Schaffer and well-foundedness of priority

Jonathan Schaffer in his famous “Monism: The Priority of the Whole” paper assumes that priority relations are well-founded. The standard definition of well-foundedness in this context would say that every non-empty set has an element that has nothing prior to it in that set.

Imagine a world w with a continuously subdivided entity E made of pointlike simples (like a real line is made of points in standard mathematical constructions) that lies between the 0 and 1 coordinates on the x-axis, and suppose that for any real number a strictly between 0 and 1 there is the part E_a of E that is composed of all the simples with x-coordinate less than a. Assume further that either proper parts are always prior to wholes or wholes are always prior to proper parts.

In either case, we don’t have well-foundedness. For let C be the chain consisting of all the objects E_a for 0 < a < 1. Since E_a is a proper part of E_b if and only if a < b, C does not have a minimal element on either of the two priority relations (the part-before-whole and the whole-before-part one).

Schaffer’s justification for assuming well-foundedness is that without it, we could have a world with no basic entities. But notice that although w violates well-foundedness, it could still have basic entities. For it could be an atomic world—E itself is made of pointlike simples—in the case of the part-to-whole priority relation and it could be a priority monist world.

Perhaps Schaffer is misusing the term “well-founded”. He glosses it as “all priority chains terminate”. This is ambiguous between two readings. On a stronger reading, any priority chain terminates inside itself: there is an item in the chain that is prior to all the other items in the chain. Assuming the Axiom of Choice, Zorn’s Lemma shows that this stronger reading is equivalent to the standard definition of well-foundedness. But on the weaker reading, the idea would be that for any priority chain C, there is an entity that is prior to all the entities in C other than perhaps itself, but without the entity being required to be itself in the chain. This is equivalent to the priority ordering being the reverse of an inductive order.

On the weaker reading, assuming the Axiom of Choice, each non-basic item has something basic prior to it. This seems to me to be what Schaffer needs for his arguments.

That said, I think there is something to be said for the standard sense of “well-founded”, even if it is incompatible with the above world with the continuously subdivided entity E and whole-to-part or part-to-whole priority. Indeed, I am now thinking that well-foundedness is a really nice condition, but assuming there are continuously subdivided entities it does rule out both kinds of priority relations. But that should be fine. And in fact Schaffer could embrace the denial of whole-to-part and part-to-whole priority. In that paper, he only commits himself to integrated wholes being prior to their parts.

Ontology matters

Among things that metaphysicians wonder about is the question of what things referred to in ordinary language exist: Are there tables, chairs, people, shadows, holes, rocks, particles, etc.?

When metaphysicians discuss this question, others are apt to wonder about a different question: Why does anyone care? After all, the metaphysics will admit that whether there are chairs or not, there are particles (or there is a wavefunction) that’s chairwise arranged, and that’s what we really care about, no?

Here are two reasons to care, assuming theism. All the things that exist are either God or lovingly created by God. Thus, in finding out that a certain kind of non-divine thing exists, we have found out that God has lovingly created the thing. And to find out that God has lovingly created a thing surely matters. Furthermore, anything that exists is either God or its being is a participation in God. And to find out that there is a case of participation in God surely matters.

If time is necessary, it's discrete

Say that a total order is locally finite provided that between any two elements there are at most finitely many elements. A locally finite total order is discrete: each non-extreme element has an immediate predecessor and successor.

Here’s a valid argument:

1. If time is necessary, the order structure of time cannot be problematically arbitrary.

2. The only possible order structures for time that are not problematically arbitrary are locally finite.

3. Thus, if time is necessary, it has to be locally finite, and in particular discrete.

Premise (1) is motivated by my and Josh Rasmussen’s research project on which fundamental and necessary things cannot be problematically arbitrary. But if time is necessary, it is very likely to be fundamental.

Why accept (2)? I am thinking like this. Suppose there are two times between which there are infinitely many times. The size of that infinity of times is arbitrary—is it countable, is it the continuum, is it something in between (if there is anything in between), is it something even bigger?

So, if time is necessary, the order of time is locally finite.

Up to isomorphism, the following are the only possible locally finite total orders:

1. ...,  − 3,  − 2,  − 1, 0, 1, 2, 3, ...: no beginning or end

2. ...,  − 3,  − 2,  − 1, 0: end but no beginning

3. 0, 1, 2, 3...: beginning but no end

4. 0, 1, 2, 3, ..., n (for some finite n): beginning and end.

Now a complex thesis:

4. If time is necessary, then either (i) necessarily time has structure A or B, or (ii) necessarily time has structure C or D.

Here’s why. If there were possible worlds where time has structure A or B and other possible worlds where time has structure C or D, it would be problematically arbitrary whether time starts off with an infinite past or a finite past.

