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.

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.

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.

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.

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.

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.)

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.

Avoiding double counting of culpabilities

Here’s an interesting double-counting problem for wrongdoing. Alice stands to inherit a lot of money from a rich uncle in Australia. Bob thinks he stands to inherit a lot of money from a rich uncle in New Zealand. Both of them know that it’s wrong to kill rich uncles for their inheritance, but each of them nonetheless hires a hitman with the instruction to kill the rich uncle. Both hitmen run off with the money and do nothing. But Bob in fact has no uncles—he was misinformed.

Here are some plausible observations:

1. Alice culpably committed two wrongs: she violated her conscience and she wronged her uncle by hiring a hitman to kill him.

2. Bob culpably committed only one of these wrongs: he violated his conscience.

3. Bob is just as morally culpable as Alice.

Here is one way to reconcile these observations. We should distinguish between something like moral failings of the will, on the one hand, and wrongdoings, on the other. It is the moral failings of the will that result in culpability. This culpability then will qualify one or more wrongdoings. But the amount of culpability is not accounted by looking at the culpable wrongdoings, but at the moral failings of the will. A being that executes unalloyed perfect justice will look only at these failings of the will. Alice and Bob each morally failed in the same way and to the same degree (as far as the stories go), and so they are equally culpable. But, nonetheless, Alice has two culpable wrongdoings—culpable through the same moral failing of the will, which should not be double counted for purposes of just punishment.

Love and deontology

Sometimes it wrongs a person to intentionally do them what is known to be in their own best interest. If by torturing you for 60 minutes I can prevent you from being tortured in the same way for 70 minutes by someone else, it may be in your best interest that I torture you. But it is still wrong for me to torture you. Cases of this sort can be multiplied, though of course only deontologists will find any of them plausible.

(One can also analyze these cases as ones where the action is wrong because it is a violation of the agent’s own human dignity. I think the actions are violations of the agent’s own dignity, but they are violations of the agent’s dignity because they wrong the other party.)

These are cases where your action wrongs someone but causes them on balance benefit. This means that to be wronged does not entail being on balance harmed.

Here is how I think we should think of these cases. The true ethics is an ethics of love: I should love everyone. But benevolence is only one of the three fundamental aspects of love, with the other two being union and appreciation. To wrong someone is to violate one or more of the three aspects of love. If I intentionally do something that is known to be in your best interest, I do not violate the benevolence aspect of love. But I may violate one of the other two aspects. In the cases I am thinking of, like torture, the act is an affront to your human dignity, and by affronting your human dignity I am directly acting against the appropriate kind of unitive relationship between human beings—hence, I violate the unitive aspect of love.

It may seem, however, that these are cases where I have a real moral dilemma. For if I refuse to do the act, then it seems I am violating the benevolence of love. But this is mistaken. To fail to be benevolent is not to oppose benevolence. Some cases are obvious. If I fail to be benevolent to you because someone just as close to me has a greater need, I may have done something not in your best interest, but I have not violated the benevolence of love. Now, if I intentionally did to you what was not in your best interest because it was not in your best interest, then I have violated love.

Alethic Platonism

I’ve been thinking about an interesting metaphysical thesis about arithmetic, which we might call alethic Platonism about arithmetic: there is a privileged, complete and objectively correct assignment of truth values to arithmetical sentences, not relative to a particular model or axiomatization.

Prima facie, one can be an alethic Platonist about arithmetic without being an ontological Platonist: one can be an alethic Platonist without thinking that numbers really exist. One might, for instance, be a conceptualist, or think that facts about natural numbers are hypothetical facts about sequences of dashes.

Conversely, one can be an ontological Platonist without being an alethic Platonist about arithmetic: one can, for instance, think there really are infinitely many pluralities of abstracta each of which is equally well qualified to count as “the natural numbers”, with different such candidates for “the natural numbers” disagreeing on some of the truths of arithmetic.

Alethic Platonism is, thus, orthogonal to ontological Platonism. Similar orthogonal pairs of Platonist claims can be made about sets as about naturals.

