch of trees that grow together to form a tower. We could likewise imagine a bunch of moving stony and metallic beings that come together to form an exact duplicate of the Empire State Building. This is ruled out by the "never changes".

Thursday, May 30, 2013

Yet another amusing ontological argument

  1. The existence of a perfect being does not entail the existence of horrendous, morally intolerable, gratuitous evil.
  2. If it is possible that there is a perfect being, then there is a perfect being. (By S5 and as a perfect being exists necessarily and is essentially perfect.)
  3. So, possibly there is a perfect being. (By (1), since an impossibility entails every proposition)
  4. So, there is a perfect being. (By (2) and (3))
The problematic (though true) premise is (1), of course.

Wednesday, May 29, 2013

A deontic-ontological argument

  1. There ought to be a perfect being.
  2. What ought to be is possible. (Ought implies can.)
  3. If a perfect being is possible, there is a perfect being. (By S5 and as a perfect being is necessarily existent and essentially perfect.)
  4. So, there is a perfect being.

Monday, May 27, 2013

Speculation on the virtuous person's emotions

Take seriously the Socratic idea that emotional awareness is a kind of perception of normative states of affairs: pain is a perception of actual ill-being, fear of potential future ill-being, joy of present good, and so on. Aristotle seems to think that the virtuous person's emotions will be perfectly in tune with reality. The courageous person will only fear what is genuinely fearsome, and so on.

But if we take the Socratic view of emotional consciousness, then we will have reason to doubt Aristotle. For the virtuous person's emotional awareness is going to be properly functioning. But it is false that properly functioning perceptions are always veridical. Indeed, sometimes, a sensory perception must be non-veridical to be properly functioning. As Descartes notes in the Meditations, any cause that produces the same effect in our body will result in the same perception. A barn and a barn facade (seen face on) produce the same effect on our retinas, and result in the same perceptions. Moreover, when this happens, our perceptions are properly functioning, and were we to see these same effects differently, our perceptions would be improperly functioning. If the barn facade didn't look like a barn, your sense of sight would be malfunctioning. (Our senses might always produce veridical results in heaven. If so, then that is a sense in which our nature is somehow transformed in heaven.) Most of the time, we see barns, not barn facades, and our senses' proper functioning is adapted to what gets us to truth for the most part (much as according to Aquinas, the proper functioning of the reproductive faculties is adapted to what leads to the good for the most part, which is why he thinks fornication is wrong even when it does not harm children, since for the most part, fornication is bad for offspring, and therefore our nature is such that fornication is not a part of our proper function.)

It would be surprising indeed if the same weren't true of emotional awareness. Sometimes, this is true simply because of sensory illusions. If you grew up happily on a farm, the convincing barn facade gives you the same nostalgic feeling as a real barn, and if you failed to feel nostalgia, something is wrong with you. But the same can be true even where there is no sensory illusion. Sometimes, virtue requires one to to knowingly (maybe even intentionally) cause pain to another. But the knowledgeable causing of pain to another quite properly makes one feel bad about what one is doing, and this should happen even when one knows that one ought to be doing what one is doing. In such a case, the emotional awareness may be non-veridical, but it is, nonetheless, properly functioning. If one did not feel bad, then one would have a malfunctioning emotional awareness, and one would not be fully virtuous.

I remember reading that St Catherine the Great Martyr, when young, was puzzled by Christ's suffering in the garden. Wouldn't Christ welcome suffering for righteousness' sake? So she supposed that his suffering was a kind of display for our benefit. But no: It wasn't a show. A virtuous person suffers fear when contemplating a terrible death, even if the virtuous person knows that this death should not be avoided. The feeling may be non-veridical, but why should Christ have been exempt from non-veridical emotions? After all, we do not think he was exempt from visual illusions: Surely a stick in the water looked broken to him. Of course, he might well judge the emotion non-veridical, much as we judge the one about the stick, but just as the stick still looks broken, Christ would still suffer.

But perhaps this is all wrong. It all depends on just how teleological proper function is.

