Composition and laws of nature

I’ve for a while wondered if mereology can’t be nomically strongly emergent, namely whether there couldn’t be laws of nature of the form:

1. If the xs satisfy F, then the xs cause there to be a y composed of the xs.

This is a species of Markosian’s brute composition theory: There is no set of finite informative conditions that are necessary and sufficient for composition.

One might think that above nomic story has an informative necessary and sufficient condition for composition:

2. The xs compose something if and only if there is only one x or there is a law of the form (1) that the xs satisfy the antecedent of.

Condition (2) wouldn’t count as an answer to van Inwagen’s Special Composition Question because it uses mereological vocabulary on its right hand side (since (1) uses the word “composed”). However, (2) would still be pretty informative. Nonetheless, (2) shouldn’t be acceptable to a theist, because the kind of thing that can happen by a law of nature could also be directly produced by God apart from any law: God can make the xs compose y.

Nomically possible branches and open future views

Some open future views rely on the concept of a nomically possible branch—a complete sequence of how things might go given the laws of nature.

The problem with the concept is this. A nomically possible branch seems to be something like an exhaustive collection of propositions about all times, specifying precisely what happens at all times, with the collection as a whole compatible with the laws of nature. But now consider a world where indeterminism never gives out on any branch: no matter how things go, at every time there will still be more branching. (Our world may well be like that.) Then on an open future view, the propositions making up a branch cannot be all true together—for at no time t can the exhaustive propositions about t’s future be true, as that would violate open futurism given that branching never gives out.

For a while I thought that a decent solution to this is to say that a branch only needs to satisfy the weaker condition that for every time t, all the propositions in the branch about times up to t can be true together with the laws of nature.

But my recent example of random transtemporal causation is problematic for this solution. Suppose that today an indeterministic event E causes a green flash of light to happen on a random future day, and that the laws guarantee that no green flashes happen for any other reason. Then a branch that contains E but no green flashes of light satisfies the weaker possibility condition: for at every time t, all the propositions in the branch about times up to t can be true together with the laws of nature, since E does not causally guarantee that a green flash will happen at or before t, but only that a green flash will happen at some time or other.

Probably the best move for the open futurist is to deny causation across temporal gaps or any other mechanism that nomically guarantees that some event will happen without guaranteeing a time by which it will happen.

Restricted composition and laws of nature

Ted Sider famously argues for the universality of composition on the grounds that:

1. If composition is not universal, then one can find a continuous series of cases from a case of no composition to a case of composition.

2. Given such a continuous series, there won’t be any abrupt cut-off in composition.

3. But composition is never vague, so there would have to be an abrupt cut-off.

Consider this argument that every velocity is an escape velocity:

4. If it’s not the case that every velocity is an escape velocity from a spherically symmetric body of some fixed size and mass, then one can find a continuous series of cases from a case of insufficiency to escape to a case of sufficiency to escape.

5. Given such a continuous series, there won’t be any abrupt cut-off in escape velocity.

6. But escape velocity is never vague, so there would have to be an abrupt cut-off.

It’s obvious that we should deny (5). There is an abrupt cut-off in escape velocity, and there is a precise formula for what it is: (2GM/r)^1/2 where G is the gravitational constant, M is the mass of the spherical body, and r is its radius. As the velocity of a projectile gets closer and closer to the (2GM/r)^1/2, the projectile goes further and further before turning back. When the velocity reaches (2GM/r)^1/2, the projectile goes out forever. There is no paradox here.

Why think that composition is different from escape velocity? Why not think that just as the laws of nature precisely specify when the projectile can escape gravity, they also precisely specify when a bunch of objects compose a whole?

My suspicion is that the reason for thinking the two are different is thinking that composition is something like a “logical” or maybe “metaphysical” matter, while escape is a “causal” matter. Now, universalists like David Lewis do tend to think that the whole is a free lunch, nothing but the “sum of the parts”, in which case it makes sense to think that composition is not something for the laws of nature to specify. But if we are not universalists, then it seems to me that it is very natural to think of composition in a causal way: when a proper plurality of xs are arranged a certain way, they cause the existence of a new entity y that stands in a composed-by relation to the xs, just as when a projectile has a certain velocity, that causes the projectile to escape to infinity.

Some may be bothered by the fact that laws of nature are often taken to be contingent, and so there would be a world with the same parts as ours but different wholes. That would bother one if one thinks that wholes are a free lunch. But if we take wholes seriously, it should no more bother us than a world where particles behave the same way up to time t₁, and then behave differently after t₁ because the laws are different.

Humeans have good reason to reject the above view, though. If the laws of composition are to match our intuitions about composition, they are likely to be extremely complex, and perhaps too complex to be part of the best system defining the laws on a Humean account of laws. But if we are not Humeans about laws, and think the simplicity of laws is merely an epistemic virtue, the explanatory power of laws of composition might make it reasonable to accept very complex such laws.

