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)?

Materialism and incompleteness

It is sometimes thought that Goedel’s incompleteness theorems yield an argument against materialism, on something like the grounds that we can see that the Goedel sentence for any recursively axiomatizable system of arithmetic is true, and hence our minds cannot operate algorithmically.

In this post, I want to note that materialism is quite compatible with being able to correctly decide the truth value of all sentences of arithmetic. For imagine that we live in an infinite universe which contains infinitely many brass plaques with a sentence of arithmetic followed by the word “true” or “false”, such that every sentence of arithmetic is found on exactly one brass plaque. There is nothing contrary to materialsim in this assumption. Now add the further assumption that the word “true” is found on all and only the plaques containing a true sentence of arithmetic. Again, there is nothing contradicting materialism here. It could happen that way simply by chance movements of atoms! Next, imagine a machine where you type in a sentence of arithmetic, and the machine starts traveling outward in the universe in a spiral pattern until it arrives at a plaque with that sentence, reads whether the sentence is true or false, and comes back to you with the result. This could all be implemented in a materialist system, and yet you could then correctly decide the truth value of every sentence of arithmetic.

Note that we should not think of this as an algorithmic process. So the way that this example challenges the argument at the beginning of this post is by showing that materialism does not imply algorithmism.

Objection 1: The plaques are a part of the mechanism for deciding arithmetic, and so the argument only shows that an infinite materialistic machine could decide arithmetic. But our brains are finite.

Response: While our brains are finite, they are analog devices. An analog system contains an infinite amount of information. For instance, suppose that my brain particles have completely precise positions (e.g., on a Bohmian quantum mechanics). Then the diameter of my brain expressed in units of Planck length at some specific time t is some decimal number with infinitely many significant figures. It could turn out that this infinitely long decimal number encodes the truth values of all the sentences of arithmetic, and a machine that measures the diameter of my brain to arbitrary precision could then determine the truth value of every arithmetical statement. Of course, this might turn out not to be compatible with the details of our laws of nature—it may be that arbitrary precision is unachievable—but it is not incompatible with materialism as such.

Objection 2: In these kinds of scenarios, we wouldn’t know that the plaques are right.

Response: After verifying a large number of plaques to be correct, and finding none that we could tell are incorrect, it would be reasonable to conclude by induction that they are all right. However, if the plaques are in fact due to random processes, this inductive conclusion wouldn’t constitute knowledge, except on some versions of reliabilism (which seem implausible to me). But it could be a law of nature that the plaques are right—that’s compatible with materialism. In any case, here the discussion gets complicated.

Supererogation and breakfast

Consider this plausible definition:

1. An action is supererogatory provided that it is good but not obligatory.

Now add this thesis:

2. We have no obligations to ourselves.

It now follows that we’re constantly doing supererogatory stuff. For instance, I just refrained from deliberately painfully burning my finger with a match. My refraining was good. But if I have no obligation to myself, it wasn’t obligatory. Hence, my refraining was supererogatory. This does not seem to be a plausible consequence.

Given the plausibility of (1), this yields some reason to deny (2). Painfully burning my finger with a match violates my obligations to myself.

But there may be other problems with (1) where allowing obligations to self will not solve the problem. I had breakfast this morning, which was good, but even if I do have obligations to self, having breakfast doesn’t seem to be one of them.

Perhaps, though, the breakfast case isn’t so damaging. The supererogation literature talks of very minor supererogatory acts, such as minor acts of politeness. Perhaps having breakfast is just one of the very minor supererogatory acts. If so, then we can save (1), as long as we reject (2) and allow for obligations to self.

The other move is to redefine (1) in a way that excludes benefits to self:

2. An action is supererogatory provided that it is good, and not just for oneself, but not obligatory.

Celibacy for the sake of the Kingdom

The Catholic Church teaches that celibacy for the sake of the Kingdom is better than marriage. Until recently, I assumed that this was celibacy which was chosen by the person as a sacrifice for the sake of the Kingdom. On this interpretation, celibacy which was not chosen by the person—say, because some internal or external factor precludes marriage—does not have that superiority.

