Saturday, April 18, 2015

Bigger and smaller infinities

Anecdotal data suggests that a number of people find counterintuitive the Cantorian idea that some infinities are bigger than others.

This is curious. After all, the naive thing to say about the prime numbers and the natural numbers is that

  1. while are infinitely many of both, there are more natural numbers than primes.
For the same reason it is also surely the obvious thing to say that:
  1. while there are infinitely many of both, there are more real numbers than natural numbers.
So there is nothing counterintuitive about different sizes of infinity. Of course, (1) is false. Our untutored intuitions are wrong about that case. And that fact should make us suspicious whether (2) is true; given that the same intuitions led us astray in the case of (1), we shouldn't trust them in case (2). However, the fact that (1) is false should not switch (2) from being intuitive to being counterintuitive. Moroever, our reasons for thinking (1) to be false—namely, the proof of the existence of a bijection between the primes and the naturals—don't work for (2).

All in all, rather than taking (2) to show us how counterintuitive infinity is, we should take (2) to vindicate our pretheoretic intuition that cardinality comparisons can take us beyond the finite, even though some of our pretheoretic intuitions as to particular cardinality comparisons are wrong.

Friday, April 17, 2015

Living in the moment, literally

Jim lives for a minute. Then he activates the time-and-space machine in his backpack, and travels to position one minute back and one meter back. Then the story repeats, giving Jim a lifespan of 80 internal years, all contained within a single minute of external time.

We could shorten that minute of external time to a second, or to any non-zero length of time, by making him jump back in time even faster.

Bold Hypothesis: We could shorten it to zero.

This works most easily if Jim is made out of ghostly matter that can overlap itself (nothing absurd about this: two photons can be in the same place at the same time), and as we shorten the time interval, we shorten the spatial distance of the jump.

The Bold Hypothesis basically says that just as one can have a time-travel machine, one can have a time-non-travel machine that keeps one in the same place in external time for all one's life.

Given the possibility of time travel, and the possibility of discrete time, it's not hard to argue for the Bold Hypothesis. Suppose at each instant of time, Jim can set the time-machine to determine where he will be in the next internal instant. Then why couldn't he set it so that in the next internal instant he will be at the same external instant as he is now.

Given the Bold Hypothesis, Jim would have a lifespan of 80 internal years, all in one moment.

All this suggests that when thinking about time, we should be careful with moving from our subjective experience of time and change--which Jim would have in his all-at-one-moment life--to claims about what external time is like.

Thursday, April 16, 2015

When would a computer feel pain?

Whether a computer could feel pain shouldn't depend on fine detail of how the CPU synchronization works or whether the CPU is implemented with electricity, or light, or gears. It's only the computational characteristics that matter if computers can be conscious.

Let's imagine a simplified picture of a computer's synchronization. There is a synchronizing clock. Each time the clock ticks, the computer very quickly executes the next instruction and enters a new state. Then it stays in its new state until the next clock tick.

Let's imagine that the dynamic stuff that is triggered by each clock tick takes places over a small portion of the time between ticks—most of the time between ticks, the computer is staying in a static state. For instance, if the computer is made up of gears, as a computer could well be (though it would be impractically big), then the picture is this. The computer is still for a while. Then the clock ticks. The gears make a quick movement to a new configuration. And then the computer is still until the next tick.

Suppose the computer feels pain. When does it do that? It is hard to believe that the computer is feeling pain when it is statically maintaining its state in between ticks. Suppose the computer got stuck between ticks—the clock broke down. Would the computer be permanently in pain?! I guess I just find it incredible that a contraption made of gears should feel something when the gears are not even moving.

So I think the best candidate for when the computer would feel pain would be when it is transitioning between states. But remember the intuition that only the computational characteristics matter to things like pain if computers can be conscious—the details as to implementation should be irrelevant. Assuming it's possible for a computer to feel pain, we should thus be able to have a possible world with a computer in pain whose state transitions are instantaneous. For a nanosecond all is still. Then a clock ticks. The computer instantaneously jumps to a new state. Another nanosecond of stillness. And so on.

The computer in this story, then, feels pain only during a series of instants. The total amount of time it spends feeling pain is zero. Yet it feels pain. Is that possible? Can there be pains that take no time at all?

Perhaps, though, the computer's subjective time doesn't line up with objective time. Maybe objectively the pains take zero time, but subjectively they take a lot more time? I don't know if this is possible.

There may be an argument against the possibility of computers feeling pain in the vicinity. In any case, there are interesting questions here.

