What numbers could be

Benacerraf famously argued that no set theoretic reduction can capture the natural numbers. While one might conclude from this that the natural numbers are some kind of sui generis entities, Benacerraf instead opts for a structuralist view on which different things can play the role of different numbers.

The argument that no set theoretic reduction captures the negative numbers is based on thinking about two common reductions. On both, 0 is the empty set ⌀. But then the two accounts differ in how the successor sn of a number n is formed:

1. sn = n ∪ {n}

2. sn = {n}.

On the first account, the number 5 is equal to the set {0, 1, 2, 3, 4}. On the second account, the number 5 is equal to the singleton {{{{{⌀}}}}}. Benacerraf thinks that we couldn’t imagine a good argument for preferring one account over another, and hence (I don’t know how this is supposed to follow) there can’t be a fact of the matter about why one account—or any other set-theoretic reductive account—is correct.

But I think there is a way to adjudicate different set-theoretic reductions of numbers. Plausibly, there is reference magnetism to simpler referrents of our terminology. Consider an as consisting of a set of natural numbers, a relation <, and two operations + and ⋅, satisfying some axioms. We might then say that our ordinary language arithmetic is attracted to the abstract entities that are most simply defined in terms of the fundamental relations. If the only relevant fundamental relation is set membership ∈, then we can ask which of the two accounts (a) and (b) more simply defines <, + and ⋅.

If simplicity is brevity of expression in first order logic, then this can be made a well-defined mathematical question. For instance, on (a), we can define a < b as a ∈ b. One provably cannot get briefer than that. (Any definition of a < b will need to contain a, b and ∈.) On the other hand, on (b), there is no way to define a < b as simply. Now it could turn out that + or ⋅ can be defined more simply on (b), in a way that offsets (a)’s victory with <, but it seems unlikely to me. So I conjecture that on the above account, (a) beats (b), and so there is a way to decide between the two reductions of numbers—(b) is the wrong one, while (a) at least has a chance of being right, unless there is a third that gives a simpler reduction.

In any case, on this picture, there is a way forward in the debate, which undercuts Benacerraf’s claim that there is no way forward.

I am not endorsing this. I worry about the multiplicity of first-order languages (e.g., infix-notation FOL vs. Polish-notation FOL).

Perfect nomic correlations

Here is an interesting special case of Ockham’s Razor:

1. If we find that of nomic necessity whenever A occurs, so does B, then it is reasonable to assume that B is not distinct from A.

Here are three examples.

1. We learn from Newton and Einstein that inertial mass and gravitational mass always have the same value. So by (1) we should suppose them to be one property, rather than two properties that are nomically correlated.

2. In a Newtonian context consider the hypothesis of a gravitational field. Because the gravitational field values at any point are fully determined by the positions and masses of material objects, (1) tells us that it’s reasonable to assume the gravitational field isn’t some additional entity beyond the positions and masses of material objects.

3. Suppose that we find that mental states supervene on physical states: that there is no difference in mental states without a corresponding difference in physical states. Then by (1) it’s reasonable to expect that mental states are not distinct from physical states. (This is of course more controversial than (A) and (B).)

But now consider that in a deterministic theory, future states occur of nomic necessity given past states. Thus, (1) makes it reasonable to reduce future states to past states: What it is for the universe to be in state S₇ at time t₇ is nothing but the universe’s being in state S₀ at time t₀ and the pair (S₇,t₇) having such-and-such a mathematical relationship to the pair (S₀,t₀). Similarly, entities that don’t exist at the beginning of the universe can be reduced to the initial state of the universe—we are thus reducible. This consequence of (1) will seem rather absurd to many people.

What should we do? One move is to embrace the consequence and conclude that indeed if we find good evidence for determinism, it will be reasonable to reduce the present to the past. I find this implausible.

Another move is to take the above argument as evidence against determinism.

Yet another move is to restrict (1) to cases where B occurs at the same time as A. This restriction is problematic in a relativistic context, since simultaneity is relative. Probably the better version of the move is to restrict (1) to cases where B occurs at the same time and place as A. Interestingly, this will undercut the gravitational field example (B). Moreover, because it is not clear that mental states have a location in space, this may undercut application (C) to mental staes.

A final move is either to reject (1) or, more modestly, to claim that the the evidence provided by nomic coincidence is pretty weak and defeasible on the basis of intuitions, such as our intuition that the present does not reduce to the past. In either case, application (C) is in question.

In any case, it is interesting to note that thinking about determinism gives us some reason to be suspicious of (1), and hence of the argument for mental reduction in (C).

Reducing goods to reasons?

In my previous post I cast doubt on reducing moral reasons to goods.

What about the other direction? Can we reduce goods to reasons?

The simplest story would be that goods reduce to reasons to promote them.

But there seem to be goods that give no one a reason to promote them. Consider the good fact that there exist (in the eternalist sense: existed, exist now, will exist, or exist timelessly) agents. No agent can promote the fact that there exist agents: that good fact is part of the agent’s thrownness, to put it in Heideggerese.

Maybe, though, this isn’t quite right. If Alice is an agent, then Alice’s existence is a good, but the fact that some agent or other exists isn’t a good as such. I’m not sure. It seems like a world with agents is better for the existence of agency, and not just better for the particular agents it has. Adding another agent to the world seems a lesser value contribution than just ensuring that there is agency at all. But I could be wrong about that.