One might also call alethic Platonism “alethic absolutism”.

I suspect causal finitism commits one to alethic Platonism.

Something close to alethic Platonism about arithmetic is required if one thinks that there is a privileged, complete and objectively correct assignment of truth values to claims about what sentence can be proved from what sentence. Specifically, it seems to me that such an absolutism about proof-existence commits one to alethic Platonism about the Σ₁⁰ sentences of arithmetic.

We aren't just rational animals

I think some Aristotelian philosophers are inclined to think that our nature is to be rational animals, so that all rational animals would be of the same metaphysical species. Here is a problem with this. Our nature—or form or essence—specifies the norms for our structure. Our norms specify that we should be bipedal: there is something wrong with us if we are incapable of bipedality. But an intelligent squid would be a rational animal, and its norms would surely not specify that it is supposed to be bipedal. So, it seems, that the hypothetical intelligent squid would have a different nature from ours.

But that was too quick. For it could be that our nature grounds conditionals like:

1. If you’re human, you should have two arms and two legs

2. If you’re a squid, you should have eight arms and two tentacles.

We have some reason to think there are such conditional normative facts even if we take our metaphysical species narrowly to be something like human or even homo sapiens, since presumably our nature grounds normative conditionals about bodily structure with antecedents specifying whether we are male or female.

But there is a hitch here: if humans and intelligent squid have the same form, what makes it be the case that for me the antecedent of 1 is true while for Alice (say) the antecedent of 2 is true? I think our best story may be that it is facts about DNA, so in fact the antecedents of 1 and 2 are abbreviations for complex facts about DNA.

That might work for DNA-based animals, which are all the animals we have on earth, but it probably won’t work for all possible animals. For surely there nomically could be animals that are not based on DNA, and it is implausible that we carry in our nature the grounds for an array of conditionals for all the nomically (at least) possible genetic encoding schemes.

I suppose we could take our nature to be rational members of the Animalia, with the assumption that the kingdom Animalia necessarily includes only DNA-based organisms (but not all of them, of course). But Animalia seems a somewhat arbitrary choice of classification to tack on to rationality. It doesn’t have the exobiological generality of animal, the earthly generality of DNA-based organism, or the specificity of human.

It seems to me that

• rational DNA-based organism, or

• rational member of genus Homo

are better options for where to draw the lines of our metaphysical species, assuming “rationality” is the right category (as opposed to, say, St. John Paul II’s suggestion that we are fundamentally self-givers), than either rational animal or rational member of Animalia.

Balancing between theism and atheism

The problem of evil consists of three main parts:

• The problem of suffering.

• The problem of evil choices.

• The problem of hiddenness (which is an evil at most conditionally on God’s existing).

The theist has trouble explaining why there is so much suffering. The atheist, however, has trouble explaining why there is any suffering, given that suffering presupposes consciousness, and the atheist has trouble explaining why there is any consciousness.

Of course, there are atheist-friendly naturalistic accounts of consciousness. But they all face serious difficulties. This parallels the fact that theists have theodical accounts of why God permits so much suffering, accounts that also face serious difficulties.

So, on the above, considerations of suffering are a net tie between theism and atheism.

The theist does not actually have all that much trouble explaining why there are evil choices. Libertarian free will does the job. Of course, there are some problems with libertarian accounts of free will. These problems are not, I think, nearly as serious as the problems that theists have with explaining why there is so much suffering or atheists have with explaining why there is consciousness. Moreover, there is a parallel problem for the atheist. Evil choices can only exist given free will. Prima facie the most plausible accounts of free will are libertarian agent-causal ones. But those are problematic for the atheist, who will find it difficult to explaining where libertarian agents come from. The atheist probably has to embrace a compatibilist theory, which has at least as many problems as libertarian agent-causalism.

So, considerations of evil choices look at best as a net tie for the atheist.

Finally, there is the problem of hiddenness for the theist. But while the theist has trouble explaining how we don’t all know something so important as the existence of God, the atheist has epistemological trouble of her own: she has trouble explaining how she knows that there is no God. After all, knowledge of the highly abstract facts that enter into arguments regarding the existence of God is not the sort of knowledge that seems to be accessible to evolved natural beings.

