Supervenience and counterfactuals

On typical functionalist views of mind, what mental states a physical system has depends on counterfactual connections between physical properties in that system. But we can have two worlds that are exactly the same physically—have exactly the same tapestry of physical objects, properties and relations—but differ in what counterfactual connections hold between the physical properties. To see that, just imagine that one of the two worlds is purely physical, and in that world, w₁, striking a certain match causes a fire, and:

1. Were that match not struck, there would have been no fire.

But now imagine another world, w₂, which is physically exactly the same, but there is a nonphysical spirit who wants the fire to happen, and who will miraculously cause the fire if the match is not struck. But since the match is struck, the spirit does nothing. In w₂, the counterfactual (1) is false. (This is of course just a Frankfurt case.)

Thus physicalist theories where counterfactual connections are essential are incompatible with supervenience of the mental upon the physical.

I suppose one could insist that the supervenience base has to include counterfactual facts, and not just physical facts. But this is problematic. Even in purely physical worlds, counterfactual facts are not grounded in physical facts, but in physical facts combined with the absence of spirits, ghosts, etc. And in worlds that are only partly physical, counterfactual connections between physical facts may be grounded in the dispositions of non-physical entities.

Something Mary doesn't know

Here is something our old friend Mary, raised in a black and white world, cannot know simply by knowing all of physics:

1. What are the necessary and sufficient physical conditions for two individuals to be in exactly the same phenomenal state?

Of course, her being raised in a black and white world is a red herring. I think nobody can know the answer to (2) simply by knowing all of physics.

Some remarks:

• Knowledge of the answer to (1) is clearly factual descriptive knowledge. So responses to the standard knowledge argument for dualism that distinguish kinds of knowledge have no effect here.

• The answer to (1) could presumably be formulated entirely in the language of physics.

• Question (1) has a presupposition, namely that there are necessary and sufficient physical conditions, but the physicalist can’t deny that.

• A sufficient conditions is easy given physicalism: the individuals have the exact same physical state.

• Dennettian RoboMary-style simulation does not solve the question. One might hope that if you rewrite your software, you can check if you have the same qualia before and after the rewrite. But the problem is that you can only really do exact comparisons of qualia that you see in a unified way, and there is insufficient unification of your state across the software rewrite.

Humanity and humans

From childhood, I remember the Polish Christmas carol “Amidst the Silence of Night” from around the beginning of the 19th century, and I remember being particularly impressed by the lines:

Ahh, welcome, Savior, longed for of old,
four thousand years awaited.
For you, kings, prophets waited,
and you this night to us appeared.

I have lately found troubling the question: Why did God wait over a hundred thousand years from the beginning of the human race to send us his Son and give us the Gospel?

The standard answer is that God needed to prepare humankind. The carol’s version of this answer suggests that this preparation intensified our longings for salvation through millenia of waiting. A variant is that we need a lot of time to fully realize our moral depravity in the absence of God. Or one might emphasize that moral teaching is a slow and gradual process, and millenia are needed to make us ready to receive the Gospel.

I think there is something to all the answers, but they do not fully satisfy as they stand. After all, a human child from 100,000 years ago is presumably roughly as capable of moral development as a modern child. If we had time travel, it seems plausible that missionaries would be just as effective 100,000 years ago as they were 1000 years ago. The intensification of longings and the realization of social moral depravity are, indeed, important considerations, but human memory, even aided by writing, only goes back a few thousand years. Thus, two thousand years of waiting and learning about moral depravity would likely have had basically the same result for the individuals in the time of the Incarnation as a hundred thousand years did.

I am starting to think that this problem cannot be fully resolved simply by considering individual goods. It is important, I think, to consider humankind as a whole, with goods attached to the human community as a whole. The good of moral development can be considered on an individual level, and that good needs a few decade rather than millenia. But the good of moral development can also be considered on the level of humankind as well, and there millenia are fitting for the development not to ride roughshod over nature. Similarly, the good of longing for and anticipation of a great good only needs at most a few decades in an individual, but there is a value in humankind as a whole longing for and anticipating on a species timescale rather an individual timescale.