Can we go further? Maybe, but I get less confident. Still, it seems like it would be weird that time could end at some point, but had to have started. Thus:

5. If time does not have to go on endlessly, there need not have been time at all.

It follows that:

6. If time is necessary, then it necessarily has structure A or it necessarily has structure C.

In other words, if time is necessary, it must be like the integers (option A) or like the natural numbers (option C).

Can we narrow it down further? Well, causal finitism suggests but does not prove that options with no beginning are problematic. That would get one down to option C. But this is a weak argument. Option A also has the disadvantage that it leaves unexplained why events in time didn’t all happen a day earlier than they did.

In any case, there is another move we can make at this point. It follows from 6 that:

7. If time is necessary, all the specifics of the order structure of time are necessary.

I am personally inclined to add:

8. The specifics of the order structure of time are not necessary.

9. So, time is not necessary.

But my intuitions for 8 aren’t that strong, so I don’t think this is a very strong argument for the contingency of time.

Evanescence

Compare two worlds:

• Evanescent World (EW): There is a rich three-dimensional reality that exists for only one moment of time.

• Timeless World (TW): There is a rich timeless three-dimensional reality.

Intuitively, EW is, well, evanescent, while TW is “permanent”.

On a B-theory of time, it is difficult to make sense of any such difference in evanescence between EW and TW. But a staunch B-theorist will say that this is a feature, not a bug.

However, an A-theorist will, I think, want to distinguish EW from TW. But it is difficult to do so.

Start with a moving spotlight theory. The difference between TW and EW seems to be that in EW we have time’s spotlight, and in Timeless we don’t. But time’s spotlight in EW is always (i.e., at all times, of which there is only one) lighting all of the three-dimensional reality. The spotlight is no more evanescent than the rest of the reality, and so adding the spotlight does not seem to make the Evanescent world be really evanescent.

Now, consider any of the three major A-theories: moving spotlight, presentism and growing block. At the one and only moment of time in EW, one could say (if one could speak in one moment):

1. It is not the case that there will be anything

2. It is not the case that there was anything.

But one could say the very same thing in TW (if one could speak timelessly, which is no harder than speaking in one moment). And in neither EW nor in TW can one say things like:

3. Reality will vanish.

For “will” cannot reach beyond time.

What can the A-theorist do? I think they can do only thing: They can say that EW is evanescent precisely because:

4. In EW there is exactly one moment of time.

But this is something that the B-theorist can say as well. In other words, the alleged dynamism of time on the A-theory does not seem to do anything to help explain why EW is evanescent.

So what? Well, if the A-theory explains anything better than the B-theory does, it better explains more ordinary evanescence, such as the evanescence of our lives—things flow out of existence with time, or something like that. But now take the limit case of ordinary evanescence: an entity that exists for only one moment, in a world like ours which begins long before the entity and ends long after it. And notice that the entity that exists for one moment of time in a world like ours is no more evanescent than every entity in EW. Thus there is something wrong with an A-theoretic explanation for evanescence that applies to the entity in a world like ours but not to all the entities of EW—it fails to really get at what evanescence is.

But if the A-theory really doesn’t explain evanescence better than the B-theory does, then I suspect it fails to be the better alternative.

Consensus gentium and double counting

Here’s a way that an atheist could undercut the consensus gentium argument (the argument for theism based on how much belief in God or gods there is in the world), building on my previous post.

Suppose that views about God or “big gods” came into the world as follows: Brilliant very early thinkers came up with plausible arguments for the existence of God. These arguments may have, for instance, included some selection taken from among cosmological, teleological, moral and religious experience arguments. These arguments were rationally taken up by their followers, but eventually supernaturalistic and maybe even theistic belief was simply passed on by testimony, with the selective advantages of supernaturalism (say, social cohesion effects) helping to keep communities where these beliefs were widespread going.

If this is right, then treating the consensus gentium argument as evidence for theism involves double counting. For the theist presumably also counts the evidential force of arguments like cosmological, teleological, moral and religious experience ones, and since on the above story theism (a) started with one or more of these, and (b) its further spread and persistence can be explained naturalistically quite well by selection effects.

Now, the above argument-focused story may not be plausible to all. But the atheist can make a further move. Dilemma. Either the above story is true or not. If it’s true, then the consensus gentium argument involves double counting. If it’s not true, then likely theism started in some way that is not rational, and then spread and persisted by naturalistic selection effects, and that’s not evidence for theism, either. So, either way, we can dismiss consensus gentium?

What do I say to this? As a theist who is convinced that the other arguments for theism do provide significant evidence, I don’t find it particularly hard to just concede the point.

But there is another move: a merely partial concession. The story about how belief in God is helpful to communities is a plausible story. But it is even more plausible on theism! For on theism, there is an additional probability that God would “go out of his way” and have rational animals exist in conditions where belief in God is helpful, precisely as a means to making belief in God persist and spread naturally. Thus, widespread of theistic belief may be probable on naturalism, but on theism is even more so. This may not be strong evidence for theism (the Bayes ratio can’t exceed 2, which is “substantial” but not “strong” on Jeffreys’ scale), but is still some evidence.