Note: When I talk of a computer feeling pain, I mean a merely material computer. As Swinburne has pointed out to me, God could give a soul to a computer. And then there could be consciousness. But the subject of the pain, I think, wouldn't be the computer. The computer would be like a body. My body never feels pain. It is I (the whole of which the body is a part) who feel the pain.

Wednesday, April 15, 2015

A paradox about prediction of belief

Sally is perfectly honest, knows for sure whether there has ever been life on Mars (she's just finished an enormous amount of NASA data analysis), and is a perfect predictor of my future beliefs. She then informs me that she knows what I will believe at midnight about whether there was once life on Mars, and she further informs me that:

  1. There was once life on Mars if and only at midnight tonight I will fail to believe that there was once life on Mars.
Moreover, I know that:
  1. I won't get any other evidence relevant to whether there was once life on Mars.
I'd love to know whether there was once life on Mars. I start off thinking:
Well, right now I have no belief either way, and I am unlikely to get any evidence before midnight. So by midnight I will also have no belief either way. And thus by Sally's information there was once life on Mars.
But of course as soon as I accept this argument, I start to believe that there was life on Mars. And I know that if I keep on believing this until midnight, then my belief is false. I quickly see the pattern, and I realize that I don't know what to think! But when I don't know what to think, I default to suspension of judgment. But this, too, leads me astray: For as soon as I think that the appropriate rational attitude for me is suspension of judgment, then I start thinking I will suspend judgment at midnight, and I then conclude that there was once life on Mars. And the circle starts again.

Now, I know I'm not perfectly rational. So I can get out of the circle by concluding that given how confusing this case is, I am probably not going to act rationally. So something non-rational will affect my beliefs by midnight, and I don't know what that will be, so I might as well not speculate until that happens. Sally knows what it will be, but I don't.

But suppose I am perfectly rational. I shall assume that a part of perfect rationality is knowing for sure you're perfectly rational, knowing for sure what you belief, and drawing all the right conclusions from one's evidence. What should I believe in the above case?

Tuesday, April 14, 2015

Truth and Dutch Books

Suppose I initially assigned probability 0.5 to p and 0.5 to ~p. Suppose p is in fact true, and my credence in p comes to be magically increased to 0.8 without my credence in ~p being changed. I thus have inconsistent probabilities: 0.8 for p and 0.5 for ~p. This is supposed to be bad: it lays me open to Dutch Books. For instance, I will accept the following pair of options:

  1. Pay $0.75 to win $1.00 if p
  2. Pay $0.45 to win $1.00 if ~p.
But if I do that, then I will pay $1.20 and get $1.00, for a net loss of $0.20.

Yes, that's an unhappy result. But note that I am actually better off than earlier when my credences were consistent. Earlier I would have rejected (1) since my credence in p was 0.5, but I would have accepted (2). So I would have paid $0.45 and got nothing to show for it. Thus my revision in the direction of truth made me be better off, even though it also led me to accept a Dutch Book.

This suggests that pragmatically and synchronically speaking what matters is truth, not probabilistic consistency. Better be inconsistent and closer to truth than consistent and further from truth.

Diachronically, of course, at least logical inconsistency could be dangerous, as it can lead to lots of absurd conclusions. But in practice we all have inconsistent beliefs and we manage to contain the inconsistency without much in the way of explosion.

So what's so bad about Dutch Books? It seems to be this: an opponent who knows (with certainty) your credences and doesn't know (at least with certainty) whether p is true can offer you a series of bets that you are guaranteed to lose money on. This is a big deal if you're playing an adversarial game against such an opponent. But such games are, I think, a special case, and while they do occur in war, business, sport and other competitive pursuits, we should not let competitive pursuits against fellow humans dictate the nature of rationality to us. And note a curious thing: consistency is not the only available strategy against such an opponent—hiding your credences will also help. If you revise your credence in the direction of truth but your opponent doesn't know about your revision, you will do at least as well as before, and quite possibly better.

Monday, April 13, 2015

Particle accretion and excretion in Aristotelian ontology

In Aristotelian ontology the matter and parts of a substance get their being from their substance. But now we have a problem: we constantly accrete (say, when eating) and excrete (say, when sloughing off skin-cells) particles. These particles seem to exist outside of us, then they exist as part of us, and then one day they come to exist outside of us again. How could their being come from our form, when they existed before they joined up with us—sometimes, presumably, even before we existed at all?

But suppose an ontology for physics on which fields are more fundamental than particles, and particles are like a bump or wave-packet in a field. Then we have a very nice solution to the problem of accretion and excretion.

