A modal approximative ontological argument

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

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

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

3. If possibly God exists, then God exists.

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

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

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

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

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

Infinite proofs

Consider this fun “proof” that 0=1:

      …

• So, 3=4

• So, 2=3

• So, 1=2

• So, 0=1.

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

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

• 0=0

• So, 0=0

• So, 0=0

• So, 0=0

      …

      …

• So, 3=4

• So, 2=3

• So, 1=2

• So, 0=1.

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

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

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

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

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

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

Gamecube controller to USB adapter

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

Loss of vice and growth of virtue

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

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

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

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

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

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

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

(But may God have mercy on us!)

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

Walking off to infinity

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

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

Here are four answers:

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

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

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

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

A causal finitist definition of the finite

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

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

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

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

Mathematical Platonist Universalism, consistency, and causal finitism

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

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

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

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

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

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

Free and responsible unconscious decisions

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

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

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

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

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

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

Two questions about sets

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

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

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

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

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

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

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

Progress report on books

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

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

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

Promising to sing infinitely many duets

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

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

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

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

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

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

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

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

A Trinitarian structure in love

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

• benevolence
• appreciation
• union.

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

The Probabilistic Counterexampler

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

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

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

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

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

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

python probab-reasoning.py "P(a,c)>P(a,b)" "P(a,b|c)>P(a,b)" 

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

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

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

python probab-reasoning.py "P(a)*P(b)==P(a&b)" "P(b)>0" "P(a,b)==P(a)" 

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

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

python probab-reason.py premise1 premise2 ... conclusions

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

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

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

python probab-reasoning.py 8 "P(a)*P(b)==P(a&b)" "P(b)>0" "P(a,b)==P(a)" 

Two ways of being vicious

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

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

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

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

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

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

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

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

Some arguments about the existence of a good theodicy

This argument is valid:

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

2. No virtuous person’s life is worthless.

3. So, a good theodicy can be given.

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

Here is an argument for 2:

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

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

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

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

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

Here is yet another argument.

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

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

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

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

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