Friday, October 20, 2017

Why my present existence can't depend on future events

I find very persuasive arguments like this:

  1. If theory T is true, then whether I exist now depends on some future events.

  2. Facts about what exists now do not depend on future events.

  3. So, theory T is not true.

For instance, some four-dimensionalist solutions to problems of fission according to which the number of people there are now depends on whether fission will are subject to this criticism.

But I’ve had a nagging worry about arguments like this, that in accepting (2), I am not being faithful to my eternalist four-dimensionalist convictions: why should the present aspects of the four-dimensional me have this sort of priority? Moreover, I didn’t really have an argument for (2). Until today.

Here is an argument for (2). Start with this.

  1. If facts about my present existence depend on future events, then facts about my present existence depend on future events that happen to me.

For instance, suppose that whether I exist now depends on whether some surgeon cuts my brain in half tomorrow. Well, then, some of the events that my present existence depends on will be events that happen entirely to someone else—for instance, whether the surgeon gets to work on time. But other events, such as the cutting or non-cutting of the brain, will happen to me. It would be absurd to think that facts about my present existence or identity depend on future events that happen entirely to something other than me.

Then add:

  1. Any events that happen to me in the future depend on my present existence.

For, such events presuppose my future existence, and my future existence is caused by my present existence.

  1. Circular dependence is impossible.

  2. So, facts about my present existence do not depend on future events.

Note that (6) is a very strong premise, and is one place the argument can get attacked. Many people think that you can have circular dependence when the dependence in the two directions is of a different sort. In the case at hand, facts about my present existence might depend constitutively on future events, while the future events depend causally on my present existence. Nonetheless, I think (6) is true, even if the dependence in the two directions is of a different sort.

Another move is to describe the future events on which my existence depends without reference to me. Don’t describe what the surgeon does as the splitting of my brain, but as the splitting of brain x. Then we could say that the future event of the surgeon’s splitting my brain does depend on my present existence, but my present existence doesn’t depend on that event. Instead, it depends on the future event of the surgeon’s splitting brain x. This objection denies (5): while the splitting of my brain depends on my present existence, the splitting of brain x does not, and yet it happens to me.

I think this is mistaken. The splitting of brain x depends on the future existence of that brain, and that brain depends on me, because parts depend on wholes—that is a deep Aristotelian premise I accept. Thus I think (5) is true. An event that happens to me is an event that involves at least a part of me, and none of my parts could exist without me. Granted, a brain like mine could exist without me. But token events are individuated in part by the things caught up in them. A splitting of a brain merely like mine would be a different event from the splitting of this particular brain. And it is a token event that my present existence is supposed to depend on.

The above argument won’t move non-Aristotelians who think that wholes depend on parts rather than parts depending on wholes. But it works for me. And hence it assuages the worry that in accepting (2), I am being unfaithful to my views about time.

All that said, I don’t really want to affirm (2) in an exceptionless way. If I am a time-traveller born in the year 2200, then my present existence does depend on what will happen in the future. But it only depends on what will happen in the external-time future not on what will happen in my internal-time future. And, crucially, I think time-travel is only possible when it doesn’t result in causal loops. So even if I am a time-traveller from the future, I cannot affect anything that is causally relevant to whether I will be born, etc. This probably means that if time-travel is possible, it is possible only in very carefully limited settings.

Thursday, October 19, 2017

Conciliationism is false or trivial

Suppose you and I are adding up a column of expenses, but our only interest is the last digit for some reason. You and I know that we are epistemic peers. We’ve both just calculated the last digit, and a Carl asks: Is the last digit a one? You and I speak up at the same time. I say: “Probably not; my credence that it’s a one is 0.27.” You say: “Very likely; my credence that it’s a one is 0.99.”

Concialiationists now seem to say that I should lower my credence and you should raise yours.

But now suppose that you determine the credence for the last digit as follows: You do the addition three times, each time knowing that you have an independent 1/10 chance of error. Then you assign your credence as the result of a Bayesian calculation with equal priors over all ten options for the last digit. And since I’m your epistemic peer, I do it the same way. Moreover, while we’re poor at adding digits, we’re really good at Bayesianism—maybe we’ve just memorized a lot of Bayes’ factor related tables. So we don’t make mistakes in Bayesian calculations, but we do at addition.

Now I can reverse engineer your answer. If you say your credence in a one is 0.27, then I know that of your three calculations, one of them must have been a one. For if none of your calculations was a one, your credence that the digit was a one would have been very low and if two of your calculations yielded a one, your credence would have been quite high. There are now two options: either you came up with three different answers, or you had a one and then two answers that were the same. In the latter case, it turns out that your credence in a one would have been fairly low, around 0.08. So it must be that your calculations yielded a one, and then two other numbers.

And you can reverse engineer my answer. The only way my credence could be as high as 0.99 is if all three of my calculations yielded a one. So now we both know that my calculations were 1, 1, 1 and yours were 1, x, y where 1, x, y are all distinct. So now you aggregate this data, and I do the same as your peer. We have six calculations yielding 1, 1, 1, 1, x, y. A quick Bayesian calculation, given the fact that the chance of error in each calculation is 0.9, yields a posterior probability of 0.997.

So, your credence did go up. But mine went up too. Thus we can have cases where the aggregation of a high credence with a low credence results in an even higher credence.

Of course, you may say that the case is a cheat. You and I are not epistemic peers, because we don’t have the same evidence: you have the evidence of your calculations and I have the evidence of mine. But if this counts as a difference of evidence, then the standard example conciliationists give, that of different people splitting a bill in a restaurant, is also not a case of epistemic peerhood. And if the results of internal calculations count as evidence for purposes of peerhood, then there just can’t be any peers who disagree, and conciliationism is trivial.

Wednesday, October 18, 2017

From the finite to the countable

Causal finitism lets you give a metaphysical definition of the finite. Here’s something I just noticed. This yields a metaphysical definition of the countable (phrased in terms of pluralities rather than sets):

  1. The xs are countable provided that it is possible to have a total ordering on the xs such if a is any of the xs, then there are only finitely many xs smaller (in that ordering) than x.

Here’s an intuitive argument that this definition fits with the usual mathematical one if we have an independently adequate notion of nautral numbers. Let N be the natural numbers. Then if the xs are countable, for any a among the xs, define f(a) to be the number of xs smaller than a. Since all finite pluralities are numbered by the natural numbers, f(a) is a natural number. Moreover, f is one-to-one. For suppose that a ≠ b are both xs. By total ordering, either a is less than b or b is less than a. If a is less than b, there will be fewer things less than a than there are less than b, since (a) anything less than a is less than b but not conversely, and (b) if you take something away from a finite collection, you get a smaller collection. Thus, if a is less than b, then f(a)<f(b). Conversely, if b is less than a, then f(b)<f(a). In either case, f(a)≠f(b), and so f is one-to-one. Since there is a one-to-one map from the xs to the natural numbers, there are only countably many xs.

This means that if causal finitism can solve the problem of how to define the finite, we get a solution to the problem of defining the countable as a bonus.

One of the big picture things I’ve lately been thinking about is that, more generally, the concept of the finite is foundationally important and prior to mathematics. Descartes realized this, and he thought that we needed the concept of God to get the concept of the infinite in order to get the concept of the finite in turn. I am not sure we need the concept of God for this purpose.

Are there multiple models of the naturals that are "on par"?

Assuming the Peano Axioms of arithmetic are consistent, we know that there are infinitely many sets that satisfy them. Which of these infinitely many sets is the set of natural numbers?

A plausible tempting answer is: “It doesn’t matter—any one of them will do.”

But that’s not right. For the infinitely many sets each of which is a model of the Peano Axioms are not isomorphic. They disagree with each other on arithmetical questions. (Famously, one of the models “claims” that the Peano Axioms are consistent and another “claims” that they are inconsistent, where we know from Goedel that consistency is equivalent to an arithmetical question.)

So it seems that with regard to the Peano Axioms, the models are all on par, and yet they disagree.

Here’s a point, however, that is known to specialists, but not widely recognized (e.g., I only recognized the point recently). When one says that some set M is a model of the Peano Axioms, one isn’t saying quite as much as the non-expert might think. Admittedly, one is saying that for every Peano Axiom A, A is true according to M (i.e., MA). But one is not saying that according to M all the Peano Axioms are true. One must be careful with quantifiers. The statement:

  1. For every Peano Axiom A, according to M, A is true.

is different from:

  1. According to M, all the Peano Axioms are true.

The main technical reason there is such a difference is that (2) is actually nonsense, because the truth predicate in (2) is ineliminable and cannot be defined in M, while the truth predicate in (1) is eliminable; we are just saying that for any Peano Axiom A, MA.

There is an important philosophical issue here. The Peano Axiomatization includes the Axiom Schema of Induction, which schema has infinitely many formulas as instances. Whether a given sequence of symbols is an instance of the Axiom Schema of Induction is a syntactic matter that can be defined arithmetically in terms of the Goedel encoding of the sequence. Thus, it makes sense to say that some sequence of symbols is a Peano Axiom according to a model M, i.e., that according to M, its Goedel number satisfies a certain arithmetical formula, I(x).