Another family of goods, though, are necessary goods. That God exists is good, but it is necessarily true. That various mathematical theorems are beautiful is necessarily true. Yet no one has reason to promote a necessary truth.

But perhaps we could have a subtler story on which goods reduce not just to reasons to promote them, but to reasons to “stand for them” (taken as the opposite of “standing against them”), where promotion is one way of “standing for” a good, but there are others, such as celebration. It does not make sense to promote the existence of God, the existence of agents, or the Pythagorean theorem, but celebrating these goods makes sense.

However, while it might be the case that something is good just in case an agent should “stand for it”, it does not seem right to think that it is good to the extent that an agent should “stand for it”. For the degree to which an agent should stand for a good is determined not just by the magnitude of the good, but the agent’s relationship to the good. I should celebrate my children’s accomplishments more than strangers’.

Perhaps, though, we can modify the story in terms of goods-for-x, and say that G is good-for-x to the extent that x should stand for G. But that doesn’t seem right, either. I should stand for justice for all, and not merely to the degree that justice-for-all is good-for-me. Moreover, there goods that are good for non-agents, while a non-agent does not have a reason to do anything.

I love reductions. But alas it looks to me like reasons and goods are not reducible in either direction.

The incompleteness of current physics

1. There is causation in the physical world.

2. Causation is irreducible.

3. Our fundamental physics does not use the concept of causation.

4. So, our fundamental physics is incomplete as a description of the physical world.

Microphysics and philosophy of mind

Much (but not all) contemporary philosophy of mind is written as if microphysics were fundamental physics. But as far as I know, only on those interpretations of quantum mechanics that disallow indeterminacy as to the number of particles can microphysics be fundamental physics. The most prominent such interpretation is Bohmianism. On most other interpretations, the most we can say about the number of particles is that we are in a superposition between states with different numbers of particles. But reality has to have determinate numbers of fundamental entities. The picture of reality we get from both relativity theory and mainstream interpretations of quantum mechanics other than Bohmianism and its close cousins is that fundamental physical reality consists of global entities such as the spacetime manifold or the wavefunction of the universe rather than microscopic entities like particles. (I am attracted to a non-mainstream interpretation on which the fundamental physical entities may include mid-sized things like dogs and trees.)

Sometimes, pretending microphysics is fundamental physics is excusable. For certain discussions, it doesn’t matter what the fundamental physics is—the arguments work equally well for global and local fundamental entities. In other cases, all that matters is relative fundamentality. Thus, facts about chemistry might be held to be more fundamental relative to biology, and facts about microphysics might be fundamental relative to chemistry, even if the microphysics facts themselves are not fundamental simpliciter, being reducible, say, to facts about global fields.

But even when the arguments do not formally rely on fundamental physics being microphysics, it is risky in a field so reliant on intuition to let one’s intuitions be guided by acting as if fundamental physics were microphysics. And doing this is likely to mis-focus one’s critical attention, say focusing one more on the puzzle of why the functioning of various neurons produces a unified consciousness than on the puzzle of how the functioning of a handful of global entities results in the existence of billions of minded persons.

A tension in some theistic Aristotelian thinkers

Here is a tension in the views of some theistic Aristotelian philosophers. On the one hand, we argue:

1. The mathematical elegance and discoverability of the laws of physics is evidence for the existence of God

but we also think:

2. There are higher-level (e.g., biological and psychological) laws that do not reduce to the laws of physics.

These higher-level laws, among other things, govern the emergence of higher-level structures from lower-level ones and the control that the higher-level structures exert over the lower-level ones.

The higher-level laws are largely unknown except in the broadest outline. They are thus not discoverable in the way the laws of physics are claimed to be, and since no serious proposals are yet available as to their exact formulation, we have no evidence as to their elegance. But as evidence for the existence of God, the elegance and discoverability of a proper subset of the laws is much less impressive. In other words, (1) is really impressive if all the laws reduce to the laws of physics. But otherwise, (1) is rather less impressive. I’ve never never seen this criticism.

I think, however, there is a way for the Aristotelian to still run a design argument.

Either all the laws reduce to the laws of physics or not.

If they all reduce to the laws of physics, pace Aristotelianism, we have a great elegance and discoverability design argument.

Suppose now that they don’t. Then there is, presumably, a great deal of complex connection between structural levels that is logically contingent. It would be logically possible for minds to arise out of the kinds of arrangements of physical materials we have in stones, but then the minds wouldn’t be able to operate very effectively in the world, at least without massively overriding the physics. Instead, minds arise in brains. The higher-level laws rarely if ever override the lower-level ones. Having higher-level laws that fit so harmoniously with the lower-level laws is very surprising a priori. Indeed, this harmony is so great as to be epistemically suspicious, suspicious enough that the need for such a harmony makes one worry that the higher-level laws are a mere fiction. But if they are a mere fiction, then we go back to the first option, namely reduction. Here we are assuming the higher level stuff is irreducible. And now we have a great design argument from their harmony with the lower-level laws.

Naturalism and property dualism

It is generally taken that a view on which there are mental properties that do not supervene on the properties of physics is a non-naturalistic view: it is a form of property dualism.

But now imagine that we find out that:

1. There are chemical properties that do not supervene on the properties physics speaks of.

That would be a really exciting discovery, but it wouldn’t be a discovery incompatible with naturalism. The new chemical properties would presumably be just as natural as the physical ones.