In other words, reflection on the waiting for Christ pushes us away from an overly individualistic view. As do, of course, other aspects of Christian theology, such as reflection on the Fall, the Church, the atonement, etc.

Am I fully satisfied? Not quite. Is the value of humankind’s more organic development worth sacrificing the goods of thousands of generations of ordinary humans who did not hear the Gospel? God seems to think so, and I am willing to trust him. There is doubtless a lot more to be said. But it helps me to think that this is yet another one of those many things where one needs to view a community (broadly understood) as having a moral significance going beyond the provision of more individualistic goods.

Two more remarks. First, a graduate student pointed out to me (if I understood them right) that perhaps we should measure individual moral achievement relative to the state of social development. If so, then perhaps there was not so great a loss to individuals, since what might matter for their moral wellbeing is this relative moral achievement.

Second, the specifically Christian theological problem that this post addresses has an analogue to a subspecies of the problem of evil that somehow has particularly bothered me for a long time: the evils caused by lack of knowledge, and especially lack of medical knowledge. Think of the millenia of people suffering and dying of in ways that could have been averted had people only known more, say, about boiling water, washing hands or making vaccines. I think there is a value in humankind’s organic epistemic development. But to employ that as an answer one has to be willing to say that such global goods of humankind as a whole can trump individual goods.

(Note that all that I say is meant to be compatible with a metaphysics of value on which the loci of value are always individuals. For an individual’s well-being can include external facts about humankind. Thus the good of humankind as a whole might be metaphysically housed in the members. The important thing, however, is that these goods are goods the human has qua part of humanity.)

Leibniz's King of China thought experiment

Leibniz famously offers this thought experiment:

Supposing that an individual were to instantly become King of China, but on the condition of forgetting what he has been, as if he was completely born again—isn’t that practically, with regard to perceivable effects, as if he were to be annihilated and a King of China were to be created in the same moment in his place? This the individual has no reason to desire. (Gerhardt IV, p. 460)

The context is that Leibniz isn’t doing metaphysics here, but supporting an ethical point that memory is needed for one to be a fit subject for reward and punishment and a theological point that eternal life requires more than mere eternal existence without the psychological features of human life. Nonetheless, some have thought that thought experiments like Leibniz’s offer support for memory theories of personal identity. I will argue that tweaking Leibniz’s thought experiment in two ways shows that this employment would be mistaken. In fact, I think the second tweak will offer an argument against memory theories.

Tweak 1: Memory theories of personal identity require a chain of memories, but not a chain of personally important memories. So all we need to ensure identity of the earlier individual with the later King of China is that the King of China remembers something really minor from the hour before the transformation, say seeing a fly buzzing around. Allowing the memory of a fly to survive the enthronement does not affect the intuition that the process is one that “the individual has no reason to desire.” The loss of personally important memories—especially of interpersonal relationships—is too high a price for the alleged benefit of ruling a great nation. Hence the intuition is not about personal identity, but—as Leibniz himself thinks—about prudential connections in a person’s life. Nor should we modify memory theories of personal identity to require the memories to be personally important, since that would make personal identity too fragile.

Tweak 2: First, suppose that in addition to the individual’s memories being wiped, the individual gets a new set of memories implanted, copied from some other living person. That so far does not affect the intuition that the process is one that one has “no reason to desire.” Second, add that the other living person happens to be one’s exact duplicate from Duplicate Earth. On memory theories of personal identity, one still perishes—the memories aren’t one’s own, even if they are exactly like one’s own. But a good chunk of the force of the thought experiment evaporates. It is, admittedly, an important thing that one’s apparent memories be real memories, and when they are taken from one’s exact duplicate, they are not. If one’s apparent memories are from one’s duplicate, then one isn’t remembering one’s friends and family, but instead is having quasi-memories of the duplicate’s friends and family, who happen to be exactly like one’s own. That is a real loss objectively speaking. But it is a much lesser loss than if one’s memories are simply wiped or replaced by those of a non-duplicate.

