Thursday, May 29, 2014

God, love and creation

Consider this argument (inspired by a comment Dan Johnson made on this post):

  1. God does everything out of love.
  2. God creates Francis.
  3. If God creates Francis out of love, he creates Francis out of self-love, out of love for Francis, or out of love for someone or something other than God or Francis.
  4. God does not create Francis out of self-love.
  5. God does not create Francis out of love for Francis.
  6. God does not create Francis out of love for anyone or anything other than himself or Francis.
  7. So, God does not create Francis out of love.
  8. Contradiction!
Premise (4) is justified by the idea that our existence doesn't benefit the infinite, self-sufficient, transcendent and triune God. Premise (6), and to some degree (4) as well, is justified by Kantian thoughts about how Francis is an end in himself.

Premise (5), on the other hand, comes from the thought that if God creates Francis out of love for Francis, then God's love for Francis is explanatorily prior to his decision to create Francis. But insofar as God is loving Francis, God must have "already" (in the explanatory order) decided to create him, and so he can't be basing his decision to create on his love for Francis.

I am not confident about (4). God's extended, non-intrinsic, well-being (the kind of well-being that is constituted by the flourishing of our friends or the success of our projects) may be promoted by creating Francis, and because God is God, Kantian worries about Francis existing for God's sake are inappropriate.

But one might also question (1). When I bestow goods on my kids, that they are my kids, that they are in the image of God and that the goods are good yield one or two sufficient reasons to bestow the good. That I love my kids, while true, is a fact about me. That fact may give me additional reason to bestow goods on them, but that additional reason doesn't seem to me to be something that should be in the forefront of my mind. Love is more focused on the beloved than on one's own status as a love. Moreover, when I bestow the goods on my kids because they are my kids, because they are in the image of God, and because the goods are good, I act lovingly. My act of will is partly constitutive of my love for them.

Likewise, then, we can say that when God creates Francis, he does not have to do that because he loves Francis. Instead, God's creation of Francis is partly constitutive of that love. Thus, we have reason to revise (1) to:

  1. Everything that God does, God does either out of love or it is partly constitutive of love.

We can now ask the interesting question: Can we procreate not out of love for the child, but in a way partly constitutive of love for that child? Notice a difference between us and God. While God's creation is essentially efficacious, our procreation is very chancy—indeed, most cases where people intend to procreate, they do not actually succeed (then). Moreover, our procreation is not tied to one person: it might be Francis who will result, but it might be someone else. So there will be worlds where Francis's parents do the same as they did in our world, but someone other than Francis comes about or even no one comes about. In those other worlds, they don't love Francis, since there is no Francis to love, and it doesn't even seem right to say that their act is partly consitutive of love for Francis (not just because it is not clear whether one can partly constitute something that doesn't obtain).

I suppose it could be the case, however, that it is a contingent matter that this particular procreative act is partly constitutive of love for Francis—in some worlds it's not partly constitutive of love for any child (maybe it's partly constitutive of love of spouse and love of God), and in others it's partly constitutive of love for another child. So, yes, perhaps the same thing can be said in our case as in God's case, with the difference that the divine creative act is essentially constitutive of love.

In an earlier post, I argued for a paradox in regard to human reproduction. Maybe the above considerations help? Maybe a couple can intentionally procreate in order to perform an act constitutive of love for a child? But I don't think that helps with the argument in that post, since performing an act constitutive of love for a child is either seen as good for the child, in which case premise 2 of that argument covers the case, or it does not (probably, it's seen as good for the couple), in which case premise 1 of that argument covers the case.

By the same token, saying that God's creative act is partly constitutive of God's love for the creature doesn't answer the question of why God creates the creature. I still think the answer that he does so for the good of the creature, which I give in the aforementioned post, can be defended. As the Catechism says, we are created to know and love God. And that's good for us.

Tuesday, May 27, 2014

Explanation modulo a fact

A notion that I find rather natural is the idea of explanation modulo a fact. This notion can do some of the work that contrastive explanation can.