So, why would we call non-supervenient mental properties non-natural, if we wouldn’t call non-supervenient chemical properties non-natural? It can’t be just because chemical properties are the province of a science, namely chemistry. For mental properties are the province of a science, too, namely psychology.

While we’re exploring this corner of logical space, consider this view:

2. Chemical properties do not supervene on physical properties, and mental properties do not supervene on physical properties either, but mental properties do supervene on, and even reduce to, physical and chemical properties.

I’ve never met an advocate of (2). It would be a very strange view. But here is one that, I think, is not actually all that strange:

3. Biological properties do not supervene on physical properties, and mental properties do not supervene on physical properties either, but mental properties do supervene on, and even reduce to, biological properties.

I think view (3) is worth thinking about. Most of the people who have tried to reduce the mental have tried to reduce it to the physical, but perhaps a reduction to an irreducible biological level would be more promising.

Qualia are not all fundamental entities if theism is true

This argument is sound. I am not sure if premise (2) is true, though.

1. If God exists, then all fundamental entities are intrinsically good.
2. Pain qualia are not intrinsically good.
3. So, pain qualia are not fundamental entities.

Eliminating or reducing parthood

Parthood is a mysterious relation. It would really simplify our picture of the world if we could get rid of it.

There are two standard ways of doing this. The microscopic mereological nihilist says that only the fundamental “small” bits—particles, fields, etc.—exist, and that there are no complex objects like tables, trees and people that are made of such bits. (Though one could be a microscopic mereological nihilist dualist, and hold that people are simple souls.)

The macroscopic mereological nihilist says that big things like organisms do exist, but their commonly supposed constituents, such as particles, do not exist, except in a manner of speaking. We can talk as if there were electrons in us, but there are no electrons in us. The typical macroscopic mereological nihilist is a Thomist who talks of “virtual existence” of electrons in us.

Both the microscopic and macroscopic nihilist get rid of parthood at the cost of ridding themselves of large swathes of objects that common sense accepts. The microscopic nihilist gets rid of the things that are commonly thought to be wholes. The macroscopic nihilist gets rid of the things that are commonly thought to be parts.

But there is a third way of getting rid of parthood that has not been sufficiently explored. The third kind of mereological nihilist would neither deny the existence of things commonly thought to be wholes nor of things commonly thought to be parts. Instead, she would deny the parthood relation that is commonly thought to hold between the micro and the macro things. Parts of the space occupied by me are also occupied by my arms, my legs, my heart, the electrons in these, etc. But these things are not parts of me: they are just substances that happen to be colocated with me. I’ll call this “parthood nihilism”.

This is compatible with a neat picture of organ transplants. If my kidney becomes your kidney, nothing changes with respect to parthood. All that changes is the causal interactions: the kidney that previously was causing certain distributional properties in me starts to cause certain distributional properties in you.

An obvious question is what about property inheritance? Whenever my hand is stained purple, I am partly purple. We don’t want this to be just a coincidence. The common-sense parts theorist has a nice explanation: I inherit being partly purple from my hand being partly purple (note that they’re only properly partly purple—they aren’t purple inside the bones, say). My partial purpleness derives from the partial purpleness of a part of me.

But the parthood nihilist can accept accept this kind of property inheritance and give an account of it: the inheritance is causal. My hand’s being partly purple causes me to be partly purple, which is a distributional property of an extended simple). I guess on the standard view, property inheritance is going to be a kind of grounding: my being partly purple occurs in virtue of my hand’s being a part of me and its being partly purple. On the present nihilism, we have simultaneous causation instead of grounding.

Here’s another difficulty: what about gravity (and relevantly similar forces). I have a mass of 77kg. If my mass is m₁ and yours is m₂ and the distance between us is r, there is a force pulling you towards me of magnitude Gm₁m₂/r². But why isn’t that force equal in magnitude to (m₁ + m₁₁ + m₁₂ + m₁₃ + ...)m₂/r², where m₁₁, m₁₂, m₁₃, ... are the masses of what common sense calls “my parts” (about five kilograms for my head, four for my left arm, four for my right arm, and so on)? After all, wouldn’t all these objects be expected to exert gravitational force?

The first two kinds of nihilists have easy answers to the problem. The microscopic nihilist says that only particles have mass as only particles exist. The macroscopic one says that I am all there is here—the head, arms, etc. don’t exist. The standard common-sense view has a slightly more complicated answer available: gravitational forces only take into account non-inherited mass. But parthood nihilist can give a variant of this: it’s a law of nature that only fundamental particles produce gravitational forces.

There is a fourth kind of view. This fourth kind of view is no longer a mereological nihilism, but mereological causal reductivism. On the fourth kind of view, for x to be a part of y just is for x to be identical with y or for x to be a proper part of y. And for x to be a proper part of y just is for a certain causal relation to hold between x’s properties and y’s properties.

Spelling out the details of this causal relation is difficult. Roughly, it just says that all of x’s properties and relations cause corresponding properties and relations of y. Thus, x’s being properly partly located in Pittsburgh causes y to be properly partly located in Pittsburgh, while x’s being wholly located in Pittsburgh causes y to be at least partly located in Pittsburgh; x’s being green on its left half causes y’s being green in the left half of the locational property that x causes y to have; and so on.

As I said, it’s difficult to spell out the details of this causal relation. But it is no more difficult than the common-sense parts theorist’s difficulty in spelling out the details of property inheritance. Wherever the common-sense parts theorist says that there is a part-to-whole inheritance between properties, our reductionist requires a causal relation.