That said, we all have reject the simple causal version of the above view, where a proper plurality composing a whole causes the whole’s existence. For instance, I am composed by a plurality of parts that includes my hair, but my hair is not a cause of my existence: I would have just as much existed had I never developed hair. So a more complex version of the causal view is needed: initial parts (maybe the DNA in the zygote that I started as) causally contribute to the existence of the whole, but the causal relation runs in a different direction with respect to later parts, like teeth: perhaps I and my teeth together cause the teeth to be parts of me.

(I don’t endorse the more complex causal view either. I prefer, but still do not endorse, an Aristotelian alternative: when y is in a certain condition, it causes the existence of all of the parts. This is much neater because the causation always runs in the same direction.)

The epistemic force of beauty in laws of nature does not reduce to simplicity

Some people think that simplicity of laws of nature is a guide to truth, and some think beauty of laws of nature is. One might ask: Is the beauty of laws of nature a guide that goes beyond simplicity? Are there times when one could make epistemic decisions about the laws of nature on the basis of beauty where simplicity wouldn’t do the job?

I think so. Here is one case. Suppose we live in a Newtonian universe, and we are discovering fundamental forces. The first one has an inverse cube law. The second has an inverse cube law. These two laws account for most phenomena, but a few don’t fit. Scientists think there is a third fundamental force. For the third force law, we have two proposals that fit the data: an inverse square law and a slightly more complicated inverse cube law. It is, I think, quite reasonable to go for an inverse cube law by induction over the laws.

There is something indeed beautiful about the idea that the same power law applies to all the forces of nature. But if we just go with simplicity, we will go for an inverse square law. However, going for the inverse cube law seems clearly reasonable, and it is what beauty suggests—but not simplicity.

Here is another thought. Sometimes a fundamental law has some particularly lovely mathematical implications. For instance, a conservative force law is connected in a lovely way with a potential. But it need not be the case that a conservative force law is simpler than a non-conservative alternative. (It is true that a conservative force is the gradient of a potential. If the potential can be particularly simply expressed, this makes it easier to express the conservative force law. But we can have a case where the potential is harder to express than the force itself.)

Laws and mathematical complexity

Over the last couple of days I have realized that the laws of physics are rather more complex than they seem. The lovely equations like G = 8πT and F = Gmm′/r² (with a different G in the two equations) seem to be an iceberg most of which is submerged in the icy waters of the foundations of mathematics where the foundational concepts of real analysis and arithmetic are defined in terms of axioms.

This has a curious consequence. We might think that F = Gmm′/r² is much simpler than F = Gmm′/r² + Hmm′/r³ (where H is presumably very, very small). But if we fill out each proposal with the foundational mathematical structure, the percentage difference in complexity will be slight, as almost all of the contribution to complexity will be in such things as the construction of real numbers (say, via Dedekind cuts).

Perhaps, though, the above line of thought is reason to think that real analysis and arithmetic are actually fundamental?

Humeanism and knowledge of fundamental laws

On a "Humean" Best System Account (BSA) of laws of nature, the fundamental laws are the axioms of the system of laws that best combines brevity and informativeness.

An interesting consequence of this is that, very likely, no amount of advances in physics will
suffice to tell us what the fundamental laws are: significant advances in mathematics will also be needed. For suppose that after a lot of extra physics, propositions formulated in sentences p₁, ..., p_n are the physicist’s best proposal for the fundamental laws. They are simple, informative and fit the empirical data really well.

But we would still need some very serious mathematics. For we would need to know there isn’t a simpler collection of sentences {q₁, ..., q_m} that is logically equivalent to {p₁, ..., p_n} but simpler. To do that would require us to have a method for solving the following type of mathematical problem:

1. Given a sentence s in some formal language, find a simplest sentence s′ that is logically equivalent to s,

in the case of significantly non-trivial sentences s.

We might be able to solve (1) for some very simple sentences. Maybe there is no simpler way of saying that there is only one thing in existence than ∃x∀y(x=y). But it is very plausible that any serious proposal for the laws of physics will be much more complicated than that.

Here is one reason to think that any credible proposal for fundamental laws is going to be pretty complicated. Past experience gives us good reason to think the proposal will involve arithmetical operations on real numbers. Thus, a full statement of the laws will require including a definition of the arithmetical operations as well as of the real numbers. To give a simplest formulation of such laws will, thus, require us to solve the problem of finding a simplest axiomatization of the portions of arithmetic and real analysis that are needed for the laws. While we have multiple axiomatizations, I doubt we are at all close to solving the problem of finding an optimal such axiomatization.