Debunking and testimony

An evolutionary debunking argument against theism can be employed defensively or offensively by an atheist. Applied defensively, it is a response to the theist’s offering an argument from the prevalence of belief in God. Applied offensively, it is suppose to show that whether or not God exists, because theistic beliefs come from a suspect source, they are not knowledge.

The focus of debunking argument tends to be on mimetic selection. The idea is that theistic (or maybe more generally supernaturalistic) beliefs confer some kind of a selective advantage on the communities or individuals exhibiting them, and hence we can explain naturalistically the prevalence of these beliefs. This explanation works whether or not God exists, and hence whether or not God exists, typical theists’ beliefs are not knowledge.

But there is a serious gap in this argument. Mimetic selection only operates on a belief once the belief is in place. But we still have to ask: Where did the belief come from in the first place? And I think that if we are to be honest, we have to say that not only is this something we don’t empirically know, but given how it concerns fine-grained details of events before written history, we have no empirical access to the answer. For any particular lineage of theistic beliefs, there was a first believer in the lineage. How did they come to believe? Well, we can make up a bunch of ideas:

1. They were philosophically inclined and came up with a good argument for theism (say, a good design argument).

2. They were philosophically inclined and came up with a bad argument for theism (say, a bad ontological argument).

3. They got an apparent perception of God from eating bad mushrooms.

4. They got an apparent perception of God from God.

5. They were bullied into believing in God by a tribal leader who claimed divine right of ruling.

6. They heard a fairy tale told by their parents and misunderstand it as history rather than fiction.

And a lot more. The point is that empirically we have no idea. This was very long ago, and we have no records of how it went. And it may be that for different mimetic lineages of theism the initial explanation is different.

Now notice that the options include ones that are friendly to theistic knowledge (1 and 4) as well as ones that are unfriendly (the others). Furthermore, if the lineage begins with one of the options friendly to theistic knowledge, then we might say that this knowledge was then testimonially transmitted. Of course, the selective facts helped with the transmission. But as long as the chain of testimony goes back to a case of genuine knowledge, the fact that the transmission was made possible by selective effects that are not as such truth conducive may not undercut the possibility that the chain actually does transmit knowledge.

Thus, once we think of the variety of ways that the lineage of theistic beliefs could begun, we see that an offensive debunking argument fails, as the offensive debunker would need to offer evidence that our lineage of theistic belief began in one of the knowledge-unfriendly ways. On the other hand, the same line of thought leaves unaffected the defensive debunker, who only needs a serious possibility that the lineage of theistic belief began in a way that gives no special credence to theism.

I think this line of thought is most plausible if you have an irreducible view of testimonial evidence.

Abstract objects and the Nicene Creed

There are Christians who believe in the existence of abstract objects, such as properties, that are neither grounded in God’s mind (as on Augustine’s view) nor created by God. One challenge on such a view is to make sense of the doctrine of creation which says that God creates everything other than himself.

A natural response is to restrict the quantifiers further. God is not the only exception to being created by God. Abstract objects are also an exception. When our Christian and Jewish forebears said that God created everything, they were only thinking of more ordinary objects than the abstract ones.

I doubt this. Take the beginning of the Creed of Nicaea:

I believe in one God, the Father almighty, maker of all things visible and invisible [poiêtên oratôn te pantôn kai aoratôn].

Now, the Fathers at Nicaea were more than aware of Platonic philosophy and of a broad variety of neo-Platonic views on which there were all sorts of uncreated entities, perhaps emanating from the One. Against that background, it seems very plausible that their talk of creation of the invisible is not only meant to tell us about angels, but is also meant to rule out such views. What invisible things reality might contain was mysterious and contentious at the time, and remains mysterious and contentious to our day, and the explicit inclusion of the invisible makes it quite implausible to think that the Creed is meant to apply to a restricted class of beings (apart from the obvious exclusion of God himself).

Furthermore, I think the use of the concrete word poiêtês (maker), as opposed to some more general philosophical term like aitia (cause) or archê (origin, principle) carries the message that God the Father almighty is engaging in intentional agency in producing the visible and the invisible. The agentiality and intentionality implied by the word seems to me to rule out—and quite possibly with the specific intention of doing so—emanationist views.

We are used to reciting the creed in the later version from the Council of Constantinople, where “of heaven and of earth” is added between “maker” and “of all things”. It may be tempting to see “of heaven and of earth” as restricting the visible and the invisible to the concrete occupiers of heaven and of earth. I think that would be a mistake. The Council of Nicaea was an ecumenical council in its own right, and its creed is still normative. If I am right in thinking that the Fathers really wanted to rule all the exotic uncreated entities of neo-Platonism, that remains binding. I would instead say that the Fathers at Constantinople added a clarification that it is not only the mundane things that God created, but also the heavenly. They weren’t narrowing the scope of creation.