  1. Why did you eat a banana?
can be precisified contrastively in ways like:
  1. Why did you eat a banana rather than something else?
  2. Why did you eat a banana rather than doing something else with it?
  3. Why did you eat a banana rather than not eating anything?
But one can also accomplish the same thing this with requests for explanation modulo a fact. Doing the work of 2 and 3 is easy:
  1. Given that you ate, why did you eat a banana?
  2. Given that you did something with a banana, why did you eat it?
Doing the work of 4 is a bit harder, but maybe this works:
  1. Given that you ate a banana if you ate anything, why did you eat a banana?

When we ask for explanation why p given q, we are asking for something that isn't just an explanation why q, and that gives us a further explanation why p when added to the fact taht q. The question presupposes there is such an additional truth. If the fact that q is the one and only complete explanation why p, then the question has a false presupposition.

It seems, however, that any request for explanation modulo a fact can be rephrased as a request for contrastive explanation. For instead of asking:

  1. Given q, why p?
one can ask:
  1. Why p and q rather than not-p and q?
This makes explanation modulo a fact not very useful, it seems. However, one can also formulate contrastive questions as requests for explanation modulo a fact. Thus the contrastive question:
  1. Why p1 rather than p2, p3, ... or pn?
can be replaced with:
  1. Given p1, p2, p3, ... or pn, why p1?

So which is more fundamental? Explanation-modulo or contrastive explanation? I don't know. The literature prefers contrastive explanation, but that may be a historical accident.

In any case, I think all this makes an important fact about contrastive explanation clearer, a fact that I learned from conversation with Dan Johnson: When you make a request for a contrastive explanation, you are also indicating a refusal to accept certain kinds of answers. In the explanation-modulo case, these are answers that merely go through the given fact. But then the defender of the Principle of Sufficient Reason should feel no embarrassment about being unable to give answers to all requests for contrastive explanation—after all, that every fact has an explanation does not make for any kind of guarantee that the explanation will be of the sort we want. Once we start ruling out certain kinds of explanations, we might well be left with none.

Friday, May 23, 2014

Thomson's lamp and the Axiom of Choice

Consider Thomson's lamp: a lamp with a pushbutton switch that toggles it on and off. The lamp starts in the off position, and then in the next half minute the button is pressed, and in the next quarter it is pressed again, and then in the neight eighth again, and so on. Then at the end of the supertask, the lamp is either on or off.

Now keep the lamp but change the story. During each of the ever shorter intervals, a coin is flipped and the switch is pressed if it lands heads, and not pressed if it lands tails. Moreover, the final state of the lamp depends on the results of the coin flips in the following ways:

  1. The results of the coin flips determine the final state of the lamp.
  2. For any sequence of coin flip results, if any one (and only one) coin flip had a different, the lamp's final state would have been different, too.
Surprisingly, the existence of a lamp that would work in this way implies a version of the Axiom of Choice. To see this, notice that if the coin flips are independent and fair, then the subset of the probability space where the lamp's final state is on is nonmeasurable.[note 1]

But of course, on some technical assumptions, the existence of a nonmeasurable set requires a version of the Axiom of Choice. So if we read the Thomson's lamp story in such a way that the final outcome is determined by which presses are made and which aren't, in such a way that changing a single press changes the final outcome, that story seems to commit us to a version of the Axiom of Choice.

Conversely, it is easy to use the Axiom of Choice for pairs to prove the existence of a function such as would be implemented by the lamp.[note 2]

Thursday, May 22, 2014

Kindle version of One Body discounted to $26

My apologies for more advertising. While I was happy to notice earlier in the week that the Kindle version of my One Body book was out, I was disappointed by its high price (I have nothing to do with setting the price). But it looks like it's now discounted to $26.