Perhaps the Humean could more modestly hope that we will at least know a part of the fundamental laws—namely the part that doesn’t include the mathematical axiomatization. But I suspect that even this is going to be very difficult, because different arithmetical formulations are apt to need different portions of arithmetic and real analysis.

A problem for probabilistic best systems accounts of laws

Suppose that we live in a Humean universe and the universe contains an extremely large collection of coins scattered on a flat surface. Statistical analysis of all the copper coins fits extremely well with the hypothesis that each coin was independently randomly placed with the chance of heads being 1/16 and that of tails being 15/16.

Additionally, there is a gold coin where you haven’t observed which side it’s on.

And there are no other coins.

On a Lewisian best systems account of laws of nature, if the number of coins is sufficeintly large, it will be a law of nature that all coins are independently randomly placed with the chance of heads being 1/16 and that of tails being 15/16. This is true regardless of whether the gold coin is heads or tails. If you know the information I just gave, and have done the requisite statistical analysis of the copper coins, you can be fully confident that this is indeed a law of nature.

If you are fully confident that it is a law of nature that the chance of tails is 15/16, then your credence for tails for the unobserved gold coin should also be 15/16 (I guess this is a case of the Principal Principle).

But that’s wrong. The fact that the coin is of a different material from the observed coins should affect your credence in its being tails. Inductive inferences are weakened by differences between the unobserved and the observed cases.

One might object that perhaps the Lewisian will say that instead of a law saying that the chance of tails on a coin is 15/16, there would be a law that the chance of tails on a copper coin is 15/16. But that’s mistaken. The latter law is not significantly more informative than the former (given that all but one coin is copper), but is significantly less brief. And laws are generated by balancing informativeness with brevity.

A dilemma for best-systems accounts of laws

Here is a dilemma for best-systems accounts of laws.

Either:

1. law-based scientific explanations invoke the lawlike generalization itself as part of the explanation, or

2. they invoke the further fact that this generalization is a law.

Thus, if it is a law that all electrons are charged, and Bob is an electron, on (1) we explain Bob’s charge as follows:

3. All electrons are charged.

4. Bob is an electron.

5. So and that’s why Bob is charged.

But on (2), we replace (3) with:

6. It is a law that all electrons are charged.

Both options provide the Humean with problems.

If it is just the lawlike generalization that explains, then the explanation is fishy. The explanation of why Bob is charged in terms of all electrons being charged seems too close to explaining a proposition by a conjunction that includes it:

7. Bob is charged because Bob is charged and Alice is charged.

Indeed both (3)–(5) and (7) are objectionably cases of explaining the mysterious by the more mysterious: the conjunction is more mysterious than its conjunct and the universal generalization is more mysterious than its instances.

On the other hand, suppose that our explanation of why Bob is charged is that it’s a law that all electrons are charged. This sounds correct in general, but is not appealing on a best-systems view. For on a best-systems view, what the claim that it’s a law that all electrons are charged adds to the claim that all electrons are charged is that the generalization that all electrons are charged is sufficiently informative and brief to make it into the best system. But the fact that it is thus informative and brief does not help it explain anything.

Moreover, if the problem with (3)–(5) was that universal generalizations are too much like conjunctions, the problem will not be relieved by adding more conjuncts to the explanation, namely that the generalization is sufficiently informative and brief.

On two problems for non-Humean accounts of laws

There are three main views of laws:

• Humeanism: Laws are a summing up of the most important patterns in the arrangement of things in spacetime.

• Nomism: Laws are necessary relations between universals.

• Powerism: Laws are grounded in the essential powers of things.

The deficiencies of Humeanism are well known. There are also deficiencies in nomism and powerism, and I want to focus on two.

The first is that they counterintuitively imply that laws are metaphysically necessary. This is well-known.

The second is perhaps less well-known. Nomism and powerism work great for fundamental laws, and for those non-fundamental laws that are logical deductions from the fundamental laws. But there is a category of non-fundamental laws, which I will call impure laws, which are not derivable solely from the fundamental laws, but from the fundamental laws conjoined with certain facts about the arrangement of things in spacetime.

The most notorious of the impure laws is the second law of thermodynamics, that entropy tends to increase. To derive this from the fundamental laws, we need to add some fact about the initial conditions, such as that they have a low entropy. The nomic relations between universals and the essential powers of things do not yield the second law of thermodynamics unless they are combined with facts about which universals are instantiated or which things with which essential powers exist.