Theoretical parsimony

As is usual, distinguish ontological parsimony, namely positing fewer beings, from ideological parsimony, namely positing fewer fundamental undefined concepts.

The principle that we should assign higher probabilities to more ontologically parsimonious theories is wacky. It violates the Principal Principle. If you are flipping a coin and planning to have children if and only if you get heads, then you should assign a slightly higher probability to tails. (Granted, this argument fails if eternalism is false. But you can tweak it by supposing that a hundred years ago Alice flipped a fair coin with that intention, and now you should again assign a slightly higher probability to tails.)

What about ideological parsimony? It initially sounds a lot better to assign higher probabilities to more ideologically parsimonious theories. Indeed, I think there has to be something right about such a principle of simplicity. But I don’t at present fully know how to formulate it in a way that escapes objection.

For suppose that we are non-reductionists about biological species, so we think that species are fundamental undefined concepts. You are working with a bacterium in the lab, and on a whim plan to flip a coin. On heads, you will irradiate the bacterium, and on tails you will do no such thing. Irradiated, the bacterium is more likely to produce a new species when it reproduces. So on ideological parsimony it seems we should assign a slightly higher probability to tails, as then it is more likely that the true theory of the world posits fewer species, and hence fewer fundamental undefined concepts.

Objection 1: Reductionism is the right theory about biological species.

Response: Maybe it is. But it’s not the case that it certainly is. And as long as there is a non-zero probability that non-reductionism about species is right, we should assign a slightly higher probability to tails—for conditionally on reductionism, the probabilities are equal, while conditionally on non-reductionism, we should assign a higher probability on tails.

Objection 2: There are so many species that the difference from positing one more is negligible.

Response: Maybe it is. But now parsimony cannot do the work it’s designed to do, and hence it is largely unmotivated. Here’s an example of the kind of work it’s designed for. You observe the world’s koalas and see no significant differences in behavior or appearance. You could suppose there are two species of koalas and you just haven’t been able to tell them apart. But that’s not parsimonious. So you suppose one species. But given how large the number of species on earth is, positing one more makes little difference—assuming the objection is correct.

Objection 3: A full story of the world will include the causal powers in the world. If the radiation has the causal power C_S to produce a new species S of bacteria, then a full description of the world will contain a description of C_S, and hence of S, even if S has no members. Thus it makes no difference to the ideological complexity of the world whether the radiation is applied or not.

Response: This seems the most powerful objection. But it might undercut the applications of parsimony. Plausibly, the world has a first cause behind everything else. Now either the first cause’s causal powers are such that their complete description includes all possible kinds or not (this may depend on questions involving the first cause’s simplicity). If it has such causal powers, then ideological parsimony does rather less work than we thought, since the total description of reality already includes the concepts of the two species of koalas as per my previous response. Suppose now the first cause’s causal powers are not such that their complete description includes all possible kinds. Imagine that you make the following plan. You will flip a fair coin. On heads you will pray that the first cause produce a new kind of thing. Now, the first cause might not fulfill your prayer—indeed, perhaps you are not even confident the first cause is a person. But it is at least slightly more likely that the first cause will make a new kind of thing if you pray than if you don’t. And that’s enough to change the epistemic probabilities for your coin flip, in an implausible way.

Six-dimensional universalism

Four-dimensional mereological universalism has been proposed as a way of letting ordinary objects into our ontology. If all pluralities have a fusion, one should expect a fusion of simples (or maybe time-slices of simples or something like that) that exactly correspond to the four-dimensional career of a chair, a horse, and an oak tree. But then some philosophers started taking seriously the observation that it’s not enough to have a fusion of the simples in an oak to have an oak. For the fusion of simples may not have the same modal properties as the oak. This led to what I call five-dimensional mereological univeralism. Say that a modal profile is a function that assigns to each world an empty or non-empty collections of simples (or time-slices of simples, etc.)

The five-dimensionalist mereologist then holds that for any modal profile that has at least one non-empty assignment there exists an object that matches that profile—an object that exists in all and only the worlds that the assignment assigns a non-empty set to, and in these worlds is a fusion of the simples in the assigned set.

I just realized another thing. The ordinary objects that the universalist wants to have in their ontology include artifacts like tables and ships. The vagueness of artifacts is deeply embedded into our thinking about them. While we could become convinced of the idea (indeed, I am convinced of it) that living things have souls, and the metaphysics of the relationship of these souls to bodies ensures that living things have non-vague spatial, temporal and modal boundaries, the analogous thesis for tables and ships seems much harder to swallow.

But now modal profiles are sharp. Thus, to model the vagueness of artifacts, we not only need every modal profile to be be filled, but we need every non-trivial vagueness profile to be filled. What’s a vagueness profile filled by a possible object x? To a first approximation, it’s a set of modal profiles, so that an object x has vagueness profile V if and only if it is definite that x fits some profile in V, and definite that x does not fit any profile outside V, and for any profile in V it’s vague or definite that x fits that profile. In other words, for mereological universalism to save all of our common sense about ordinary objects, it needs to be a six-dimensional universalism.