The reductionism changes the order of explanation. Suppose my hand is the only green part of me and it gets amputated. According to the common-sense parts theorist, I am no longer partly green because the green hand has stopped being a part of me. According to the reductionist, on the other hand, the hand’s no longer contributing to my greenness makes it no longer a part of me.

The reductionist and parthood nihilist, however, have an extra explanatory burden. Why do all these causal relations cease together? Why is it that when my right hand stops causing me to be partially green, my right hand also stops causing me to have five right fingers? The common-sense parts theorist has a nice story: when the part stops being a part, all the relevant grounding relations stop because a portion of the ground is the fact that the part is a part.

But there is also a causal solution. The common-sense parts theorist has to give a story as to when it is that certain kinds of causal interaction—say, a surgeon using a scalpel—cause a part to stop being a part. For each such kind of causal interaction, the reductionist and parthood nihilist can say that there is a cessation of all the causal relations that the common-sense parts theorist would say go with inheritance.

All in all, I think the reductionist has a simpler fundamental ideology than the standard common-sense inheritance view: the reductionist can reduce parthood to patterns of causation. Her theory is overall not significantly more complicated than the common-sense inheritance theory, but it is more complicated than either microscopic or macroscopic nihilism. But she gets to keep a lot more of common-sense than the nihilists do. In fact, maybe she gets to keep all of common-sense, except for pretty theoretical claims about the direction of explanation, etc.

The parthood nihilist has most of the advantages of reductionism, but there is some common-sense stuff that she denies—she denies that my arm is a part of me, etc. Overall parthood nihilism is not significantly simpler than reductionism, I think, because the parthood nihilist’s account of how all the relevant causal relations cease together will include all the complications that the reduction includes. So I think reductionism is superior to parthood nihilism.

But I still like macroscopic nihilism more than reductionism.

Location, causation and transsubstantiation

Here’s a fun thought experiment. By a miracle (say) I am sitting in my armchair in Waco but my causal interaction with my environment at the boundaries of my body would be as if I were in Paris. There is a region of space in Paris shaped like my body. When a photon hits the boundary of that region, it causally interacts with me as if I were in Paris: I have the causal power to act at a distance to reflect Parisian photons as if I were in that region in Paris. Alternately, that photon might be absorbed by me: I have the causal power to absorb Parisian photons. As a result, it looks to Parisians like I am in Paris, and as I look around, it looks to me like Paris is all around me. The same is true for other interactions. When my vocal cords vibrate, instead of causing pressure changes in Texan air, they cause pressure changes in chilly France. As I walk, the region of space shaped like my body in Paris that is the locus of my interaction with Parisians moves in the usual way that bodies move.

Furthermore, my body does not interact with the environment in Waco at all. Wacoan photons aimed at my body go right through it and so I am invisible. In fact, not just photons do that: you could walk right through my body in Waco without noticing. My body is unaffected by Texan gravity. It is simply suspended over my sofa. As I wave my hand, my hand does in fact wave in Texas, but does not cause any movement of the air in Texas—but in Paris, the region of space in which I interact with the Parisians changes through the wave, and the air moves as a result. When I eat, it is by means of Parisian food particles that come to be incorporated into my Wacoan body.

To me, to Wacoans and to Parisians it looks in all respects like I am in Paris. But I am in Waco.

Or am I? There is a view on which the causal facts that I’ve described imply that I am in Paris, namely the view that spatial relationships reduce to causal relationships. It is an attractive view to those like me who like reductions.

But this attractive view threatens to be heretical. Christ’s body is here on earth in the Eucharist, as well as in heaven in the more normal way for a body to be. But while the body is surely visible in heaven and interacts with Mary and any other embodied persons in heaven, it does not interact physically with anything on earth. Granted, there is spiritual interaction: Christ’s presence in the Eucharist is a means of grace to recipients. But that probably isn’t the sort of interaction that would ground spatial location.

There is, however, a way to modify the causal reduction of location that handles the case of the Eucharist. Actual causal interactions do not seem to be enough to ground location. The reduction very likely needs needs dispositional causal interactions that typically ground causal counterfactuals like:

1. If Parisians were to shine a flashlight into that dark alley, they’d see me.

However, dispositions can be masked. For instance, sugar is still soluble even if God has promised to miraculously keep it from dissolving when it is placed in water. In such a case, the sugar still has the disposition to dissolve in water, but fails to ground the counterfactual:

2. The lump would be dissolved were it placed in water.

We might, thus, suppose that when the Mass is being celebrated in Waco, Christ comes to have the dispositional causal properties that would ordinarily be contitutive of his being present in Waco, such as the disposition to reflect Texan photons, and so on. But by miracle, all these dispositions are masked and do not result in actual causal interaction. The unmasked dispositions are those corresponding to spiritual interaction.

Here’s an interesting lesson. The kind of causal-reductive view of location that I’ve just considered seems to be one of the least transsubstantiation-congenial views of location. But, nonetheless, the transsubstantiation can still be made sense of on that view when the view is refined. This gives us evidence that transsubstantiation makes sense.

And we can now go back to the story of my being in Waco while interacting in Paris. The story was underspecified. I didn’t say whether I have the dispositions that go with being in Waco. If I do, these dispositions are being miraculously masked. But they may be enough to make me count as being in Waco. So on the story as I’ve told it, I might actually be both in Waco and in Paris.

Final question: Can external temporal location be similarly causally grounded? (Cf. this interesting paper.)