A less obvious example of an impure law seems to be conservation of energy. The necessary relations between universals will tell us that in interactions between things with precisely such-and-such universals energy is conserved. And it might well be that the physical things in our world only have these kinds of energy-conserving universals. But things whose universals don’t conserve energy are surely metaphysically possible, and the fact that such things don’t exist is a contingent fact, not grounded in the necessary relations between universals. Similarly, substances with causal powers that do not conserve energy are metaphysically possible, and the non-existence of such things is at best a contingent fact. Thus, to derive the law of conservation of energy, we need not only the fundamental laws grounded in relations between universals or essential powers, but we also need the contingent fact that conservation-violators don’t exist.

Finally, the special sciences (geology, biology, etc.) are surely full of impure laws. Some of them perhaps even merely local ones.

One might bite the bullet and say that the impure laws are not laws at all. But that makes the nomist and powerist accounts inadequate to how “law” gets used in science.

The Humean stands in a different position. If they can account for fundamental laws, impure laws are easy, since the additional grounding is precisely a function of patterns of arrangement. The Humean’s difficulty is with the fundamental laws.

There is a solution, and this is for the nomist and powerist to say that “law of nature” is spoken in many ways, analogically. The primary sense is the fundamental laws that the theories nicely account for. But there are also non-fundamental laws. The pure ones are logical consequences of the fundamental laws, and the impure ones are particularly important consequences of the fundamental laws conjoined with important patterns of things in nature. In other words, impure laws are to be accounted for by a hybrid of the non-Humean theory and the Humean theory.

Now let’s come back to the other difficulty: the necessity worry. I submit that our intuitions about the contingency of laws of nature are much stronger in the case of impure laws than fundamental laws or pure non-fundamental laws. It is not much of a bullet to bite to say that matching charges metaphysically cannot attract—it is quite plausible that this is explained by thevery nature of charge. It is the impure laws where contingency is most obvious: it is metaphysically possible for entropy to decrease (funnily enough, many Humeans deny this, because they define the direction of time in terms of the increase of entropy), and it is metaphysically possible for energy conservation to be violated. But on our hybrid account, the contingency of impure laws is accounted for by the Humean element in them.

Of course, we have to check whether the objections to Humeanism apply to the hybrid theory. Perhaps the most powerful objection to a Humean account of laws is that it only sums up and does not explain. But the hybrid theory can explain, because it doesn’t just sum up—it also cites some fundamental laws. Moreover, it may be the case that the patterns that need to be added to get the impure laws could be initial conditions, such as that the initial entropy is law or that no conservation-violators come into existence. But fundamental law plus initial conditions is a perfectly respectable form of explanation.

On the plurality of bestnesses

According to the best-systems account of laws (BSA), the fundamental laws of nature are the axioms of the system that are true and optimize a balance of informativeness and brevity in a perfectly natural language (i.e., the language cuts reality perfectly at the joints). There are some complications in probabilistic cases, but those will only make my argument below more compelling.

Here is the issue I want to think about: There are many reasonable ways of defining the “balance of informativeness and brevity”.

First, in the case of theories that rule out all but a finite number of worlds, we can say that a theory is more informative if it is compatible with fewer worlds. In such a case, there may be some natural information-theoretic way of measuring informativeness. But in fact, we do not expect the laws of nature to rule out all but a finite number of worlds. We expect them to be compatible with an infinite number of worlds.

Perhaps, though, we get lucky and the laws place restrictions on the determinables in such a way that provides for a natural state space. Then we can try to measure what proportion of that state space is compatible with the laws. This is going to be technically quite difficult. The state space may well turn out to be unbounded and/or infinite dimensional, without a natural volume measure. But even if there is a natural volume measure, it is quite likely that the restrictions placed by the laws make the permitted subset of the state space have zero volume (e.g., if the state space includes inertial and gravitational mass, then laws that say that inertial mass equals gravitational mass will reduce the number of dimensions of the state space, and the reduced space is apt to have zero volume relative to the full space). So we need some way of comparing subsets with zero volume. And mathematically there are many, many tools for this.

Second, brevity is always measured relative to a language. And while the requirement that the language be perfectly natural, i.e., that it cut nature at the joints, rules out some languages, there will be a lot of options remaining. Minimally, we will have a choice point about grouping, Polish notation, dot notation, parentheses, and a slew of other options we haven’t thought of yet, and we will have choice points about the primitive logical operators.

Finally, we have a lot of freedom in how we combine the informativeness and brevity measures. This is especially true since it is unlikely that the informativeness measure is a simple numerical measure, given the zero-volume issue.

We could suppose that there is some objective fact, unknowable to humans, as to what is the right way to define the informativeness and brevity balance, a fact that yields the truth about the laws of nature. This seems implausible. Absent such a fact, what the laws are will be relative to the choice of informativeness and brevity
measure ρ. We might have gotten lucky, and in our world all the measures yield the same laws, but we have little reason to hope for that, and even if this is correct, that’s just our world.