A (paradoxical?) argument that intentional reproduction is wrong


  1. We cannot permissibly intend to produce a person for reasons that do not include the specific person's own good.
  2. We cannot intend to produce a person for reasons that include the specific person's own good.
  3. We cannot permissibly intend to produce a person without reasons.
  4. Necessarily, if we intend to produce a person, we do so for no reason, or for reasons that include the person's own good, or for reasons that do not include the person's own good.
  5. We cannot permissibly intend to produce a person. (1-4)
Premise (1) is due to Kantian considerations: persons are ends not means. I will argue for premise (2) shortly. Perhaps the easiest way to argue for premise (3) is that we simply cannot intend—permissibly or not—without reasons. We intend things because they are good, either as means or as ends, and in either case we have reasons. But one might also argue that if it is wrong to produce persons for reasons that do not include the person's own good, it is a fortiori wrong to produce them wantonly, for no reason at all, as it were on a whim. Or one might think that being rational animals, it is wrong for us to intentionally act in a non-reasnable way, and to act intentionally but without reasons is to do that.

Now on its face, premise (2) is false. Surely people do procreate for the child's own good. But I don't think so. They may be acting for the good of whatever child results from the reproduction, but there is no specific child for whose good they are acting. And when they act for the good of whatever child results, the specific child's good ends up being a constitutive means to the good they are seeking, so they do not escape from the Kantian criticism. The good of a person is an incommunicable good: it is that specific person's good. But the existence and identity of the child depends on the couple's decision in a way that the couple is unable to figure out beforehand. Thus the couple cannot be deciding in light of the identity of the child, and hence cannot be acting for the good of that specific person.

Note that God does not suffer from the cognitive limitations that give rise to (2): he can know our identity before he decides to create us, and can decide to create us for our own good.

Now, when a couple engages in the marital act, they have a reason to engage in that act apart from reproduction: the act is good in itself, being an embodiment of marital union. Thus they can act so as to unite, and accept the child as a gift from God that goes beyond their intention. Note that even given my argument they can permissibly rationally consider the reproduction in their decision whether to make love, for instance as a defeater to various defeaters (being tired, etc.) to the marital reasons for lovemaking. On the other hand, in non-coital methods of reproduction like IVF the couple is specifically intending reproduction, and that is wrong if the argument succeeds.

I am not myself entirely convinced of (1), because I am not entirely convinced of the Kantian autonomy framework. We aren't ends in ourselves: we exist as constitutive glorifications of God. Thus it does not seem contrary to the dignity of a person to be produced for the greater glory of God. Kantianism is what you get when you remove God from the story. If that's right, then we get the surprising result that only theists can permissibly intend to produce a child. Atheists, to be consistent, will need to have the Kantian attitude, and while they can permissibly reproduce, they cannot do so with the intention of reproducing, if everything else in the argument works.

Wednesday, May 21, 2014

A metaphysical argument for a version of the Axiom of Choice

I will argue for the Axiom of Choice for sets of real numbers (ACR). ACR states that:

  • Given a set U of non-empty sets of real numbers, there is a function f on U such that f(A)∈A for every AS.
ACR is sufficiently strong to generate all the interesting paradoxes about the Axiom of Choice such as the ones linked here.

  1. There is a physically possible causally isolated situation in which there is a physical process R generating exactly one maximal semi-infinite (with beginning and no end) sequence of independent fair indeterministic coin tosses, and such that every combination of coin toss results can occur.
  2. For any possible physical process, and any cardinality K, it is possible that there are K causally isolated situations in each of which an instance of that process runs.
  3. If there is a set S of causally isolated situations, a set E of event types, and a function g from S to T such that for each sS an event of type g(s) is causally possible in s, then it is metaphysically possible that for each sS, an event of type g(s) occurs in s.
  4. If U is a set of non-empty sets of real numbers and possibly possibly there is a function f on U such that f(A)∈A for every AU, then there is a function f on U such that f(A)∈A for every AU.

Premise (1) is highly intuitive. Premise (2) is plausible, though finitists will deny it.

Premise (3) has the dubious form of saying that if each of some set of propositions—in this case, the propositions that g(s) as s ranges over S—is possible, then these propositions can all be true at once. Of course, this is false in general. But (3) limits this claim by making the propositions not just be metaphysically possible, but causally possible, and by saying that the propositions report what happens in different causally isolated situations. And individually causally possible events in different causally isolated situations should be compossible.

