Wednesday, May 16, 2018

Possibly giving a finite description of a nonmeasurable set

It is often assumed that one couldn’t finitely specify a nonmeasurable set. In this post I will argue for two theses:

  1. It is possible that someone finitely specifies a nonmeasurable set.

  2. It is possible that someone finitely specifies a nonmeasurable set and reasonably believes—and maybe even knows—that she is doing so.

Here’s the argument for (1).

Imagine we live an uncountable multiverse where the universes differ with respect to some parameter V such that every possible value of V corresponds to exactly one universe in the multiverse. (Perhaps there is some branching process which generates a universe for every possible value of V.)

Suppose that there is a non-trivial interval L of possible values of V such that all and only the universes with V in L have intelligent life. Suppose that within each universe with V in L there runs a random evolutionary process, and that the evolutionary processes in different universes are causally isolated of each other.

Finally, suppose that for each universe with V in L, the chance that the first instance of intelligent life will be warm-blooded is 1/2.

Now, I claim that for every subset W of L, the following statement is possible:

  1. The set W is in fact the set of all the values of V corresponding to universes in which the first instance of intelligent life is warm-blooded.

The reason is that if some subset W of L were not a possible option for the set of all V-values corresponding to the first instance of intelligent life being warm-blooded, then that would require some sort of an interaction or dependency between the evolutionary processes in the different universes that rules out W. But the evolutionary procesess in the different universes are causally isolated.

Now, let W be any nonmeasurable subset of L (I am assuming that there are nonmeasurable sets, say because of the Axiom of Choice). Then since (3) is possible, it follows that it is possible that the finite description “The set of values of V corresponding to universes in which the first instance of intelligent life is warm blooded” describes W, and hence describes a nonmeasurable set. It is also plainly compossible with everything above that somebody in this multiverse in fact makes use of this finite description, and hence (1) is true.

The argument for (2) is more contentious. Enrich the above assumptions with the added possibility that the people in one of the universes have figured out that they live in a multiverse such as above: one parametrized by values of V, with an interval L of intelligent-life-permitting values of V, with random and isolated evolutionary processes, and with the chance of intelligent life being warm-blooded being 1/2 conditionally on V being in L. For instance, the above claims might follow from particularly elegant and well-confirmed laws of nature.

Given that they have figured this out, they can then let “Q” be an abbreviation for “The set of all values of V corresponding to universes wehre the first instance of intelligent life is warm-blooded.” And they can ask themselves: Is Q likely to be measurable or not?

The set Q is a randomly chosen subset of L. On the standard (product measure) understanding of how to probabilistically make sense of this “random choice” of subset, the event of Q being nonmeasurable is itself nonmeasurable (see the Sawin answer here). However, intuitively we would expect Q to be nonmeasurable. Terence Tao shares this intuition (see the paragraph starting “Intuitively”). His reason for the intuition is that if Q were measurable, then by something like the Law of Large Numbers, we would expect the intersection of Q with a subinterval I of L to have a measure equal to half of the measure of I, which would be in tension with the Lebesgue Density Theorem. This reasoning may not be precisifiable mathematically, but it is intuitively compelling. One might also just have a reasonable and direct intuition that the nonmeasurability is the default among subsets, and so a “random subset” is going to be nonmeasurable.

So, the denizens of our multiverse can use these intuitions to reasonably conclude that Q is nonmeasurable. Hence, (2) is true. Can they leverage these intuitions into knowledge? That’s less clear to me, but I can’t rule it out.

Monday, May 14, 2018

Simultaneous and diachronic causation

The main problem with the idea that all causation is simultaneous is to make sense of the obvious fact of diachronic causation, as when setting an alarm in the evening causes it to go off in the morning. Here is a theory that has both simultaneity and diachronicity that bears further examination:

  • All causation between substances is simultaneous

  • There is diachronic causation within a substance.

We now have a model of how setting the alarm works, on the simplifying assumption that the alarm clock is a substance. In the evening, by simultaneous causation, I cause the clock to have a certain state. A sequence of diachronic causal interactions within the clock—accidents of the clock causing other accidents of the clock, say—then causes the alarm to go off. The alarm’s going off then, by means of simultaneous causation between substances, causes particles in the air to move, etc. In other words, the diachronicity of the causation is all internal to the substances.

An even more interesting theory would hold that:

  • All causation between substances is simultaneous

  • All causation within a substance is diachronic.

If we were willing to swallow this, then we would have a very elegant account of the internal time of a substance as constituted by the causal relations within the substance (presumably, the causal relations between the accidents of the substance).

Why are there infinitely many abstracta rather than none?

It just hit me how puzzling Platonism is. There are infinitely many abstract objects. These objects are really real, and their existence seems not to be explained by the existence of concreta, as on Aristotelianism. Why is there this infinitude of objects?