Now, non-standard models of the naturals—i.e., models other than our “normal” model—will contain infinite naturals. Some of these infinite naturals will intuitively correspond, via Goedel encoding, to infinite strings of symbols. In fact, given a non-standard model M of the naturals, there will be infinite strings of symbols that according to M are Peano Axioms—i.e., there will be an infinite string s of symbols such that its Goedel number gs is such that I(gs). But then we have no way to make sense of the statement: “s is true according to M” or Ms. For truth-in-a-model is defined only for finite strings of symbols.

Thus, there is an intuitive difference between the standard model of the naturals and non-standard models:

  1. The standard model N is such that all the numbers that according to N satisfy I(x) correspond to formulas that are true in N.

  2. A non-standard model M is not such that all the numbers that according to M satisfy I(x) correspond to formulas that are true in M.

The reason for this difference is that the notion of “true in M” is only defined for finite formulas, where “finite” is understood according to the standard model.

I do not know how exactly to rescue the idea of many inequivalent models of arithmetic that are all on par.

Tuesday, October 17, 2017

Approximate truth and the very recent past

Suppose I say that Jim yelled in delight at 12:31. But in fact he did so at 12:32. Then I said something false but approximately true.

Now, suppose that I hear Jim giving a loud yell of delight about 300 meters away. While I am listening to that yell, I think that Jim is yelling. But in the last second of my hearing, Jim is no longer yelling, but the sound waves are still traveling to me. No big deal. My belief that Jim is yelling is false, but approximately true. Or so I want to say.

And it’s important to say something like this, for it allows us to preserve the idea that our sense give us approximate truth. The case of sound from 300 meters away is particularly strong, but the point goes through in all our sensation, as none of it travels faster than the speed of light. Now, granted, often when we become aware of a stimulus, our sensory organs are still undergoing it. But nonetheless it is strictly speaking false to say that this very part of the stimulus that we are now aware of is in fact going on. So our senses seem to lead us slightly astray. But at most very slightly. It is approximately true that this part of the stimulus is going on now, because it is in fact going on a fraction of a second earlier. Or, perhaps, it is a part of our common sense knowledge of the world that the data of the senses is only meant as an approximation to the truth, and so there is no straying at all.

Now imagine that I say that Jim actually yelled in delight at 12:31, but he was actually completely silent all day, although in a very nearby possible world he did yell in delight at 12:31. Then what I said is not approximately true. In ordinary contexts, the modal difference between the actual and the merely possible vitiates approximate truth, no matter how nearby the merely possible world is.

So now on to one of my hobby horses: presentism. If presentism is true, then the difference between what is happening now and what happened earlier is relevantly like the difference between the actual and the possible. In both cases, it is a difference between a neat and clean predication and a predication in the scope of a modal operator, pastly or possibly, respectively. If this is right, then if presentism is true, I cannot say what I said about its being approximately true that Jim is yelling if Jim has actually stopped. That difference is a very deep modal difference. That the time when Jim is yelling is in a nearby past no more suffices for the approximate truth of “Jim is yelling now” than that Jim is yelling in a nearby possible world is enough for the approximate truth of “Jim is actually yelling”. The ontological gulf between the actual and the possible is vast; so would be the ontological gulf between the present and the past if presentism were true.

Thus, the presentist cannot say that the senses tend to deliver approximate truth.

Objection: We know to correct the data of the senses for the delay.

Response: We know. But that's a recent development.

Hope vs. despair

A well-known problem, noticed by Meirav, is that it is difficult to distinguish hope from despair. Both the hoper and the despairer are unsure about an outcome and they both have a positive attitude towards it. So what's the difference? Meirav has a story involving a special factor, but I want to try something else.

If I predict an outcome, and the outcome happens, there is the pleasure of correct prediction. When I despair and predict a negative outcome, that pleasure takes the distinctive more intense "I told you so" form of vindicated despair. And if the good outcome happens, despite my despair, then I should be glad about the outcome, but there is a perverse kind of sadness at the frustration of the despair.

The opposite happens when I hope. When the better outcome happens, then even though I may not have predicted the better outcome, and hence I may not have the pleasure of correct prediction, I do have the pleasure of hope's vindication. And when the bad outcome happens, I forego the small comfort of the vindication of despair.

The pleasures of correct prediction and the pains of incorrect prediction are doxastic in nature: they are pleasures and pains of right and wrong opinion. But hope and despair can, of course, exist without prediction. But when I hope for a good outcome, then I dispose myself for pleasures and pains of this doxastic sort much as if I were predicting the good outcome. When I despair of the good outcome, then I dispose myself for these pleasures and pains much as if I were predicting the bad outcome.

We can think of hoping and despairing as moves in a game. If you hope for p, then you win if and only if p is true. If you despair of p, then you win if and only if p is false. In this game of hoping and despairing, you are respectively banking on the good and the bad outcomes.

But this banking is restricted. It is in general false that when I hope for a good outcome, I act as if it were to come true. I can hope for the best while preparing for the worst. But nonetheless, by hoping I align myself with the best.

This gives us an interesting emotional utility story about hope and despair. When I hope for a good outcome, I stack a second good outcome--a victory in the hope and despair game, and the pleasure of that victory--on top of the hoped-for good outcome, and I stack a second bad outcome--a sad loss in the game--on top of the hoped-against bad outcome. And when I despair of the good outcome, I moderate my goods and bads: when the bad outcome happens, the badness is moderated by the joy of victory in the game, but when the good outcome happens, the goodness is tempered by the pain of loss. Despair, thus, functions very much like an insurance policy, spreading some utility from worlds where things go well into worlds where things go badly.

If the four goods and bads that the hope/despair game super-adds (goods: vindicated hope and vindicated despair; bads: frustrated hope and needless despair) are equal in magnitude, and if we have additive expected utilities with expected utility maximization, then as far this super-addition goes, you are better off hoping when the probability of the good outcome is greater than 1/2 and are better off despairing when the probability of the bad outcome is is less than 1/2. And I suspect (without doing the calculations) that realistic risk-averseness will shift the rationality cut-off higher up, so that with credences slightly above 1/2, despair will still be reasonable. Hope, on the other hand, intensifies risks: the person who hoped whose hope was in vain is worse off than the person who despaired and was right. A particularly risk-averse person, by the above considerations, may have reason to despair even when the probability is fairly high. These considerations might give us a nice evolutionary explanation of why we developed the mechanisms of hope and despair as part of our emotional repertoire.

However, these considerations are crude. For there can be something qualitatively bad about despair: it makes one not be as single-minded. It aligns one's will with the bad outcome in such a way that one rejoices in it, and one is saddened by the good outcome. To engage in despair on the above utility grounds is like taking out life-insurance on someone one loves in order to be comforted should the person die, rather than for the normal reasons of fiscal prudence.

This suggests a reason why the New Testament calls Christians to hope. Hope in Christ is part and parcel of a single-minded betting of everything on Christ, rather than the hedging of despair or holding back from wagering in neither hoping nor despairing. We should not take out insurance policies against Christianity's truth. But when the hope is vindicated, the fact that we hoped will intensify the joy.

I am making no claim that the above is all there is to hope and despair.

Friday, October 13, 2017

An excessively simple theory of pain

A physicalist research program is to identify physical state types that underlie mental state types.

Here is an overly naive physicalist-friendly theory of pain:

  1. Pain is what occurs in the triggering of a damage-detector state that is linked to aversive behavior.

This theory is simple and elegant. Arguably, all actual instances of pain fits with the theory. So if there is an extensional problem with (1), it is that it classifies as pains states that aren’t pains. Plants feel pain on (1), and a program that monitors the health of a hard drive and relocates data away from damaged areas feels pain.

For the above reasons, I assume nobody will find (1) plausible.

The physicalist research program needs to be based on fitting physical stories to the data about mental states. This data has to be data about where a mental state type occurs and where a mental state type does not occurs. For as (1) shows, it is too easy to find physical stories that simply fit data about where mental states do occur. In fact, we can do even better than (1) if our only constraint is catching all cases of pain by giving this story:

  1. To be in pain is to be physical.

We have a nice source of data about where pain does occur: our own experience, the reports of other persons, and the behavior of animals similar to us. But do we have data about where pain does not occur?

We could say this: I now know that I am not in pain. So (2) is refuted directly: I am a physical being, but I am not in pain. Slightly more subtly, I can refute (1) as follows. No doubt as I am writing this, some of my cells are being damaged by some factors in the environment, and my body is doing something aversive about it. But I am not in pain.

But I think this argument against (1) is not as evidentially strong as it seems. The leading physicalist theory is functionalism. If functionalism is true, then pain is some sort of a functional state. I exhibit this functional state in the brain. But my body could mutate so that my stomach would host that sort of functional state, without any connection between that state and my brain. When my stomach would host the state, then on the functionalist theory I would be in pain. But it would be a pain that I am incapable of reporting, because the state would not be connected to the speech centers in the brain. This would be a case where either there are two conscious things—I and my stomach—or a case where my consciousness is divided into a brain-based and a stomach-based consciousness, and only the brain-based consciousness is able to drive action. (Similar phenomena seem to happen with split-brain patients.)