Imagine two ropes. Rope A is tied by one end to a hook on the wall and the other end of rope A is tied to the end of rope B. And you're holding the other end of rope B. You rapidly move your end of the rope up and down. A wave starts traveling along rope B, then over the knot, and finally along rope A. We are quite untroubled by this description of this ordinary phenomenon.

In particular, it is correct to say that the same wave was traveling along rope A as along rope B. Yet surely the being of a wave in a medium comes from the medium and its movement. So we have a very nice model. Rope A has excreted the wave and rope B has accreted the wave. (You might object that in Aristotelian ontology, ropes aren't substances. Very well: replace them with strings of living kelp.) If the knot is negligible enough, then the shape of the wave will seamlessly travel from rope A into rope B.

I think one reason an Aristotelian is apt to be untroubled by the description is because we don't take waves in a rope ontologically very seriously, just as we shouldn't take kings in chess very seriously. They're certainly not fundamental. Perhaps they don't really exist, but we have merely adopted a mode of speech on which it's correct to talk as if they existed.

However, if a field ontology is correct, we shouldn't take particles any more seriously than waves in a rope. And then we can start with the following model. Among the substances in the world, there are fields, gigantic objects that fill much of spacetime, such as the electromagnetic field. And there are also localized substances, which are tiny things like an elephant or a human or a bacterium. The fields have holes in them, holes perfectly filled by the localized substances. The localized substances exist within the fields much like a diver exists in the ocean—the diver exists in a kind of hole in the ocean's water.

Next, pretty much the same kinds of causal powers that are had by the fields are had by the localized substances. Thus, while strictly speaking there is no electromagnetic field where your body is found, you—i.e., the substance that is you—act causally just as the electromagnetic field would. A picture of the field you might have is of a string whose central piece had rotted out and was seamlessly replaced with a piece of living kelp that happened to have the same material properties as the surrounding string. But you don't just do duty for the electromagnetic field. You do duty for all the fundamental fields.

Because you have pretty much the same kinds of causal powers as the fields that surround you, waves can seamlessly pass through you, much as they can through a well-installed patch in a rubber sheet. You accrete the waves and then excrete them. Some wave packets we call "particles".

Objection: When I digest something, it becomes a part of me. But when a radio wave passes through me, it doesn't become a part of me even for a brief period of time.

Response 1: We shouldn't worry about this. In both cases we're talking about non-fundamental entities. There are many ways of talking. For practical reasons, it's useful to distinguish those wave packets that stick around for a long time from those that pass in and out. So we say that the former are denizens of us and the latter are visitors.

Response 2: Perhaps that's right. Maybe we don't exist in holes in the fields, but rather the fields overlap us. However, when the fields are in us, we take over some, but not all, of their causal powers. The radio wave that travels through me does so by virtue of the electromagnetic field's causal powers, while the particles of the piece of cheese that I digest and which eventually slough off with my dead skin travel through me by virtue of my causal powers. The picture now is more complicated.

Friday, April 10, 2015

Big or small?

Isn't it interesting that we are currently don't know whether the fundamental physical entities are the tiniest of things—particles—or the largest of things—fields—or both?


It sure seems that:

  1. A good human life is an integrated human life.
But suppose we have a completely non-religious view. Wouldn't it be plausible to think that there is a plurality of incommensurable human goods and the good life encompasses a variety of them, but they do not integrate into a unified whole? There is friendship, professional achievement, family, knowledge, justice, etc. Each of these constitutively contributes to a good human life. But why would we expect that there be a single narrative that they should all integrally fit into? The historical Aristotle, of course, did have a highest end, the contemplation of the gods, available in his story, and that provides some integration. But that's religion (though natural religion: he had arguments for the gods' existence and nature).

Nathan Cartagena pointed out to me that one might try to give a secular justification for (1) on empirical grounds: people whose lives are fragmented tend not to do well. I guess this might suggest that if there is no narrative that fits the various human goods into a single story, then one should make one, say by expressly centering one's life on a personally chosen pattern of life. But I think this is unsatisfactory. For I think that the norms that are created by our own choices for ourselves do not bear much weight. They are not much beyond hobbies, and hobbies do not bear much of the meaning of human life.

So all in all, I think the intuition behind (1) requires something like a religious view of life.

Thursday, April 9, 2015

Can something material become immaterial?

Two angels are playing chess. They are immaterial, but have the causal powers of moving physical pieces on the board. Along comes a big snake and swallows the board. No worries: the angels keep on playing, but now the positions of the chess pieces are kept track of in their minds instead. So the king, say, was first a material object. But the king then became an object wholly constituted by the angels' thoughts, and hence immaterial. And it is the same king. While in chess you can get a new queen by promoting a pawn, you don't get a new king within a game.

