Role kinds

Some kinds of roles succeed in imposing norms on those in roles of that kind. For instance, friends should help each other because they are friends, and spouses should be faithful to each other because they are spouses. Other kinds of roles do not succeed in imposing any norms. If I hurt you, I cannot rationalize my action by saying that I did that because I was your enemy. That may be an explanation, but being an enemy imposed no norm on me. At most it imposes merely apparent norms (before Socrates, the Greeks thought you should treat your enemies badly--they were wrong). What makes the difference? What gives some roles the genuine normative power they have?

Here is a natural law hypothesis I am attracted to. There are some natural human roles, and they each impose norms on those who fill them. I've in effect argued that spouse is one of those roles. Some other plausible examples: parent, child, friend, authority, subject of authority. Our nature specifies a potentiality to such roles, and gives them their normative power. There is a classificatory hierarchy between the natural roles. Spouse is a sub-role of friend, for instance. (An interesting and controversial question: are husband and wife natural sub-roles of spouse? If so, there will be further norms to being a husband and to being a wife.) And then all roles that have normative power are either natural roles or sub-roles specialized on the scaffolding of one or more natural roles, inheriting all their normative power from the natural roles. For instance, the roles of president and monarch are socially constructed sub-roles of authority, and their normative power over those in the role entirely comes from the role of authority. What about the normative power of the role over other people? That I suspect actually comes from the other people having the roles of citizen or subject, respectively, which are both sub-roles of the natural role of subject of authority. Being a monarch spouse is, on the other hand, a constructed sub-role of two different roles: monarch and spouse. (Think here of object oriented languages with multiple inheritance.) And while spouse is natural monarch is a constructed sub-role of authority.

In the above, I was talking about roles considered as general types, like spouse or parent or monarch. These types have tokens: spouse of Bill, parent of Joey, monarch of Canada. One can think of these token roles as also sub-roles, specializations of the role. An interesting question: is there any token role that is natural? That would be a token role whose normative force comes not just from the type role that it is a token of, as when being a spouse of Bill gets its normative force from being a spouse of someone-in-general. I think one plausible case is when the role is with respect to God. Being a subject of God may be a natural role. (What about friend of God? Perhaps that, too, but there the relevant nature may be a grace-nature.)

We can ask some interesting structural questions. Is there a highest level natural role? Perhaps being human or being a person. I have in the past speculated that it might be friend, but I now think that's mistaken, because friend roles are tied to particular individuals, while some of the other roles are not: an authority need not change qua authority when subjects are replaced by others (through conception, death and migration), but a friend does change in respect of friendship when friends come and go.

And how is this all tied to love? I suspect like this. Love isn't itself a role. But the roles determine which form one's love should take. Maybe in fact that is how they exercise their normative force, and that is what explains why there can't be a natural role such as enemy.

Cerebrums, animalism and teleology

Suppose you are essentially an animal. If your cerebrum were transplanted into a vat and the rest of your body—including, of course, the brain-stem—were to maintain circulation, nutritive functioning, muscle tone and so on, where would you go? Would you go with the cerebrum in the vat, with the rest of the body, or would you be just plain dead?

Here’s a line of thought. There are more primitive animals that don’t have a cerebrum, but still have a circulatory system, a nutritive system, etc. Thinking about these animals makes on think that the survival of an animal has to do with maintenance of the lower level homeostatic functions. So, we go with the parts of the body responsible for such things.

But an Aristotelian animalist can resist the analogy to primitive animals on teleological grounds. For instance, our circulatory system’s physical resemblance to the circulatory systems of primitive animals misses out on a crucial metaphysical difference: our circulatory system has the support of the life of the mind as its central telos, and it supports the life of the mind by supporting the cerebrum. The teleological structure of primitive animals and human animals is different: functions that are close to the teleological center of the life of a primitive animal are further from the teleological center of human life. It may be the case that when an animal is divided, it goes with the parts that are teleologically more central. If so, then in the initial thought experiment, you would go with the cerebrum.

[By the way, this post represents a new workflow. I am using John MacFarlane pandoc, writing the post as a text file, and then running a script that does pandoc -S filename | iconv -f utf-8 -t utf-16le | clip and pasting it in. This should make math less painful to type.]

Pleasure and pain are discrete

Alice, Bob and Chuck each come into existence at 1 o'clock.

• Alice lives for one hour. She feels continuous and unchanging morally innocent pleasure during that hour with no pain.
• Bob lives for two hours. During the first hour his experiences are exactly like Alice's; during the second hour, these experiences re-run.
• Chuck has the same internal stream of subjective experiences as Bob, but is accelerated by a factor of two relative to external time, so he lives only for one hour.

Now, let's add some axioms about hedonic value:

1. If x and y have the same internal stream of subjective experiences, though perhaps at different external rates, their lives are hedonically equally.
2. If x and y live during the same period of external time, and at each moment experience the same pleasure or pain, their lives are hedonically equal.
3. If x and y live hedonically equal lives, and y and z live hedonically equal lives, then x and z live hedonically equal lives (hedonic equality is transitive).
4. The same pleasant experience lived twice is hedonically better than when lived once.

Note that (4) needs to be carefully understood. Of course, a longer stint of a pleasant experience can get boring. But if one experiences boredom, that's not the same experience then.

Now, we have a contradiction. For by (1), Chuck and Bob's lives are hedonically equal. But by (2), Alice's and Chuck's lives are hedonically equal. Here's why. Take any time t between 1 and 2 o'clock, i.e., any time during the lives of Alice and Chuck. Because the pleasure is constant during that hour-long period, Alice's pleasure at t is the same as her pleasure at (say) 1:30. And for the same reason, Chuck's pleasure at t is the same as his pleasure at (say) 1:15. But Chuck's state at 1:15 is the same as Bob's state at 1:30, since Chuck lives the same life that Bob does, but twice as fast. And Bob's state at 1:30 is the same as Alice's state at 1:30. So, Chuck's pleasure at t is equal to Alice's pleasure at t, and hence by (2) Alice's and Chuck's lives are hedonically equal. Hence, by transitivity, Alice's and Bob's lives are hedonically equal. And this contradicts (4).