Likewise, then, if some of my individual cells were currently being damaged, and my body detected that damage and engaged in something aversive, then it shouldn’t be expected that I would report pain, even if (1) were true. Rather, if (1) were true, then in a scenario like this, either there would be two conscious things, one located in and around the particular cells and the other in the brain, or else I would have a divided consciousness. In neither case would the absence of pain to the brain-based consciousness be a refutation of (1).

So it seems I cannot refute (1) by observing my lacks of pain, because the pains predicted by (1) could be occurring in a different conscious thing found in my body or in a consciousness divided from the one that is driving my paradigmatically human activity.

I think the best way to refute (1) is to rely on intuitions like that plants aren’t conscious. But if naturalism were true, I wonder if there would be any reason to think such intuitions are truth-conducive.

Thursday, October 12, 2017

Consciousness in transitions

We can think of a digital computer processor as doing two things: Transitioning between states and remaining in a constant state between the transitions. How long the processor remains in a constant state depends on the clock rate: after the processor has done a flurry of computation (“combinatorial logic”) in a clock cycle, it will stay in a constant state until it’s time for the next flurry. If the clock rate is low, it will be able to stay in that constant state for a significant portion of the time, which is great, because presumably then the processor will be cooling off.

Suppose the computer is conscious by virtue of computation (as opposed to, say, being conscious by virtue of the functioning of a soul that God creates for it). When is it conscious? Is it during the transitions between the states or while remaining in a state? Intuitively, it should be during the transitions. After all, while it was remaining in a state, we could suddenly lower the computer’s temperature to near absolute zero. That wouldn’t disturb the computer’s remaining-in-a-state.

(Granted, it would disturb the computer’s clock. But the clock seems something extrinsic to the conscious system. One could in principle run a processor—very slowly—on a clock signal produced by a human being tapping a telegraph key, and surely that wouldn’t make the human’s hand a part of the conscious system.)

But the frozen state is functionally very much like the processor’s regular holding of a state when it waits for the next clock pulse. So just as it is implausible to think that a physical system like a computer that is frozen near absolute zero would continue to be conscious, it is implausible to think that the computer would be conscious while simply holding a state.

Thus, if a digital computer is conscious by virtue of computation, that consciousness occurs in and through transitions between states.

So what? I don’t know. I’m just trying to figure out what the best functionalist view would be like.

And note the contrast between this picture of consciousness-in-transitions and classical theism, according to which consciousness occurs in a timeless state.

A materialist intuition against materialism

The following argument is valid:

  1. It is metaphysically impossible for us to become wholly immaterial.

  2. If we are wholly material, then functionalism is true.

  3. If functionalism is true, then it is metaphysically possible for us to become wholly immaterial.

  4. So, we are not wholly material.

I think premise 1 is false, but intuitively 1 is pretty plausible—especially to a materialist.

Premise 2 is made plausible by the way functionalism solves serious problems in other materialist theories.

Premise 3 can be argued for: it is metaphysically possible for an immaterial being to have the same functional properties as I do, and furthermore for the immaterial being’s isomorphic functional states to be caused by my functional states at the last moment of my body’s existence in such a way that the immaterial being is a continuant of me given functionalism.

Wednesday, October 11, 2017

MIDI fruit piano

My daughters and I saw a Makey-Makey banana piano at a local fair, and they thought it was cool. So I made an Arduino(clone) fruit piano, using capacitive sensing, and a Python program on a computer that plays polyphonic music. It's super-simple, as it uses the ADCTouch library which doesn't need any electronic components besides the Arduino(clone), and it's better than the banana piano as it doesn't require the user to be grounded.

While tweaking the project, I learned that MIDI format is really simple, so now the fruit-piano sends notes to the computer via MIDI-over-serial-over-USB, and so one can presumably use the fruit-piano as a keyboard for various kinds of desktop music software.

Instructions are here. Code is here.

If you look at the picture carefully, you'll see that I cheated. We didn't have the eight oranges for the C major scale that my eldest daughter thought we should have, so two of the keys are soda cans.

Tuesday, October 10, 2017

Infinity book progress

I've just sent off the final contracted-for manuscript of Infinity, Causation and Paradox.

Attempts at wrongdoing

It is a common intuition, especially among Christians, that attempts at immoral actions—say, attempted murder or attempted adultery—are just as bad as the completion of the actions.

But in practice the situation is rather more complicated. Suppose Samantha is about to murder Fred. She is sitting on the rooftop with her rifle, has measured the windspeed, has made the corrections to her sights, is putting Fred in her cross-hairs and is getting ready to squeeze the trigger at an opportune moment. Then suddenly a police officer comes up and grabs Samantha’s rifle before she can do anything.

Samantha has performed actions whose end was Fred’s death. She is an attempted murderer. But I think there is an immoral act that she has been saved from. For imagine three versions of how the story could end:

  1. The police officer comes up and grabs her rifle at time t1 before she squeezes the trigger.

  2. At time t1, Samantha decides not to squeeze the trigger and not commit the murder.

  3. At time t1, Samantha decides to squeeze the trigger.

In all three cases, by the time of t1, Samantha is already an attempted murderer. But in version 2, Samantha has done at least one less bad thing than in version 3. As of t1, Samantha still has a decision to make: to go through with the action or not. In case 3, she decides that wrongly. In case 2, she decides that rightly.

In case 1, the police officer prevents her from making that decision. It seems clear that Samantha’s moral state in case 1 is less bad in than in case 3. For in case 3, Samantha makes a morally wrong decision that has no parallel in case 1. So the police officer has not only saved Fred’s life, but he has decreased the number of wrongs done by Samantha.

Of course, timing and details matter here. Suppose that the police officer grabs Samantha’s rifle at a moment when the bullet is already traveling through the barrel, making the shot go wide. Then Samantha is an attempted murderer, but the amount of wickedness on her conscience is the same as in case 3.

So there is a moral distinction to be made between Samantha in cases 1 and 3, but the distinction isn’t the distinction between attempt and success. Rather, the issue is that a typical wrong action involves multiple acts of will, many of which may well come with the possibility of stopping. Each time one does not will to stop, while being capable of willing to stop, one does another wrong. If one is prevented from completion of the act after the last of these acts of will, then one is not better off in terms of one’s moral guilt state. (Though one is better off in terms of how much restitution one owes and similar considerations.) But if one is stopped earlier, then one is better off.

This means that counting counts of sin is tricky. Suppose Fred had decided on committing adultery with Samantha’s sister Patricia. He texted Patricia offering to meet with her in a hotel room. He is already an attempted adulterer. But then he makes a number of decisions each of which could be a stopping point. He decides to get in his car. To drive to the hotel. To enter the room. Etc. At each of these points, Fred could have stopped, I assume. But at each point he chose adultery instead. So by the time he is in the room, he has committed adultery in his will many times.

But when we count wrongs, we don’t count like that. We count the number of murders, the number of adulteries or the number of thefts—not the number of times that one could have stopped along the way. We act as if the person who murdered five is worse than the person who murdered one, even if the person who murdered the one had to drive ten times as far.

Maybe the reason we count as we do is just a pragmatic matter. We don’t know just how many times one’s will is capable of stopping one, and how much a person just acts on auto-pilot, having set a course of action.

Or maybe the responsibility for the choose-not-to-stop decisions is much lower than for the initial decision?

I don’t know.

Monday, October 9, 2017

Preventing someone from murdering Hitler

You are a secret opponent of the Nazi regime, and you happen to see Schmidt sneaking up on Hitler with an axe and murderous intent. You know what’s happening: Schmidt believes that Hitler has been committing adultery with Mrs. Schmidt, and is going to murder Hitler. Should you warn Hitler’s guards?

  1. Intuition: No! If Hitler stays alive, millions will die.

  2. Objection: You would be intending Schmidt to kill Hitler, a killing that you know would be a murder, and you are morally speaking an accomplice. And it is wrong to intend an evil to prevent more evil.

There is a subtlety here. Perhaps you think: “It is permissible to kill an evil tyrant like Hitler, and so Schmidt is doing the right thing, but for the wrong reasons. So by not warning the guards, I am not intending Schmidt to commit a murder, but only a killing that is objectively morally right, albeit I foresee that Schmidt will commit it for the wrong reasons.” I think this reasoning is flawed—I don’t think one can say that Schmidt is doing anything morally permissible, even if the same physical actions would be morally permissible if they had other motive. But if you’re impressed by the reasoning, tweak the case a little. All this is happening before Hitler has done any of the evil tyrannical deeds that would justify killing him. However, you foresee with certainty that if Hitler is not stopped, he will do them. So Schmidt’s killing would be wrong, even if Schmidt were doing it to prevent millions of deaths.