Of course, we can say that this is just a necessary fact. And maybe it’s just brute and unexplained why necessarily there is this infinitude of objects. But isn’t it puzzling?

Augustinian Platonism, on which the abstract objects are ideas in the mind of God, offers an explanation of the puzzle: the infinitely many objects exist because God thinks them. That still raises the question of why God thinks them. But maybe there is some hope that there is a story as to why God’s perfection requires him to think these infinitely many ideas, even if the story is beyond our ken.

I suppose a non-theistic Platonist could similarly hope for an explanation. My intuition is that the Augustinian’s hope is more reasonable.

Friday, May 11, 2018

Fun with desire satisfaction

Suppose that desire satisfaction as such contributes to happiness. Then it makes sense to pay a neuroscientist to induce in me as many desires as possible for obvious mathematical truths: the desire that 1+1=2, that 1+2=3, that 1+3=4, etc.

Or if desire has to be for a state of affairs in one’s control, one can pay the neuroscientist to induce in me as many desires as possible for states of affairs like: my not wearing a T-shirt that has a green numeral 1, my not wearing a T-shirt that has a green numeral 2, etc. Then by not wearing a T-shirt with any green numerals, I fulfill lots of desires.

Thursday, May 10, 2018

Provability and numerical experiments

A tempting view of mathematics is that mathematicians are discovering not facts about what is true, but about what is provable from what.

But proof is not the only way mathematicians have of getting at truth. Numerical experiment is another. For instance, while we don’t have a proof of Goldbach’s Conjecture (each even number bigger than two is the sum of two primes), it has been checked to hold for numbers up to 4 ⋅ 1018. This seems to give significant inductive evidence that Goldbach’s Conjecture is true. But it does not seem to give significant evidence that Goldbach’s Conjecture can be proved.

Here’s why. Admittedly, when we learned that that the conjecture holds for some particular number n, say 13, we also learned that the conjecture can be proved for that specific number n (e.g., 13 = 11 + 2 and 11 and 2 are prime, etc.). Inductively, then, this gives us significant evidence that for each particular number n, Goldbach’s conjecture for n is provable (to simplify notation, stipulate Goldbach’s Conjecture to hold trivially for odd n or n < 4). But one cannot move from ∀n Provable(G(n)) to Provable(∀n G(n)) (to abuse notation a little).

The issue is that the inductive evidence we have gathered strongly supports the claim that Goldbach’s Conjecture is true, but gives much less evidence for the further claim that Goldbach’s Conjecture is provable.

The argument above is a parallel to the standard argument in the philosophy of science that the success of the practice of induction is best explained by scientific realism.

Monday, May 7, 2018

Heaven and materialism: The return of the swollen head problem

Plausibly, there is a maximum information density for human brains. This means that if internal mental states supervene on the information content of brains and there is infinite eternal life, then either:

  1. Our head grows without bound to accommodate a larger and larger brain, or

  2. Our brain remains bounded in size and either (a) eventually we settle down to a single unchanging internal mental state (including experiential state) which we maintain for eternity, or (b) we eternally move between a finite number of different internal mental states (including experiential states).

For if a brain remains bounded in size, there are only finitely many information states it can have, because of the maximum information density. Neither of options 2a and 2b is satisfactory, because mental (intellectual, emotional and volitive) growth is important to human flourishing, and a single unchanging internal mental state or eternal repetition does not fit with human flourishing.

Note, too, that on both options 2a and 2b, a human being in heaven will eventually be ignorant of how long she’s been there. On option 2b, she will eventually also be ignorant of whether it is the first time, the second time, or the billionth that she is experiencing a particular internal mental state. (I am distinguishing “internal mental states” from broad mental states that may have externalist semantics.) This, too, does not fit with the image of eternal flourishing.

This is, of course, a serious problem for the Christian materialist. I assume they won’t want to embrace the growing head option 1. Probably the best bet will be to say that in the afterlife, our physics and biology changes in such a way as to remove the information density limits from the brain. It is not clear, however, that we would still count as human beings after such a radical change in how our brains function.

The above is also a problem for any materialist or supervenientist who becomes convinced—as I think we all should be—that our full flourishing requires eternal life. For the flourishing of an entity cannot involve something that is contrary to the nature of a being of that sort. But if 2a and 2b are not compatible with our flourishing, and if 1 is contrary to our nature, then our flourishing would seem to involve something contrary to our human nature.

This is a variant of the argument here, but focused on mental states rather than on memory.

Friday, May 4, 2018

Medical and spacecraft ventilators

Some thinking that to turn off a patient’s ventilator would not be to kill but “to let die”. But it seems obvious that to turn off a spacecraft’s ventilation system would be to kill the astronauts through suffocation.