Thus, the story implies that for any reasonable informativeness and brevity measure ρ, we have a concept of a law_ρ. This in itself sounds a bit wrong. It makes the concept of a law not sound objective enough. Moreover, corresponding to each reasonable choice of ρ, it seems we will have a potentially different way to give a scientific explanation, and so the objectivity of scientific explanations is also endangered.

But perhaps worst of all, what BSA had going for it was simplicity: we don’t need any fundamental law or causal concepts, just a Humean mosaic of the distribution of powerless properties. However, the above shows that there is enormous complexity in the account of laws. This is not ideological complexity, but it is great complexity nonetheless. If I am right in my preceding post that at least on probabilistic BSA the fact that something is a law actually enters into explanation, and if I am right in this post that the BSA concept of law has great complexity, then this will end up greatly complicating not just philosophy of science, but scientific explanations.

On probabilistic best-systems accounts, laws aren't propositions

According to the probabilistic best-systems account of laws (PBSA), the fundamental laws of nature are the axioms of the system that optimizes a balance of probabilistic fit to reality, informativeness, and brevity in a perfectly natural language.

But here is a tricky little thing. Probabilistic laws include statements about chances, such as that an event of a certain type E has a chance of 1/3. But on PBSA, chances are themselves defined by PBSA. What it means to say “E has a chance of 1/3” seems to be that the best system entails that E has a chance of 1/3. On its face, this is circular: chance is defined in terms of entailment of chance.

I think there may be a way out of this, but it is to make the fundamental laws be sentences that need not express propositions. Here’s the idea. The fundamental laws are sentences in an formal language (with terms having perfectly natural meanings) and an additional uninterpreted chance operator. There are a bunch of choice-points here: is the chance operator unary (unconditional) or binary (conditional)? is it a function? does it apply to formulas, sentences, event tokens, event types or propositions? For simplicity, I will suppose it’s unary function applying to event types, even though that’s likely not the best solution in the final analysis. We now say that the laws are the sentences provable from the axioms of our best system. These sentences include the uninterpreted chance(x) function. We then say stuff like this:

1. When a sentence that does not use the chance operator is provable from the axioms, that sentence contributes to informativeness, but when that sentence is in fact false, the fit of the whole system becomes  − ∞.

2. When a sentence of the form chance(E) = p is provable from the axioms, then the closeness of the frequency of event type E to p contributes to fit (unless the fit is  − ∞ because of the previous rule), and the statement as such contributes to informativeness.

I have no idea how fit is to be measured when instead of being able to prove things like chance(E) = p, we can prove less precise statements like chance(E) = chance(F) or chance(E) ≥ p. Perhaps we need clauses to cover cases like that, or maybe we can hope that we don’t need to deal with this.

An immediate problem with this approach is that the laws are no longer propositions. We can no longer say that the laws explain, because sentences in a language that is not fully interpreted do not explain. But we can form propositions from the sentences: instead of invoking a law s as itself an explanation, we can invoke as our explanation the second order fact that s is a law, i.e., that s is provable from the axioms of the best system.

This is counterintuitive. The explanation of the evolution of amoebae should not include meta-linguistic facts about a formal language!

Laws of nature are hyperintensional

Are the laws of nature hyperintensional? I.e., if p and q are logically equivalent, could it be that one of them is a law of nature and the other is not?

I am inclined to think so.

Argument 1: The laws of nature in our world do not make reference to particular substances. But if p is a law of nature, then let q be the proposition that p is true and either Biden is president or Biden is not presiden. Then p and q are logically equivalent, but q is not a law as it makes reference to a particular substance.

Argument 2: The laws of nature in our world are first-order. But any first-order proposition p is logically equivalent to the second-order proposition that p is true.

Argument 3: Plausibly, the values of fundamental constants like the fine-structure constant α are a part of the laws of nature. But now imagine that it turns out that the infinitely many significant digits of α express the infinite list of all arithmetical propositions and their truth values in some specific simple encoding scheme. There are two possibilities. Supposing that it is a law of nature that the digits of α have this curious property, then after verifying this property for a sufficiently large number of digits, we could know which of the remaining arithmetical propositions are true simply by measuring α to a high degree of precision. But if the law of nature is simply the brute fact that the digits are 0.007297352569…, and it just happens that these digits encode arithmetical truths in that encoding scheme, then we wouldn’t know truths by just measuring α. (Compare: Imagine a machine where you input an arithmetical proposition, and the machine flips a coin to yield an output of “True” and “False”. Even if we are so lucky that the machine always gives the right answer, that answer wouldn’t be knowledge. It would be just luck.) This means that there is a difference between having a law that says that the digits of α are determined by the arithmetical truths according to that encoding scheme and having an infinite law that simply states the digits, even though the two laws are logically equivalent (assuming the truths of arithmetic are logically necessary; if not, replace the truths of arithmetic by any sequence of hard to know logically necessary truths).