Note further that in the case where one’s memories are replaced by those of a duplicate, if enough benefits are thrown into the King of China scenario, the whole thing might actually become positively worthwhile. Suppose you are a lonely individual without significant personal relationships, but as King of China you would have a fuller and more interpersonally fulfilling life, despite the inevitable presence of flatterers and the mind-numbing work of ruling a vast empire. Or suppose creditors are hounding you night and day. Or you have a disease that can only be cured with the resources of a vast empire. When we note this, we see that the modified thought experiment provides evidence against the memory theory. For on the memory theory, it makes no difference to one’s identity whether the memories will come from a duplicate or not, as long as they don’t come from oneself, and what benefits the King of China will receive is largely prudentially irrelevant.

Objection 1: If the King of China gets memories from your duplicate, then the King of China will have your values and will promote your goals with all of the power of an empire. That could be prudentially worth it and provides some noise for the tweaked thought experiment.

Response: We can control for this noise. Distinguish your goals into two classes: those where your existence is essential to the goal and those where your existence is at most incidental to the goal. We can now suppose that you are a selfish individual who has no goals of the second type. Or we can suppose that all your goals of the second type are such that you think that being King of China will not actually help with them. (Perhaps world peace is a goal of yours, but like Tolstoy you think individuals, including emperors, are irrelevant to such goals.)

Objection 2: If you know that the duplicate has the exact same memories as you do, then copying memories from the duplicate at your behest maintains a counterfactual connection between the final memory state and your pre-transformation memories. If the latter were different from what they are, you wouldn’t have agreed to the copying.

Response: There is nothing in Leibniz’s thought experiment about your consent. We can suppose this just happens to you. And that it is a complete coincidence that the subject from whom memories are taken and put into you is your duplicate.

The hand and the moon

Suppose Alice tells me: “My right hand is identical with the moon.”

My first reaction will be to suppose that Alice is employing some sort of metaphor, or is using language in some unusual way. But suppose that further conversation rules out any such hypotheses. Alice is not claiming some deep pantheistic connection between things in the universe, or holding that her hand accompanies her like the moon accompanies the earth, or anything like that. She is literally claiming of the object that the typical person will identify as “Alice’s hand” that it is the very same entity as the typical person will identify as “the moon”.

I think I would be a little stymied at this point. Suppose I expressed this puzzlement to Alice, and she said: “An oracle told me that over the next decade my hand will swell to enormous proportions, and will turn hard and rocky, the exact size and shape of the moon. Then aliens will amputate the hand, put it in a giant time machine, send it back 4.5 billion years, so that it will orbit the earth for billions of years. So, you see, my hand literally is the moon.”

If Alice isn’t pulling my leg, she is insane to think this. But now I can make some sense of her communication. Yes, she really is using words in the ordinary and literal sense.

Now, to some dualist philosophers the claim that a mental state of feeling sourness is an electrochemical process in the brain is about as weird as the claim that Alice’s hand is the moon. I’ve never found this “obvious difference” argument very plausible, despite being a dualist. Thinking through my Alice story makes me a little more sympathetic to the idea that there is something incredible about the mental-physical identity claim. But I think there is an obvious difference between the hand = moon and quale = brain-state claims. The hand and the moon obviously have incompatible properties: the colors are different, the shapes are different, etc. Some sort of an explanation is needed how that can happen despite identity—say, time-travel.

The analogue would be something like this: the quale doesn’t have a shape, while the brain process does. But it doesn’t seem at all clear to me that the quale positively doesn’t have a shape. It’s just that it is not the case that it positively seems to have a shape. Imagine that qualia turned out to be nonphysical but spatially extended entities spread through regions of the brain, kind of like a ghost is a nonphysical but spatially extended entity. There is nothing obvious about the falsity of this hypothesis. And on this hypothesis, qualia would have shape.

To be honest, I suspect that even if qualia don’t have a shape, God could give them the additional properties (say, the right relation to points of space) that would give them shape.

Computationalism and subjective time

Conway’s Game of Life is Turing complete. So if computationalism about mind is true, we can have conscious life in the Game of Life.

Now, consider a world C (for Conway) with three discrete spatial dimensions, x, y and z, and one temporal dimension, t, discrete or not. Space is thus a three-dimensional regular grid. In addition to various “ordinary” particles that occupy the three spatial dimensions and have effects forwards along the temporal dimension, C also has two special particle types, E and F.