Assuming our hedonic axioms (1)-(4) are correct, the story about Alice, Bob and Chuck leads to a contradiction. So what's wrong with the story? I think it's the assumption that it's possible for pleasure to be continuous. Instead, I submit, temporally extended pleasure has to be discrete, made up of a finite number of pieces of pleasure. These pieces might be instantaneous or temporally undivided but extended. And of course the argument can be run with pain in place of pleasure as well.

Baylor's PhD program in philosophy

We've just finished a promotional video for our graduate program. Enjoy!

Marriage is a natural kind

I just posted the slides from my SCP Midwest talk on marriage as a natural kind.

Pursuit of victory

I wonder if it's rational to choose to play a game precisely in order to win. Winning is a practice-internal goal. Does this goal make sense when one isn't already decided on playing the game? Of course people do choose to play in order to win. But I wonder if those aren't cases where the victory is a means to something else, like money, fame or satisfaction?

A Copenhagen interpretation of classical mechanics

One can always take an indeterministic theory and turn it deterministic in some way or other while preserving empirical predictions. Bohmian mechanics is an example of doing that with quantum mechanics. It's mildly interesting that one can go the other way: take a deterministic theory and turn it indeterministic. I'm going to sketch how to do that.

Suppose we have classical physics with phase space S and a time evolution operator T_t. If the theory is formulated in terms of a constant finite number n of particles, then S will be a 6n-dimensional vector space (three position and three momentum variables for each particle). The time evolution operator takes a point in phrase space and says where the system will be after time t elapses if it starts at that point. I will assume that there is a beginning to time at time zero. The normal story then is that physical reality is modeled by a trajectory function s from times to points of S, such that T_t(s(u))=s(u+t).

Our indeterministic theory will instead say that physical reality is modeled by a (continuous) sequence of probability measures P_t on the phase space S for times t≥0. These probability measures should be thought of as something like a physical field, akin to the wavefunction of quantum mechanics--they represent physical reality, and not just our state of knowledge of it. Mirroring the consciousness-causes-collapse version of the Copenhagen interpretation of quantum mechanics, we now say this. If from time u (exclusive) to time t+u (inclusive) no observation of the system was made, then P_t+u(A)=P_t(T_u⁻¹[A]). I.e., the probability measure is just given by tracking forward by the time-evolution operator in that case.

On the other hand, suppose that at time t an observation is made. Assume that observations are binary, and correspond to measurable subsets of phase space. Intuitively, when we observe we are checking if reality is in some region A of phase space. (It's easy to generalize this to observations having any countable number of possible outcomes.) Suppose P_t^* is the value that P_t would have had there been no observation at t by the no-observation evolution rule. Then I suppose that with objective chance P_t^*(A) we observe A and with objective chance 1−P_t^*(A) we observe not-A, with the further supposition that if one of these numbers is zero, the corresponding observation physically cannot happen. Then the probability measure P_t equals the conditionalization of P_t^* on the observation that does in fact occur. In other words, if we observe A, then P_t(B)=P_t^*(B|A) and otherwise P_t(B)=P_t^*(B|not-A). And then the deterministic evolution continues as before until the next observation.

As far as I can see, this story generates the same empirical predictions as the original deterministic classical story. Also note that while in this story, collapse was triggered by observation, presumably one can also come up with stories on which collapse is triggered by some other kind of physical process.

So what? Well, here's one thought. Free will is (I and others have argued) incompatible with determinism. One thought experiment that people have raised is this. If you think free will incompatible with determinism, and suddenly the best physics turned out to deterministic, what would you do? Would you deny free will? Or would you become a compatibilist? Well, the above example shows that there is a third option: give an indeterministic but empirically adequate reinterpretation of the physics. (Well, to be honest, this might not entirely solve the problem. For it might be, depending on how the details work out, that past observations narrow down the options for brain states so much that they become deterministic. But at least there would be hope that one wouldn't need to give up on libertarianism.)

The above way of making free will compatible with physical determinism is functionally similar to Kant's idea that our free choices affect the initial conditions of the universe, but without the freaky backwards-like (not exactly backwards, since the noumenal isn't in time) causation.

Here's another thought. Any indeterministic theory can be reinterpreted as a deterministic multiverse theory with traveling minds, while maintaining empirical adequacy. The multiverse traveling minds theory allows for causal closure of a deterministic physics together with robust alternate-possibilities freedom. Combining the two reinterpretations, we could in principle start with a deterministic physics, then reinterpret it in a Copenhagen way, and then impose on top of that the traveling minds interpretation, thereby gaining an empirical equivalent theory with robust alternate-possibilities freedom and no mental-to-physical causation. I bet a lot of people thought this can't be done.

More on competitive sports and other games

I wonder how psychologically feasible it would be to generally engage in competitive sports or other games with one's intention being that the one's competitor win against as strong an opposition as possible. This is not all that difficult to achieve when competing with one's child: one may want the child to beat one, and to beat one when one is playing at one's best. The psychological difficulty is that one's intention that one's competitor win may well weaken one's playing. If one could play excellently with such an intention, wouldn't it be a laudable way to play?

To be as strong an opposition as possible, it would help to have the intention to win. I wonder if it would be possible to have two clearly logically incompatible ends at the same time: (a) that my competitor win against as strong an opposition as possible and (b) that I win. This isn't as problematic as intending p and not p at the same time. Maybe you can't do that, because any action that furthers not p impedes p. But actions that promote (b) can promote (a) by making the opposition as strong as possible, and vice versa. So it might be that incompatible ends like (a) and (b) can be both held together, though it is uncomfortable to do so.

Competitive sports

We think of competitive team sports as involving two groups of people, with cooperation within each group but competition between the groups. However, there is a better picture. We can think of the two teams as part of a larger cooperating group, which is subdivided into two subgroups. The two subgroups cooperate with each other for the goods that the sport achieves. The means by which the two subgroups cooperate for the goods of the sport is competition, much as when lawyers for two sides (normatively speaking) cooperate for the sake of truth and justice by competitively each giving the best rendition of one side of the case. Central among the goods in the sport case will presumably be athletic excellence (I am grateful to Dan Johnson for pointing out this good to me), but there will be other goods such as health, fun, entertainment of others, etc.