The most fundamental and what matters most

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

But here is a plausible hypothesis or at least heuristic:

• Fundamental predicates apply primarily to fundamental entities, and derivatively to other entities.

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

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

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

Induction, naturalness and physicalism

Something is grue provided that it is now before the year 3000 and it is green or it's the year 3000 or later and it's blue. From:

1. All observed emeralds were grue

we should not infer that all emeralds will be grue. But from

2. All observed emeralds were green

we should infer that all emeralds will be green. A standard thought (e.g., Sider in his Book book) is that the relevant difference between (1) and (2) is that "green" carves reality more at the joints, is more natural, than "grue".

Suppose that we understand naturalness in a Lewisian way: a concept is more unnatural the longer its expression in a language whose bits refer to perfectly natural stuff. And suppose we think that among the sciences only the terms of fundamental physics refer to perfectly natural stuff. Now consider:

3. All observed electrons were nesitively charged

where an object is nesitively charged provided it's negatively charged and it's before the year 3000 or it's positively charged and it's 3000 or later. We had better not infer that all electrons will be nesitively charged. But "nesitively charged" is an order of magnitude more natural than "green". Consider this beginning of an account of "green":

4. in electromagnetic radiation of the 484-789 THz range, reflecting or transmitting primarily that in the 526-606 THz range.

And this account is not finished. To make this be in terms of the perfectly natural stuff, we'd need to specify the units (terahertz) in microphysical terms, presumably in terms of Planck times or something like that, and we'll get quite messy numbers. Moreover, we need an account of reflection and transmission. I suspect that we can more easily give an account of nesitive charge: "positive" and "negative charge" seem to already be perfectly natural or close to it; the year 3000 is a bit tricky, but we can count it (or maybe just some other "neater" date) in Planck times from the Big Bang.

If naturalness then correlates with brevity of microphysical expression, "green" is not more natural, and probably is less natural, than "nesitive charge". And so we had better not base induction on naturalness.

I think the lesson of this is that we either shouldn't think of degrees of unnaturalness as distance from the perfectly natural, or we shouldn't limit the perfectly natural (even in the concrete realm) to the microphysical. The latter gives us reason to accept some kind of antireductionism about the special sciences and ordinary language.

A nominalist reduction

Suppose that there were only four possible properties: heat, cold, dryness and moistness. Then the Platonic-sounding sentences that trouble nominalists could have their Platonic commitments reduced away. For instance, van Inwagen set the challenge of how to get rid of the commitment to properties (or features) in:

1. Spiders and insects have a feature in common.

On our hypothesis of four properties, this is easy. We just replace the existential quantification by a disjunction over the four properties:

2. Spiders and insects are both hot, or spiders and insects are both cold, or spiders and insects are both dry, or spiders and insects are both moist.

And other sentences are handled similarly. Some, of course, turn into a mess. For instance,

3. All but one property are instantiated

becomes:

4. Something is hot and something is cold and something is dry but nothing is moist, or something is hot and something is cold and something is moist but nothing is dry, or something is hot and something is dry and something is moist but nothing is cold, or something is cold and something is dry and something is moist but nothing is hot.

Of course, this wouldn't satisfy Deep Platonists in the sense of this post, but that post gives reason not to be a Deep Platonist.

And of course there are more than four properties. But as long as there is a finite list of all the possible properties, the above solution works. But in fact the solution works even if the list is infinite, as long as (a) we can form infinite conjunctions (or infinite disjunctions—they are interdefinable by de Morgan) and (b) the list of properties does not vary between possible worlds. Fortunately in regard to (b), the default view among Platonists seems to be that properties are necessary beings.

The value of punishment

Boethius gives a striking thesis:

The wicked are happier in undergoing punishment than if no penalty of justice chasten them. And I am not now meaning what might occur to anyone—that bad character is amended by retribution, and is brought into the right path by the terror of punishment, or that it serves as an example to warn others to avoid transgression [...] .
Surely, then, the wicked, when they are punished, have a good thing added to them, the punishment which by the law of justice is good [...]. (Consolation, IV)

The brief argument seems to be:

1. What justice calls for is good.
2. Justice calls for punishment.
3. So, punishment is good.

Unfortunately, the conclusion that we want is not (3) but:

4. Punishment is good for the person undergoing it.

Can we fill in some plausible steps between (3) and (4)?

Perhaps Boethius takes it as clear that justice does not call for punishment simply because "bad character is amended by retribution ... or as an example to warn others" or in any other easy reductive account that "might occur to anyone" (e.g., Nietzsche's "account" on which punishment gives compensatory pleasure to the victims, or accounts on which criminals are simply taken out of circulation by being jailed, for the protection of society). Such benefits are there, but they aren't the benefit that justice is primarily aimed at.

Plausibly, the benefits of justice are to persons. Well, the relevant persons seem to be:

• the criminal
• the victims
• other members of society
• the punisher
• God.

Which of these are such that justice's primary aim is at a benefit for them?

Let's start by ruling out options from the bottom of the list. Everyone benefits, in an extrinsic but important way, when those they love benefit. God loves all. So any benefit to anyone is an extrinsic benefit to God. God is love and is immutable and simple. It plausibly follows that all contingent benefits to God are such extrinsic benefits. Thus, a contingent benefit to God will require a benefit to someone else. So choosing the option of God doesn't get us out of the puzzle. Moreover, when we aim at a benefit for God in this way, we should also be aiming non-instrumentally at a benefit for the creature.