Finally, premise (4) says that the truth of mathematical propositions about the existence of numerically valued functions on sets of sets of real numbers does not vary across possible worlds, or even possibly possible worlds (given S4, the two would be be the same).

Now on to the argument. Let r be a function from semi-infinite sequences of heads-tails to real numbers, such that every real number is in the range of r. (For instance, one can let the coin toss sequence define a binary fraction between 0 and 1, and then apply some function to scale that up to all of (−∞,∞).) Let K be the cardinality of our set U of non-empty sets of real numbers. By (2) there is a possible world w1 containing a set S of K causally isolated instances of our random toss process R. Suppose now that w1 is actual. Let h be a bijection from S to the set U in ACR. Let g(s) be the event type of the random toss process in generating a sequence x of tosses such that r(x)∈h(s). Then g(s) is a causally possible event type, since every heads-tails sequence can occur by means of R and every real number can be generated by applying r to some heads-tails sequence. By (3), there is a possible world w2 at which all of this happens and each event g(s) occurs. But at w2, we can then let f(A) for AU be equal to r(x) where x is the result of R in the unique situation s such that h(s)=A. Then f(A)∈A since g(s) occurs.

We have thus shown that at w2, there is a choice function f. Since w2 is possible at w1, and w1 is actually possible, by (4) there is a choice function f, and the argument is complete.

I've given a version of this argument before, but this version identifies the assumptions more clearly, especially premise (3) about the conglomeration of causal possibilities across isolated scenarios.

Tuesday, May 20, 2014

One Body, now for Kindle

Finally my One Body book has a Kindle version!

The completed infinite

It is very hard to deny that it is logically possible that every rabbit has at least one offspring and there are no loops (no rabbit is its own ancestor). But in that situation, there will be infinitely many rabbits.

"Not so fast!" say the defenders of the distinction between the potential and the completed infinite. This is a case of a potential but not a completed, or actual, infinite. But why? On the scenario in question, there are infinitely many humans. A standard answer is to embrace a theory of time, like growing block or presentism, on which there are no future entities, and then to say something like this about the scenario:

  • Infinitelymany rabbits will come into existence, but there are only finitely many rabbits and at any given future time there will only be finitely many then.
Not only does this require abandoning eternalism, which the correct theory of time, but it requires further work to explain why it couldn't also be the case that every rabbit has the property that its offspring take half as long as it did to produce offspring.[note 1] But in that case, after a finite amount of time there would be infinitely many rabbits. Further, this approach requires either nominalism or a special story about why concrete objects like rabbits can't form a complete infinity, while mathematical entities like prime numbers can (not a big problem for me, since I'm not a Platonist).

My proposal is that we should see the denial of a completed infinite differently. Rather than seeing it ontologically as saying that there are not infinitely many of anything, we should see it causally. A student has completed a class provided that the class is available for her to build on, either in her future thinking and work or as a prerequisite for other classes. Likewise, a process is completed provided that its product is available for other processes to build on.

A completed infinity, I propose, is the sort of infinity that can be causally built upon. The rabbits in my initial scenario cannot be built upon: they aren't all causally available to anyone. In that initial scenario, there is a plausible explanation about this in terms of time: there is no time at which there are infinitely many rabbits, so there is no time at which you can build on all of them.

But my causal finitism suggests that the same is true on my modified scenario where the rabbits breed faster and faster. Maybe that scenario can produce an infinite number of rabbits in a finite amount of time. But nonetheless, only finitely many of the rabbits will irreducibly work together causally. (I wonder whether irreducibility rules out overdetermination. Worth thinking about...) Let's say you cast a glance at that infinity of rabbits. You will only see finitely many at a time—your field of view is only finitely large and finitely sharp. Only finitely many of them will eat up your garden. And so on.

If we see a "completed infinity" as a causal notion, then we have no worries about Platonist mathematics. For mathematical entities are typically taken to be causally inert, and even if for some epistemological reason we do not take them so, we could still think that only finitely many are involved in any one causal interaction.