The causal powers of the E and F particles have effects simultaneous with their causes, and follow a spatialized version of the rules of the Game of Life. Say that a particle at coordinates (x₀,y₀,z₀) has as its “neighbors” particles at the eight grid points with the same z coordinate z₀ and surrounding (x₀,y₀,z₀). Then posit these causal powers of an E (for “empty”) or F (for “full”) particle located at (x₀,y₀,z₀):

• If it’s an F particle and has less two or three F neighbors, it instantaneously causes an F particle at (x₀,y₀,z₀+1).

• If it’s an E particle and has exactly three F neighbors, it instantaneously causes an F particle at (x₀,y₀,z₀+1).

• Otherwise, it instantaneously causes an E particle at (x₀,y₀,z₀+1).

Furthermore, suppose that along the temporal axis, the E and F particles are evanescent: if they appear at some time, they exist only at that time, perishing right after.

In other words, once some E and F particles occur somewhere in space, they instantly propagate more E and/or F particles to infinity along the z-axis, all of which particles then perish by the next moment of time.

Given the Turing completeness of the Game of Life and a computational theory of mind, the E and F particles can compute whatever is needed for a conscious life. We have, after all, a computational isomorphism between an appropriately arranged E and F particle system in C and any digital computer system in our world.

But because the particles are evanescent, that conscious life—with all its subjective temporal structure—will happen all at once according to the objective time of its world!

If this is right, then on a computational theory of mind, we can have an internal temporally structured conscious life with a time sequence that has nothing to do with objective time.

One can easily get out of this consequence by stipulating that mind-constituting computations must be arranged in an objective temporal direction. But I don’t think a computationalist should add this ad hoc posit. It is better, I think, simply to embrace the conclusion that internal subjective time need not have anything to do with external time.

Heavier objects fall sooner

We like to say that Galileo was right that more massive objects don’t fall any faster than lighter ones, at least if we abstract away from friction.

But it occurred to me that there is a sense in which this is false. Suppose I drop an object from a meter above the moon, and measure the time until impact. If the object is more massive, the time to impact is lower. Why? Because there are two relevant gravitational accelerations that affect the time of impact: the moon pulls the object down, but simultaneously additionally the object pulls the moon up. The impact time is affected by both accelerations, and the more massive the object, the greater the upward acceleration of the moon, even though the object's acceleration is unaffected by its mass.

Of course, if we are dropping a one kilogram ball, the gravitational acceleration it induces on the moon is about 1/10²³ of the gravitational acceleration the moon induces on it. It’s negligible. But it’s still not zero. :-) A heavier object of the same size will impact sooner.

If all this is unclear, think about the extreme case where we are “dropping” a black hole on the moon.

Physicalism, consciousness and history

Many physicalists think that conscious states are partly constituted by historical features of the organism. For instance, they think that Davidson’s swampman (who is a molecule-by-molecule duplicate of Davidson randomly formed by lightning hitting a swamp) does not have conscious states, because swampman lacks the right history (on some views, one just needs a history of earlier life, and on others, one needs the millenia of evolutionary history).

I want to argue that probably all physicalists should agree that conscious states are partly constituted by historical features.

For if there is no historical component to the constitution of a conscious state, and physicalism is true, then conscious states are constituted by the simultaneous arrangement of spatially disparate parts of the brain. But consciousness is not relative to a reference frame, while simultaneity is.

Here’s another way to see the point. Suppose that conscious states are not even partly constituted by the past. Then, surely, they are also not even partly constituted by the past. In other words, conscious states are fully constituted by how things are on an infinitesimally thin time-slice. On that view, it would be possible for a human-like being, Alice, to exist only for an instant and to be conscious at that instant. But now imagine that in inertial reference frame F, Alice is a three-dimensional object that exists only at an instant. Then it turns out that in every other frame than F, Alice’s intersection with a simultaneity hyperplane is two-dimensional—but she also has a non-empty intersection with more than one simultaneity hyperplane. Consequently, in every frame other than F, Alice exists for more than an instant, but is two-dimensional at every time. A two-dimensional slice of a human brain can’t support consciousness, so in no frame other than F can Alice be conscious. But then consciousness is frame-relative, which is absurd.