Of course, something similar happens in competitive individual sports: the individuals cooperate with each other in order that they achieve the goods of the sport.

From this high vantage point, all competitive sports--as well as other games--are a cooperative human activity. I think one can feel this particularly well when one wants to play a sport or another kind of game and an opponent becomes only available after some difficulty. There is a gratitude one has to the opponent for making the game possible.

Yet, paradoxically, the cooperation can involve each pursuing an incompatible end: their own victory. But ideally each pursues that end because the pursuit (but not necessarily achievement) of that end is what makes the joint goods possible.

The meaning of life

The meaning of life is a contingent constant of the story about the same as the particles.

The state of the basic sense of the following options are responsible for an infinite sense of evil is a member of the probability of the fact that the above argument for the sentence for the probability that what is a table of the second sense of the above theory of a lot of the particles in the way that the problem of the subject of the probability of the responsibility is probably the second problem of moral states of a sentence is that it is a non-empty actions of P(A)=P(A) is a property of an argument for the virtue of a controversial and more about the conditional to the constraints of the best ordinary and more probability of the content of the conclusion of the case of the healthy of a positive sex in the state in A is the interesting to the action to the property of the person who does not allow that the answer to include the heart with the ...

What's that? Well, I had some fun running 6mb worth of my blog posts through this character-based recurrent neural network trainer which is based on this. I then followed an online suggestion and prompted it with "The meaning of life is " and let it generate text mimicking my posts (temperature = 0.4). I find the first sentence kind of interesting. What's cool is that the system starts out without any knowledge of language. It works character by character, eventually figuring out how to make sequences of characters that look like words, how to punctuate, capitalize, insert HTML formatting, etc. The grammar often leaves something to be desired, and it has a tendency to get stuck generating giant sentences with lots of clauses (guess whose writing taught it that?).

Some other quotations, with the temperature value listed (the higher the temperature, the more randomness in the output):

If this is something like the same as the proposition that p is true that the moral reason to be sufficient to do any necessity of the probability that it is the problem of the same probability that the principle and sex and the probability of the same reason to do the reason to be a difference between the problem of the probability of the problem of the same explanation of a sentence is a concept of the probability to the problem of the probability that the probability of a proposition that the contradiction is that the same thing that is the same thing to say that the probability that the proposition that the concept of a property of the proposition p to the probability that it is true and the argument is the argument is that the same thing in the above existence is that the other conditions are not the probability of the same probability of the state of the same as the probability that the same sentence that are not true and the same token, we can also be a stronger than an object of B is a proposition that the probability of the same time [0.3]

The problems think that a promise of the predicates are not human beings and interesting assumption that the presentist stands the correct conscience can be in Case, the strangers are more conversion of the world, not true, for short where P(B_K(x₁ be the only simple condition to be a physical probability of the like right. </li></ol> where the only quite a very organism is measurable acts should insisted to exist. [0.7]

A relations are 1/2)=(√X₂)=/titiely, at all view is that when I said because the person is the substance "p.

Whether I fill out that by the following decided the borning and God wrong, which can presuppose on indeed more pothing" was clearly open the explanan of ohn suggests that they are surprising how we learned lacks effort, if Q; <0) (A>), where this society (e.g., generally distinguish science is said to see that they should say the sake that generates it that each would freely-valiel was intrinsically basically good response holding worlds x as one stating chasted to the following problems, when as the chosen biological speaking only: [1.0]

Quantum mechanics and points in space

There are two different pictures of space: space is made of points or space is made of regions. I will argue that quantum mechanics naturally suggests the latter view.

Consider a single-particle system. Its quantum state can be representated by a wavefunction Ψ, which is a square-integrable complex-value function. This seems to go nicely along with the idea that there is such a thing as space, and each point in space has the property of having such-and-such a value of the wavefunction Ψ. (And in a multi-particle system, tuples of points are related to a value.) But that misses a subtlety. While a square-integrable function Ψ represents a quantum state, any two square-integrable functions Ψ₁ and Ψ₂ that agree outside of a set of measure zero are taken to represent the same quantum state. The measure of a (set consisting of a) single point is going to be zero. So there is no a physical fact of the matter as to the value of the wavefunction is a particular point.

This seems to me to cohere a little better with a view on which space is built up out of extended regions rather than points. While there will be no fact of the matter as to what "the" value of the wavefunction is at a point x, for any extended (at least in the sense of being nonzero-measure) but bounded region R of space, say a ball or cube, there will be a fact about what the average value of the wavefunction over R is (functions that are square-integrable are locally integrable). Moreover, one can recover the value of the quantum state from the values of such averages over, say, all balls. (If space is potentially subdivisible but not actually subdivided, some of these balls will be potential regions of space, and the average of the wavefunction over such a ball may be a dispositional property--the average it would have if space were divided so as to have that ball as a region.)

This is not a knock-down argument against views on which space is made of points. One could say that space is made of points but deny that quantum states have values at single points. Or one could say that wavefunctions that differ on a set of zero measure represent different in principle empirically indistinguishable quantum states. The latter is, I think, unattractive. We should avoid positing an infinite number of empirically unobservable degrees of freedom in physics.

The problem of sleep

Consider this natural law argument:

1. Rational functioning is a basic good.
2. One may never intentionally act against a basic good.
3. In intending to fall asleep, one intends to stop rational function (i.e., thought).
4. Therefore, it is wrong to intend to fall asleep.

One could, I suppose, embrace the conclusion and say something like this: At night, we foresee but do not intend sleep. At night we lie down in bed, accepting but not intending the evil of sleep, much like a person who foresees death might lie down to face death in comfort. But this just won't explain all our practices. First of all, we often lie down and close our eyes to sleep hours before we would expect sleep to overtake us were we to stay up. It seems clear that we lie down and close our eyes in order to accelerate the sleep process. And sometimes, with good reason, we may take medication to help us fall asleep. To condemn such practices would be highly counterintuitive. In fact, one might take the anti-sleep argument as a reductio ad absurdum of natural law reasoning, which appears to be committed to premises (1) and (2).

It is tempting to dismiss the argument by saying that we need sleep to be rational. But that doesn't touch the argument. There are circumstances where the only way to survive is by killing an innocent person--but the end does not justify such a means. Likewise, if (1)-(3) are true, even if the only way we can maintain rational functioning is by sleeping, such a means is impermissible.