Monday, May 19, 2014

The temporal insurpassability of heaven

Heavenly bliss lasts infinitely long. (Some theologians think of heaven as timeless, but that fits poorly with the doctrine of the resurrection of the body.) But wouldn't it be better to have a second heavenly life, after the first infinite one? And then instead of the usual order type ω for one's future days (1st future day, 2nd future day, 3rd future day, ...) one would have order type ω·2 (1st day, 2nd day, 3rd day, ..., infinitieth day + ωth day, (ω+1)st day, (ω+2)nd day, ...). And why stop there? Why not future days of order type ω·3? Or ω2? Or ωω? No temporal infinity seems insurpassable, so it seems that there could always be a longer afterlife.

Not so if my causal finitist thesis is true. For while the causal finitist thesis does not by itself deny the possibility of a longer infinite afterlife, it denies the possibility that any event could essentially depend on an earlier infinity of events. In particular, it means that if one had an infinite afterlife, and then continued to exist after that, it would be impossible to integrate that infinite afterlife in memory. But it is an important feature of the sort of creatures that we are that we integrate our past in our memory. Thus, given causal finitism, an afterlife whose events went beyond order type ω would be a disintegrated afterlife, unfitting for the sorts of beings we are.

This solves the third of the theological questions here.

By the same token, causal finitism makes implausible the following variant on universalism: "While hell is infinitely long, everyone who goes to hell is eventually saved (after that infinite time)." For presumably the salvation on that variant would be a result of a purification process in the infinitely long sojourn in hell, thereby being very likely to violate causal finitism.

Thursday, May 15, 2014

Popular devotions

It is good to participate in a popular devotion because the devotion is popular (of course, this is defeasible: theological unsoundness would be a particularly important defeater). When one participates in a popular devotion because it is popular, one is thereby united in will with the community in which the devotion was popular. This is true even if the devotion involves something kitschy or a little garish, and such cases highlight the need to see the devotion from the point of view of the community which gives the devotion its life.

When the devotion is centered on a saint, that deepens the community aspect by extending it beyond death.

From this point of view, I think I can now understand the ways in which we pay respect to Mary under many appellations like "Our Lady of Czestochowa", "Our Lady of Mount Carmel" and "Our Lady of Perpetual Help." For the different appellations connect one with the different overlapping communities (ethnic, monastic, etc.) that are inspired by that aspect of our Lady's character and life. And part of the

richness of the life of a large vibrant community like the Church (or a nation, for that matter) are the overlapping smaller synchronic and diachronic communities found within it. Just as it is good to have particular friends, it is good to identify with multiple particular communities. All if this fulfills us as the social animals we are.

Thus, those Christians, especially Catholics, who focus on the horizontal aspects of the Christian life, who take the notion of community as central, should love popular devotions. (One thinks here of Fr. Andrew Greeley as an example of this love.)

Tuesday, May 13, 2014

More fun with infinite fair lotteries

Imagine an infinite sequence of games such that you are nearly certain to win each one, but you're also certain to lose all but finitely many of them. This seems really absurd. But given an infinite fair lottery, it can be easily arranged. Suppose a secret natural number N is chosen in our infinite fair lottery. Let Gn be the following game, for n a natural number:

  • You win if N>n and you lose otherwise.
Your probability of winning Gn is P(N>n). But P(Nn) is zero or infinitesimal for each finite n. So, your probability of winning is within an infinitesimal of one. But it is guaranteed that you will win at most N−1 games. So, indeed, for each game, you're nearly certain to win it, and you're certain to lose all but finitely many.

Imagine placing bets on this game. If it costs a penny to play and the payoff is a dollar, you'll think it's a great deal: after all, you're nearly certain you will win. But if you play all the games, you will make only finitely many dollars, and lose infinitely many pennies.

Conclusion? I suppose the best one is that infinite fair lotteries are impossible.