While the punisher may receive some pleasure or the good of excellence in a job well done, surely that's not what justice aims for. It seems clear that just punishment does not require other members of society. Imagine someone has killed every other member of her society. She deserves punishment—say, from another society, or from a self-imposed life of penance.

The victims? There is an intuition that punishment is a way of honoring victims, perhaps posthumously. One problem with this is that criminal punishment appears just even when the criminal was forgiven by the victims. But when the criminal was forgiven by the victims, the criminal should not be punished for the sake of the victims, since by forgiveness the victims relinquish claims to punishment on their behalf. But I worry that this argument is not sufficient. After all, it could be that all of society counts as a victim in the case of a crime, since society's laws have been unjustly violated, so forgiveness by the more particular victims is not yet forgiveness by all of society. (This also shows how "victimless" crimes aren't victimless.)

Here is perhaps a more telling counterexample to the victim theory. Suppose Patricia notices a planet with rudimentary unicellural life but where she has very good reason to think intelligent life will evolve. She leaves behind a device which will kill off the intelligent life on the planet as soon as there are a million intelligent beings. A week later, Darth Vader tests the Death Star on the planet. Patricia has committed attempted murder, but there are no victims: there never was and never will be any intelligent life on this planet. Yet Patricia is clearly deserving of punishment as having committed a species of attempted genocide. Moreover, even if she broke no society's law, there is some sense in which justice calls for her punishment—this is the sort of thing that there ought to be a law against, and this "ought" is an "ought" of justice.

Another worry about locating the benefit in the victims is that may seem problematic to impose such great intrinsic burdens on the living for the sake of what appear to be merely extrinsic benefits to the dead (apart from somewhat dubious theories of the afterlife on which the dead rejoice in seeing their malefactors punished).

That leaves the criminal as the recipient of the benefits of punishment.

I think the most serious challenge to the argument is one on which justice leads to goods that are not the goods to any persons. Maybe justice benefits "the moral order of the universe". I am sceptical, though, of goods that are not derivative from goods to fundamental entities, and on my ontology the universe and its moral order are not fundamental entities.

Reducing asserting to promising

Suppose I had a language that had only one kind of speech act: promising. Could I communicate information in the way we do in assertion? Yes! For, suppose I want to communicate that it's raining. Then I could say "I promise to immediately exclaim 'just kidding!' unless it's raining" without exclaiming "just kidding!"

We could even imagine that over time this would get abbreviated to: "It's raining."

Thus, one can reduce assertion, or something very much assertion-like, to promising. Moreover, if we say that this really is normatively equivalent to assertion, then we get an account of the wrongness of lying and a reduction of the normativity of assertion to moral normativity.

One cannot, however, reduce all speech acts to promises. For while promises generate reasons for self, requests (including questions) and commands generate reasons for others.

Deep Thoughts XXXIV

H₂O is nothing but water.

Reducing sets

I find sets to be very mysterious candidates for abstract entities. I think it's their extensionality that seems strange to me. And anyway, if one can reduce entities to entities that we anyway want to have in our ontology, ceteris paribus we should. I want to describe a three-step procedure—with some choices at each step—for generating sets. I will use plural quantification quite a lot in this. I am assuming that one can make sense of plural quantification apart from sets.

Step 1: The non-empty candidates. The non-empty candidates, some of which will end up counting as sets in the next step, will be entities that stand in "packaging" relation to a plurality of objects, such that for any plurality, or at least for enough pluralities, there is a candidate that packages that plurality. There are many options for the non-empty candidates and the packaging relation.

Option A: Plural existential propositions, of the form <The Xs exist>, where a plural existential proposition p packages a plurality, the Xs, provided that it attributes existence to the Xs and only to the Xs.

Option B: Plural existential states of affairs (either Armstrong or Plantinga style), i.e., states of affairs of the Xs existing, where a plural existential state of affairs e packages a plurality, the Xs, provided that it is a state of affairs of the Xs existing. I got this option from Rob Koons.

Option C: This family of options generates the candidates in two sub-steps. The first is to have candidates that stand in a packaging relation to individuals, such that each candidates packages precisely one individual. Call these "singleton candidates". For brevity if x is a singleton candidate that packages y, I will say x is a singleton of y. The second step is to take our non-empty candidates to be mereological sums of singleton candidates, and to say that a mereological sum m packages the Xs if and only if m is a mereological sum of Ys such that each of the Ys is a singleton of one of the Xs and each of the Xs is packaged by exactly one of the Ys. We need the singleton packaging relation to satisfy the condition (*) that a mereological sum of singletons of the Xs has no singletons as parts other than the singletons of the Xs. (In particular, no singleton of y can be a part of any singleton of x if x and y are distinct.)

We get different instances of Option C by considering different singleton candidates. For instance, we could have the singleton candidates be individual essences, and a singleton candidate then packages precisely the entity that it is an individual essence of. I got this from Josh Rasmussen. Or we might use variants of Options A and B here: maybe a proposition attributing existence to x or a state of affairs of x existing will be our candidate singleton. (Whether the state of affairs option here differs from Option B depends on whether the state of affairs of a plurality existing is something different from the mereological sum of the states of affairs of the individuals in the plurality existing.)

There are many other ways of packaging pluralities.