Aquinas discusses the question whether sex can be permissible in light of the fact that sex involves such an "excess of pleasure" that "it is incompatible with the act of understanding" (he attributes the latter claim to Aristotle). His answer is that sex can be done in accordance with reason, and what is done in accordance with reason is not sinful. He then says: "For it is not contrary to virtue, if the act of reason be sometimes interrupted for something that is done in accordance with reason, else it would be against virtue for a person to set himself to sleep." Unfortunately, Aquinas doesn't tell us which premise of the anti-sleep argument is false. It is not even clear that he has the same argument clearly in mind. In the case of sex, after all, the hampering of rational function looks like a side-effect (it's interesting that Aquinas doesn't just use Double Effect here) which need not be intended, while in sleep the lack of thought seems central.

For years I've struggled with the anti-sleep argument (but lost no sleep over it). I have two responses. Both of them leave (1) and (2) intact, but query (3). The first response is that in intending to fall asleep one intends to put off one's rational functioning rather than to stop it. A philosopher who leaves his office to walk around the beautiful campus intends that his rational functioning occur outdoors rather than in his office. Likewise, one might intend that one's rational functioning occur in the morning rather than late at night. And the reasons can be similar. The rational functioning outdoors or in the morning is likely to be fresher than in the office or late at night. The analogy here is strongest if one accepts a B-theory of time (and in fact, it may be an argument for a B-theory of time that it makes it easier to justify sleep).

The second answer is that sleep is not actually a cessation of rational function. It is very plausible that unconscious mental processes occur during asleep (it is clear that brain processes do!)--and an important part of sleep involves consciousness anyway. Sleep seems to be an important part of our rational functioning rather than an interruption. Clearly it is not an action against a basic good to switch from one kind of rational functioning to another, say turning one's mind from practical to abstract matters.

A difficulty with the second answer is that some people may not know that rational function continues in sleep. Yet surely such ignorance doesn't make it wrong to fall sleep. I agree. But we can also say that such a person may not intend the cessation of rational functioning. She may simply intend sleep, a particular natural human organic process. And if I am right that sleep is not constituted by a cessation of rational function, then we cannot even say that she "implicitly" acts against rationality or anything like that.

So, the anti-sleep argument fails, and natural lawyers can sleep with a sound conscience.

More on comparing infinities

Some people don't want to say that there are just as many even natural numbers as natural numbers. But suppose that you and I will spend eternity singing numbers in harmony. You will sing every natural number in sequence: 1, 2, 3, ..., with a long pause for applause in between. And while you sing n, I will sing 2n. We will vary the speed of our singing to ensure that we take equal amounts of time. Clearly:

1. The number of natural numbers = the number of your performances.
2. The number of your performances = the number of my performances.
3. The number of my performances = the number of even natural numbers.
4. So, the number of natural numbers = the number of even natural numbers.

Premises (1)-(3) are obviously true, and I don't understand what "the same number" relation could mean if it's not transitive.

Mathematics without proof

My 11-year-old complained to me that his mathematics teacher tells them things without proof. This made me realize that the sorts of things that he mentioned as given without proof--say, the distributive law and maybe some facts about prime factorization (maybe the Fundamental Theorem of Arithmetic? I can't remember)--were things that somehow no one ever showed me a proof of, either, despite getting an undergraduate degree in mathematics and then a PhD. So I can't just say: "Hold on, one day they will give you the proof of this."

Does the size of an organism matter morally?

One might with pull a small plant from one's garden with little thought. But one wouldn't do that to a full grown tree. Of course it's harder to pull out a tree, but that doesn't seem to be all that's going on. The tree seems more significant.

Part of that is that the tree has been growing for a longer time. Temporal size definitely seems to matter. We would think a lot harder about cutting down a tree that hundreds of years old rather than one that's five years old. (Interestingly, we tend to have the opposite judgment in the case of people: it is perfectly understandable when an older person lays down their life for a child. Maybe this is because people have an irreplaceability that plants do not.)

But what about pure spatial size? Does that matter? I once thought about this case. We kill insects for minor reasons. But would we do that if the insects were our size? I thought at the time that we would have more hesitation to kill the large insects for minor reasons (we might not hesitate on self defense), but that this was an irrational bias.

But I now think there might be a justification to thinking of spatially larger organisms as having more value. The larger organisms have more cells, and that makes for a complex system, just like a castle made of ten thousand Legos is more complex, other things being equal, than one made of a thousand.

In the case of people, I guess we will have a duty of justice to bracket reasons arising from the number of cells. So we shouldn't save the fatter person just because he has more cells.

But what about dogs, say. Is it really the case that if a Chihuahua and a Great Dane are drowning, other things being equal we should try to save the Great Dane?

Maybe the differences due to the number of cells are on a logarithmic scale, and hence are only significant given an order of magnitude difference? But a Great Dane is an order of magnitude heavier than a Chihuahua, and so I'd guess it has an order of magnitude more cells.

Maybe the moral difference requires several orders of magnitude? Or maybe it runs on a loglog scale?

Or maybe I'm barking up the wrong tree and spatial size doesn't matter morally at all.

If size doesn't matter morally at all, we have a nice argument that the parts of a substance are never substances. For if the parts of a substance are ever substances, the cells of a multicellular organism will surely qualify. But if the cells are substances, then they are living substances. But surely an order of magnitude difference in the number of living substances destroyed makes a moral difference.

How a blog radically changes the world forever

On any given day, one in 30,000 Americans will conceive a child. So, roughly, there is a one in 60,000 chance that someone you (I'll just assume you're in the US for convenience) are interacting with will be conceiving a child later that day. Any interaction you have with a person who will be conceiving a child later that day is likely to affect the exact time of conception, and it seems very likely that varying the time of conception will vary the genetic identity of the child conceived. However, there might be some "resetting" mechanisms throughout the day, mediated by the way our days are governed by times of meetings and so on, and so not every interaction will change the time of conception. So let's say that one in four interactions with someone who will be conceiving a child later that day will vary who will be conceived (or whether anyone will be). That means that one in 240,000 interactions we have with people affects who will be conceived on that day.