What’s behind (2) is the thought that Double Effect forbids you to intend an evil, even if it’s for the purpose of preventing a greater evil.

But here is the fascinating thing. Double Effect forbids you from warning the guards. The action of warning the guards is an action that has two effects: (i) prevention of a murder, and (ii) the foreseen deaths of millions. Double Effect has a proportionality condition: it is only permissible to do an action with a good and a bad effect when the bad effect is proportionate to the good effect. But millions of deaths are not proportionate to the prevention of one murder. So Double Effect forbids you from warning the guards.

Now it seems that we have a conflict between Double Effect and Double Effect. On the one hand, Double Effect seems to say that you may not warn the guards, because doing so will cause millions of deaths. On the other hand, it seems to say that you may not refrain from warning the guards in order to save millions because in so doing you are intending Schmidt to kill Hitler.

I know of three ways out of this conflict.

Resolution 1: Double Effect applies only to commissions and not omissions. It is permissible to omit warning the guarads in order that Schmidt may have a free hand to kill Hitler, even though it would not be permissible to help Schmidt by any positive act. One may intend the killing of Hitler in the context of one’s omission but not in the context of one’s commission.

Resolution 2: This is a case of Triple Effect or, equivalent, of a defeater-defeater. You have some reason not to warn the guards. Maybe it’s just the general moral reason that you have not to invoke the stern apparatus of Nazi law, or the very minor reason not to bother straining one’s voice. There is a defeater for that reason, namely that warning the guards will prevent a murder. And there is a defeater-defeater: preventing that murder will lead to the deaths of millions. Thus, the defeater to your initial relatively minor moral reason not to warn guards—viz., that if you don’t, a murder will be committed—is defeater, and so you can just go with the initial moral reason. On this story, the initial Objection to the Intuition is wrong-headed, because it is not your intention to save millions—that is just a defeater to a defeater.

Resolution 3: Your intention is simply to refrain from acting in ways that have a disproportionately bad effect. We should simply not perform such actions. You aren’t refraining as a means to the prevention of the disproportionately bead effect, as the initial Objection claimed. Rather, you are refraining as a means to prevent oneself from contributing to a disproportionately bad effect, namely to prevent oneself from defending the life of the man who will kill millions.


While Resolution 1 is in some ways attractive, it requires an explanation why intentions for evils are permissible in the context of omissions but not of commissions.

I used to really like something like Resolution 2. But now it seems forced to me, because it claims that your primary intention in the omission can be something so very minor—perhaps as minor as not straining one’s voice in some versions of the story. That just doesn’t seem psychologically realistic, and it seems to trivialize the goods and evils involved if one is focused on something minor. I still think the Triple Effect reasoning like has much to be said for it, but only in those cases where there is a significant good at stake in the initial intention.

I find myself now pulled to Resolution 3. The worry is that Resolution 3 pulls one towards the consequentialist justification of the initial intuition. But I think Resolution 3 is distinguishable from consequentialism, both logically and psychologically. Logically: the intention is not to contribute to an overwhelmingly bad outcome. Psychologically: one can refrain from warning the guards even if one wouldn’t raise a finger to help Schmidt. Resolution 3 suggests that there is an asymmetry between commission and omission, but it locates that asymmetry more plausibly than Resolution 1 did. Resolution 1 claimed that it was permissible to intend evils in the context of omissions. That is implausible for the same reason why it is impermissible to intend evils in the context of comissions: the will of someone who intends evil is a corrupt will. But Resolution 3 is an intuitively plausible non-consequentialist principle about avoiding being a contributor to evil.

In fact, if one so wishes, one can use Resolution 3 to fix the problem with Resolution 2. The initial intention becomes: Don’t be a contributor to evil. Defeater: If you don’t warn, a murder will happen. Defeater-defeater: But millions will die. Now the initial intention is very much non-trivial.

Friday, October 6, 2017

Practice-internal goods

I’m hereby instituting a game: the breathing game. My score in the game ranges from 0 to 10. I get 0 points if I hold my breath for a minute. Otherwise, my score equals the number of breaths I took during the minute, up to ten (if I took more than eight breaths, my score is still ten).

It is good to do well at games. And I am really good at the breathing game, as are all other healthy people. For every game, there is a practice-internal good of victory. Thus, by choosing to play the breathing game, my life is enriched by a new practice-internal good, the good of winning the breathing game over and over. And of course there is an immense practice-external good at stake: this is a game where victory is life, as the Jem’Hadar say.

There is something absurd about the idea that I have significantly enhanced my life simply by deciding to be a player of the breathing game and thus attaining victory about 1440 times a day.

It is widely thought that there can be significant practice-internal goods in practices we institute. The breathing game’s practice-internal good of victory is not significant. Why not? Maybe because the game hasn’t caught on: I am the only one playing it. (Everybody is else is just breathing.) But if the good of victory would become significant were the game to catch on, then we have good consequentialist reason to promote the breathing game as widely as we can, so that as many people as possible could get a significant good 1440 times a day, thereby brighting up many drab lives. But that’s silly. It’s not that easy to improve the lot of humankind.

An intuitive thing to say about the breathing game is that it’s not very challenging. Healthy people can win without even trying. It’s a lot harder to get a score of 0 than a score of 10. The lack of challenge certainly makes the game less fun. But fun is a practice-external good. Does the lack of challenge make the practice-internal good less?

Maybe, but I am dubious. Challenge really seems rather external, while practice-internal goods are supposed to be instituted. Maybe, though, the story is this. When I institute a game, I am filling out a template provided by a broader social practice, the practing of playing games, and the broader social practice includes a rule that says that unchallenging victory is not worth much. I can’t override that rule while still counting as instituting a game.

That may be. But if the larger social practice, the one of games in general, is itself one that we have instituted, then we have good moral reason to institute another social practice, a practice of shgames. The practice of shgames is just like the practice of games, except that the practice-internal good of victory is stipulated as being a great good even when victory comes easily. We have very good reason to institute the practice of shgames, as this would allow everybody to play the breathing shgame (which has the same rules as the breathing game), and thus enrich their lives by 1440 valuable victories a day.

That’s absurd, too.

Here’s where the line of thought is leading me: We have significant limits on our normative power to set the value of the practice-internal goods of the practices we create. In particular, the practice-internal goods that are entirely our creation are only of little value—like the value of victory in a game.

One might think that this is just an artifact of games and similar practices, which are not very significant practices. Perhaps in our political practices, we can institute great practice-internal goods. I don’t think so. The state can bestow a title on everyone who scores a perfect ten in the breathing game, but the state cannot by mere stipulation make that title have great value of a practice-internal sort. Otherwise, it’s too easy to create value. (One may think that the issues here are related to why the state can’t just print more money to create wealth. But I think this is quite different: the reason the state can’t just print more money to create wealth is that wealth is defined partly in terms of practice-external goods, and mere printing doesn’t affect those. But purely internal goods can be broadcast widely.)

I am not claiming that there are no great practice-internal goods. There are great internal goods in marriage, for instance. But here is my hypothesis: wherever there are great practice-internal goods, these goods derive their value from a practice we do not instituted. For instance, if there are great internal goods in marriage, that is because either we have not institute marriage or because marriage is itself the filling out of a template provided by a broader practice that we have not instituted (I think the former is the case).

If we could institute practices with great practice-internal value, we should, just for the sake of the practice-internal value. But that is wrong-headed. In fact, I think that when we institute practices, it is for the sake of goods that we are not instituting. We get the practice-internal goods, then, but they are just icing on the cake, and not a good icing even.

Perhaps I am misunderstanding practice-internal goods, though. Maybe they have the following property: they provide reasons to pursue them for those who participate in the practice, but they do not provide any reasons for those who do not participate in the practice. On this picture, one could have a great practice-internal good, one that provides very significant reasons, but it would provide no reason at all to a non-participant, and hence it would provide no reason to institute the practice. This seems wrongheaded. Only real goods provide real reasons. If practice-internal goods were to provide real reasons to the participants, they would have to be real goods. But if something—say, the institution of a practice—would result in the existence of real goods to people, that does provide a reason to bring about the something. That’s part of what is true in consequentialism. Moreover, even if one removes the absurdity of thinking that there is reason to institute the breathing game, one does not remove the absurdity of thinking that people who play the breathing game are racking up much good.

Thursday, October 5, 2017

4D, 3D and 2D

No three-dimensionalist has this ludicrous picture of the human being:

  1. The human being is a three-dimensional whole whose functioning derives from the functioning of many two-dimensional slices that make her up.

But some four-dimensionlists have this picture of the human being:

  1. The human being is a four-dimensional whole whose functioning derives from the functioning of many three-dimensional slices that make her up.

But it seems to me that (2) is not much better than (1). Just as we should be very dubious that there are any such things as two-dimensional slices of us—infinitely thin slices—we should be very dubious that there any such things as three-dimensional slices of us. And even if there are such slices, they are more like abstractions than components from which we derive our functioning.

So just as all three-dimensionalists deny 2D-slicism, all four-dimensionalists should deny 3D-slicism.