Monday, May 20, 2013

Emotional perception

On Friday I was feeling somewhat poorly and in the interests of public health (not that I minded!) I opted out of participation in the PhD graduation dinner and Saturday's commencement. Sunday I felt somewhat worse, and today rather worse (nothing serious, just the usual "flu-like" symptoms, and both of our big kids had such in the preceding two weeks). But there is one piece of relief: It is good to have been right about the fact that I was getting sick.

It is not particularly bad to have suffered physically and similarly it is not particularly good to have enjoyed physical pleasure. But it is bad (or at least it feels bad--which is evidence for its being bad!) to have been wrong and similarly it is good to have been right. The value here isn't the value of being recognized by others as having been right or wrong. Nor is it the value of oneself presently recognizing oneself as right or wrong. For one hopes (though perhaps not simpliciter, if the prospect is particularly nasty) that one be right and that one not be wrong, not just that one recognize oneself or be recognized as right or wrong. The recognition is just the icing or mould on the top of a good or bad cake.

Eternalists have a difficulty with the fact that it doesn't seem bad to have suffered physically (bracketing any present suffering from painful memories, of course), even though past suffering is just as real as present suffering. Presentists have a difficulty with the fact that it seems to be bad to have been wrong and to be good to have been right.

I think eternalists can make a better go of it, though. Feelings like the pleasure of having been right or the pain of having been wrong are a kind of perception of normative features of the world. But not all truth is equally perceived. I am now visually aware that I have two hands, and properly so. But were I now visually aware that you have a head, my visual system would be malfunctioning. For although, dear reader, you do have a head, your head is not presently within my field of view. It is thus a part of the correct functioning of my visual apparatus that I be presently aware of my hands but not your head, even though all three parts (my two hands and your one head) are equally real.

Likewise, then, some goods and bads are appropriately within my emotional field of view--e.g., my having been right about getting sick--and some goods and bads are not appropriately within my emotional field of view--e.g., the unpleasantness of the last time I had a cavity filled. These goods and bads may be equally real (assuming that pain itself really is bad--there is room for discussion here, but it is at least extrinsically bad), but it could be (I am not sure about the first one, actually) that my having been right is appropriately within my emotional field of view while my having suffered (not at all severely--he really is an excellent dentist) at a past dental visit is not.

But we sometimes mistake absence of perception for perception of absence, like an infant who cries that the parent has left the room or the adult who sees no objection to an action and all too hastily concludes the action is permissible. Not emotionally seeing the past pain as bad--i.e., a not being pained by the past pain--is mistaken by us for seeing the past pain as not being bad.

The eternalist should thus say that the past physical pains and pleasures are bad or good, in the same way that present ones are, but we do not see their badness or goodness. Thus, the eternalist attributes to the agent a misinterpretation of absence of perception. The presentist, however, should say that having been right or wrong is not presently good or bad (though maybe it was good or bad), but we misperceive it as such. The eternalist thus attributes more correctness to our emotional perception, while attributing a well-known generalized cognitive error in explaining what went wrong. The presentist has to say our emotional perception is just wrong. I prefer the eternalist explanation.

A similar issue comes up for Christ's suffering on the cross. With a number of theologians, I take the center of our Savior's suffering not to be the horrific suffering of nails ripping through his flesh, but his deep emotional awareness of the horribleness of the totality of our sins (perhaps with the help of the hypostatic union or beatific vision bringing the particularities of all of humankind's sins to him). But this leads to a query: Why did Christ only have this awareness on the cross? We do not see him constantly and equally weighed down by this suffering earlier in life? Was he failing to have a correct emotional awareness? But now we can say: Not at all. It is the salient goods or bads that are within the field of view of correct emotional perception. And it is on the cross, at the high point of the sacrifice for our salvation from these sins (the high point: for all his life was such a sacrifice), that this became fully salient, in such a way that this perfect man--who is also true God--emotionally bore the full weight of our sin.

Note, too, that this is a story about Christ's sufferings that is difficult for the presentist to give. For it is difficult for the presentist to explain why earlier and later, and hence then-unreal, sins were bad at the time of Christ's crucifixion. Perhaps the presentist has to say that Christ's suffering came from an erroneous emotional perception of past and future sins as then-bad?