Once we have established:

1. If physicalism is true, conscious states are partly constituted by historical features,

it is tempting to add:

2. Conscious states are not even partly constituted by historical features.

3. So, physicalism is not true.

But I am not very confident of (2).

If materialism is true, we can't die in constant pain

Here is an unfortunate fact:

1. The last minute of your life can consist of constant conscious pain.

Of course, I think all pain is conscious, but I might as well spell it out. The modality of the “can” in this post will be something fairly ordinary, like some sort of nomic possibility.

Now say that a reference frame is “ordinary for you” provided that it is a reference frame corresponding to something moving no more than 100 miles per hour relative to your center of mass.

Next, note that switching between reference frames should not turn pain into non-pain: consciousness is not reference-frame relative. Thus:

2. If the last minute of your life consists of constant conscious pain, then in every reference frame that is ordinary for you, in the last half-minute of your life you are in constant conscious pain.

Relativistic time-dilation effects of differences between “ordinary” frames will very slightly affect how long your final pre-death segment of pain is, but will not shorten that segment by even one second, and certainly not by 30 seconds.

Next add:

3. If materialism is true, then you cannot have a conscious state when you are the size of a handful of atoms.

Such a small piece of the human body is not enough for consciousness.

But now (1)–(3) yield an argument against materialism. I have shown here that, given the simplifying assumption of special relativity, in almost every reference frame, and in particular in some ordinary frames, your life will end with you being the size of a handful of atoms. If materialism is true, in those frames towards the very end of your life you will have to exist without consciousness by (3), and in particular you won’t be able to have constant conscious pain (or any other conscious state) for your last half-minute.

Second-order awareness

Here is a plausible Cartesian thesis:

1. There is a kind of second-order awareness of first-order conscious states that is never mistaken: whenever you have the awareness, then you have the first-order conscious state.

I don’t mean this to be true by packing factiveness into awareness.

But also plausibly:

2. A veridical awareness of an event is caused by the event it is the awareness of.

And:

3. Causation always involves a time-delay.

Given (1)–(3), suppose that you are having the second-order awareness, which is being caused byt he first-order conscious states, and then suddenly the first-order conscious state ends. Since causation always involves a time-delay, it follows that the second-order awareness lags after the end of the first-order conscious state—that there is a time at which you have the second-order awareness even though the first-order conscious state has ended. And this contradicts (1).

I am inclined to deny (1).

But there are other paths out. One could deny (2) and say that in some cases, awareness is not caused but partly constituted by the event it is the awareness of. That’s how God’s awareness of the world has to work by divine simplicity. So the argument provides a tiny bit of evidence for that picture of divine awareness.

Or one might deny (3). Apart from some phenomena like collapse of entangled quantum states or a particle’s position causing a field at the precise position of the particle, which phenomena don’t seem relevant to the case at hand, the best way out here would be to deny naturalism, allowing that a non-physical first-order awareness could instantly cause a non-physical second-order awareness.

Consciousness and plurality

One classic critique of Descartes’ “cogito ergo sum” is that perhaps there can be thought without a subject. Perhaps the right thing to say about thought is feature-placing language like “It’s thinking”, understood as parallel to “It’s raining” or “It’s sunny”, where there really is no entity that is raining or sunny, but English grammar requires a subject so we through in an “it”.

There is a more moderate option, though, that I think deserves a bit more consideration. Perhaps thought has an irreducibly plural subject, and in a language that expresses the underlying metaphysics better, we should say “The neurons are (collectively) thinking” or maybe even “The particles are (collectively) thinking.” On this view, thought is a relation that holds between a plurality of objects, without these objects making up a whole that thinks. This, for instance, is a very natural view for physicalist who is a compositional nihilist (i.e., thinks that only simples exists).

It seems to me that it is hard to reject this view if one’s only data is the fact of consciousness, as it is for Descartes. What kills the three-dimensionalist version of this view, in my opinion, is that it cannot do justice to the identity of the thinker over time, since there would be different pluralities of neurons or particles engaged in the thinking over time. And a four-dimensionalist version cannot do justice to the identity of the thinker in counterfactual scenarios. However, this data isn’t quite as self-evident as what Descartes wants.