Step 2: The empty candidate. We also need an empty candidate, which will be some entity that differs from the non-empty candidates of Step 1. Ideally, this will be an entity of the same sort as the non-empty candidates. For instance, if our non-empty candidates are propositions, we will want our empty candidate to be a proposition, say some contradictory proposition.

Step 3: Pruning the candidates. The basic idea will be that x is a member of a candidate y if and only if y is one of the non-empty candidates and y packages a plurality that has x in it. But the above is apt to give us too many candidates for them all to be sets. There are at least two reasons for this. First, on some of the options, there won't be a unique candidate packaging any given plurality. For instance, there might be more than one proposition attributing existence to the same plurality. Thus, the propositions <The Stagirite and Tully exist> and <Aristotle and Cicero exist> will be different propositions if Millianism is false, but both attribute existence to the same plurality. Second, some of the candidates will be better suited as candidates for proper classes than for sets and some candidates may be unsuitable either as sets or as proper classes. For instance, there might be a proposition that says that the plural existential propositions exist. Such a proposition packages all the candidates, including itself, and will not be a good set or proper class on many axiomatizations.

Spinoza and reductionistic determinism

According to some presentist theories of time, facts about the future are grounded in facts about the present and in the laws of nature. What grounds the fact, if it is a fact, that tomorrow the sun will rise is that the present conditions together with the laws of nature entail that the sun will rise tomorrow. Alan Rhoda played with a similar view in regard to the past: facts about the past are grounded in facts about God's present memories.

Suppose determinism holds and there is an initial time t₀. Let L be the laws. Then we can imagine a view which we might call initialism in the place of presentism. According to initialism, facts about what happens at a time t>t₀ reduce to facts about what the laws are and what the initial conditions are. More precisely, if I is the initial conditions of the world at t₀, according to initialism, what it is for a state of affairs to obtain at a time t>t₀ is for I and L to jointly entail that it obtains at t. Thus, what it is for there to be humans in the world is for the world to have had initial conditions and laws such as to guarantee the arising of humans.

According to initialism, none of us are substances, because facts about our existence reduce to facts about the initial conditions and laws. In Spinozistic terminology, we are modes of laws and initial conditions or of whatever grounds the laws and initial conditions.

Initialism has some obvious problems. It assumes that determinism holds and that there is an initial time t₀. But determinism is in tension with quantum mechanics, and probably the best interpretation of the Big Bang is that although the universe has finite age, there was no initial moment.

There is a strong resemblance between initialism and Spinoza's metaphysics. To make the resemblance closer, we will make some modifications.

Modification 1: Take time to discrete. Thus, there is a finite number of moments of time between t₀ and the present. If we do this, we can get a nested view closer to Spinoza's. Instead of reducing the conditions at time t_n to the laws and the conditions at t₀, we reduce them to the conditions at t_n−1 and the laws. Now our present time slices are modes of modes of ... modes of the initial conditions and laws.

The second move we can make is to remove the initial time t₀. Instead, there is a doubly infinite sequence of times ...,t₋₂,t₋₁,t₀,t₁,t₂,.... How things are at each time reduces to the laws and how they were at the preceding time. Thus, in Spinozistic terminology, we are modes of modes of modes of ....

The third move is to reintroduce something outside of the whole sequence of modes, in which the sequence of events is grounded. After all, the idea of a sequence of modes without any substance seems absurd. One move would be to take that which is outside the sequence to be the lawmaker of L—that entity in virtue of which L is law, the truthmaker of the proposition that L is law. We may perhaps call this entity "Natura Naturans", nature naturing, or if we are pantheistically inclined like Spinoza, "Deus sive Natura" (though the latter identification would be taking a stand on whether Spinoza's Deus is Natura Naturans or the whole shebang of nature, in favor of the former). If we like, we can call the mereological sum of the modes "Natura Naturata", nature natured. The Natura Naturans, then, is the substance of which the temporal modes are ultimately (though with an infinite chain intervening) are modes.

The final move, to make the view be more like Spinoza's, is to take out the reference to times. Instead, we just have a sequence of entities—objects and/or events—that are each reduced to previous ones.

I think one puzzle about this view is how the Natura Naturans is related to the sequence of temporally qualified, "determinate", modes. We could take this relationship to be one of reduction once again: the whole infinite sequence of times reduces to the laws. This fits with much of what Spinoza says. It is, however, in some tension with Spinoza's idea that from the idea of God qua eternal, and it is this which seems to fit best with this eternal lawmaker, temporally determinate facts do not follow.

This exegetical difficulty can perhaps be overcome.

Here is one way. Accept a relationist B-theory of time, and then say that something is determinate insofar as we can delineate the times of its beginning and end. But on a relationist B-theory, sub specie aeternitatis, we just have a doubly infinite sequence without time-as-a-container, and no non-relative, non-arbitrary way of identifying times like "November 15, 2011". Of course, we can stipulate names for beginning and end times of some events, and then with this stipulative delineation in hand, we can delineate temporally when other events will happen. Thus, if a match struck just before noon, it will come on fire just after noon. Thus, to derive facts about when events happen we need facts about when other events happen. We cannot derive when-facts from eternal laws. Spinoza is clear on his view that times are the product of human beings divisions of duration.

If all there was to being a determinate mode was having a beginning and end time, I think that would be a satisfactory answer. But I think temporally determinate modes may be prior on his view to times. Perhaps, though, his thought is this. What we can derive from L is the whole sequence of things, but considered as an undivided sequence, and all divisions and delineations in the sequence are due to us. And from a delineated cause—say, a match's being struck, which is delineated from what comes before (the movement of the match) and what comes after (the fire)—there can be derived a delineated effect. Again, on this reading, the division in the modes is arbitrary.