Quick air-powered rockets

Here's a very quick slightly weekend-morning educational project to do with kids. To make it more educational, one can try to have kids figure out a method for measuring how high the rockets go. (Two methods off-the-top of my head: use a quadrant--or phone app that provides similar functionality--and trigonometry, or measure how far the shadow of the rocket goes from the launch zone, and compare the shadow length of something of known height.)

Wednesday, May 15, 2013

Random numbers and their sequences

Bear with a simple and standard bit of mathematics: the mathematics may give us lessons about God and evolution, frequentism, single-case chances and Humean views of causation.

Consider the following standard one-to-one and onto map between the interval [0,1] and the space [0,1]ω of infinite sequences of numbers from that interval. The map starts with a single decimal number x=0.d1d2d3... in [0,1][note 1] and generates an infinite sequence ψ(x)=(ψ1(x),ψ2(x),...) by taking every second digit of x after the decimal point and letting that define ψ1(x), then discarding these digits, taking every second digit of what remains and letting that define ψ2(x), and so on. Thus, ψ1(x)=0.d1d3d5..., ψ2(x)=0.d2d6d10..., ψ3(x)=0.d4d12d20..., and so on.

Interestingly, ψ not only shows that [0,1] and [0,1]ω have the same cardinality, but if we equip [0,1] with a uniform probability measure and [0,1]ω with an infinite product of uniform probability measures, i.e., let [0,1]ω be the probability space modeling infinite independent choices of uniformly distributed numbers in [0,1], then it turns out that ψ is a probability-preserving isomorphism. Hence, the two probability spaces are probabilistically isomorphic. There is, thus, "nothing more" to choosing an infinite sequence of uniformly distributed numbers in [0,1] than there is to choosing a single such number.

And of course what goes for [0,1] and [0,1]ω also goes for finite sequences: the probability-preserving isomorphism between [0,1] and [0,1]n is even easier to construct.

There are some potential philosophical consequences of this isomorphism: it shows that there is no principled difference between single-case and sequences, when we're willing to deal with continuous outcomes (there is when we have a finite outcome space).

Lesson 1: Anybody who believes in the utter impossibility of single-case chances or probabilities, including for continuous-valued events like decay times or darts thrown at boards, should believe in the utter impossibility of chances or probabilities in the case of infinite sequences as well.

Thus, Lesson 2: Frequentism is dubious.

Lesson 3: If probabilistic causation with continuous-valued outcomes is possible, single-case probabilistic causation should be possible, and in particular single-case causation should be possible. For there is in principle no difference between single-case and sequential probabilities.

Thus, Lesson 4: Humeanism about causation is dubious.

Lesson 5: Given that it is plausible that if God intentionally and specifically chooses just a single real number in [0,1] with full precision, that real number isn't genuinely random in the sense scientists like biologists or quantum physicists mean, neither will an infinite sequence of divine choices embody randomness. Hence, reconciliations between random evolution and exhaustive divine planning of every particular event fail.

Monday, May 13, 2013

Copy and paste

A colleague just asked me which dining halls were open. I pasted text from the Dining Services website into an email in response, without quotation marks or other indication of quotation. (In an academic context, the lack of quotation marks would make it plagiarism, but not in this very informal context.) The text I pasted into the email constituted assertions whose content I did not believe, simply because by the time I pasted it, I had forgotten what it said (I have no memory for times and the like). But I believed, maybe even knew, that the assertions were all true, and there was no dishonesty.

In a similar way, it is possible to assert something you know to be true, and yet be lying. Suppose I am writing my chair to convince him of some point of policy and I have an unscrupulous colleague who emails me a complex sentence that he says is false but will convince the chair. I trust the savvy of my unscrupulous colleague and paste the sentence into my email without bothering to read it. It turns out that my unscrupulous colleague was mistaken, and the sentence is true, and in fact it expresses a truth that I believe. I then send the email to the chair. In so doing, it seems that I lied to the chair, even though I asserted something which I knew to be the case.

One might think that this isn't lying. If it's not, it's something morally equivalent to it, and our category of "lying" is artificially constricted.

Sunday, May 12, 2013

Blogger to LaTeX book converter

Someone I know wanted to print a book from a Blogger blog. We tried the software from one book printing site, but it kept crashing. After some searching, I found a perl script that converts to Adobe InDesign. But InDesign is expensive. So I modified the script to put out LaTeX.