Now, let’s turn to three-dimensionalists. The following proposition is crazy:

  1. The human being is two-dimensional.

But some think:

  1. The human being is three-dimensional.

Why think (1) is false? The best reason is that the characteristic functioning of a human being requires more than just an infinitely thin section. Any thin section—whether infinitely thin or of non-zero thickness—through a human being is going to be quite unnatural, being intimately connected causally to other sections that are just as important to the characteristic functioning of a human being.

But the same reason applies against (2). Any temporally thin section—whether infinitely thin or of non-zero temporal thickness—through a human being is going to be quite unnatural, being intimately connected causally to other sections that are just as important to the characteristic functioning of a human being.

A secondary reason against (1) is that it is implausible that are privileged 2D slices. But likewise it is implausible (though maybe a bit less so) that there are privileged 3D slices.

Where does all this leave us? We should be at least four-dimensionalists (there might turn out to be more dimensions), and we should not think of ourselves as derivative from 3D or thinner slices. We should, indeed, be sceptical of the existence of such slices.

The argument from highly intelligent saints who are Christians

  1. There have been many highly intelligent saints who were Christians.

  2. If there have been may highly intelligent saints who were Christians, then probably (insofar as the above evidence goes) the central doctrines of Christianity are true.

  3. So, probably (insofar as the above evidence goes), the central doctrines of Christianity are true.

(An interesting variant is to replace “are true” in (2) and (3) with “are approximately true”, and then to combine the conclusion with my previous post.)

I do not plan to defend 1. That’s too easy. Note, though, that while easy, it’s not trivial. I am not claiming that there were many highly intelligent people who were canonized “Saints” by the Catholic or Orthodox Church, though that’s true. Nor am I claiming that there were many highly intelligent people who were Christian saints. I am claiming that there are may highly intelligent people who were saints simpliciter, as well as being Christian.

What is a saint like? Saints are deeply morally good people who, insofar as it depends on them, lead a deeply flourishing human life. Their lives are meaningful and when seen closely—which may be difficult, as many saints are very unostentatious—these lives are deeply compelling to others. Saints tightly integrate the important components of their lives. In particular, those saints who are highly intelligent—and not all saints are intelligent, though all are wise—integrate their intellectual life and their moral life. Highly intelligent saints are reflective. They have an active and humble conscience that is on the lookout for correction, and this requires integration between the intellectual life and the moral life.

An intelligent saint who is a Christian is also a Christian saint. For Christianity is not the sort of doctrine that can be held on the peripheries of a well-lived life. Someone who is a Christian but to whose life Christianity is not central is neither a saint simpliciter nor a Christian saint. For a central part of being Christian is believing that Christianity should be central to one’s life, and an intelligent saint—in either sense—will see this and thus either conscientiously act on such a belief, making Christianity be central to her life, or else conclude that Christianity is false.

Now, the existence of a highly intelligent saint who is a Christian is evidence for coherence between central moral truths and the truth of Christianity. For if they were not coherent, the reflectiveness of the highly intelligent saint would likely have seen the incoherence, and her commitment to morality would have led to the rejection of Christianity. But it’s not just that the moral truths and the truth of Christianity cohere: the truths of Christianity support and motivate the moral life. For the saint who is a Christian is, as I just argued, also a Christian saint. And a Christian saint is motivated in the moral life by considerations central to Christianity—the love of God as shown in creation and in the incarnate Son’s sacrificial death on the cross.

It is difficult to have a coherent theory that includes in a highly integrated way deeply metaphysical beliefs and correct moral views in a way where the metaphysical beliefs support the moral ones. That a theory is such is significant evidence for the theory’s truth. More generally and loosely, I think that a person whose life is deeply compelling is likely to be right in those central beliefs of her that are tightly interwoven with what makes her life compelling. But the saint’s moral life is compelling, and if she is a Christian, then her central Christian beliefs are tightly interwoven with her moral life.

Hence, 2 is true.

Of course, the above is not all the evidence there is. What about highly intelligent saints who are not Christians? The existence of such may well weaken the argument. But at least, I think, the argument makes Christianity an intellectually serious option.

And there may be something we can say more specifically on a case by case basis about saints outside of Christianity. Crucial to my argument was that one cannot be a saint and a Christian and have the Christianity be peripheral to one’s moral life. But one can be a saint and an atheist and have the atheism be peripheral to one’s moral life. Atheism is a negative doctrine, after all. If one turns it into a positive motivational doctrine, one gets something like Russell’s “A Free Man’s Worship”. But that is too proud, too haughty, too cold, too dark to be the central motivational doctrine of a saint. A saint who is an atheist is, I suspect, not as likely to be an atheist saint as a saint who is a Christian is to be a Christian saint.

Eastern religions have their saints, but there is an obvious tension between the irrealism to which Eastern religions tend and moral truths about the importance of love of others, of corporal care for the needs of others. One can adhere to an irrealist philosophy and despite this live a life of service to others, but it is unlikely that the service to others be central to one’s life in the way that moral sainthood requires.

What about Jewish and Muslim saints? Well, it may be that many of the motivationally central parts of Judaism and Islam are shared by Christianity—though the converse is not true, given the motivational centrality of the Incarnation to Christianity. One might object that the transcendence and simplicitly of God as taught in Judaism and Islam is motivationally central. But classical Christian theism embraces the transcendence and simplicity of God—and the Incarnation and Trinity, too.

A modal approximative ontological argument

Here is an ontological argument that I haven’t seen:

  1. Possibly, it is approximately true that God exists.

  2. Necessarily, if it is approximately true that God exists, then it true that God exists.

  3. If possibly God exists, then God exists.

  4. So, possibly, it is true that God exists. (1 and 2)

  5. So, God exists. (3 and 4)

Premise 1 is an interesting weakening of the familiar possibility premise from modal ontological arguments.

Premise 3 is also familiar, going back at least to Mersenne. We can say that God is the sort of being that couldn’t exist merely contingently: he either exists necessarily or he can’t exist at all—there is no room for mere possibilities in the case of God’s existence.

The thought behind 2 is rather similar to that behind 3: God is a kind of infinity that cannot be approximated. It is not possible for there to be a state of affairs merely approximating the existence of God.

Tuesday, October 3, 2017

Infinite proofs

Consider this fun “proof” that 0=1:


  • So, 3=4

  • So, 2=3

  • So, 1=2

  • So, 0=1.

What’s wrong with the proof? Each step follows from the preceding one, after all, and the only axiom used is an uncontroversial axiom of arithmetic that if x + 1 = y + 1 then x = y (by definition, 2 = 1 + 1, 3 = 1 + 1 + 1, 4 = 1 + 1 + 1 + 1 and so on).

Well, one problem is that intuitively a proof should have a beginning and an end. This one has an end, but no beginning. But that’s easily fixed. Prefix the above infinite proof with this infinite number of repetitions of “0=0”, to get:

  • 0=0

  • So, 0=0

  • So, 0=0

  • So, 0=0



  • So, 3=4

  • So, 2=3

  • So, 1=2

  • So, 0=1.

Now, there is a beginning and an end. Every step in the proof follows from a step before it (in fact, from the step immediately before it). But the conclusion is false. So what’s wrong?

The answer is that there is a condition on proofs that we may not actually bother to mention explicitly when we teach logic: a proof needs to have a finite number of steps. (We implicitly indicate this by numbering lines with natural numbers. In the above proof, we can’t do that: the “second half” of the proof would have infinite line numbers.)

So, our systems of proof depend on the notion of finitude. This is disquieting. The concept of finitude is connected to arithmetic (the standard definition of a finite set is one that can be numbered by a natural number). So is arithmetic conceptually prior to proof? That would be a kind of Platonism.

Interestingly, though, causal finitism—the doctrine that nothing can have an infinite causal history—gives us a metaphysical verificationist account of proof that does not presuppose Platonism:

  • A proof is a sequence of steps such that it is metaphysically possible for an agent to verify that each one followed by the rules from the preceding steps and/or the axioms by observation of each step.

For, given causal finitism, only a finite number of steps can be in the causal history of an act of verification of a proposition. (God can know all the steps in an infinite chain, but God isn’t an observer: an observer’s observational state is caused by the observations.)

Friday, September 29, 2017

Gamecube controller to USB adapter

I wanted to make an adapter that lets us use our Wii dance pads and Gamecube gamepads with PC-based games. It seemed like it would be fun, for instance, to play Tetris using one's feet on a dance pad. One can do this project for under $4 or so using an stm32f1 development board. Here are instructions.

Loss of vice and growth of virtue

Here is a pattern in the moral life. People have a conversion experience and puts away “the gross sins of the flesh” like robbery, drug abuse, violence or fornication. And then they struggle for decades with faults sins like laziness, unkindliness, vanity, impatience or judgmentality. What’s going on? It seems like at the outset they put away the bigger moral faults, and then they were left with smaller ones. Why is it that it takes so much longer to fight off the smaller ones? Why doesn’t it get easier, given that the faults are smaller? This is frustrating!