But six-dimensional universalism is crazy. Consider the vagueness profile R corresponding to Rover the dog in England and the vagueness profile F corresponding to Felix the cat in Canada. Then take their union: R ∪ F. If this vagueness profile is filled, it’s filled by an entity such that it is vague that it barks and vague that it meows, but definite that it clearly barks or clearly meows. En entity such that it is vague that it is in England and vague that it is wholly in Canada, but definite that it is in exactly one of England and Canada. And so on.

(Things are even worse if theism is true. Take God’s vagueness profile and mine. The union of these two profiles would be filled by an entity that is vaguely divine and vaguely a sinner.)

So, the argument goes like this. Mereological universalism only fully saves common sense if it is extended to six-dimensional universalism. The latter is absurd. So we’re not going to fully save common sense by going universalist.

I suppose one might be satisfied with stopping at five dimensions, at modal profiles, and ignoring our judgments of vagueness about artifacts. Maybe.

Some arguments against panpsychism

1. If pansychism is true, all organisms are always conscious.

2. We are not always conscious.

3. We are organisms.

4. So, some organisms are not always conscious.

5. So, panpsychism is not true.

We can get some more specific conclusions. It is very plausible that:

6. We are not conscious during deep sleep.

7. If we are not conscious during deep sleep, primitive plants are not conscious.

8. If primitive plants are not conscious, neither are particles.

9. So, neither primitive plants nor particles are conscious.

Why think (7)? Because our lower-brain functioning in deep sleep is surely still way more sophisticated than the coordinational functioning of primitive plants, and it is plausible there is a correlation between degree of functioning and level of consciousness. A similar line of thought yields (8).

Suppose it’s disputed that we are not conscious during deep sleep. Maybe we are conscious then, but don’t remember. Then I have another argument for the non-consciousness of primitive plants.

10. Right now I am now not directly conscious of the activity of my autonomous nervous system.

11. If right now I am not directly conscious of the activity of my autonomous nervous system, primitive plants are not conscious.

12. So, primitive plants are not conscious.

To deny (10) would to fall into wacky skepticism. It would posit that I may have conscious states that, as it were, I am completely unaware of. Note that I am often (always?) aware of various effects of the autonomous nervous system’s functioning, such as my breathing.

Premise (11) is where I think the hard work happens. But the thought is again about sophistication of coordinational functioning, and my guess is that my autonomous nervous system is more sophisticated than its analogues in primitive plants.

I think the general line of argument yields something potentially interesting. By knowing that we are not aware of what’s going on in our autonomous nervous system, we know how much coordinational sophistication is insufficient for consciousness.

I am not a biologist. Don’t ask me what primitive plants are. I just made that concept up, and I could be wrong in thinking that anything in reality corresponds to it.

An argument that organisms aren't wholly material

Here’s an argument for non-materialism about all organisms.

Start with these premises:

1. Organisms exist.

2. If organisms exist, it is possible to have two organisms that have all their matter in common.

3. Antisymmetry of parthood: Two distinct things can’t each be a part of the other.

4. Strong supplementation: If y isn’t a part of x, then y has a part that doesn’t overlap x.

Now suppose that x and y are two organisms that have all their matter in common. By antisymmetry of parthood, they can’t both be mutual parts of each other. Swapping labels if necessary, suppose y isn’t a part of x. Then by strong supplementation, y has a part z that doesn’t overlap x. But any material part of y is a material part of x by (2). Thus, y has a part that isn’t a material part. Hence:

5. It is possible to have an organism with a non-material part.

But this is weak in two ways. First, it doesn’t show that any actual organism has a non-material part. Second, even if we showed that, that wouldn’t show that all organisms have non-material parts. Moreover, I have given no argument for premise 2.

I will fix all these problems together. In a recent post, I gave a plausible construction of a world where there is a dog-like organism and a cat-like organism with all their matter in common. The same construction would give the following conclusion: there is a world where there is a mushroom-like organism and a grass-like organism thathave all their matter in common. Plugging that into the above argument yields:

5. Possibly a mushroom-like or a grass-like organism has a non-material part.

The likeness here is functional: the way of life of the organism is like that of mushroom or a grass plant. Then add:

6. If possibly a mushroom-like organism has a non-material part, all organisms have non-material parts.

7. If possibly a grass-like organism has a non-material part, all organisms have non-material parts.

From 5-7, we get:

8. All organisms have non-material parts.

Why believe 6 and 7? Here’s a thought. You might think that conscious animals have to have souls or something like that. But if a mushroom- or grass-like organism can have a non-material part, probably it actually does, and if it has a non-material part, probably all organisms do.

There is a lot in this argument that doesn’t personally convince me. In fact, I am skeptical of premises 2 and 4, and my main reason for accepting 6 and 7 is that I already accept anti-materialism about all organisms. But I think the argument has some force.