Of course, there are differences between the two cases. One difference is that the medical ventilator is more intimately connected to the patient. This difference, however, would seem to make turning off the ventilator be more of a killing.

A perhaps more promising difference is that when the patient’s ventilator is turned off, the patient dies from a disease that renders unassisted breathing impossible, while the astronauts die from the turning off of the air system. Maybe there is something to this, but I am doubtful. For we can also say that just as the patient would die from a disease, the astronauts would die from the airlessness of space. It is true that one of these is a disease and the other is an environmental condition, but why should that make a difference with respect to what is a killing?

Moreover, if an engineer turns off the ventilation system on the spacecraft before an astronaut reveals that the technician’s doctoral dissertation was plagiarized, that’s murder. And similarly if a doctor turns off a ventilator before the patient reveals that the doctor cheated in medical school, that’s clearly murder, too.

Similarly, if the death penalty is ever permissible, it could in some cases be administered by disconnecting a ventilator—and it would clearly still be an execution, and hence a killing.

But what if the doctor turns off the ventilator for some reason other than to cause the patient’s death, say to prevent an electrical overload to the hospital’s system which would kill many other patients? Changing the intentions with which an act of killing is done can change whether the act is an intentional killing, whether the act is wrong and whether the act is a murder, but I do not think it changes whether the act is a killing. Thus, the doctor who turns off the ventilator for a reason other than to cause death is still killing, but not intentionally.

Nor does it make a difference with respect to killing whether the disconnection is thought of as causing or hastening death. The doctor who turns off the ventilator to prevent the doctor’s medical school cheating from coming to light could think of the activity as hastening death—making the patient die before revealing the secret. But it’s still murder, and hence it’s still killing. Similarly, the plagiarist engineer would be a murderer even if the air system on the spacecraft were failing and the astronauts would die anyway within a week.

Of course, the judgment that turning off the ventilator is killing does not imply that it is murder or even impermissible. But if we grant that it is always murder to intentionally kill the innocent, the turning off a ventilator in order to cause or hasten death is murder.

Wednesday, May 2, 2018

Time as the measure of change?

Aristotle says that time is the measure of change.

Suppose a pool of liquid changes from fragrant to putrid. We can quantify or measure such features as:

  • the spatial extent of the change

  • the value (in multiple senses) of the change

  • the probability of the change

  • the temporal extent of the change.

Obviously, when we talk of time as the measure of change, we have in mind the last of these four. But to define time as the temporal measure of change is blatantly circular. So the Aristotelian needs to non-circularly specify the sort of measurement of change that time provides. (I am not saying this can’t be done. But it is a challenge.)

Tuesday, May 1, 2018

Time and clocks

Einstein said that time is what clocks measure.

Consider an object x that travels over some path P in spacetime. How long did the travels of x take? Well, if in fact x had a clock traveling with it, we can say that the travels of x took the amount of time indicated on the clock.

But what if x had no clock with it? Surely, time still passed for x.

A natural answer:

  • the travels of x took an amount of time t if and only if a clock would have measured t had it been co-traveling with x.

That can’t be quite right. After all, perhaps x would have traveled for a different amount of time if x had a clock with it. Imagine, for instance, that x went for a one-hour morning jog, but x forgot her clock. Having forgot her clock, she ended up jogging 64 minutes. But had she had a clock with her, she would have jogged exactly 60 minutes.

That seems, though, a really uncharitable interpretation of the counterfactual. Obviously, we need to fix the spacetime path P that x takes. Thus:

  • the travels of x over path P took an amount of time t if and only if a clock would have measured t had it been co-traveling with x over the same path P.

But this is a very strange counterfactual if we think about it. Clocks have mass. Like any other massive object, they distort spacetime. The spacetime manifold would thus have been slightly different if x had a clock co-moving with it. In fact, it is quite unclear whether one can make any sense of “the same path P” in the counterfactual manifold.

We can try to control for the mass of the clock. Perhaps in the counterfactual scenario, we need to require that x lose some weight—that x plus the clock have the same mass in the counterfactual scenario as x alone had in the actual scenario. Or, more simply, perhaps we can drop x altogether from the counterfactual scenario, and suppose that P is being traveled by a clock of the same mass as x.

But we won’t be able to control for the mass of the clock if x is lighter than any clock could be. Perhaps no clock can be as light as a single electron, say.

I doubt one can fix these counterfactuals.

Perhaps, though, I was too quick to say that if x had no clock with it, time still passed for x. Ordinary material substances do have clocks in them. These clocks may not move perfectly uniformly, but they still provide a measure of length of time. Alice’s jog took 396,400 heartbeats. Bob’s education took up 3/4 of his childhood. Maybe the relevant clocks, then, are internal changes in substances. And where the substances lack such internal changes, time does not pass for them.