But now it has occurred to me that there are two senses in which celibacy can be for the sake of the Kingdom. First, the celibate person may choose it for the sake of the Kingdom. But the second way is that God may choose it for the person for the sake of the Kingdom. Understood in the second way, an involuntary celibacy can still count as for the sake of the Kingdom.

The same point would apply to such things as poverty and obedience. Some choose poverty and obedience to better witness to the Kingdom of God. But for some, God chooses it. And the poverty and obedience can still be for the sake of the Kingdom.

The above is especially true if the calling is embraced with gratitude and love. In that case, we can have a genuine sacrifice of something that, paradoxically, may not even have been available to one. Think here of two early followers of St Francis who joyfully embrace the poverty that he preached: one came from a rich family, and sold all he had, and the other was very poor, and had nothing to give away. It would be problematic if the formerly-rich Franciscan had a permanent superiority in his poverty. Instead, I think, we can say that the always-poor Franciscan is still making a sacrifice by embracing the poverty, by renouncing griping, by rejoicing in God’s gift. The same can be true of a eunuch who embraces celibacy.

Berkeley and authority over bodies

A friend asked me how Berkeley can be refuted. I am fond of ethical insights as epistemically primary. Here is an ethical insight: I have a special authority over my body. But on Berkeley’s view, my body is co-constituted by my ideas and everybody else’s ideas. My ideas are a part of me, yours are a part of you, and so my body is partly constituted by my parts and partly by your parts, Now it is difficult to see why I have special authority over it.

One might say that I have more in the way of perceptions of my body than you do, because I have kinesthetic sensation, introspection, etc. While that is typically true, it is only typically true. If you are doing neurosurgery on me, and I am unconscious, then my body is not constituted at all by my ideas at the time, and you have a lot of perceptions of my body that I never do.

Presentism, evil and privation

Suppose at 8 am, I promised you to call you before noon, and then I didn’t, even though I have no excuse. That’s an evil. When did this evil happen?

If time is continuous, there is no good candidate for the time of this evil. For the omission of calling happened before noon, so noon or later are not when the evil happened. But at any time before noon today, it wasn’t yet true that the promise was unfulfilled, since, if time is continuous, there was always a little bit more time (though, granted, once that time got short enough, it would have taken a miracle to call).

If time is discrete, there is exactly one somewhat plausible candidate for the time of the evil: the very last moment of time time before noon, call it t₁₂₋. It was then that the promise became unfulfilled, and yet that time was itself a time at which the promise was being broken. But even so, even though t₁₂₋ is a somewhat plausible candidate for the time of the evil, it’s not really a great candidate. For the omission didn’t just happen at the very end of the interval of times. It happened throughout the interval.

It seems that the right way to temporally locate the evil is to say that it happened on the time interval between 8 and 12. But note that this is interval-valued temporal location is intuitively different from the case of a headache that one might have from 10 to 11. For we can think of the whole evil of the headache as a sum of evils that are located at shorter intervals or even moments. But it seems the promise-breaking isn’t a sum of evils located at shorter intervals or moments, because the only shorter interval or moment that contains a relevant evil is an interval or moment that contains t₁₂₋ (and even that only if time is discrete). Rather, the promise-breaking is essentially spread over the interval from 8 to 12.

This provides a counterexample to the combination of presentism with a privation theory of evil. For on a privation theory of evil, each evil is constituted by a privation—a lack of something that should be there. But on presentism, things can only exist at specific times, and likewise privations can only be found at specific times. But the evil of promise-breaking is not at a time.

Identity and quantification

One way of posing the question of diachronic identity is to ask for an explanation of facts like

1. The xs compose the very same object at t₁ as the ys compose at t₂

where we do not use sameness or identity or similar concepts in the explanation.

This task turns out to be quite easy. The following is logically equivalent to (1) and does not use sameness or identity:

2. There is an object z such that the xs compose z at t₁ and the ys compose z at t₂.

This is a variant of the point made here.

Two brutenesses

Some philosophers think identity over time is brute: diachronic identity facts are not further explained. Markosian proposed (though later rejected) a view on which composition is brute: whether a bunch of simples make up an object at a time has no further ground.

It seems to me that it would be very natural to hold the two views together in the case of objects composed of simples. Consider the thesis:

• When we have a plurality of particles at t₁ and a plurality of particles at t₂, it is brute whether there is one object which both pluralities compose.