(Of course, I suspect that the true ontology doesn't include artifacts like chess kings.)

Wednesday, April 8, 2015

The equal weight view

Suppose I assign a credence p to some proposition and you assign a different credence q to it, even though we have the same evidence. We learn of each other's credences. What should we do? The Equal Weight View says that:

  1. I shouldn't give any extra weight to my own credence just because it's mine.
It is also a standard part of the EWV as typically discussed in the literature that:
  1. Each of us should revise the credence in the direction of the other's credence.
Thus if p>q, then I should revise my credence down and you should revise your credence up.

It's an odd fact that the name "Equal Weight View" only connects up with tenet (1). Further, the main intuition behind (1) is a thought that I shouldn't hold myself out as epistemically special, and that does not yield (2). What (1) yields is at most the claim that the method I should use for computing my final credence upon learning of the disagreement should be agnostic as to which of the two initial credences was mine and which was yours. But this is quite compatible with (2) being false. The symmetry condition (1) does nothing to force the final credence to be between the two credences. It could be higher than both credences, or it could be lower than both.

In fact, it's easy to come up with cases where this seems reasonable. A standard case in the literature is where different people calculate their share of the bill in a restaurant differently. Vary the case as follows. You and I are eating together, we agree on a 20% tip and an equal share, and we both see the bill clearly. I calculate my share to be $14.53 with credence p=0.96. You calculate your share to be $14.53 with credence q=0.94. We share our results and credences. Should I lower my confidence, say to 0.95? On the contrary, I should raise it! How unlikely it is, after all, that you should have come to the same conclusion as me if we both made a mistake! Thus we have (1) but not (2): we both revise upward.

There is a general pattern here. We have a proposition that has very low prior probability (in the splitting case the proposition that my share will be $14.53 surely has prior credence less than 0.01). We both get the same evidence, and on the basis of the evidence revise to a high credence. But neither of us is completely confident in the evaluation of the evidence. However, the fact that the other evaluated the evidence in a pretty similar direction overcomes the lack of confidence.

One might think that (2) is at least true in the case where the two credences are on opposite sides of 1/2. But even that may be wrong. Suppose that you and I are looking at the results of some scientific experiment and are calculating the value of some statistic v that is determined entirely by the data. You calculate v at 4.884764447, with credence 0.8, being moderately sure of yourself. But I am much less confident at my arithmetical abilities, and so I conclude that v is 4.884764447 with credence 0.4, We're now on opposite sides of 1/2. Nonetheless, I think your credence should go up: it would be too unlikely that my calculations would support the exact same value that yours did.

One might worry that in these cases, the calculations are unshared evidence, and hence we're not epistemic peers. If that's right, then the bill-splitting story standard in the literature is not a case of epistemic peers, either. And I think it's going to be hard to come up with a useful notion of epistemic peerhood that gives this sort of judgment.

I think what all this suggests is that we aren't going find some general formula for pooling our information in cases of disagreement as some people in the literature have tried (e.g., here). Rather, to pool our information, we need a model of how you and I came to our conclusions, a model of the kinds of errors that we were liable to commit on this path, and then we need to use the model to evaluate how to revise our credences.

Tuesday, April 7, 2015

Plot Armor

When Bob is the lead protagonist of a work, his presence is essential to the plot. Accordingly, the rules of the world seem to bend around him. The very fact that he's the main character protects him from death, serious wounds, and generally all lasting harm (until the plot calls for it). Even psychological damage can be held at bay by Bob's suit of Plot Armor. -tvtropes
It's natural to think of Plot Armor as a bad thing, a kind of invulnerability with no in-world explanation.

But I think it's not as bad as it seems at first sight. Suppose the possible world where Bob's story happens were actual. There is a selection effect as to which people we want to hear a long real-life story about. First, their life has to be interesting. One way for a life to be interesting is for the person to face a lot of danger. Second, their life needs to be sufficiently long to tell a long story about. Third, we don't want to hear too many depressing stories, so we don't want a story about someone whose life completely falls apart. All of this makes it likely that even in the real world, stories like Bob's are going to be likely to be told.

In a world with billions of people, we expect some to have multiple unlikely hair's-breadth escapes. And we'd like to hear stories about them. It's unlikely that escapes this narrow happen to Bob, but not so unlikely that they happen to someone.