Instead, the experience sometimes seems to be like when you take a piece of unstretchable rope by two ends and pull. It is easy to get the initial big sag out. But as the sag gets smaller, it gets harder and harder to get it out.

Here is a thought. There are two ways of quantifying one’s moral state: the degree of vice and the degree of virtue. And the two are not related in a simple way, with one being the negation of the other. In ordinary circumstances, it doesn’t take much virtue to exclude murder from one’s life. But it does take a lot of virtue to exclude vanity from one’s life. To cease murdering is to lose much vice but is not to gain much virtue. To cease being vain, though, is not to lose much vice but it is to gain much virtue (is this true if one is still murdering, though?).

The “ordinary decent person” is perhaps not much more vicious than St. Teresa of Calcutta or St. Francis. But the “ordinary decent person” is far less virtuous.

Note that this is true even if we limit the discussion to what one might call “obligatory virtue”, i.e., the virtue that is opposed to vice rather than supererogatory virtue. The virtue involved in eliminating vanity is an obligatory virtue, though the virtue involved in giving up property for the sake of God and the poor, as St. Teresa and St. Francis did, is supererogatory. Yet the ordinary person is far less obligatorily virtuous than St. Teresa or St. Francis.

There may be an inverse relationship between vice and obligatory virtue: the more you have of the one, the less you have of the other. But a small increase of vice can correspond to large losses of virtue, and vice versa. It’s be a bit like that between the amount of sag in the rope and the horizontal force you need to push the ends with to balance the rope. As the sag goes to zero, the horizontal force goes to infinity.

The frustration I mentioned in the first paragraph then may be misplaced. For while it may seem like the moral life stalls after an initial burst of energy, the stalling may only be there if we measure the progress by the amount of vice. But if we measure by the amount of virtue, there might be steady increase throughout, just as a rope may linearly increase in tension, even though the sag seems not to be changing much.

(But may God have mercy on us!)

(While pushing metaphors perhaps too far, note that on the other hand the sag can’t go to infinity if the rope is unstretchable, since eventually we run out of rope. Likewise, perhaps, there is a limit to our vice, set by our nature. This fits with the idea of evil as a privation of the good.)

Thursday, September 28, 2017

Walking off to infinity

This is a simplified version of a paradox Josh Rasmussen sent me (“Rasmussen’s Rod”). Suppose that Laika is in a spaceship in a Euclidean non-relativistic space, and in one second she flies a kilometer, in the next half second another kilometer, and in the next quarter second another, and so on, all in exactly the same direction.

What will happen to Laika and the spaceship in two seconds?

Here are four answers:

  1. Causal finitism: The story is impossible, as the outcome has infinitely many accelerations as causes.

  2. Space is constituted by the relations between things in them rather than being a container. After two seconds, Laika will be infinitely far away from us. Where is that? It’s a place that didn’t exist until Laika got there, a place constituted by Laika’s being there and her distance from us.

  3. Laika and the spaceship will leave space, and will exist as objects that aren’t externally spatial. (They might be internally spatial.)

  4. Dogs and spaceships depend on space for their existence, and hence upon leaving they will cease to exist.

Tuesday, September 26, 2017

A causal finitist definition of the finite

Causal finitism says that nothing can have infinitely many causes. Interestingly, we can turn causal finitism around into a definition of the finite.

Say that a plurality of objects, the xs, is finite if and only if it possible for there to be a plurality of beings, the ys, such that (a) it is possible for the ys to have a common effect, and (b) it is possible for there to be a relation R such that whenever x0 one of the xs, then there is exactly one of a y0 among the xs such that Rx0y0.

Here's a way to make it plausible that the definition is extensionally correct if causal finitism is true. First, if the definition holds, then clearly there are no more of the xs than of the ys, and causal finitism together with (a) ensures that there are finitely many of the ys, so anything that the definition rules to be finite is indeed finite. Conversely, suppose the xs are a finite plurality. Then it should be possible for there to be a finite plurality of persons each of which thinks about a different one of the xs in such a way that each of the xs is thought about by one of the ys. Taking being thought about as the relation R makes the definition be satisfied.

Of course, on this account of finitude, causal finitism is trivial, for if a plurality of objects has an effect, then they satisfy the above definition if we take R to be identity. But what then becomes non-trivial is that our usual platitudes about the finite are correct.

Monday, September 25, 2017

Mathematical Platonist Universalism, consistency, and causal finitism

Mathematical Platonists say that sets and numbers exist. But there is a standard epistemological problem: How do we have epistemic access to the sets to the extent of knowing some of the axioms they satisfy? There is a solution to this epistemological problem, mathematical Platonist universalism (MPU): for any consistent collection of mathematical axioms, there are Platonic objects that satisfy these axioms. MPU looks to be a great solution to the epistemological problems surrounding mathematical Platonism. How did evolved creatures like us get lucky enough to have axioms of set theory or arithmetic that are actually true of the sets? It didn’t take much luck: As soon as we had consistent axioms, it was guaranteed that there would be a plurality of objects that satisfied them, and if the axioms fit with our “set intuitions”, we could call the members of any such plurality “sets” while if they fit with our “number intuitions”, we could call them “natural numbers”. And the difficult questions about whether things like the Axiom of Choice are true are also easily resolved: the Axiom of Choice is true of some pluralities of Platonic objects and is false of others, and unless we settle the matter by stipulation, no one of these pluralities is the sets. (The story here is somewhat similar to Joel Hamkins’ set theoretic multiverse, but I don’t know if Hamkins has the kind of far-reaching epistemological application in mind that I am thinking about.)

This story has a serious problem. It is surely only the consistent axioms that are satisfied by a plurality of objects. Axioms are consistent, by definition, provided that there is no proof of a contradiction from them. But proofs are themselves mathematical objects. In fact, we’ve learned from Goedel that proofs can be thought of as just numbers. (Just write your proof in ASCII, and encode it as a binary number.) Hence, a plurality of axioms is consistent if and only if there does not exist a number with a certain property, namely the property of encoding a proof of a contradiction from these axioms. But on MPU there is no unique plurality of mathematical objects deserving to be called “the numbers”. So now MPU faces a very serious problem. It said that any consistent plurality of axioms is true of some plurality of Platonic objects, and there are no privileged pluralities of “numbers” or “sets”. But consistency is itself defined by means of “the numbers”. And the old epistemological problems for Platonism resurface at this level. How do we have access to “the numbers” and the axioms they satisfy so as to have reason to think that the facts about consistency of axioms are as we think they are?

One could try making the same move again. There is no privileged notion of consistency. There are many notions of consistency, and for any axioms that are consistent with respect to any notion of consistency there exists a plurality of Platonic satisfiers. But now this literally threatens incoherence. But unless we specify some boundaries on the notion of consistency, this is going to literally let square circles into Platonic universalism. And if we specify the boundaries, then epistemological problems that MPU was trying to solve will come back.

At my dissertation defense, Robert Brandom offered a very clever suggestion for how to use my causal powers account of modality to account for provability: q can be proved from p provided that it is causally possible for someone to write down a proof of q from p. This can be used to account for consistency: axioms are consistent provided that it is not causally possible to write down a proof of a contradiction from them. There is a bit of a problem here, in that proofs must be finite strings of symbols, so one needs an account of the finite, and a plurality is finite if and only if its count is a natural number, and so this account seems to get us back to needing privileged numbers.

But if one adds causal finitism (the doctrine that only finite pluralities can together cause something) to the mix, we get a cool account of proof and consistency. Add the stipulation that the parts of a “written proof” need to have causal powers such that they are capable of together causing something (e.g., causing someone to understand the proof). Causal finitism then guarantees that any plurality of things that can work together to cause an effect is finite.

So, causal finitism together with the causal powers account of modality gives us a metaphysical account of consistency: axioms are consistent provided that it is not causally possible for someone to produce a written proof of a contradiction from them.

Friday, September 22, 2017

Free and responsible unconscious decisions

  1. Whether a decision to do A is free and responsible does not depend on anything explanatorily posterior to the decision.

  2. Our consciousness of x is always explanatorily posterior to x.

  3. Hence, whether our decision to do A is free and responsible does not depend on our consciousness of having decided to do A.

  4. If whether our decision to do A is free and responsible does not depend on our consciousness of having decided to do A, then it is possible to have a free and responsible unconscious decision to do A.

  5. So, it is possible to have a free and responsible unconscious decision.

Let me, though, clarify something. This argument does not establish that the deliberation itself can be unconscious. It only establishes that one can be unconscious of the outcome of the deliberation. I suspect the deliberation can be unconscious as well, but I don't have as good an argument.

Two questions about sets

Here are two curious philosophical questions about set theory and its applicability outside mathematics.

Question 1: Suppose that every person has a perfectly well-defined mass. Is there a set of everybody mass, say the set of all real numbers x such that x is someone’s mass in kilograms?

The standard ZFC axioms are silent on this. They do say that for any predicate F in the language of set theory there is a set of all real numbers x satisfying F. But "mass" and "kilogram" are not parts of the language of set theory.