If the pluralities are the same and t₁ = t₂, then as a special case we get the bruteness of composition. And if we presuppose that both pluralities compose, then we get the following bruteness of identity for objects composed of simples: namely, it being brute whether the object at t₁ composed of the xs is identical with the object at t₂ composed of the ys. Since bruteness of diachronic identity for simples is itself extremely plausible, and it is very plausible (pace gunkiness) that everything is either simple or composed of simples, it follows that the above thesis very plausibly yields bruteness of diachronic identity in general.

While one could hold to bruteness of diachronic identity without holding to bruteness of composition or vice versa, it does not seem very natural to me to do so.

Chance and intention

If I intend for an event to happen, I had better intend for my action to raise the chance of the event happening. And most of the time I raise the chance of an event happening precisely in order that the event happen.

But I can also intend to raise the chance of an event happening without intending the event to happen. Thus, when testing one’s product, one uses the product in more extreme ways that deliberately raise the chance of failure. But one isn’t intending failure. One raises the probability of failure with the hope that despite the raised chance, the product does not fail.

Alligator statue

Waco riverwalk, Yashica 12, HPS5+

Impairment of parts and wholes

Aristotelians tend to find the following argument plausible:

1. Every disability includes the impairment of the proper function of one or more body parts or aspects.

2. If a part or aspect is impaired, the whole is impaired.

3. It is bad for one to be impaired.

4. Hence, every disability includes something that is bad for one.

The argument is compatible with a disability not being on balance bad for one (e.g., because maybe it is instrumentally good, for instance because it might contribute to personal growth in various ways, or make one a member of a particularly valuable community, etc.), but it implies that a disability is always bad for one in some respect. I don’t know that I have seen the argument expressly formulated as above, but I think something like this argument has always been at the back of my mind when thinking about disability.

There is philosophical literature challenging premise (3). But (3) is central to Aristotelian eudaimonism, so challenges to it did not seem very plausible to me.

However, it occurred to me yesterday that premise (2) may well be false. Think about cases of redundancy. Suppose toothies are a species of organisms that, unlike humans, have a variable number of teeth. The minimum number of teeth for the first-order proper functioning of a toothy is 60. With fewer teeth, a toothy can’t chew its food sufficiently well to function properly. However, typical healthy toothies exhibit significant redundancy in their dentation, and have anywhere between 70 and 100 teeth. Toothies that have from 60 and 69 teeth either do not have any redundancy in their dentation (at 60) or have insufficient redundancy (from 61 to 69). We can think of redundancy as a second-order proper function. Furthermore, toothies that have more than 100 teeth have the teeth crowded too much in the jaw, which isn’t good for them.

Toothies constantly grow need teeth and wear out old teeth. The worn-out teeth become flaky and weak, and eventually break and fall out. Now suppose that Alice is a toothy with 85 teeth, one of which is well on its way to wearing out. That tooth is impaired. However, Alice is not impaired by virtue of having an impaired tooth, because any number of teeth between 70 and 100 is sufficient for proper first-order (chewing) and second-order (redundancy) functioning of the organism. When that tooth falls out completely, Alice won’t be impaired, and when the teeth has partial function, as it does now, she isn’t impaired either. This, premise (2) is false.

We might suppose that even if Alice isn’t impaired by having an impaired tooth, she would be better off if that tooth weren’t impaired. But that need not be true. For it need not be true that having more teeth is better for one. Having more teeth makes for more redundancy but it also makes for more crowding in the jaw. When the tooth is wearing out, crowding may be decreased (the tooth may be thinner), even though redundancy is also decreased. So it need not be the case that the tooth impairment is in any way bad for Alice.

Now, it may seem that typical impairments of human bodies or aspects are unlike Alice’s tooth impairment. However, this is not clear. Consider intellectual aptitudes. These include reasoning aptitudes in the domains of the spiritual, moral, emotional, intuitive, interpersonal, spatial, logical, arithmetical, artistic, linguistic, kinaesthetic, etc., etc. But different humans have different social roles. Perhaps what is normal for humans is proper functioning of a sufficient number of these aptitudes, not of all of them, so that each human being can find a good niche in society. Furthermore, the sufficient number of these aptitudes may be one that is sufficient to ensure redundancy. In that case, if someone has more than enough redundancy, a severe impairment of one of the aptitudes need not imply a lack of proper function of the human as a whole. But we might, nonetheless, count someone with such a severe impairment of an intellectual aptitude as disabled. If so, being disabled in that respect need not imply being impaired on whole, or badly off in any respect.