Here is my script.

It's not doubt buggy, and you will probably need to make some manual adjustments to the output. Post comments here if it doesn't work for you.

This may also work for non-Blogger blogs that use Atom xml. Let me know if it does.

Here are some quick instructions.

Download an xml backup of your blog. Then edit config.cfg to point to that file and set conversion options. Edit header.tex to set title, author and similar information, as well as to customize formatting. Since much of the formatting is done via macros, you can customize a lot of it.

After ensuring you have all the needed perl packages--see here for perl and package information--run:


Then process output.tex (or whatever you specified as the output in config.cfg) with a modern LaTeX that fetches needed packages.

You can edit which posts and comments are included by editing output.tex. A post begins with

\begin{blogpost}{1}{other stuff}

The {1} means the post is included. To uninclude it, just change to {0}.

Similar things can be done with comments.

Friday, May 10, 2013

Homonymous synonyms

Suppose that in Elbonian word-order is unimportant and "xyzzy" can mean either "stands" or "the zebra" while "zoxas" can mean either "the zebra" or "stands". Sam says "Xyzzy zoxas" in Elbonian, meaning the first word to be a noun and the second a verb. Martha misunderstands his sentence as having the verb first and the noun second. But both versions express the same proposition, that the zebra stands. Thus, Martha misunderstands Sam--she takes him to be uttering a homonymous synonym of what he actually uttered--but she gets right the proposition he asserted.

But what if Sam has no particular intention as to which word is the noun and which one is the verb?

Thursday, May 9, 2013

Punctured vanity

I noticed that my Principle of Sufficient Reason book was 52nd in Metaphysics on Amazon. That sounded kind of nice, though not amazing, until I noticed that it was right behind We Are Our Ancestors, a book on reincarnation. Great title, though. :-)

What is location?

I will first argue that location is a multiply-realizable—i.e., functional—determinable. Then I will offer a sketch of what defines it.

A multiply-realizable determinable is one such that attributions of its determinates are grounded in different ways in different situations. For instance, running a computer program is multiply realizable: that something is running some algorithm A could be at least partly made true by electrical facts about doped silicon, or by mechanical facts about gears, or by electrochemical facts about neurons. Moreover, computer programs can run in worlds with very different laws from ours.

As a result, a multiply-realizable determinable is not fundamental. But location seems fundamental, so what I am arguing for seems to be a non-starter. Bear with me.

Consider a quantum system with a single particle z. What does it mean to say that z is located in region A at time t?[note 1] It seems that the quantum answer is: The wavefunction (in position space) ψ(x,t) is zero for almost all x outside A. And more generally, quantum mechanics gives us a notion of partial location: x is in A to degree p provided that p=∫A|ψ(x,t)|2dx, assuming ψ is normalized. On these answers, being located in A is not fundamental: it is grounded in facts about the wavefunction.

But it is also plausible that objects that do not have wavefunction can have location. For instance, there may be a world governed by classical Newtonian mechanics, and objects in that world have locations but no wavefunctions. (And even in a world with the same laws as ours, it is possible that some non-quantum entity, like an angel, might have a location, alongside the quantum entities.) Thus, location is multiply-realizable.

Very well. But what is the functional characterization of location? What makes a determinable be a location determinable? A quantum particle is located in A provided that ψ vanishes outside A. But a quantum particle also has a momentum-space wavefunction, and we do not want to say that it is located in A provided that the momentum-space wavefunction vanishes outside A? Why is the "position-space" wavefunction the right one for defining location? Why in a classical world is it the "position" vector that defines location, rather than, say, the momentum vector or an axis of spin or even the electric charge (a one-dimensional position)?

I want to suggest a simple answer. Two objects can have very similar electric charges, very similar spins or very similar momenta, and yet hardly be capable of interacting because they are too far apart. In our world, distance affects the ability of objects to interact with one another. Suppose we say that this is the fundamental function of distance. Then we can say that a determinable L is a location-determinable to the extent that L is natural and the capability of objects to interact with one another tends to be correlated with the closeness of values of L. This requires that L have values where one can talk about closeness, e.g., values lying in a metric space. In a quantum world without too much entanglement and with forces like those in our world, the wavefunction story gives such a determinable. In a classical world, the position gives such a determinable.