Question 2: What does it mean to say that there are finitely many horses?

An obvious answer is that if H is the set of all horses, then H is in one-to-one correspondence with some natural number. But the standard ZFC axioms only give us sets of sets, not sets of physical things like horses. If the correct set theory has ur-elements, elements that aren’t sets, maybe there is a set of all horses—but maybe not even then.

I suppose we could go metalinguistic. Begin by describing the set S of first-order logic sentences (sentences can be thought of as sets, even if sets are pure, i.e., have only sets as members) that say "There are no horses", "There is at most one horse", "There are at most two horses",.... And then say, using language beyond set theory, that at least one sentence in S is true.

But the metalinguistic approach won’t solve the seemingly related problem of what it means to say that there are countably many horses.

Progress report on books

My Necessary Existence book with Josh Rasmussen is right now in copyediting by Oxford.

I am making final revisions to the manuscript of Infinity, Causation and Paradox, with a deadline in mid October. As of right now, I've finished revising five out of ten chapters.

I am toying with one day writing a book on the ethics of love.

Thursday, September 21, 2017

Promising to sing infinitely many duets

Suppose you and I are going to live forever in heaven. I promise you that I will play sing a duet with you infinitely many times. Is this a valid promise?

Here is an argument that it is not. It seems that if the promise is valid, it generates reasons to sing duets with you. But it doesn’t. The reasons generated by a promise are reasons to do things that contribute to the fulfillment of the promise. But singing a duet with you does not contribute to the fulfillment of the promise. Here is one way to see this. Suppose I am considering whether to sing the duet with you on Wednesday, September 1, 2060. Consider now these two potential promises that I could imagine myself to have made:

  1. I will sing a duet with you on infinitely many of the days that are not September 1, 2060.

  2. I will sing a duet with you on infinitely many days.

Then singing the duet with you on September 1, 2060 does nothing to promote the fulfillment of promise (1). But (2) is logically equivalent to (1)! For I sing a duet with you on infinitely many days if and only if I sing a duet with you on infinitely many days that are not September 1, 2060. So my singing the duet on September 1, 2060 will no more promote the fulfillment of (2) than it will promote the fulfillment of (1). So, the promise doesn’t generate reasons to sing duets.

But things aren’t so simple. For while it doesn’t generate reasons to sing duets, it could generate reasons to do other things that bring about my singing duets with you on infinitely many days that are not September 1, 2060. For instance, here is something I could do: I could promise you to sing a duet with you every Wednesday for eternity. Making that promise will promote both (1) and (2). For the promise to sing duets on Wednesdays does unproblematically generate a reason to sing a duet on every Wednesday, and this generation of reasons is likely to contribute to my singing a duet with you on infinitely many days.

Of course, there are other promises I could make you that would make (1) and (2) likely. I could promise to sing a duet with you every January 1. Or every January 1 of a prime-numbered year. It’s a difficult question which of these promises I should make. But I have reason to make some such promise, or do something else that is likely to motivate me infinitely often, say inculcate a habit in myself.

So the answer to the initial question is plausibly positive. But it is only plausible if there is something other than singing duets that one can do in fulfillment of the promise. If all I am facing are the individual daily choices whether to sing a duet or not, without any habituation, I cannot validly promise to sing the duet on infinitely many occasions, as it would not generate any reasons.

A Trinitarian structure in love

On my view, love has a three-fold structure:

  • benevolence
  • appreciation
  • union.

This three-fold structure has certain Trinitarian parallels. The Father is the benefactor: he gives being to the Son and thereby to the Holy Spirit. The Son admires the Father, is the Logos that reflects upon the Father’s goodness. The Holy Spirit unites the Father and the Son.

Wednesday, September 20, 2017

The Probabilistic Counterexampler

Every so often someone asks me if some piece of probabilistic reasoning works. For instance, today I got a query from a grad student whether

  1. P(A|C)>P(A|B) implies P(A|B ∨ C)>P(A|B).

Of course, I could think about it each time somebody asks me something. But why think when a computer can solve a problem by brute force?

So, last spring I wrote a quick and dirty python program that looks for counterexamples to questions like that simply by considering situations with three dice, and iterating over all the possible combinations of subsets A, B and C of the state space (with some reduction due to symmetries).

The program is still quick and dirty, but at least I made the premises and conclusions not be hardcoded. You can get it here.

For instance, for the query above, you can run:

python "P(a,c)>P(a,b)" "P(a,b|c)>P(a,b)" 

(The vertical bars are disjunction, not conditional probability. Conditional probability uses commas.) The result is:

a={1}, b={1, 2}, c={1}
a={1}, b={1, 2, 3}, c={1}
a={1}, b={1, 2, 3}, c={1, 2}
a={1}, b={1, 2, 3}, c={1, 3}
a={1}, b={1, 2, 3}, c={1, 4}

So, lots of counterexamples. On the other hand, you can do this:

python "P(a)*P(b)==P(a&b)" "P(b)>0" "P(a,b)==P(a)" 

and it will tell you no counterexamples were found. Of course, that doesn’t prove that the result is true, but in this case it is.

The general operation is that you install python (either 2.7 or 3.x) and use a commandline to run:

python premise1 premise2 ... conclusions

You can use any single letter variables for events, other than P, and the operations & (conjunction), | (disjunction) and ~ (negation) between the events. You can use conditional probability P(a,b) and unconditional probability P(a). You can use standard arithmetical and comparison operators on probabilities. Make sure that you use python’s operators. For instance, equality is ==, not =. You should also use python’s boolean operations when you are not working with events: e.g., “P(a)==1 and P(b)==0.5”.

Any premise or conclusion that requires conditionalization on a probability zero event to evaluate automatically counts as false.

You can use up to five single-letter variables and you can also specify the number of sides the die has prior to listing the premises. E.g.:

python 8 "P(a)*P(b)==P(a&b)" "P(b)>0" "P(a,b)==P(a)" 

Monday, September 18, 2017

Two ways of being vicious

Many of the times when Hitler made a wrong decision, his character thereby deteriorated and he became more vicious. Let’s imagine that Hitler was a decent young man at age 19. Now imagine Schmitler, who lived a life externally just like Hitler’s, but on Twin Earth. Until age 19, Schmitler’s life was just like Hitler. But from then on, each time Schmitler made a wrong choice, aliens or angels or God intervened and made sure that the moral deterioration that normally follows upon wrong action never occurred. As it happens, however, Schmitler still made the same choices Hitler did, and made them with freedom and clear understanding of their wickedness.

Thus, presumably unlike Hitler, Schmitler did not morally fall, one wrong action at a time, to the point of a genocidal character. Instead, he committed a series of wrong actions, culminating in genocide, but each action was committed from the same base level of virtue and vice, the same level that both he and Hitler had at age 19. This is improbable, but in a large enough universe all sorts of improbable things will happen.

So, now, here is the oddity. Since Schmitler’s level of virtue and vice at the depth of his moral depradations was the same as at age 19, and at age 19 both he and Hitler were decent young men (or so I assume), it seems we cannot say that Schmitler was a vicious man even while he was committing genocidal atrocities. And yet Schmitler was fully responsible for these atrocities, perhaps more so than Hitler.

I want to say that Schmitler is spectacularly vicious without having much in the way of vices, indeed while having more virtue than vice (he was, I assume, a decent young man), even though that sounds like a contradiction. Schmitler is spectacularly vicious because of what he has done.

This doesn’t sound right, though. Actions are episodic. Being vicious is a state. Hitler was a vicious man while innocently walking his dog on a nice spring day in 1944, even when not doing any wrongs. And we can explain why Hitler was vicious then: he had a character with very nasty vices, even while he was not exercising the vices. But how can we say that Schmitler was vicious then?

Here’s my best answer. Even on that seemingly innocent walk, Schmitler and Hitler were both failing to repent of their evil deeds, failing to set out on the road of reconciliation with their victims. A continuing failure to repent is not something episodic, but something more like a state.

If this is right, then there are two ways of being vicious: by having vices and by being an unrepentant evildoer.

(A difficult question Robert Garcia once asked me is relevant, though: What should we say about people who have done bad things but suffered amnesia?)

Some arguments about the existence of a good theodicy

This argument is valid:

  1. If no good theodicy can be given, some virtuous people’s lives are worthless.

  2. No virtuous person’s life is worthless.

  3. So, a good theodicy can be given.

The thought behind 1 is that unless we accept the sorts of claims that theodicists make about the value of virtue or the value of existence or about an afterlife, some virtuous people live lives of such great suffering, and are so far ignored or worse by others, that their lives are worthless. But once one accepts those sorts of claims, then a good theodicy can be given.

Here is an argument for 2:

  1. It would be offensive to a virtuous person that her life is worthless.

  2. The truth is not offensive to a virtuous person.

  3. So, no virtuous person’s life is worthless.

Perhaps, too, an argument similar to Kant’s arguments about God can be made. We ought to at least hope that each virtuous person’s life has value on balance. But to hope for that is to hope for something like a theodicy. So we ought to hope for something like a theodicy.