Monday, May 12, 2014

Simplicity and divine decisions

One of the most difficult problems for divine simplicity are how to square it with creation and divine knowledge of free actions. On its face, there are at least four distinct states of God:

  1. God's essential nature
  2. God's contingent decisions
  3. God's knowledge of his contingent decisions
  4. God's knowledge of creatures' free responses to his contingent decisions.
Calvinists can reduce (4) to (3) (say, by grounding (4) in (3), and holding that if state B is grounded in state A, that does not really multiply states in a way contrary to divine simplicity), thereby reducing the number of distinct states from four to three. Thomists, and presumably some Calvinists as well, can reduce (3) to (2): God's decision is identical with his knowledge of his decision. Even if we make both of these controversial moves, we still have the distinction between God's essential nature and his contingent decisions (which are then identical with his knowledge of the decisions and his knowledge of creatures' responses thereto).

My own preferred sketch of a solution to these problems is here. The solution proceeds by making the contingent aspects of (2)-(4) be extrinsic to God.

For those Christians who are unimpressed by the strength of the traditional commitments (in the pre-Reformation tradition, but also in people like Calvin and Turretin) to divine simplicity, and the arguments for divine simplicity, the natural solution will appear to be to deny divine simplicity, and then not worry about the problem.

They should still worry about the problem. For if one denies divine simplicity and holds that God has at least the two distinct constituents: his essential nature, N, and his contingent decisions, D, then one has to say something about the relationship between these two. Clearly, D is in some way explained by N: God acts as he does in part because of his essentially perfectly good character. The explanation is not a grounding-type explanation—to make it be a grounding-type explanation would be to hold on to a version of a divine simplicity explanation. In creatures, the corresponding explanation of decisions would be causal: the character causes (deterministically or not) the decision. So it seems that we have something very much like a causal relationship between N and D. And this in turn makes D be very much like a creature, indeed perhaps literally a creature. Since D is a constituent of God, it follows that a constituent of God is very much like a creature, perhaps literally a creature. But this surely contradicts transcendence!

Now perhaps one can insist that the relationship between N and D while being akin to causation is sufficiently different from it that D is sufficiently different from a creature that we have no violation of transcendence. Maybe, but I am still worried.

So if I am right, even if one denies divine simplicity, a version of the problem remains. And so the problem may not be a problem specifically for divine simplicity.

Friday, May 9, 2014

The most fundamental and what matters most

What matters most are things like people, love, understanding, courage, friendship, beauty, etc. According to many contemporary metaphysicians, what is most fundamental are things like sets, points, photons, charge, spin, the electromagnetic field, etc. It's almost as if the metaphysicians took the fact that something matters to be evidence that it isn't fundamental.

But here is a plausible hypothesis or at least heuristic:

  • Fundamental predicates apply primarily to fundamental entities, and derivatively to other entities.
While a table can have mass or be charged, it has mass or is charged derivatively. It is particles that primarily have mass or are charged. Now, some value predicates like "matters" or "is valuable" are fundamental. (Of course, this is the controversial assumption.) Thus we have reason to think the kinds of things they primarily apply to are themselves fundamental, and they apply only derivatively to non-fundamental things. But the value of a person is not derivative from the value of the person's constituents like fields or particles, and the way in which a person matters does not derive from the ways in which fields or particles matter.

Thus, either persons will be themselves fundamental, and primary bearers of value, or else persons will be partly constituted by something fundamental which is a primary bearer of value. The best candidate for this valuable constituent is the soul. Hence, either persons are fundamental or they have souls that are fundamental.

In fact, I would conjecture that we should turn on its head the correlation between fundamentality and not mattering that we find in much contemporary metaphysics. The more something matters, the more reason we have to think it is fundamental, I suspect. This may lead to a metaphysics on which there are fundamental facts about persons, their psychology and their biology, a realist metaphysics with a human face.

Thursday, May 8, 2014

No event can irreducibly depend on infinitely many things

I am now thinking the following principle is likely to be true:

  • (NoInfDep) No event can irreducibly depend on infinitely many things.
This might be true even if we drop "irreducibly", but the "irreducibly" makes me more confident. When A causes B in such a way that B is determined to cause C, then I say that C reducibly depends on B. No additional information is given about how C came about by describing B than is already implicit in a sufficiently rich description of A. In other words, reducible dependence is dependence on a merely and totally instrumental cause, one that doesn't make any contribution truly of its own. We can then see the debate between compatibilists and incompatibilists as a debate on whether an agent's free actions have to irreducibly depend on something in the agent.