(One could also have an obvious relationalist variant, where we try to define the notion of being spatially related instead. The same points should go through.)

Notice that on this story, it may be vague whether in a world some determinable is location. That seems right.

I think this story fits well with common-sense thought about distance and location, and helps explain why we maintained these concepts across radical changes in physical theory.

This story also reminds me a little of what Aquinas says about the location of angels: An angel is at location x if and only if the angel is causally interacting with something at x. But there is a difference. While Aquinas defines location of an angel in terms of actual interaction, I define location at two removes from interaction: I only talk of the capability for interaction and I do not define location in terms of that, but in terms of a fairly natural determinable closeness in respect of which tends to correlate with capability for interaction.

We have two non-philosophical tests for a theory of location. One is whether it coheres with science and the other is whether it coheres with theology, and especially with transsubstantiation. It is no surprise that this theory coheres with science, since it was designed to. What about transsubstantiation (John Heil referred to it as a supercollider for metaphysical theories in medieval times)?

I think it coheres with transsubstantiation quite well. The Catholic tradition tends not to talk about physical but sacramental presence. This is a real presence of course. There are multiple ways of being located for Aquinas: physically, by power (as angels are where they act and as God is everywhere), as well as this sacramental presence that he has an elaborate metaphysical theory of (see my paper here). Our above theory allows for a multiplicity of location determinables, all taking values in a common space, and so there might be some determinable that would give an account of sacramental presence.

But one could also defend the physical presence of Christ's body in the Eucharist coherently with my view. Christ's body could have multiple wavefunctions, each defining a different location. Or maybe space could curve in on itself as I suggest (but do not endorse as my preferred view) in the above-cited paper so that the location of the consecrated host and the location of heaven are literally the same.

So while I didn't design the view initially with the transsubstantiationon in mind, I think it passes the transsubstantiation test.

Tuesday, May 7, 2013

"Using as"

I can use a fork as a backscratcher or my thumb and forefinger as the prongs of a slingshot.

I claim that when I do so, there isn't a backscratcher or a set of prongs that comes into existence when I do so.

For consider the three possibilities on which it is correct to say that prongs come into existence:

  1. The thumb and forefinger cease to exist and prongs come into existence, made out of the former digits.
  2. A set of prongs comes into existence in exactly the space occupied by the thumb and forefinger, and are made out of the same matter as the prongs.
  3. The thumb and forefinger are both a thumb and forefinger and a pair of prongs after the transformation.

The first option is obviously false.  I didn't temporarily come to have only eight fingers when I did it for the purposes of the photo.

The second option doesn't match the how we talk.  I would say: "I used my fingers as the prongs of a slingshot."  But according to (2), I had the prongs of a slingshot right there in the very same region of space occupied by my fingers--why didn't I use them as the prongs of a slingshot, since they are surely at least as usable for that purpose.  Or did I use both my digits and the prongs as prongs?  But I need only two things for slingshot prongs, not four.

Moreover, as I am typing with both hands, surely the prongs no longer exist.  When did they cease to exist?  Right after the shot?  But when I took the picture, I didn't actually take a shot--I only used my fingers as prongs for show.  (I did take a shot on other occasions, shooting a little fuzzy ball from the kids' craft drawer.)  When I relaxed the fingers?  But why not, instead, think of the relaxed fingers as folded prongs?  A slingshot could, after all, fold.  It's not like I destroyed the prongs when I relaxed my fingers--they're ready for convenient use at any other time.  Yet if they do continue to exist, do I have forty-four other pairs of prongs on my hands (granted, 25 of the pairs--the ones with one finger from one hand and the other from the other--can only be used by having a friend pull back the pocket or by pulling the pocket back with the teeth) if I form the odd ambition to use a different pair every day for the next forty-four days?  And if the prongs ceased to exist, will the very same pair of prongs be resurrected the next time I use my thumb and forefinger as prongs?  These questions seem silly, and their silliness suggests that they are predicated on a mistake.