The above arguments may not be all that compelling. But at least they counter the argument in the other direction, that it is offensive to say that someone’s sufferings have a theodicy.

Here is yet another argument.

  1. That there is no good theodicy is an utterly depressing claim.

  2. One ought not advocate utterly depressing claims, without very strong moral reason.

  3. There is no very strong moral reason to advocate that there is no good theodicy.

  4. So, one ought not advocate that there is no good theodicy.

The grounds for 8 are pragmatic: utterly depressing claims tend to utterly depress people, and being utterly depressed is very bad. One needs very strong reason to do something that causes a very bad state of affairs. I suppose the main controversial thesis here is 9. Someone who thinks religion is a great evil might deny 9.

Let's not exaggerate the centrality of virtue to ethics

Virtues are important. They are useful: they internalize the moral law and allow us to make the right decision quickly, which we often need to do. They aren’t just time-savers: they shine light on the issues we deliberate over. And the development of virtue allows our freedom to include the two valuable poles that are otherwise in tension: (a) self-origination (via alternate possibilities available when we are developing virtue) and (b) reliable rightness of action. This in turn allows our development of virtue reflect the self-origination and perfect reliability in divine freedom.

But while virtues are important, they are not essential to ethics. We can imagine beings that only ever make a single, but truly momentous, decision. They come into existence with a clear understanding of the issues involved, and they make their decision, without any habituation before or after. That decision could be a moral one, with a wrong option, a merely permissible option, and a supererogatory option. They would be somewhat like Aquinas’ angels.

We could even imagine beings that make frequent moral choices, like we do, but whose nature does not lead them too habituate in the direction of virtue or vice. Perhaps throughout his life whenever Bill decides whether to keep an onerous promise or not, there is a 90% chance that he will freely decide rightly and a 10% chance that he will freely decide wrongly, a chance he is born and dies with. A society of such beings would be rather alien in many practices. For instance, members of that society could not be held responsible for their character, but only for their choices. Punishment could still be retributive and motivational (for the chance of wrong action might go down when there are extrinsic reasons against wrongdoing). I think such beings would tend to have lower culpability for wrongdoing than we do. For typically when I do wrong as a middle-aged adult, I am doubly guilty for the wrong: (a) I am guilty for the particular wrong choice that I made, and (b) I am guilty for not having yet transformed my character to the point where that choice was not an option. (There are two reasons we hold children less responsible: first, their understanding is less developed, and, second, they haven’t had much time to grow in virtue.)

Nonetheless, while such virtue-less beings woould be less responsible, and we wouldn’t want to be them or live among them, they would still have some responsibility, and moral concepts could apply to them.

Saturday, September 16, 2017

Adding a USB charging port to an elliptical machine

Last night I added a USB charging port to our elliptical machine, using a $0.70 buck converter, so that we can exercise while watching TV on a tablet even when running out of batteries. Here are instructions.

Note, too, how the tablet is held in place with 3D printed holders. My next elliptical upgrade project will be to make it be a part of a USB game controller (the other part will be a Wii Nunchuk) so that one can control speed in games with speed of movement.

Friday, September 15, 2017

Four-dimensionalism and caring about identity

In normal situations, diachronic psychological connections and personal identity go together. A view introduced by Parfit is that when the two come apart, what we care about are the connections and not the identity.

This view seems to me to be deeply implausible from a four-dimensional point of view. I am a four-dimensional thing. This four-dimensional thing should prudentially care about what happens to it, and only about what happens to it. The red-and-black four-dimensional thing in the diagram here (up/down represents time; one spatial dimension is omitted) should care about what happens to the red-and-black four-dimensional thing, all along its temporal trunk. This judgment seems completely unaffected by learning that the dark slice represents an episode of amnesia, and that no memories pass from the bottom half to the upper half.

Or take a case of symmetric fission, and suppose that the facts of identity are such that I am the red four-dimensional thing in the diagram on the right. Suppose both branches have full memories of what happens before the fission event. If I am the red four-dimensional thing, I should prudentially care about what happens to the red four-dimensional thing. What happens to the green thing on the right is irrelevant, even if it happens to have in it memories of the pre-split portion of me.

The same is true if the correct account of identity in fission is Parfit’s account, on which one perishes in a split. On this account, if I am the red four-dimensional person in the diagram on the left, surely I should prudentially care only about what happens to the red four-dimensional thing; if I am the green person, I should prudentially care only about what happens to the green one; and if I am the blue one, I should prudentially care only about what happens to the blue one. The fact that both the green and the blue people remember what happened to the red person neither make the green and blue people responsible for what the red person did nor make it prudent for the red person to care about what happens to the green and blue people.

This four-dimensional way of thinking just isn’t how the discussion is normally phrased. The discussion is normally framed in terms of us finding ourselves at some time—perhaps a time before the split in the last diagram—and wondering which future states we should care about. The usual framing is implicitly three-dimensionalist: what should I, a three-dimensional thing at this time, prudentially care about?

But there is an obvious response to my line of thought. My line of thought makes it seem like I am transtemporally caring about what happens. But that’s not right, not even if four-dimensionalism is true. Even if I am four-dimensional, my cares occur at slices. So on four-dimensionalism, the real question isn’t what I, the four-dimensional entity, should prudentially care about, but what my three-dimensional slices, existent at different times, should care about. And once put that way, the obviousness of the fact that if I am the red thing, I should care about what happens to the red thing disappears. For it is not obvious that a slice of the red thing should care only about what happens to other slices of the red thing. Indeed, it is quite compelling to think that the psychological connections between slices A and B matter more than the fact that A and B are in fact both parts of the same entity. (Compare: the psychological connections between me and you would matter more than the fact that you and I are both parts of the same nation, say.) The correct picture is the one here, where the question is whether the opaque red slice should care about the opaque green and opaque blue slices.

In fact, in this four-dimensionalist context, it’s not quite correct to put the Parfit view as “psychological connections matter more than identity”. For identity doesn’t obtain between different slices. Rather, what obtains is co-parthood, an obviously less significant relation.

However, this response to me depends on a very common but wrongheaded version of four-dimensionalism. It is I that care, feel and think at different times. My slices don’t care, don’t feel and don’t think. Otherwise, there will be too many carers, feelers and thinkers. If one must have slices in the picture (and I don’t know that that is so), the slices might engage in activities that ground my caring, my feeling and my thinking. But these grounding activities are not caring, feeling or thinking. Similarly, the slices are not responsive to reasons: I am responsive to reasons. The slices might engage in activity that grounds my responsiveness to reasons, but that’s all.

So the question is what cares I prudentially should have at different times. And the answer is obvious: they should be cares about what happens to me at different times.

About the graphics: The images are generated using mikicon’s CC-by-3.0 licensed Gingerbread icon from the Noun Project, exported through this Inkscape plugin and turned into an OpenSCAD program (you will also need my tubemesh library).

Thursday, September 14, 2017

Agents, patients and natural law

Thanks to Adam Myers’ insightful comments, I’ve been thinking about the ways that natural law ethics concerns natures in two ways: on the side of the agent qua agent and on the of the patient qua patient.

Companionship is good for humans and bad for intelligent sharks, let’s suppose. This means that we have reasons to promote companionship among humans and to hamper companionship among intelligent sharks. That’s a difference in reasons based on a difference in the patients’ nature. Next, let’s suppose that intelligent sharks by nature have a higher degree of self-concern vs. other-concern than humans do. Then the degree to which one has an obligation to promote the very same good–say, the companionship of Socrates–will vary depending on whether one is human or a shark. That’s a difference in reasons based on a difference in the agents’ nature.

I suspect it would make natural law ethics clearer if natural lawyers were always clear on what is due to the agent’s nature and what is due to the patient’s nature, even if in fact their interest were solely in cases where the agent and patient are both human.

Consider, for instance, this plausible thesis:

  • I should typically prioritize my understanding over my fun.

Suppose the thesis is true. But now it’s really interesting to ask if this is true due to my nature qua agent or my nature qua patient. If I should prioritize my understanding over my fun solely because of my nature qua patient, then we could have this situation: Both I and an alien of some particular fun-loving sort should prioritize my understanding over my fun, but likewise both I and the alien should prioritize the alien’s fun over the alien’s understanding, since human understanding is more important than human fun, while the fun of a being like the alien is more important than the understanding of such a being. On this picture, the nature of the patient specifies which goods are more central to a patient of that nature. On the other hand, if I should prioritize my understanding over my fun solely because of my nature qua agent, then quite possibly we are in the interesting position that I should prioritize my understanding over my fun, but also that I should prioritize the alien’s understanding over the alien’s fun, while the alien should prioritize both its and my fun over its and my understanding. For me promoting understanding is a priority while for the alien promoting fun is a priority, regardless of whose understanding and fun they are.

And of course we do have actual and morally relevant cases of interaction across natures:

  • God and humans

  • Angels and humans

  • Humans and brute animals.