Irreducibly joint activity

I’ve long wondered what it means to say that a bunch of things are “irreducibly jointly producing an effect.” One place this comes up in contemporary metaphysics is strongly non-reductivist materialist theories of mind on which the particles of our bodies working together produce thought.

This irreducible production—if there is such a thing—is supposed to be different from the reducible way of how, say, four rocks can depress an analog spring-based scale. Yet I’ve always found it hard to grasp the difference. After all, no single one kilogram rock has the power to depress the scale to the four kilogram point. You need all four. Why is that not an irreducibly joint production of an effect?

A paradigm example of irreducible production would be something like this, however. Suppose the laws of nature ensure that when you arrange four spherical objects of equal size in a tetrahedron, they begin to whistle (I am adapting an example by Trenton Merricks). You arrange three or five or any other number of marbles, nothing like that happens. What’s the difference?

In the case of the rocks and the scale, it is tempting to say that the outcome is just a matter of “summing forces”. But as we learn from Nancy Cartwright and by reflecting on Hume’s insights, summing forces is not as innocent as it sounds. The world could have been so that when multiple forces impinge, they average, or the greatest force carries the day, or the weakest does. Thus the laws of nature seem to have to include some clause like

1. When a number of objects act on an object with forces, the latter accelerates in proportion to the vector sum of the forces.

But now consider.

2. When a number of spherical objects are arranged, whistling results with an intensity equal to one if the objects are four in number, equal in size, and tetrahedrally arranged, and intensity zero otherwise.

The main difference seems to be that “vector sum” is a simpler and more mathematically natural description than “one if the objects are four in number, equal in size, and tetrahedrally arranged, and zero otherwise”. And that’s just a difference of degree, while the difference between reductively and non-reductively joint action is not a difference of degree.

Here is a thought. Perhaps the worry arises out of a mistaken way to think about the case of the scale. The force on the scale’s platform (subsequently transmitted to the spring) is not the sum of the pressures imposed by the four rocks. In the Newtonian setting, each particle of the platform is affected by gravitational attraction to every other particle with mass in the universe, as well as affected by interaction via other forces (e.g., electromagnetic) with countless particles. There are no small closed systems. Perhaps, then, we should rethink the ordinary case: it isn’t a case of reducibility either. On the contrary, it is a case of an irreducible interaction with, basically, everything else—or, in a relativistic context, everything else in its past light cone.

The difference between the whistling tetrahedron of balls and the scale’s depression then isn’t that the whistling is irreducibly jointly effected by multiple things and the scale’s depression isn’t. It is, rather, that the scale’s depression is jointly effected by vastly more things, most of which have only a slight effect, while the magical-seeming whistling is produced only by the four things.

On this picture, what would reducible joint influence be like? It would have to be a case of independent production, where maybe object x produces property P in object z while object y independently produces property Q in z, so that x and z reducibly jointly produce P ∧ Q in z. But it’s not clear that this ever actually happens in our world—all we have are better and worse approximations to such causally independent production.

Some non-reductivist materialists will want to say that our particles, through irreducibly joint activity, produce mental states. Can we accommodate this in the above story? Yes, but only if we suppose that the mental states are in no way directly affected by anything outside our body.

Here is a worry about what I said. Suppose that the four rocks are sufficient to push the scale to a position where the indicator needle is right around the middle of the 4 kg dashmark, while the sum total of the other forces in the universe is insufficient to shift the needle to the outside of the 4 kg dashmark. Then while the exact position of the needle isn’t effected by the four rocks alone, the fact that the needle points somewhere within the 4 kg dashmark on the scale seems to be accounted for by the four rocks alone. Thus, with respect to the coarse grained event of the needle being within the 4 kg dashmark, the four rocks are sufficient to produce the effect. And if that’s right, then we’ve lost the distinction between the four rocks depressing the scale and four marbles causing whistling.

Conjecturally, I answer as follows. The coarse-grained event of a needle’s being in some range (given by the thickness of the 4 kg dashmark) has an identity dependence on fine-grained event of the needle’s being exactly where it is. Thus, had the fine-grained event been different, the coarse-grained event would have been a numerically different event. Hence the coarse-grained event does actually depend on vast numbers of things.

Completely overlapping lives

Van Inwagen thinks that the very same particles cannot have their activities constitute two life events. I am not so sure.

Pretend for the sake of simplicity that our world is made of fundamental particles with well-defined positions in three-dimensional space, with n basic quantities Q₁, ..., Q_n (mass, charge, etc.), otherwise interchangeable, and with no particles coming in and out of existence. And suppose we have dogs and cats (or something like them).

Suppose, further, that our world contains Rover the dog and Felix the cat. No particle that is ever a part of one of them is a part of the other. Finally, we have a very odd coincidence. There is a one-to-one correspondence ϕ between the particles that are ever a part of Rover and those that are ever a part of Felix such that particle c is a part of Rover at time t if and only if ϕ(c) is a part of Felix at that time. This last condition can be assured if we assume that Rover and Felix don’t change in what particles they have and they have the same number, but we don’t need such a strong assumption.