Argument 4: Laws of nature figure in explanations, but explanation is hyperintensional. The correct explanation of why the apple fell down is not that F = Gm₁m₂/r² and either Biden is president or Biden is not president, but simply that F = Gm₁m₂/r².

Argument 5: One of our best accounts of laws of nature is the Lewis-Ramsey best-systems model. But on that model it is very natural to identify the laws of nature with the axioms of the best system, and not just with propositions equivalent to the axioms of the best system.

Final note: I wonder, though, whether there is a unique proposition that expresses any given law of nature. Is there really a fact of the matter whether the law is F = Gm₁m₂/r² or F = m₁m₂(G/r²)?

The structure of morality

In physics, we hope for the following unification: there is a small set of simple laws, and all the rest of physics derives logically from these laws and the contingencies of the arrangement of stuff.

In ethics, a similar ideal has often manifested itself. While I have a hope for the ideal being realized in physics, I have come to be more pessimistic about the ideal in ethics. Instead, I think we can have a looser unificatory structure. We can have a multilevel hierarchy of more general laws, and then more specific laws that specify or implement the more general laws.

I suspect the looser structure is what we have in Aquinas’s Natural Law. At the highest level we have the general law that the good is to be pursued and the bad to be avoided. This is then specified into three laws about promoting the goods of existence, species-specific life and reason. These three laws, I think, are then further specified.

There is thus a structure to the moral law, but it is not a deductive structure. The higher level laws make the lower level laws fitting, but do not necessitate them.

Humeans laws and constants

On Mill-Ramsey-Lewis accounts of laws of nature, the laws are the propositions that best balance informativeness and brevity (in a language that cuts nature precisely at the joints).

Now, the laws of nature include constants, such as the fine-structure constant whose current best measured value is 1/137.035999206. Now, we might be lucky, and it might turn out that the fine-structure constant will have some neat and elegant precise value. There is a history of speculation that it has such a value—for a while, there was hope it was exactly 1/137, and then other guesses took over. But suppose we don’t get so lucky. Suppose it just is some messy number with no simple expression. That should, after all, be a serious possibility.

In that case, the exact value of the fine-structure constant cannot be a part of the Mill-Ramsey-Lewis “world in a nutshell” system of laws, since the system would then be infinitely long, and we lose our hope of defining laws in terms of brevity.

So we have two options. First, the system of laws might not include any specific information on the value of the fine-structure constant, but might instead be of the form ∃αF(α) where F(α) says nothing about what α is, except maybe that it’s real-valued and positive. If we go for this option, then we have to say that all the things that depend on the actual value of the fine-structure constant—and that apparently includes all of chemistry—are not in fact laws of nature. This will likely fail to yield some counterfactuals that we want, and while the laws will be briefer, they will be far less informative than if they had something to say about the value of α.

So that moves us to the second option, which is that the laws are of the form ∃αF(α) and F(α) includes some constraints on α, such as that it lies between 1/137.04 and 1/137.03. These constraints are sufficiently tight to generate the nomic implications we need for chemistry and biology. But while this result seems a better fit for science, it is metaphysically very strange. For it is very strange to think that the laws allow the fine-structure constant to have any of an infinite number of values, but these values must lie in a narrow range.

Furthermore, the exact narrow range for α would be determined by fine details (I am not sure if the pun is intended) of exactly how informativeness and brevity are balanced in the definition of the laws.

The same issue comes up for other constants in the laws of nature. Either Mill-Ramsey-Lewis laws do not include anything about the values of constants or else they include oddly specific, but not completely specific, ranges.

Failures of supervenience on Lewis's system

Start with the concept of “narrowly physical” for facts about the arrangement of physical entities and first-order physical properties such as “charge” and “mass”.

Here are two observations I have not seen made:

1. On Lewis-Ramsey accounts of laws, laws of nature concerning narrowly physical facts do not supervene on narrowly physical facts.

2. On Lewis’s account of causation, causal facts about narrowly physical events do not supervene on narrowly physical facts.

This means that in a Lewisian system we have at least four things we could mean by “physical”:

3. narrowly physical

4. grounded in the laws of narrowly physical facts and/or the narrowly physical facts themselves

5. grounded in the causal facts about narrowly physical events and/or the narrowly physical facts themeselves

6. grounded in the causal facts about narrowly physical events, the laws concerning narrowly physical facts and/or the narrowly physical facts themselves.

Here’s a corollary for the philosophy of mind:

7. On a Lewisian system, we should not even expect the mental properties of purely narrowly physical beings to supervene on narrowly physical facts.