Once one has affected who will be conceived that day, as long as the human race survives long enough, eventually just about everyone's genetic identity will be affected by one's actions. For, obviously, that conceived individual's own children's genetic identity will be affected. But that individual will interact with others, affecting the romantic decisions or at least times of conception of others, for instance. It seems quite safe to suppose that that individual's interactions over a lifetime will affect the genetic identity of ten individuals. Given an interconnected world like we have, it seems reasonable to suppose that in 20 generations, almost everyone's genetic identity will be affected (maybe there will be some isolated communities that won't be affected--but I think this is unlikely).

Counting a generation as 30 years, a blog that has 240,000 hits per year, running over a single year, will affect the genetic identity of almost everyone in 600 years. And this, in turn, will affect all vastly morally significant things where individuals matter: the starting of wars, the inventing of medical treatments, etc.

It is very likely, then, that the long-term effects of such a blog in terms of reshaping the world population vastly exceed whatever good and ill the blog does to the readers in the way proper to blogs. After all, one more or one less warmongering dictator and we have millions people killed or not killed. So the kinds of considerations one brings to bear on the question whether to have a blog--how will it affect my readers, etc.--are swamped by the real variation in consequences. (Assuming Judgment Day is still hundreds of years away.)

Not to be paralyzed in our actions, we need to bracket such great unknowns, even though we know they are there and that they matter more than the knowns on the basis of which we make our decisions!

A defense of the five minute hypothesis (given a certain false assumption)

Plausibly:

1. If the universe came into existence either for no cause at all or randomly, it is a priori more likely that it came into existence in a higher-entropy state rather than a lower-entropy one.
2. If the universe came into existence fully-formed five minutes ago ("five minute hypothesis"), it came into existence in a higher-entropy state than if it came into existence 13.8 billion years ago ("scientific orthodoxy").
3. So, if the universe came into existence either for no cause at all or randomly, the five minute hypothesis would be a priori more likely than scientific orthodoxy.
4. But there is no a posteriori evidence for scientific orthodoxy over the five minute hypothesis.
5. So, if we think the universe came into existence either for no cause at all or randomly, it is not rationally consistent for us to believe scientific orthodoxy over the five minute hypothesis.

I personally think premise (1) is dubious: I doubt there are meaningful probabilities for the universe to come into existence for no cause at all or randomly. But if there are no meaningful probabilities, then it is not a priori more likely that it come into existence as scientific orthodoxy claims, and the rest of the argument should continue to go through.

Of course, I think we should believe scientific orthodoxy over the five minute hypothesis, so we should reject the no-cause and randomness hypotheses in (1).

An amusing rhetorical way to present the argument is that if the universe came into existence for no cause at all or randomly, we shouldn't prefer scientific orthodoxy to certain young earth views.

A damselfly

Some of the cool animals in our local zoo are not part of the zoo. But the space they occupy is part of the zoo.

Artifacts like zoos thus have space as part of them. But what is this "space"? The zoo's space orbits the earth's axis once per 24 hours. It's a movable space. It's something like what I've called the internal space of a substance (except that a zoo isn't a substance).

Are the laws of nature first order?

I think it's a pretty common to think that the laws of nature should be formulated in a first-order language. But I think there is some reason to think this might not be true. We want to formulate the laws of nature briefly and elegantly. In a previous post, I suggested that this might require a sequence of stipulations. For instance, we might define momentum as the product of mass and acceleration, and then use the concept of momentum over and over in our laws. If each time we referred to the momentum of an object a we had to put something like "m(a)⋅dx(a)/dt", our formulation of the laws wouldn't have the brevity and elegance we want. It is much better to stipulate the momentum p(a) of a as "m(a)⋅dx(a)/dt" once, and then just use p(x) each time.

But our best-developed logical formalism for capturing such stipulations is the λ-calculus. So our fundamental laws might be something like:

• ∀p(p=λa(m(a)⋅dx(a)/dt)→(L₁(p)&...&L_n(p)))

instead of being a rather longer expression which contains a conjunction of n things in each of which "m(a)⋅dx(a)/dt" occurs at least once. But the λ-calculus is a second-order language. In fact, it seems very plausible that encoding stipulation is always going to use a second-order tool, since stipulation basically specifies a rewrite rule for a subsequent sentence.

So what if the language of science is second order? Well, two things happen. First, Leon Porter's argument against naturalism fails, since it assumes the language of science to be first-order. Second, I have the intuition that this line of thought supports theism to some degree, though I can't quite justify it. I think the idea is that second-order stuff is akin to metalinguistic stuff, and we would expect the origins of this sort of stuff to be an agent.

Parthood and composition

I had a very long conversation yesterday with one of our graduate students regarding Weak Supplementation: he finds Weak Supplementation plausible while I find it implausible. Anyway, I felt this was one of those really good philosophical conversations where you get to something at the root of the issue. So, here's where I felt we got to: It is crucial to how one thinks about parthood whether one sees a close connection between parthood and composition. In particular, it is crucial whether one accepts this thesis:

1. Necessarily, if an object has proper parts, it is composed of them.

There are, I think, two kinds of reasons for accepting (1). You could think that proper parthood is defined by composition, say because you think:

2. x is a proper part of y if and only if (and if so, because) there are zs that compose y such that x is one of the zs.

Or you might think that proper parthood is more fundamental than composition and think that there is a way of defining composition in terms of proper parthood, e.g.:

3. The zs compose y if and only if (and if so, because) every one of the zs is a part of y and every part of y overlaps at least one of the zs.

I think Weak Supplementation is pretty plausible if you accept (1). I also think that acceptance of (1) neatly goes along with the kind of bottom-up thinking that we get in van Inwagen's "special composition question": When do things compose a thing?

On the other hand, I am quite sceptical of (1). I think of parts as derivative from the whole. Instead of wanting to know when things compose a thing, I want to know when a thing has proper parts. One way to put the question, in the case of material substances, is this: Given a thing and a region of space (or spacetime), under what circumstances is there a part that exactly fills that region? And a tempting answer, somewhat reminiscent of van Inwagen's "life" answer, is that this happens when the thing has a function fulfilled precisely in that region.