Now imagine a six-dimensional world made of fundamental particles, with 2n basic quantities U₁, ..., U_n, V₁, ..., V_n. Now suppose:

1. The first three coordinates of the particle positions and the quantities U₁, ..., U_n over time are physically related by laws exactly like the ones that govern the three coordinates of the particle positions and (respectively) the quantities Q₁, ..., Q_n in our world are.

Thus, if we project the six-dimensional space of this world to our three-dimensional space by ignoring coordinates 4–6, and relabel U_i as Q_i, the behavior of the particles will look just like the familiar behavior of particles in our world.

Then add:

2. The last three coordinates of the particle positions and the quantities V₁, ..., V_n over time are physically related by laws exactly like the ones that govern the three coordinates of the particle positions and (respectively) the quantities Q₁, ..., Q_n in our world are.

In other words, likewise if we ignore coordinates 1–3, and relabel V_i as Q_i, the behavior of the particles will be like that of our world’s particles.

Note that the U_i and Q_j quantities don’t interact, and the particle positions with respect to coordinates 1–3 and coordinates 4–6 don’t interact. This is like two non-interacting universes in one, except that the particles are shared between them.

Now suppose that there is a one-to-one correspondence f between the particles of our world and those of the six-dimensional world such that:

3. If at time t, particle c is at (x,y,z) and has basic quantity values Q₁ = a₁, ..., Q_n = a_n, then at t in the six-dimensional world, particle f(c) is at (x,y,z,u,v,w) for some u, v and w, and has basic quantity values U₁ = a₁, ..., U_n = a_n.

And there is also also another one-to-one correspondence g such that:

4. If at time t, particle c is at (x,y,z) and has basic quantity values Q₁ = a₁, ..., Q_n = a_n, then at t in the six-dimensional world, particle g(c) is at (u,v,w,x,y,z) for some u, v and w, and has basic quantity values V₁ = a₁, ..., V_n = a_n.

In theory, these correspondences f and g could be the same. But they are not. There is a twist:

5. If c is ever a particle of Rover, then g(c) = f(ϕ(c)) and if c is ever a particle of Felix, then g(c) = f(ϕ⁻¹(c))

where ϕ is our correspondence between Rover and Felix’s particles. This twist implies that g maps Felix’s and Rover’s particles to particles that f respectively maps Rover’s and Felix’s to.

We can mathematically arrange all this.

Now, fix a time t. If the cs are Rover’s particles at t, let the es consist of the particles corresponding to them under the mapping f. Note that the es are also the particles corresponding to Felix’s particles under the mapping g.

Here’s the fun thing. With respect to their first three coordinates and the quantities U₁, ..., U_n, the es behave just like Rover’s particles. With respect to their last three coordinates and the quantities V₁, ..., V_n, the es behave just like Felix’s particles.

It seems right to say this: the es in our six-dimensional world have two lives, a canine-style life just like Rover’s and a feline-style life just like Felix’s. There is no interaction or unity between these two lives other than due to their being realized in the very same particles.

Suppose someone insists that there must be only one canine-feline hybrid life here because the particles are the same. Modify the mappings slightly, so that the Rover-like life is realized by particles e₁, ..., e_N while the Felix-like life is realized by particles e₂, ..., e_N + 1. Now the particles are not the same, but they are mostly the same. Now we have the same life event but no longer the same particles (just an overlap, like in the case of conjoint twins). But two particles should make little difference here. The slight change in mapping shouldn’t make a difference between having a canine-type life and a feline-type life on the one hand, and having a hybrid canine-feline life.

Note that I am not doing Aristotelian metaphysics here. By “life events”, I mean the kind of empirical event a biologist might talk about, not a form or anything like that. I doubt that the same matter could have two different substantial forms.

Myth of passage

Donald Williams in his famous “The Myth of Passage” argued that time cannot flow, because if it did, then it would require either a meta-time to flow in or would flow at a rate of 1 (i.e., one second per second = one hour per hour = etc.) I’ve always been puzzled by what’s supposed to be wrong with the second option. Is it that if something flows at some rate, then it could have flowed at a different rate? But light appears to be a counterexample to that—it cannot flow but at c.

Here’s another take on what’s wrong with time flowing at a rate of 1. The flow of time is supposed to be intrinsically directional. But there is nothing intrinsically directional with a flow at a rate of one. Imagine that, somehow, our universe is divided into two halves, and in one of them time flows in one direction and in the other in the other direction. In our half, from 1 to 3 pm, it flows at a rate of 2 hrs / 2 hrs = 1. In the other half, from their 3 pm to their 1 pm it flows at a rate of  −2 hrs / −2 hrs = 1. So regardless of the direction that time flows, the rate is 1: it is never −1, say. And that makes the flow of time as measured against time itself a kind of triviality that cannot capture the myth of passage.