Argument for (1): The laws are the optimal systematization of particular facts. But now imagine a possible world where there is just a coin that is tossed a trillion times, and with no discernible pattern lands heads about half the time. In the best systematization, we attribute a chance of 1/2 to the coin landing heads. But now imagine a possible world with the same narrowly physical facts, but where there is an angel that thought about ℵ₃ about a million times—each time, with a good prior mental explanation of the train of thought—and each of these times was a time just before the coin landed heads. Then the best systematization of the coin tosses will no longer make them simply have a chance of 1/2 of landing heads. Rather, they will have a chance 1/2 of landing heads when the angel didn’t just think about ℵ₃.

Argument for (2): Add to the world in the above argument some cats and suppose that on any day when the fattest cat in the world eats n mice, that leads the angel to think about ℵ_n, though there are other things that can get the angel to think about ℵ_n. We can set things up so that the fattest cat’s eating three mice in a day causes the coin to land heads on the Lewisian counterfactual account of causation, but if we subtract the angel from the story, this will no longer be the case.

A pedagogical universe

Our science developed over milennia, progressing from false theory to less false theory. Why did we not give up long ago? I take it this is because the false theories, nonetheless, had rewards associated with them: although false, they allowed for prediction and technological control in ways that were useful (in a broad sense) to us.

Thus, the success of our science depends not just on a “uniformity of nature” on which the correct fundamental scientific theories are elegant and uniform. Most of our historical progress in physics has not involved correct scientific theories—and quite possibly, we do not have any correct fundamental theories in physics yet. The success of our science required low-hanging fruit for us to pick along the way, fruit that would guide us in the direction of truth.

We can imagine worlds where the ultimate physics requires an enormous degree of sophistication (much as we expect to be the case in our world) and there is little in the way of low-hanging fruit (except maybe for the lowest level of low-hanging fruit, involving the regularities needed to enable evolution of intelligence in the first place) in the form of approximately true theories that rewards us with prediction and control so that beings like us would just give up on science. Our world is better than that.

Indeed, our world seems to be pedagogically arranged for us, arranged to gradually teach us science (and other things), much as we teach our children, with intellectual and practical rewards. There is a design argument for the existence of God from this (closely related to this one).

Formas aristotélicas y leyes de la naturaleza

Rafael Abuchedid kindly translated my piece on Aristotelian forms and laws of nature into Spanish.

Existential inertia and relativity

According to the doctrine of existential inertia, objects have a metaphysical tendency to continue existing absent interference. Existential inertia differs from ordinary physical inertia, in that existential inertia is supposed to come from the metaphysics of time and existence, while physical inertia comes from the laws of nature.

Let’s now imagine that Bob is a physical object that pops into existence at some point z₀ in spacetime, in a universe with nothing that would annihilate Bob (and with Bob lacking any means of self-annihilation). Then according to existential inertia, Bob will continue to exist. But what does that mean in a relativistic setting? Times correspond to spacelike hypersurfaces. A time is future with respect to z₀ provided that it corresponds to a spacelike hypersurface intersecting the future light-cone centered on z₀. Thus, what we get is:

1. If Bob exists at z₀, then for every spacelike hypersurface H that intersects the future light-cone of z₀, Bob exists at some location or other on H.

This is strange on two counts. First, it seems odd that for every spacelike hypersurface that intersects the future light-cone of z₀, we have some sort of a metaphysical guarantee that Bob is somewhere on it—not at any particular place, mind you, but somewhere or other on it. Maybe it doesn’t seem as odd to you as it does to me, though.

Perhaps more seriously, however, (1) describes a metaphysical tendency that makes crucial reference to the speed of light.

In fact, (1) seems to somehow make speed-of-light limits have some sort of metaphysical force. For suppose that Bob faster-than-light travels from z₀ to some point z₁ outside the light-cone of z₀. Then after that episode of faster-than-light travel, Bob could continue living a normal slower-than-light life, while violating (1) with respect to a number of hypersurfaces (that intersect the future cone of z₀). Thus, while violations of existential inertia are supposed to come from destructive powers, a violation of (1) can come from simple faster-than-light travel.

If we suppose that the metaphysics of time requires a privileged reference frame, the above problems disappear. They also disappear if we think that objects come along with a privileged internal time sequence, and that existential inertia is defined with respect to that internal time sequence.

Nomic functionalism

Functionalism says that of metaphysical necessity, whenever x has the same functional state as a system y with internal mental state M, then x has M as well.

What exactly counts as an internal mental state is not clear, but it excludes states like thinking about water for which plausibly semantic externalism is true and it includes conscious states like having a pain or seeing blue. I will assume that functional states are so understood that if a system x has functional state S, then a sufficiently good computer simulation of x has S as well.

A weaker view is nomic functionalism according to which for every internal mental state M (at least of a sort that humans have), there is a law of nature that says that everything that has functional state S has internal mental state M.