The third option fits better with our "use as" talk.  I used my fingers as prongs, and I used the prongs as prongs, but there aren't four things there, because the fingers were prongs.  But we get the wrong modal properties.  For suppose that I decided to reinforce the prongs by supergluing steel rods to them.  The steel rods would come to be a part of the prongs, but they wouldn't come to be a part of the fingers.  Hence the fingers are not identical with the prongs, by Leibniz's Law.  

All this fits with common sense.  I used fingers as slingshot prongs or a fork as a backscratcher, and there were no slingshot prongs or a backscratcher there.

But can this line be maintained?  Suppose I cease to use the fork as a fork, and start to use it exclusively as a backscratcher.  Suppose in our culture, everybody owns a backscratcher, as our greeting ritual is a light scratching of each other's backs.  And backscratchers look just like American forks.  Surely what I would have would be a backscratcher.  Yet, surely, whether a backscratcher comes into existence shouldn't depend on how permanently it is used as such.  Still, that seems to be how we talk.  If all we are doing is descriptive metaphysics, we may stop here.

But if we want to do more gutsy metaphysics, we might at this point question the initial intuition that I had a fork there.  Perhaps the fundamental concepts are not of backscratchers or slingshots (or prongs thereof) or even forks, but of using some thing or things (particles, say) as backscratcher, slingshot (or prongs thereof) or fork. To use as a backscratcher is like to dance a waltz--if we want to do serious metaphysics, we shouldn't ask where the token backscratcher is in the using or where the token waltz is in the dancing.

Rob Koons has defended the idea that artifacts are token social practices. What I am saying is quite similar, except that I do not want to identify the artifacts with social practices. But all the reality there is in artifacts is the reality of things used as, or meant to or designed to be used as something or other.

Friday, May 3, 2013

When a non-observation is an observation

Consider a standard Stern-Gerlach setup. An electron with mixed up/down spin is sent through a magnetic field. Then there are electron detectors that detect whether it went up or down.

Now, the up detector is connected to a very loud bell. The down detector is connected to a dim light. On the standard consciousness-causes-collapse (ccc) theory, if there is an observer who can both hear and see, she collapses the wavefunction, with probabilities given by the Born rule.

But now suppose that our observer has fallen asleep. The bell would wake the observer. The light wouldn't. What happens?

I think that if we accept ccc, we should also accept that the wavefunction collapses in this case. Thus, sometimes the wavefunction collapses in favor of up and a bell, and sometimes it collapses in favor of down and a light. In the latter case, there is no observation made—the observer is unconscious. Thus, on this solution, while an unconscious observer is capable of collapsing a wavefunction.

Perhaps one disagrees that the wavefunction collapses here. Then the observer in the lab is in a superposition of awake and asleep states. This by itself seems unacceptable. It seems that the whole point of ccc was to ensure that we did not have to worry about the weirdness of superpositions between different conscious states. But a superposition between a conscious and a non-conscious state is just as weird. Moreover, supposing that our observer is in a superposition of awake and asleep states, we can imagine a second observer coming into the lab. As soon as the second observer notes whether the first is asleep, we will have collapse. If that happens, then suddenly the first observer comes to be in a pure state. Some of the time, that pure state will be one of remembering being woken up by a bell. But that memory is false: she was never woken up by a bell, but was in a superposition of woken and non-woken states. So this interpretation leads to us having to attribute false memories to observers. And we should avoid that.

This line of thought suggests that we should treat non-observation as a kind of observation. Collapse happens whether the bell is observed or not. Collapse occurs always to exclude superpositions between different observational states, where non-observation counts as an observational state.

Suppose we do this outside of quantum mechanics? Well, this could have some implications for Sleeping Beauty. It would suggest that the Sleeping Beauty problem where one is woken on Monday and Tuesday on tails (with amnesia induced in between) and only on Monday on heads is equivalent to the variant where on heads one is also woken on Tuesday and informed that there was heads. For the non-observation on heads on Tuesday in the first problem should be thought of as a kind of observation. I think thirding is quite plausible on the variant on standard Bayesian grounds, so this supports thirding in the original.

We also get a variant of that lighthearted answer to verificationist worries about Christianity. The lighthearted answer is that whether Christianity is true is verifiable. Just wait until you're dead, and you'll see. Well, not quite, goes the riposte: you will see if there is an afterlife, but if there is no afterlife, you won't. But if we treat non-observation as a kind of observation, then you in effect do "observe"—through genuine observation if there is an afterlife and without one otherwise.