However, in the intellectual aptitude case, shouldn’t we say that having more of the aptitudes is better? It isn’t like teeth, where having too many can be harmful, is it? Well, that isn’t completely clear. After all, it can be harder for a kid with many talents to specialize. But even if we grant that one is better off for having more of the aptitudes, this does not mean that lacking one or more is bad. It can be just less good. If Alice’s having 70 teeth is enough for her toothy nature, but 71 is better, then having only 70 isn’t bad, just less good.

That said, there are doubtless some parts or aspects of a human being such that proper function of the part or aspect is necessary for the proper function of the whole. The most obvious cases are the moral and spiritual: someone whose moral or spiritual aspects are impaired is indeed impaired as a person.

Acknowledgment: I feel that some of what I say is influenced in various ways by conversations I had with my superb student Hilary Yancey, but where I have failed to absorb her ideas at the time.

Forced choice and deontology

Suppose the only way to save five innocent people is by killing one innocent person. The deontologist says: you must refrain.

But what you have a forced choice between killing one and killing five?

How could that be? I know of two ways.

First, a psychological block. Perhaps you are brainwashed into killing, but it’s left to your free will whether you are to choose one victim or five.

Second, you can tie the outcomes to cases where effort is required to maintain the status quo but where at least mental effort is needed to stop maintaining it.

As a little introspective experiment, I held my breath for 20 seconds. It took a mild to moderate amount of effort to do so, increasing towards the end of the time period. As our language of “holding” indicates, holding one’s breath is an action. But at the same time, it was clear at all the times that breathing would also be an action—a deliberate interruption of the holding that would also take a mental and physical effort.

Similarly, imagine you’re holding an extremely heavy suitcase. The on-going holding is an effort. But at the same time, to let the suitcase go would also be an effort: you would need to bend your knees to lower it to the ground, or at least move your fingers to release your grip.

In both the breath and suitcase cases, there is no such thing as refraining from action. Holding is an action and letting go is an action.

Very well, now imagine that an evildoer has set things up as follows. They informed you that if you don’t kill the one innocent, five innocents will die. And then they set up a machine that will shoot the one innocent if you let go of the suitcase in the next thirty seconds. What should you do?

If you hold on for thirty seconds, then your effort will ensure that overall four people will die. Even if we grant that you are not intending this tragic consequence, it is wrong to act in a way that produces such a consequence. Think about this in terms of Double Effect (I am grateful to one of our grad students for the connection): holding on to the suitcase has an evil consequence that is disproportionate to whatever goods are involved in holding on.

If you let go, however, then only one person will die. This seems better. But if that’s why you let go, then you are letting go in order that that one person’s death should prevent the deaths of the five. And that violates deontology.

Here is a tentative suggestion. Standard deontological principles have an unstated presupposition: refraining from action is possible. If refraining is impossible, the principles apply at best in modified form.

More film photography

I acquired a Yashica 12 medium-format TLR camera from the 1960s. (The lens hood is 3D printed, though.)

I'm still learning how to use it, but here are some pictures on the Waco river walk. The bridge was taken with Arista 400 and the other pictures with HPS5. I need to learn to focus better. 

Double Effect and trivially good ends

Suppose I am facing the classic trolley situation: a trolley is hurtling towards five persons on the left track, but I can redirect the trolley onto the right track where there is only one person.

A normal decent person will redirect the trolley in order to save the five persons, tolerating the unintended death of the one person as a side-effect. But now imagine that Alice redirects the trolley solely in order to enjoy the feeling of moving the railway switch. She realizes that this spells the death of the person on the right track. But she reasons as follows:

I am not intending the death of that person. I am intending the pleasant feeling of moving the railway switch. I foresee the death of one person on the right track. However, the overall foreseen consequences of the action are positive: five live, one dies and I have fun, instead of five dying and one living. Thus the Principle of Double Effect applies. I am aiming at a good (fun), with the bad effect (death of one) not intended as either an end or as a means, and the good effect is proportionate to the overall consequences, since the overall consequences are positive.