So it's false to say that Plot Armor has no in-world explanation. If we imagine the story as being told by an in-world narrator (perhaps an implied one), we can give an in-world explanation in terms of selection by the narrator.

Of course, when the unlikeness of the escapes reaches the point that we wouldn't expect anyone to have them even with the population being as large as the story portrays it to be (Science Fiction about a whole populated galaxy will have more latitude here due to a much larger population to work with), this is problematic.

Monday, April 6, 2015

More against neo-conventionalism about necessity

Assume the background here. So, there is a privileged set N of true sentences from some language L, and N includes, among other things, all mathematical truths. There is also a provability-closure operator C on sets of L-sentences. And, according to our neo-conventionalist, a sentence p of L is necessarily true just in case pC(N).

Moreover, this is supposed to be an account of necessity. Thus, N cannot contain sentences with necessity operators and C must have the property that applying C to a set of sentences without necessity operators does not yield any sentence of the form Lp, where L is the necessity operator (It may be OK to yield tautologies like "Lp or ~Lp" or conjunctions of tautologies like that with sentences in the input set, etc.) If these conditions are not met, then we have an account of necessity that presupposes a prior understanding of necessity.

Now consider an objection. Then not only is L(1=1) true, but it is necessarily true. But now we have a problem. For C(N) by the conditions in the previous paragraph contains no Lp sentences. Hence it doesn't contain the sentence "L(1=1)".

But this was far too quick. For the neo-conventionalist can say that "L(1=1)" is short for something like "'1=1'∈C(N)". And the constraints on absence of necessity operators is compatible with the sentence "'1=1'∈C(N)" itself being a member of C(N).

This means that the language L must contain a name for L, say "N", or some more complex rigidly designating term for it (say a term expressing the union of some sets). Let's suppose that "N" is in L, then. Now, sentences are mathematical objects—finite sequences of symbols in some alphabet. (Or at least that seems the best way to model them for formal purposes.) We can then show (cf. this) that there is a mathematically definable predicate D such that D(y) holds if and only if y is the following sentence:

  • "For all x, if D(x), then ~(xN)."
But if y is this sentence, then y is a mathematical claim. If this mathematical claim isn't true, then y is not a member of N. But then y is true. On the other hand, if y is true, then being a mathematical claim it is a member of N, and hence y is false. (This is, of course, structurally like the Liar. But it is legitimate to deploy a version of the Liar against a formal theory whose assumptions enable that deployment. That's what Goedel's incompleteness theorems do.)

To recap. We have an initial difficulty with neo-conventionalism in that no sentences with a necessity operator ends up necessary. That difficulty can be overcome by replacing sentences with a necessity operator with their neo-conventionalist analyses. But doing that gets us into contradiction.

(It's perhaps formally a bit nicer to formulate the above in terms of Goedel numbers. Then we replace Lp with nC*(N*) where n is the Goedel number of p, and C* and N* are the Goedel-number analogues of C and N. Diagonalization then yields a contradiction.)

One place where I imagine pushback is my assumption that C doesn't generate Lp sentences. One might think that C embodies the rule of necessitation, and hence in particular it yields Lp for any theorem p. But I think necessitation presupposes necessity, and so it is illegitimate to use rules that include necessitation to definite necessity. However, this is a part of the argument that I am not deeply confident of.

Sunday, April 5, 2015

Happy Easter!

Happy Easter to all my readers! Christ has indeed risen, turning despair into hope, shining light where there was none.

Saturday, April 4, 2015

Vows to God and expectational views of promises

On Scanlon's expectational view of promises, a crucial part of why a promise is binding is that it creates an expectation of performance in the promisee. But if Sam vows something to God, then that doesn't create any expectation of performance on the part of God. For if Sam will perform the action, God has always known that. And if Sam won't, God's always known that. But if there were a God, one could make promises to him. So the expectational view is false.

Weak promises

Commanding is meant to create an obligating reason for another, while requesting is meant to create a non-obligating one. Promising is meant to create an obligating reason for self. There is a natural spot in illocutionary space, then, for a speech act meant to create a non-obligating reason for self, a speech act type that stands to promising as requesting does to commanding.

We would expect that when I have a normative power, I also have the corresponding weaker powers. If a legislature can bind under pain of ten years' imprisonment, they can bind under pain of a week's imprisonment. If I can create an obligating reason for myself, I can create a non-obligating reason for myself. That's another reason to think that we would have the "weak promise" speech act that creates non-obligating reasons.

I am not sure we have good phrases to express weak promises. We can approximate the force of a weak promise by weaselly promissory wordage like "I'll try to do this" or "I'll take your needs into account".