Why think NoInfDep is true? The general line of argument is this. There are a number of paradoxes that NoInfDep rules out. Now in the case of each paradox, there is a narrower modal principle that could rule out the paradox, but the narrower principle is ad hoc in a way that NoInfDep isn't, and so our best explanation as to why the paradoxes are ruled out is (1).

Here are the paradoxes I currently have in mind:

  1. Thomson's Lamp
  2. Grim Reapers
  3. Coin sequence guessing
  4. Infinite fair lotteries resulting from infinitely many fair coin tosses (see the discussion in one of my comments of the paradoxicality)
  5. Satan's Apple and some other decision-theoretic paradoxes (e.g., the game where we have dollar bills numbered 1,2,3,... and you start with dollar bill #1, and in each round you give me your lowest numbered bill, and I give you two bills with higher numbers; at the end you have nothing)
  6. Realizations of the Banach-Tarski Paradox and maybe even things relating to nonmeasurable sets.
There are other solutions for some of these. (I've never been that impressed by Thomson's lamp, but it's a freebie here.) But NoInfDep provides an elegantly uniform solution. Moreover, NoInfDep expresses the intuition that there cannot be a "completed infinity" without committing one to a dubious presentist or growing block ontology. Indeed, NoInfDep shows that the "no completed infinity" intuition goes beyond considerations of time: it is about dependency.

I want to say something about the Banach-Tarski case. The paradox there is purely mathematical. But to realize this paradox in real life--to actually decompose a solid ball (if there were such a thing) into two of equal size--you would need to make something like a choice function, which would require infinitely many data points, and those would require, I suspect, irreducibly infinitely many events to generate.

And now we have the Kalaam argument.

Tuesday, May 6, 2014

Infinite regress explanations

Consider Thomson's toggle lamp—each time the button is pressed, the lamp toggles between on and off—but suppose it existed from eternity and every January 1 the switch has been pressed once, and only then. Why is the lamp on now? Consider the regress explanation: It's on in 2014 because it was off in 2013 and toggled on January 1, 2014. And it was off in 2013 because it was on in 2012 and toggled on January 1, 2013. And so on.

Hume will say that this is a complete explanation. But surely not. Surely the whole story does not explain why the lamp is on in even numbered years and off in odd numbered years.

Notice an interesting thing. The following are perfectly fine explanations:

  1. The lamp is on in 2014 because it was off in 2013 and toggled at the beginning of 2014.
  2. The lamp is on in 2014 because it was on in 2012 and toggled at the beginnings of 2013 and 2014.
  3. The lamp is on in 2014 because it was off in 2011 and toggled at the beginnings of 2012, 2013 and 2014.
And as we go down this list of explanations, our explanations get more and more ultimate. However, we can't take this to infinity. Each of the explanations in the list has wo conjuncts: a fact about the state of the lamp in year n, and then facts about the lamp being toggled in successive years. The facts about the lamp being toggled in successive years can be taken to infinity, but aren't enough to explain it. The following clearly isn't enough to give us an ultimate explanation of why the lamp was on in 2014:
  1. The lamp was toggled at the beginnings of ..., 2010, 2011, 2012, 2013 and 2014.
Can we take the first conjunct in explanations (1)-(3) to infinity? Well, we certainly can't in general say that the lamp was on, or that it was off, in year −∞, since even if such a year existed, dubious as that is, the lamp need not have existed then—it need only be supposed to exist in all finite-numbered years. So what can we say? Well, we could let the lamp state in year n be L(n)—0 being off and 1 being on—and then say:
  1. The limit of L(2n) is 1 as n→−∞ and the limit of L(2n+1) is 0 as n→−∞ (both limits over the integers only).
So if we think about how to complete our regressive explanation, it seems that it will need to be something like this:
  1. The lamp is on in 2014 because of (4) and (5).
Very good. But even if (4) were to be ulitimately explained (maybe there is some mechanism where each toggling is caused by the preceding, which according to Hume would give an ultimate explanation of (4)), it is clear that (5) calls out for an explanation as well, and so the regressive explanation just isn't ultimate explanation.