How far do we take the principle that non-observation can count as observation? Do non-observations by non-existent persons count? Quite possibly. Suppose that a lab apparatus is set up so that a conscious being is produced when the up electron detector is triggered and not otherwise. Similar reasoning to the above suggests that we should have collapse here, even if no conscious being is produced. The mere possibility of producing one triggers collapse.

Of course, ccc may be false.

Thursday, May 2, 2013

A curiosity

Suppose that L is an unknown natural number (1,2,3,...) and as far as we know, all positive numbers are equally likely candidate for L. The number L is input into a machine, and pressing a button generates a natural number between 1 and L, which number is shown.

Consider the probabilistic distribution for L. Initially, there is no well-defined distribution. The information we have is that L is any number between 1 (inclusive) and infinity (exclusive), but there is no meaningful uniform measure on that set.

You press the button once and get a number x1. You press it again and get x2.

After pressing the button once, you know that Lx1. But you still don't have a well-defined posterior distribution of values of L. (You can try to generate one by supposing L is chosen at random between 1 and M and then taking the limit as M goes to infinity. This won't work. You get the unwelcome conclusion that for all y we have P(L>y)=1.)

But once you get the second data point, you do have a well-defined posterior distribution of values of L. Approximately (for large x, x1 and x2) the posterior probability that Lx will be max(x1,x2)/x, assuming that max(x1,x2)≤x (of course, if x is less than one of the xi, the probability that Lx is 1). Thus, with probability approximately 1/2, we can say that L is no more than twice as large as the larger of x1 and x2.

This is curious. You don't have a well-defined posterior distribution with one data point, but with two you get one. And then as you gather more and more data points, you get standard Bayesian convergence. With enough data points, you can be fairly confident that L is pretty close to the largest of your data points.

I suppose this is yet another one of those phenomena where the unconditional probabilities are undefined, but the conditional ones are defined.

Life expectancy

Suppose I think of my present conscious state as uniformly randomly chosen out of all of my conscious states. Now, suppose the following two hypotheses both start off as having equal probability:

  • NoAfterlife: I will live for about 80 years.
  • Afterlife: I will live for an infinite number of years.
I now observe myself as 40-years-old. On the uniform random choice of conscious state assumption, assuming for simplicity that all my life is conscious, P(observe 40 | NoAfterlife) = 1/80 and P(observe 40 | Afterlife) = 1/infinity. In other words, the observation that I am 40-years-old seems to extremely strongly support the hypothesis that there is no afterlife, so strongly as to assign zero probability to that hypothesis.

The issue here isn't about infinity. Suppose that on the Afterlife hypothesis I get 8000 years. Then, if the two hypotheses start with equal probability, the observation that I am 40 leads me to assign something like 99% probability to the NoAfterlife hypothesis.

When I first thought about this argument, it perturbed me significantly. But now I see that it is fallacious because of the following parallel. Let's suppose (contrary to fact, like the rest of the story) that we know for sure there is no afterlife, and let's say that you and I are healthy 19-year-olds, the day before our 20th birthday, and if nothing goes wrong we will live until 80. But we've been captured and our captors have just flipped a fair coin out of our sight in order to decide which of us to kill tomorrow. They've given us no sign as to which of us is to die, but we know for sure they will kill one of us. I have two hypotheses:

  • IDie: I will die at age 20.
  • YouDie: I will die at age 80.
Initially the two hypotheses have equal probability. But I observe that I am now 19. P(observe 19 | IDie) = 1/20 and P(observe 19 | YouDie) = 1/80. Plugging into Bayes' Theorem, I conclude that P(IDie) = 4/5. (And, just to make this more fun, you do the same calculation and you conclude that the probability that you will die tomorrow is 4/5.)

But it is absurd that in this case I should think that probably they chose me to die. So by the same token, the argument against the Afterlife hypothesis fails.

Still, what goes wrong? I think it's the assumption that one should treat one's present conscious state as uniformly distributed over a life. But this does lead to an interesting question. Once we drop this assumption, and reject the arguments for NoAfterlife and IDie, as we should, can we still be thirders in Sleeping Beauty?