I then see no reason why the whole would have to be composed of its proper parts. I think I can imagine a stories like the following being true. There is something that looks like a normal human hand. It has five fingers as proper parts. And that's all. There is no such part as the fingerless palm, since the fingerless palm just doesn't have the right kind of functional unity, and I shall suppose that in this scenario there are no cells or particles. The hand then has six parts: the hand itself (an improper part) and the five fingers. Since the hand isn't composed of the fingers, the hand isn't composed of its proper parts. More interestingly, perhaps, I might be persuaded to think of the tropes of an immaterial substance as its proper parts, but I wouldn't want to say that the substance is composed of its tropes--that would be a bundle theory--nor would I want to say that the substance is composed of its tropes and a bare particular. So, then, an immaterial substance might have proper parts, but it wouldn't be composed of them.

Evolutionary knowledge formation

Suppose a very large group of people who form their scientific beliefs at random. And then aliens kill everybody whose scientific beliefs aren't largely true. Surely the survivors don't know their scientific beliefs to be true, assuming they don't know about the aliens killing off those who were wrong. After all, they didn't know their beliefs prior to the massacre, and the massacre of the erring didn't give them knowledge. (This case may be a problem for certain kinds of reliabilism.)

Suppose now that we came to believe certain truths--whether mathematical or empirical or moral--"for evolutionary reasons". In other words, those who had these beliefs survived and reproduced, and those who didn't didn't. Let's even suppose that there is a tight connection between the truth of the beliefs and their fitness value. Nonetheless I am not sure that this story is sufficiently different from the story about aliens to make the beliefs into knowledge. The crucial difference, I guess, is that the story about the aliens is a single-generation story, while the evolutionary story is a multi-generation story. But I am not sure that matters at all. Suppose, for instance, that we modify the alien story to add a new generation who takes their scientific beliefs from the survivors of the first generation. Surely if you get your scientific beliefs from people who don't know, and no additional evidence is injected, you don't know either.

Intuitive moral knowledge

People intuitively know that stealing is wrong. Maybe stealing is wrong because it violates the social institution of property which is reasonably and appropriately instituted by each community. Maybe stealing is a violation of the natural relation that an agent has to an object upon mixing her labor with it. Maybe stealing violates a divine command. But people's intuitive knowledge that stealing is wrong does not come from their knowledge of such reasons for the wrongness of stealing. So how is it knowledge?

It's not like when the child knows Pythagoras' Theorem to be true but can't prove it. For she knows the theorem to be true because she gets her belief from the testimony of other people who can prove it. But that's not how the knowledge that stealing is wrong works. People can intuitively know that stealing is wrong without their belief having come directly or indirectly from some brilliant philosopher who came up with a good argument for its wrongness.

Perhaps there is some evolutionary story. Communities where there was a widespread belief that stealing is wrong survived and reproduced while those without the belief perished, and there was no knowledge at all the back of the belief formation. However, perhaps, it came to be knowledge, because this evolutionary process was sensitive to moral truth. However, it is dubious that this evolutionary process was sensitive to moral truth as such. It was sensitive to the non-moral needs of the community, and sometimes this led to moral truth and sometimes to moral falsehood (as, for instance, when it led to the conviction that it is right to enslave members of other communities). So if this is the story where the belief came from, it's not a story about knowledge. At best, the intuitive conviction that stealing is wrong, on this story, is a justified true belief, but it's Gettiered.

This, I think, is an interesting puzzle. There is, presumably, a very good reason why stealing is wrong, but the intuitions that we have do not seem to have the right connection to that reason.

Unless, of course, we did ultimately get the knowledge from someone who has a very good argument for the wrongness of stealing. As I noted, it is very implausible that we got it from a human being who had such an argument. But maybe we got it from a Creator who did.

Presentists can't reduce time

Historically, some philosophers have attempted to reduce time to something else, say change or the causal nexus. It's interesting to note that this cannot be done if presentism is true. For to reduce time to something else requires giving a nontemporal account of the "wholly earlier" relation between events. But if presentism is true, there never exist two events one of which is wholly earlier than the other. For if one of them is present, the other is not. And only present events exist given presentism.

Now, of course, the presentist has a way of talking about non-present events. She can, for instance, use the temporal modal operators WAS and WILL. However, she cannot do so in the course of reducing time to something else, since WAS and WILL already presuppose time. The presentist must take time to be primitive, thus.

In particular, this means that Aristotle, who attempted to reduce time to change, cannot be consistent if he is a presentist.

Two ways of acting wrongly through ignorance

Case 1: Petr shoots and kills Anna while hiking, thinking that Anna is a dangerous bear (she was wearing a fur coat).

Case 2: Ivan the NKVD agent shoots and kills Natasha for speaking out against the state, thinking it is right to kill dissidents.

Both Petr and Ivan acted in ignorance. If their ignorance was inculpable, then they might well be inculpable for their respective killings. However, even if neither is culpable, it seems that Ivan is now a murderer while Petr is an accidental killer.

Wherein lies the difference? Both killed someone who shouldn't have been killed. Both did something that would have been a case of culpable murder if they killed while knowing all the relevant facts.

One difference is this. Petr acts under this description: "Killing something that appears to be a dangerous bear" or maybe even: "Killing a dangerous bear." Ivan acts under this description: "Killing someone who spoke out against the state." The description that Petr acts on is a description that it is permissible to act on (regardless of the version we choose). The description that Ivan acts on is a description that it is always impermissible to act on. Both Petr and Ivan have deficient knowledge. But Petr's intention is acceptable while Ivan's is corrupt.

I am somewhat inclined to go even further. Of the two, Petr did nothing morally wrong. Ivan, however, acted wrongly, albeit perhaps inculpably. This works best if we take Petr's intention to be "Kill a dangerous bear." In that case, we can say that Petr's action was a permissible attempt to kill a dangerous bear. But Petr's action was unsuccessful. On the other hand, it seems that Ivan's intention to kill a dissident made his action impermissible but successful.

Maybe. But what if Ivan's intention was this: "Permissibly kill Natasha for being a dissident"? In that case, Ivan failed, too, since his killing wasn't permissible. But now it seems we have a close parallel between Petr and Ivan. Petr failed to fulfill his intention to kill a dangerous bear. Ivan failed to fulfill his intention to permissibly kill Natasha. Can't we, in fact, say that just Petr accidentally killed a human, Ivan accidentally killed someone that it was impermissible to kill? So can we really say that Ivan was a murderer but Petr wasn't?