A typical nomic functionalist admits that it is metaphysically possible to have S without M, but thinks that the laws of nature necessitate M given S.

I am a dualist. As a result, I think functionalism is false. But I still wonder about nomic functionalism, often in connection with this intuition:

1. Computers can be conscious if and only if functionalism or nomic functionalism is true.

Here’s the quick argument: If functionalism or nomic functionalism is true, then a computer simulation of a conscious thing would be conscious, so computers can be conscious. Conversely, if both computers and humans can be conscious, then the best explanation of this possibility would be given by functionalism or nomic functionalism.

I now think that nomic functionalism is not all that plausible. The reason for this is the intuition that a computer simulation of a cause normally only produces a computer simulation of the effect rather than the effect itself. Let me try to be more rigorous, though.

First, let’s continue from (1):

2. Dualism is true.

3. If dualism is true, functionalism is fale.

4. Nomic functionalism is false.

5. Therefore, neither functionalism nor nomic functionalism is true. (2–4)

6. So, computers cannot be conscious. (1, 5)

And that’s really nice: the ethical worries about whether AI research will hurt or enslave inorganic persons disappear.

The premise I am least confident about in the above argument is (4). Nomic functionalism seems like a serious dualist option. However, I now think there is good inductive reason to doubt nomic functionalism.

7. No known law of nature makes functional states imply non-functional states.

8. So, no law of nature makes functional states imply non-functional states. (Inductively from 7)

9. If functionalism is false, mental states are not functional states.

10. So, mental states are not functional states. (2, 3, 9)

11. So, no law of nature makes functional states imply mental states. (8 and 10)

12. So, nomic functionalism is false. (11 and definition)

Regarding (7), if a law of nature made functional states imply non-functional states, that would mean that we have multiple realizability on the left side of the law but lacked multiple realizability on the right side. It would mean that any accurate computer simulation of a system with the given functional state would exhibit the particular non-functional state. This would be like a case where a computer simulation of water being heated were to have to result in actual water boiling.

I think the most promising potential counterexamples to (7) are thermodynamic laws that can be multiply realized. However, I think tht in those cases, the implied states are typically also multiply realizable.

A variant of the above argument replaces “law” with “fundamental law”, and uses the intuition that if dualism is true, then nomic functionalism would have to have fundamental laws that relate functional states to mental states.

Set theory and physics

Assume the correct physics has precise particle positions (similar questions can be asked in other contexts, but the particle position context is the one I will choose). And suppose we can specify a time t precisely, e.g., in terms of the duration elapsed from the beginning of physical reality, in some precisely defined unit system. Consider two particles, a and b, that exist at t. Let d be the distance between a and b at t in some precisely definable unit system.

Here’s a question that is rarely asked: Is d a real number?

This seems a silly question. How could it not be? What else could it be? A complex number?

Well, there are at least two other things that d could be without any significant change to the equations of physics.

First, d could be a hyperreal number. It could be that particle positions are more fine-grained than the reals.

Second, d could be what I am now calling a “missing number”. A missing number is something that can intuitively be defined by an English (or other meta-language) specification of an approximating “sequence”, but does not correspond to a real number in set theory. For instance, we could suppose for simplicity that d lies between 0 and 1 and imagine a physical measurement procedure that can determine the nth binary digit of d. Then we would have an English predicate M_d(n) which is true just in case that procedure determined the n binary digit to be 1. But it could turn out that in set theory there is no set whose members are the natural numbers n such that M_d(n). For the axioms of set theory only guarantee the existence of a set defined using the predicates of set theory, while M_d is not a predicate of set theory. The idea of such “missing numbers” is coherent, at least if our set theory is coherent.

It seems reasonable to say that d is indeed a real number, and to say similar things about any other quantities that can be similarly physically specified. But what guarantees such a match between set theory and physics? I see four options:

1. Luck: it’s just a coincidence.

2. Our set theory governs physics.

3. Physics governs our set theory.

4. There is a common governor to our set theory and physics.

Option 1 is an unhappy one. Option 4 might be a Cartesian God who freely chooses both mathematics and physics.

Option 2 is interesting. On this story, there is a Platonically true set theory, and then the laws of physics make reference to it. So it’s then a law of physics that distances (say) always correspond to real numbers in the Platonically true set theory.

Option 3 comes in at least two versions. First, one could have an Aristotelian story on which mathematics, including some version of set theory, is an abstraction from the physical world, and any predicates that we can define physically are going to be usable for defining sets. So, physics makes sets. Second, one could have a Platonic multiverse of universes of sets: there are infinitely many universes of sets, and we simply choose to work within those that match our physics. On this view, physics doesn’t make sets, but it chooses between the universes of sets.