In any case, I think this is a view that non-naturalists like me need to take pretty seriously.

Unifying Separation and Choice

Let's round out Axiom of Choice Week. :-)

It’s occurred to me that there is a somewhat pleasant way to integrate the Axioms of Separation and Choice into one axiom schema.

Let’s say that a formula F(x,y) is a partial equivalence (is that the right term?) provided that it’s symmetric and transitive. Now consider this schema (understood to be universally closed over all free variables in F other than x and y):

• If F(x,y) is a partial equivalence, then for any set a there is a subset b such that for every x ∈ b we have F(x,x), and for any x ∈ a such that F(x,x), there is a unique y ∈ b such that F(x,y).

We might call this the Axiom (Schema) of Representatives.

To get the Axiom of Separation, given a formula G(x), let F(x,y) be the formula G(x) ∧ y = x. To get the Axiom of Choice, if c is a set of nonempty disjoint sets, let F(x,y) be ∃d ∈ c(x∈d∧y∈d) and let a = ⋃c (so we need the Union Axiom).

So what?

Nothing earthshaking.

But, first, while there is an advantage to keeping axioms separate for purposes of proving independence results, the more unified our axiomatic system is, the less ad hoc it looks. Unifying Separation and Choice can make us less suspicious about Choice, for instance.

Second, the Axiom Schema of Representatives has nice analogues in some other contexts than set theory. It seems to directly generalize to classes, for instance. Moreover, it extends very nicely to plural quantification to integrate Plural Comprehension with a version of Choice:

• If F(x,y) is a partial equivalence, then there are bs such that (i) for every x among the bs we have F(x,x), and (ii) for any x such that F(x,x), there is a unique y among the bs such that F(x,y).

I don’t know if there is a natural way to extend this to mereology.

One might complain that partial equivalence is less natural than equivalence. I don’t think so. First, it is defined by two instead of three conditions, which makes it seem more natural. Second, examples of partial equivalence relations tend to be more natural than examples of full equivalence relations if our domain is all of reality. For instance, “same color”, “same shape”, “same size”, “same species”, etc., are all partial equivalence relations, since only things with color are the same color as themselves, only things with shape are the same shape as themselves, etc. To form full equivalences, one needs to stipulate awkward relations like “same color or both colorless”.

Fusion and the Axiom of Choice

Assume classical mereology. Then for any formula that has a satisfier, there is a fusion of all of its satisfiers. More precisely, if ϕ is a formula with z not a free variable in ϕ, then the universal closure of the following under all free variables is true:

1. ∃xϕ → ∃zF_ϕ, x(z)

where F_ϕ, x(z) says that z is a fusion of the satisfiers of ϕ with respect to the variable x (there is more than one account of what exactly the “fusion” is). This is the fusion axiom schema.

Stipulate that a region of physical space is a fusion of points.

Question: Is there a nonmeasurable region of (physical) space?

Assuming the language for formulas in our classical mereology is sufficiently rich, the answer is positive. For simplicity, suppose that physical space is Euclidean (the non-Euclidean case is handled by working in a small neighborhood which is diffeomorphic to a neighborhood of a Euclidean space). Let ψ be the isomorphism between the points of physical space and the mathematical space R³. Let ϕ be the formula ψ(x) ∈ y. Applying (1), we conclude that for any subset a of R³, there is a set of points of physical space that correspond to a under ψ. If we let a be one of the standard nonmeasurable subsets of R³, we get an affirmative answer to our question.

But now we have an interesting question:

2. What grounding or explanatory relation is there between the existence of a nonmeasurable region of physical space and the existence of a nonmeasurable subset of mathematical space?

The two simplest options are that one is explanatorily prior to the other. Let’s explore these.

Suppose the existence of a nonmeasurable physical region depends on the existence of the nonmeasurable set. Well, it is a bit strange to think of a concrete object—a region of physical space—as partly grounded in the existence of a set. This doesn’t sound quite right to me.

What about the other way around? This challenges the fairly popular doctrine that complex things entities are a free lunch given simples. For if the existence of the nonmeasurable region is prior to the existence of an abstract set, it seems that we actually have quite a significant metaphysical “effect” of this complex object.