There is something with Alice’s failure to intend to save the lives of the five. What is wrong? Here is one suggestion:

1. In the proportionality condition of the Principle of Double Effect, we do not simply compare the good and the bad effects, but we compare the intended good effects to all the foreseen bad effects.

In the case of a normal decent person, saving the five persons is intended, and proportionality holds. But Alice doesn’t intend to save the five and so there is no proportionality.

I am inclined to think the suggested proportionality condition asks for too much. Suppose Bob is an agent much like in the original trolley situation, except that he is tied down, near the switch, in such a way that his leg protrudes onto the left track after the switch. Bob is (rightly) terrified of getting his leg amputated, and decides to redirect the trolley. He then notices that on the right track there is a person, and so if he redirects the trolley, that person will die. He is about to resign himself to loss of the leg, when he looks at the left track and notices that there are five people there. He reasons that while a person will die, on balance the consequences of redirecting are good, and redirects the trolley solely in order to save his leg.

While we would prefer it if Bob intended to save the five people on the left track, I do not think Bob did anything wrong. What was wrong with Alice’s action was that her end, the pleasure of flipping the switch, was execrably trivial in comparison to the death of the person on the right track. Bob’s end is far from trivial. Thus, I suggest:

2. In the proportionality condition of the Principle of Double Effect, we do two comparisons. First, we ensure that the intended good effects are not trivial in comparison to all the foreseen bad effects. Second, we ensure that the foreseen good effects are proportionate to the foreseen bad effects.

Might there have been less randomness earlier?

In my previous post, I noted that a branching view of possibility, when continued into an infinite past, leads to the counterintuitive consequence that there is less and less randomness the further back we go.

In this post I want to note that this counterintuitive consequence may in fact be right even with a finite past, given a certain interpretation of quantum mechanics.

Start with the naive consciousness causes collapse (ccc) interpretation of quantum mechanics. On naive ccc, at each moment of time, the laws of nature prevent the world to evolve into a superposition of states that differ with respect to consciousness. Thus, there cannot be a superposition between one’s feeling hot and one’s not feeling hot, or between a cat being aware of its surroundings and a cat being asleep or dead. This is assured by constant collapse with respect to a global consciousness operator C.

Unfortunately, as it stands this is untenable, because it corresponds to a setup where there is constant observation of C, and constant observation of an observable precludes change with respect to that observable by the quantum Zeno effect. In other words, if we had naive ccc, then conscious states would never change, which is empirically absurd.

Here is one way to fix this problem. Suppose that there are special moments in time, which I’ll poetically call “cosmic heartbeats”. Collapse with respect to C only occurs at cosmic heartbeats. If the cosmic “heart rate” is not very fast (i.e., the spacing between the heartbeats is big enough), then the quantum Zeno effect will be negligible, and we needn’t worry about it. And we hypothesize that consciousness only occurs at cosmic heartbeats.

But now let’s consider the history of our universe. In the early universe, the only way to get a non-empty consciousness state is by some ridiculously unlikely feat of quantum tunnelling generating a Boltzmann brain or the like. Thus the only randomness we will have in the early universe will be that induced by pruning away the components of the global wavefunction corresponding to such ridiculously unlikely feats. And that is only a tiny bit of randomness. But as things evolve, we get components of the wave function with significant weight corresponding to the evolution of various conscious critters. Now the periodic collapse will be “deciding” between states of comparable likelihood (e.g., life on earth versus life on some other planet formed from some of the same materials orbiting the sun) rather than just pruning away extremely unlikely options.

One would need to know a lot more physics (and perhaps neuroscience?) to figure out what the cosmic heartrate needs to be to make the theory work. An upper bound is given by the quantum Zeno effect: if the cosmic heartrate is too fast, then we could predict a slowdown of consciousness. A lower bound is given by introspection: the cosmic heartrate had better be at least as fast as the speed at which our conscious states are observed to change.

I wonder if a similar decrease of randomness in the past wouldn’t be predicted by GRW collapse theories.