It seems that only when we measure the flow against a meta-time do we have a non-trivial flow. That said, perhaps Williams is wrong that we need a regress of meta-times. Maybe time’s flow can be measured against a meta-time that itself does not flow.

Occurrent memory of color experience

Many people when occurrently remembering what a tomato looks like have a mental image, with a degree of vividness and precision differing from person to person, of a red round thing. But not everyone. Visual aphantasiacs don’t have visual images at all in their memories. Presumably, too, there are partial cases: some may have visual images of shapes but without color. Nonetheless, it seems correct to say that such people still know what red is like.

I conjecture that most people have some kinds of experience where they know what the experience is like, but when they remember what the experience is like their memory does not contain a phenomenal copy of the experience. Maybe for you it’s certain kinds of tastes, or smells, or feeling hot water on your body.

For definiteness, suppose that for Alice it’s redness. She has seen red tomatoes. She knows what it’s like to see red, and she remembers what a red tomato looks like, but when she makes her memory occurrent, it does not contain a phenomenal copy of the experience.

What’s going on? Here is one possibility. Even though the memory of a colored object has no visualized color in it, it has some other distinctive phenomenal quality that encodes the specific color. Thus, while neither remembering a red tomato nor a blue plum involves visualizing a color, remembering a red tomato involves a non-color phenomenal quality that encodes the redness, while remembering a blue plum involves a different non-color phenomenal quality that encodes the blueness. It is by having that encoding non-color quale that one knows what the encoded color quale is. The encoding non-color quale is presumably not a sound, or a smell, or any other sensory quale, but a distinctive memory-of-color quale.

(One might conjecture that this is so even for those aphantasiacs who deny any phenomenal quality to their memories, because perhaps it is possible to mistake not having sensory qualia in one’s memories for not having any qualia in them.)

(One might also wonder if it goes to meta levels. If you remember remembering a red tomato, do you have a memory-of-memory-of-red quale, rather than just a memory-of-red quale? I don’t think it needs to do that. If you’re capable of having a memory-of-red quale, that can just figure in your memory of remembering.)

It seems very plausible that on this suggestion, the memory-of-X qualia can differ from person to person, even if the X qualia do not. One reason to think this is that some people do visualize colors, and for them a memory-of-red quale will just be a red quale (or a visualized-red quale, which might not be exactly the same). But a more interesting question is this: Can two people be such that one person’s memory-of-X quale is just like the other person’s memory-of-Y quale? The answer could be negative even if the memory-of-X qualia differ from person to person, for it could be that nonetheless there is a person-independent way to recover the X quale from its encoding in the memory-of-X quale. There can be multiple ways of encoding the same thing while yet each one can be uniquely decoded.

Maybe. But what if it turns out that there is no person-independent mapping from a memory-of-X quale to an X quale? I think this endangers the idea that having a memory-of-red quale in conjunction with one’s remembering is what makes one count as occurrently knowing what a ripe tomato looks like.

Is there a different way to account for what happens when one occurrently remembers what it’s like to see a red tomato when one cannot visualize colors, without positing some kind of memory-of-X qualia?

Here’s a suggestion inspired by David Bernhardt’s comment (but perhaps with I misunderstood the comment). Our thoughts can make de re reference to objects and events and the like. Perhaps when I think “Ripe tomatoes look like this”, the “this” makes a de re reference to my past experience of red. But just as the word “this” does not reflect the specific character of its referent, and indeed sounds and looks the same regardless of what it refers to, neither do the intrinsic features of the thought reflect the specific character of the past experience of red. Imagine you’re an almost exact twin of mine, except that you always wore glasses that swapped red and blue. Then you could have an intrinsically exactly similar “Ripe tomatoes look like this” thought to mine, but in your case the “this” would refer to an experience of blue.

Could something like this be how we have occurrent memories of how tomatoes look, a mere demonstrative de re thought? If so, then the knowledge argument against materialism is in trouble. For Mary, raised in her black and white room, could have a de re thought that ripe tomatoes look like this, where her “this” refers de re not to her own experience but to mine! After all, if you and I are both pointing to the same horse, and you think “This animal is lovely” and I think “This animal is lovely”, we seem to be thinking the same thought, precisely because our “This” has the same referent. So if Mary thinks “Tomatoes look like this” and I think “Tomatoes look like this”, and the “this” in both cases refers to my experience of red, then we seem to be thinking the same thing.

This suggests to me that we have three options:

1. People who can’t visualize colors don’t know what colors are like when they aren’t currently perceiving colored things.

2. They know it by means of memory-of-color qualia, which have an unambiguous mapping onto color qualia.

3. They know by means of de re thoughts that get their content from past experiences that are not reflected in the intrinsic character of the thought.

I think (1) and (2) leave the knowledge argument against materialism in place, but (3) endangers the argument.

I don’t know what the right story is.