I don't know. There is, nonetheless, this difference. The intention "Kill a dangerous bear" is one that it is possible to succeed at. The intention "Permissibly kill Natasha for being a dissident" is one that it is not possible to succeed at. (Admittedly, there may be cases where it's permissible to kill a dissident. But even in those cases, it's not permissible to kill the dissident for being a dissident. Rather, the cases of permissibility are ones where there is some reason for the killing over and beyond the dissident's dissidence.) But while this is a significant difference, it doesn't seem to be a morally significant difference. After all, just as "Permissibly kill Natasha for being a dissident" is impossible to succeed at, so too "Find a counterexample to Fermat's Last Theorem" is impossible to succeed at. However, there is nothing morally wrong with someone who tries to find a counterexample to Fermat's Last Theorem in ignorance of the impossibility.

Contingent pure sets?

It is widely held that some sets exist contingently. The standard examples are sets that have contingent entities among their members (or the members of their members or ...), such as the singleton set of me or the set of all actual cows. I wonder if such examples exhaust the contingently existing sets. Could there be contingently existing pure sets, sets whose members all the way down are sets?

Well, they're not going to be sets whose existence can be proved from the axioms of set theory, if these axioms are necessary truths. But one interesting class of potential candidates could be sets defined in non-set-theoretic terms. For instance, suppose that in the actual world a coin is actually tossed an infinite number of times, with the occasions numbered 1,2,3,.... Then, if probability theory is to be applicable to the real world, we need to suppose something like the hypothesis that there is a set of all natural numbers corresponding to occasions when the coin landed heads. But would that set exist if the coin had landed in a radically different sequence or not been tossed at all? I used to assume that of course the answer to a question like that would be affirmative. I still think it's likely to be affirmative. But the matter is far from clear to me now.

The Axiom of Separation

The Axiom of Separation in Zermelo-Fraenkel (ZF) set theory implies that, roughly, for any set A and any unary predicate F(x), there is a subset B of all the x in A such that F(x). But only roughly. Technically the axiom only implies this for predicates definable in the language of set theory. We philosophers tend to forget that technical fact when we use set theory, much as we tend to blithely extend set theory to allow for ur-elements (elements that are not themselves sets). But if we are going to be realists about sets (which I am not saying we should be), we should have a real worry about what predicates can be legitimately used in the Axiom of Separation. (That's one of the lessons of this post.)

Consider the predicate L(x) which holds if and only if someone likes x. This is definitely not formulated in the language of set theory, so ZF set theory gives us no guarantee that there is, say, the set of all real numbers that satisfy L(x). If it turns out that there are only finitely many numbers that are liked, then we have no worries: for any real numbers x₁,...,x_n, there is a set that contains them and only them (this follows from the Axiom of Pairs plus the Axiom of Union). There will be other special cases where things work out, say when all but finitely many numbers are liked. But in general there is no guarantee from the axioms of ZF that there is a set of all liked numbers.

One might use this to try to get out of some paradoxes of infinity, by limiting the applicability of set theory. That's a strategy worth exploring further, but risky. For the above observations also severely limit the physical applicability of set theory. Suppose, for instance, that at each of infinitely many points in spacetime there is a well-defined temperature. It is usual then to suppose that there is a function T from the spacetime manifold to the real numbers such that T(z)=u if and only if the temperature at z (or, more precisely, at the point of spacetime corresponding to the point z in the mathematical manifold that models spacetime) is u. And we need there to be such a function T to be able to make physical predictions.

One solution is to extend the Axiom of Separation to include some or even all predicates not in the language of set theory. This is the solution that is typically implicitly used by philosophers. The Axiom of Separation has a lot of intuitive force thus extended, but we need to be careful since we know that the incoherent Axiom of Comprehension also had a lot of intuitive force.

A second option would be to have physics make set-theoretic claims. Thus, a theory positing that at each point of spacetime there is a temperature would also posit that there exists a corresponding function from the mathematical manifold that models spacetime to the real numbers. I think this would be quite an interesting option: it would mean that physics actually places constraints on what the universe of set theory is like.

Perhaps if we are not Platonists about sets, things are easier. But I am not sure. Things might just be murkier rather than easier.

Spirograph

My three-year-old has been enjoying the vintage Spirograph set we have. I thought it would be fun to make a javascript/svg version. Code is public domain (just View Source in your browser). What can we deduce about humankind from the fact that we appreciate patterns like these?

Stationary circle size:
Mobile circle size:
Fractional distance of pen from center of mobile circle:
Rotate around outside of stationary circle:

Might all infinities be the same size?

A lot of people find Cantor's discovery that there are different infinities paradoxical. To be honest, there are many counterintuitive things involving infinities, but this one doesn't strike me as particularly counterintuitive. Nonetheless, I want to explore the possibility that while Cantor's Theorem is of course true, it doesn't actually show that infinity comes in different sizes. Cantor's Theorem says that if A is a set, then there is no pairing (i.e., bijection) between the members of A and those of the power set PA. It follows that there are different (cardinal, but in an intuitive rather than mathematical sense) sizes of infinity given this Pairing Principle:

1. PP: Two sets A and B have the same size if and only if there is a pairing between them.

Given PP and Cantor's Theorem, if A is an infinite set, then A is a different size from PA. But of course PA is infinite if A is, so there are infinite sets of different size.

A number of people have disputed the sufficiency part of PP, since it gives rise to the counterintuitive consequence that the set of primes and the set of integers have the same size as you can pair them up. But you really shouldn't both complain that there are different infinities and that PP makes the primes and the integers have the same infinite size. I am going to leave the sufficiency of PP untouched, but suggest that the pairing condition might not be necessary for sameness of size, and I will offer an alternative. That alternative seems to leave open the possibility that all infinities are the same.