So infinite regresses aren't enough for ultimate explanations, pace Hume.

Monday, May 5, 2014

Induction and eccentricity

There is reason to be a conventional person. For the more conventional one is, the more accurate will be people's inductive arguments about one's behavior and character. Being understood by others is a good thing. It is good not only because it is good that people possess the truth, but it is good for one in that relationships with one are more likely to be based on truth.

Of course, there may be defeaters.

Thursday, May 1, 2014

Truth by convention

I stipulate that "It xyzzies" is true. Clearly I have failed to make "It xyzzies" meaningful. My stipulation is compatible with "It xyzzies" meaning that 2+2=4, but also with its meaning that everything is round or non-round. So stipulating a sentence to be true isn't going to be sufficient to introduce the sentence into our language. But if I say anything more about the meaning of the sentence, I risk that no true sentence might fit what I say, and we lose the point of truth by convention. Nor does it at all help to stipulate a family of sentences at once, e.g., stipulating that whenever S and T are true, so is "S*T", and when "S*T" is true, S is true, and when "S*T" is true, T is true. That still fails to introduce a connective "*", unless we say more about the meaning. The stipulation I gave is compatible with too many things. For instance, "S*T" could mean "God believes S and God believes T", or it could mean "(S or S) and (T and T)". And if I say more about which one I mean, I risk the stipulating being unsuccessful. So that's that for truth by convention. It's fun to drive nails in the coffins of dead theories.

Functional characterizations of pain

Functionalists are committed to functional characterizations of pain. The difficulty with a functional characterization is that if it is too specific, it will be very plausible that there could be—or even are!—beings where a pain plays a somewhat different functional role, and that if it is broad, then some things that aren't pain will count as pain. In other words, while functionalism was introduced to help with the multiple-realizability problem of simple type-type identity theories, multiple-realizability comes back, though in milder form.

Think first of the great variety of functions played by pains in our own mental lives. First, it is not very plausible that there would be a single function that is played by both physical and emotional pain. When I feel pain after I touch a hot stove, that motivates avoidance. But when I am in pain that someone I cared about died, that doesn't motivate anything like avoidance. Of course, everything is similar to everything else in some way, so there will be a level of functional description which will capture both kinds of pain, but it is very likely that the description will capture lots of things other than pains as well.

Now maybe there isn't too much cost to saying that physical and emotional pain are different kinds of things that happen to have the same word applied to them, much as we call both nephrite and jadeite "jade". (If one has a hedonist theory of wellbeing, there will be a cost, as now there will be two sources of illbeing. But one shouldn't have a hedonist theory.)

The same issue, though, I think will come up between different kinds of emotional pains. It is very dubious whether there is a sufficiently robust characterization of the function of emotional pain that captures grief, guilt, terror, disappointment and boredom, but doesn't also capture things that aren't pains at all. The roles of these emotional pains are very different. But perhaps there is something to be said for the thought that these negative emotions are not of a piece, that they too shouldn't be classed together. However, at this point the theory is becoming more costly.

Let's now stick to physical pain. Plausibly, in some animals physical pain leads directly to avoidance behavior. But in humans it does not. (The instinctive jerking back from a hot stove occurs before you have pain.) So the functional role is very different. And it is dubious whether one can give a characterization of this functional role that goes beyond something very vague like "motivation to avoidance", which will include way too many things that aren't pains at all, such as the causes of aversive behavior in bacteria.

One might, of course, try to give a functional characterization that is closer to the actual functioning of our brains. Thus, one might describe the kinds of functional interconnections that happen in our brains. And it might be that sufficeintly similar interconnections happen in the brains of other vertebrates. But if the description is too close to neural structures, then we get the conclusion that aliens whose neural analogues cause very similar adaptive behavior as our brains do not have pain.