Actually, I am not sure that Spinoza's mode-to-haver relationship is reductive. But I think it gives an illuminating reading.

A reduction of spatial relations to an outdated physics

Consider a Newtonian physics with gravity and point particles with non-zero mass. Take component forces and masses as primitive quantities. Then we can reduce the distance at time t between distinct particles a and b as (m_am_b/F_ab)^1/2, where F_ab is the magnitude of the gravitational force of a on b at t, and m_a and m_b are the masses at t of a and b respectively (I am taking the units to be ones where the gravitational constant is 1); we can define the distance between a and a to be zero. For every t, we may suppose that by law that the forces are such as to define a metric structure on the point particles.

If we want to extend this to a spatiotemporal structure, rather than just a momentary temporal structure, we need to stitch the metric structure into a whole. One way to do that is to abstract a little further. Let S be a three-dimensional Euclidean space. Let P be the set of all particles. Let T be the real line. For each object a in P, let T_a be the set of times at which a exists, and let m_a(t) be the mass of a at t. For any pair of objects a and b and time t in both T_a and T_b, let F_ab(t) be the magnitude of the gravitational force of a on b at t. Let Q be the set of all pairs (a,t) such that t is a member of T_a. Say that a function f from Q to S is an admissible position function provided that:

1. If t is a member of both T_a and T_b, then F_ab(t)=m_a(t)m_b(t)/|f(b,t)−f(a,t)|².
2. f''(a,t) is equal to the sum over all particles b distinct from a of (f(b,t)−f(a,t))F_ba(t)/(m_a(t)|f(b,t)−f(a,t)|).

The laws can then be taken to say that the world is such that there is an admissible position function. We can then relativize talk of location to an admissible position function, which plays the role of a reference frame: the location of a relative to f at t is just f(a,t).

The above account generalizes to allow for other forces in the equations.

So, instead of taking spatial structure to be primitive, we can derive it from component forces, masses and objects, taking the latter trio as primitive.

I don't know how to generalize this to work in terms of a spatiotemporal position function instead of just a spatial position function.

Of course, component forces are hairy.

Perhaps the method generalizes to less out-of-date physics. Perhaps not. But at least it's a nice illustration of how spatial relations might be non-fundamental, as in Leibniz (though Leibniz wouldn't like this particular proposal).

Reduction and translation

These are very rough notes for myself.

The translatability of B-talk to A-talk as either a necessary or a sufficient condition for a reduction of Bs to As is generally rejected. Translatability can be symmetric, so it obviously can't be a sufficient condition. And it is generally thought that translations are so hard to come by, even in cases where it is very plausible that there is a reduction, that we shouldn't ask the reductionist for a translation. As an example, it seems pretty plausible that being oval is not a fundamental property. But the hopes of a reduction of being oval to more fundamental geometric concepts are pretty slim. We can start: An oval is a convex domain with a twice differentiable boundary approximating a non-circular ellipse. But if we try to explain the respects in which the oval approximates the ellipse, I expect at some point we would have to throw up our hands and say: "In the way definitive of an oval!"

It should not surprise us if there were no good translations. Words are rarely precisely redundant, and I suspect that cases of non-trivial synonymy are pretty rare. Certainly, few of the things listed in a thesaurus are genuine synonyms, i.e. words expressive of the same concept. Similarly, translation between different languages is rarely exactly right. For instance, "Il neige" and "It is snowing" are unlikely to express the same proposition. Here is one reason to think this. The boundaries of "neiger" and "to snow" are vague, and the behavior of the corresponding concepts near the boundaries will be determined by use. But different linguistic communities occupy different physical and social environments, and it is unlikely that the boundaries will be exactly the same. The same is likely to be true for most ordinary sentences, though the effect is probably decreasing with globalization.

However, I think there is a somewhat neglected option for translation. Instead of translating to an actual language, one can translate to a counterfactual language. And for purposes of testing hypotheses about ontological commitment, that should be enough.

We could imagine a community that has practices that outwardly and normatively resemble our practices of artifact production, use and possession. But they never say anything that commits them to the existence of these. They have other ways of talking. Maybe they say "It is chairing here" in circumstances that correspond to those in which we say "There is at least one chair here." They also describe the intensity of a chairing with a non-negative integer: "It is chairing here with intensity three" corresponds to our "There are three chairs here." They have some ways of talking that correspond to our possession practices. "It is Smithly chairing here with intensity three" corresponds to our "Smith owns three chairs here." They also have ways of talking that correspond to our talk of clear identity. Thus, they say "It is t₀ly chairing with intensity two at t₁" correspondingly to our "Two chairs that existed at t₀ exist at t₁."

Now here is a move that I like. It might turn out that some of our sentences have no corresponding sentences in that community. This will be a problem for the reductionist, unless those very sentences are ones that lead to logical problems in our community. For instance, it might turn out that one cannot translate all diachronic identity sentences about chairs. But that could be an asset if the untranslatable sentences are precisely the ones that lead to ship of Theseus problems. And this could, further, provide an asymmetry that could help fix the direction of reduction: in our language we can get paradoxes, while in theirs maybe we can't. We could, then, simply say that the untranslatable sentences (or maybe now we should call them "quasi-sentences") in our language are nonsense.