Moreover, if the existence of the nonmeasurable region is not grounded in the existence of nonmeasurable set, whether or not there is grounding running the other way, we have a difficult question of why there is in fact a nonmeasurable region. Without relying on nonmeasurable sets, it doesn’t seem we can get the nonmeasurable region out of the axioms of classical mereology. It seems we need some sort of a mereological Axiom of Choice. How exactly to formulate that is difficult to say, but one version that is enough for our purposes would be that given any formula ρ(x,y) that expresses a non-empty equivalence relation on the simples satisfying ϕ, there is an object z such that if ϕ(x) then there is exactly one simple x′ such that ρ(x,x′) and x′ is a part of z, and every object that meets z meets some simple satsifying ϕ.

But my intuition is that a mereological Axiom of Choice would badly violate the doctrine that complex objects are a free lunch. If all we had in the way of complex-object-forming axioms were reflexivity, transitivity and fusion, then it would not be crazy to say that complex objects are a fancy way of talking about simples. But the “indeterministic” nature of the Axiom of Choice does not, I think, allow one to say that.

Modality and the Axiom of Choice

Suppose that the set theory of our world is a Solovay model, where we don’t have the Axiom of Choice (AC), and where every subset of the reals is Lebesgue measurable. Now imagine that God picks out a line in space, and defines the Vitali equivalence relation for points on that line (where two points are equivalent if and only if the distance between them is a rational number). It is then surely within God’s power to create a particle of some unexemplified type T at exactly one point in every equivalence class. There is nothing incoherent about that! But if God did that, then there would surely be a set of the points containing a particle of type T. And that set would be a nonmeasurable Vitali set.

So what?

Well, prima facie, there are three possibilities about the existence of nonmeasurable sets:

1. Necessarily, there are no nonmeasurable sets.

2. Necessarily, there are nonmeasurable sets.

3. It is contingent whether there are nonmeasurable sets.

My argument strongly suggests that if there are no nonmeasurable sets, it is nonetheless possible that there are nonmeasurable sets. Hence, (1) is ruled out.

So we have an argument for the disjunction of (2) and (3).

Now, I think a lot of people have the intuition that mathematical facts are necessary. If so, then (3) is ruled out. They will see this as an argument for (2).

I don’t see it that way myself: I am quite open to contingent mathematical truths.

More generally, the argument shows that:

4. For any set of disjoint nonempty subsets of the reals, it is possible that there is a choice function.

Again, if the existence of pure sets is not a contingent matter, we conclude AC is true for all subsets of the reals.

An odd thought about ZFC

The axioms of ZFC set theory can be divided into (a) the positive axioms, that say that a set with certain properties exist, and (b) two negative axioms that deny the existence of certain sets (Extensionality: given any set, there is no other set with the same members; Regularity: no irregular sets).

The positive axioms divide further into two classes: (i) those that are obvious special cases of naive set theory’s Axiom of Comprehension, and (ii) the Axiom of Choice.

Here is an alternate intellectual history thought experiment. Suppose we never discovered the contradiction in naive set theory or anything like it, maybe because we had a psychological block against thinking about non-self-membered sets, applying Cantor’s Theorem to the universal set, etc. The Axiom of Choice would continue to have an intuitive plausibility, and the “mathematical need” for it, say in the case of the Hahn-Banach Theorem, would likely still arise. And so we would be pulled to adopt it.

This makes me think this. The other positive axioms of ZFC (i.e., the positive axioms of ZF) have an ad hoc feel to them. They are special cases of Comprehension, carefully chosen to both give enough applications of Comprehension and to avoid contradiction (we hope). I feel that much of the plausibility of the other positive axioms of ZFC comes from their being special cases of the highly intuitive—but incoherent—Axiom of Comprehension. And that’s a little suspicious.

Normally one thinks of the Axiom of Choice as the most suspicious of ZFC’s axioms. But here we have a source of suspicion for axioms of ZFC that does not affect Choice.

Well, maybe. Maybe an enemy of Choice could say that both Choice and Comprehension are the fruit of the poisonous tree of principles of plenitude.