To think about this, start with this thought experiment. Imagine that there is a possible world w that has some but not all of the actual world's sets, but that it still has enough sets to satisfy the ZFC axioms just as (I shall suppose) the sets of the actual world do. The set membership relation in w will be the same as in the actual world in the sense that if A is a set that exists both w and the actual world, then A has exactly the same members in both worlds (and in particular, all the actual world members of A exist in w). Then here is something that might well happen. We have two sets A and B that exist both in the actual world and in w. In the actual world, there is a pairing f between A and B. A pairing is just a set of ordered pairs satisfying some additional constraints (the first element is always from A and the second is always from B, and each element of A occurs as the first element of exactly one pair, and each element of B occurs as the second element of exactly one pair). It might, then, be the case that although A and B exist in w, f does not--it exists in the actual world but not in the impoverished world w. It might even be the case that no pairing between A and B exists in the impoverished world. In that case, we have something very interesting: A and B satisfy the pairing condition in PP in the actual world but fail to satisfy it in w. If we are to satisfy the ZFC in w, this can only happen if both A and B are infinite.

Things might go even further. We might suppose that w only contains sets that are countable in the actual world. The mathematical (much less metaphysical!) possibility of such a scenario cannot be proved from ZFC if ZFC is consistent, but it follows from the Standard Model Hypothesis which a lot of set theorists find plausible. If w only contains sets that are actually countable, then any infinite sets in w will have a pairing in the actual world. There is, thus, an important sense in which from the broader point of view of the actual world, all infinite sets in w have the same size. But w is impoverished. There are pairings that exist in the actual world but don't exist in w, and so applying PP inside w will yield the conclusion that the infinite sets in w come in different sizes. However, intuitively, it still seems true to say that these sets in w are all the same size, but w just doesn't have enough pairings to see this.

Here's one way to argue for this interpretation of the hypothesis. Plausibly:

2. Pairing-Sufficiency: If there is a pairing between sets A and B, they are the same size.
3. Absolute-Size: If two sets are the same size in one possible world, they are the same size in any world in which they both exist.

Pairing-Sufficiency is one half of PP. Given Pairing-Sufficiency and Absolute-Size, if two sets have a pairing in any possible world, including the actual one, they have the same size in every world, including w. Thus, in w all the infinite sets in fact have the size, but you wouldn't know that if your tools were restricted to the pairings in w.

Thinking about the above scenario suggests a modification of PP to a Possible Pairing Principle:

4. PPP: Two sets A and B have the same size if and only if possibly there is a pairing between them.

Given that the members of a set cannot vary between possible worlds, I think PPP is at least as plausible as PP. Moreover, I think that if there are any cases (like my hypothesis that a world like w is possible) where PPP and PP come apart, we should side with PPP. Here's why. I think we go for PP as an abstraction from our general method of comparing the sizes of pluralities by pairing. (One imagines a pre-numerate people trading goats and spears in 1:1 ratio by lining up each goat with a spear.) But the natural abstraction from our general method is that if one could pair up the two sets, then and only then they are the same size. So PPP is the natural hypothesis. The only reason to go for PP is, I think, acceptance of PPP plus an additional hypothesis such as that what pairings there are doesn't vary between possible worlds.

If PPP (or just (2) and (3)) is true and my w hypothesis is a genuine metaphysical possibility, then it is metaphysically possible that all infinite sets are the same size--i.e., it could be that the actual world is relevantly like w. Furthermore, we clearly don't have relevant empirical evidence to the contrary. So, if all this works, it is epistemically possible that all infinite sets are of the same size. (Of course, the most controversial part of all this is the idea that what sets there are might differ--even in the case of pure sets--between worlds.)

But perhaps this won't satisfy the people who find size differences between infinities paradoxical. For they might find it paradoxical enough that there could be infinities of different sizes, something that was definitely a part of my story (remember that I started with two worlds, one in which there were differently sized infinities and an impoverished w with all infinities of the same size according to PPP).

I think I might be able to do something to satisfy them, while at the same avoiding the biggest problem with the above story, namely the assumption that what pure sets there are differs between worlds. Here's my trick. In the above, I assumed that pairings were all sets. But in line with the Platonism suffusing all of the above arguments, let's try something. Let's allow that there are pairings that aren't sets. Those pairings would be binary relations satisfying the right formal axioms. But here I mean "relations" in the Platonic philosopher's sense, not in the mathematician's sense where a relation is a set of ordered pairs. Let's suppose, further, that corresponding to any set of ordered pairs, there is a relation which relates all and only those pairs which are found in the set. In my earlier story, I made sense of the idea that two sets in w might not have a pairing in w and yet might be the same size by adverting to pairings that exist in another world (the actual one--and then at the end I flipped things around so that w was actual). But now we do the same thing by distinguishing between mathematical pairings--namely, sets of ordered pairs satisfying the right axioms--and metaphysical pairings--namely, Platonic binary relations satisfying analogous axioms. If my earlier story is coherent (I mean that to be a weaker condition than "metaphysically possible"), then so is this one: In w, there are infinite sets that do not have a mathematical pairing, but every pair of infinite sets possibly has a metaphysical pairing. But now this story doesn't rely on varying what pure sets exist between worlds. The story appears compatible with the idea that pure sets are the same in every world. But there are, nonetheless, metaphysical pairings that do not correspond to mathematical pairings, and PPP should be interpreted with respect to the metaphysical pairings, not just the mathematical ones. Note, too, that what metaphysical pairings hold between sets might differ between possible worlds, without any variation in sets. For some of the pairings may correspond to extrinsic relations. Here is an extrinsic relation that could turn out to be a metaphysical pairing, depending on what I actually was thinking yesterday: x is related to y if and only if x and y came up in one of my thoughts yesterday in this order.

We can now suppose that this story works in every possible world. Thus, assuming the coherence of the Standard Model Hypothesis, we have a mathematically coherent story--whether the metaphysics works is another question (the story is too Platonic for my taste, and I don't share the motivation anyway)--on which (a) all infinite sets are really of the same size (and hence of the same size as the natural numbers), (b) what pure sets there are does not differ between worlds, and (c) Cantor's Theorem and all the axioms of ZF or ZFC are true. If we were to go for such a view, we would want to distinguish between sets being of the size metaphysically speaking and their having the same mathematical cardinality. The latter relationship would be defined by a version of the PP with "pairings" restricted to the mathematical ones. And then mathematics could go on as usual.