So, considerations of knowledge of the existence or non-existence of God look as a net tie.

The problem of evil, however, exhausts the powerful arguments for atheism. But the above considerations far from exhaust the powerful arguments for theism.

The above reasoning no doubt has difficulties. But I want to propose it as a strategy for settling disputes in cases where it's hard to assign probabilities. For even if it's hard to assign probabilities, we can have good intuitions that two considerations are a wash, that they provide equal evidence. And if we can line up arguments in such a way, being more careful with issues of statistical dependence than I was above, then we can come to a view as to which way some bunch of evidence points.

A tweak to the ontomystical argument

In an old paper, I argued that we do not hallucinate impossibilia: if we perceive something, the thing we perceive is possible, even if it is not actual. Consequently, if anyone has a perception—veridical or not—of a perfect being, a perfect being is possible. And mystics have such experiences. But as we know from the literature on ontological arguments, if a perfect being is possible, then a perfect being exists (this conditional goes back at least to Mersenne). So, a perfect being exists.

I now think the argument would have been better formulated in terms of what two-dimensional semanticists like Chalmers call “conceivability”:

1. What is perceived (perhaps non-veridically) is conceivable.

2. A perfect being is perceived (perhaps non-veridically).

3. If a perfect being is conceivable, a perfect being is possible.

4. A perfect being is possible.

5. If a perfect being is possible, a perfect being exists.

6. So, a perfect being exists.

Premise (3) follows from the fact that the notion of a perfect being is not twinearthable, so conceivability and possibility are equivalent for a perfect being (Chalmers is explicit that this is the case for God, but he concludes that God is inconceivable). Premise (1) avoids what I think is the most powerful of Ryan Byerly’s four apparent counterexamples to my original argument: the objection that one might have perceptions that are incompatible with necessary truths about natural kinds (e.g., a perception that a water molecule has three hydrogen atoms).

Non-instrumental pursuit

I pursue money instrumentally—for the sake of what it can buy—but I pursue fun non-instrumentally.

Here’s a tempting picture of the instrumental/non-instrumental difference as embodied in the money fun example:

1. Non-instrumental pursuit is a negative concept: it is instrumental pursuit minus the instrumentality.

But (1) is mistaken for at least two reasons. The shallower reason is an observation we get from the ancients: it is possible to simultaneously pursue the same goal both instrumentally and non-instrumentally. You might have fun both non-instrumentally and in order to rest. But then lack of instrumentality is not necessary for non-instrumental pursuit.

The deeper reason is this. Suppose I am purely instrumentally pursuing money for the sake of what it can buy, but I then remove the instrumentality, either by ceasing to pursue things that can be bought or by ceasing to believe that money can buy things, without adding any new motivations to my will. Then clearly the pursuit of money rationally needs to disappear—if it remains, that is a clear case of irrationality. But if non-instrumental pursuit were simply an instrumental pursuit minus the instrumentality, then why wouldn’t the removal of the instrumentality from my pursuit of money leave me non-instrumentally and rationally pursuing money, just as I non-instrumentally and rationally pursue fun?

There is a positive element in my pursuit of fun, a positive element that would be lacking in my pursuit of money if I started with instrumental pursuit of money and took away the instrumentality and somehow (perhaps per impossibile) continued (but now irrationally) pursuing money. It is thus more accurate to talk of “pursuit of a goal for its own sake” than to talk of “non-instrumental pursuit”, as the latter suggests something negative.

The difference here is somewhat like the difference between the concepts of an uncaused being and a self-existent being. If you take away the cause of a brick and yet keep the brick (perhaps per impossibile), you have a mere uncaused being. That’s not a self-existent being like God is said to be.

Affronts to human dignity

Some evils are not just very bad. They are affronts to human dignity. But those evils, paradoxically, provide an argument for the existence of God. We do not know what human dignity consists in, but it isn’t just being an agent, being really smart, etc. For human dignity to play the sort of moral role it does, it needs to be something beyond the physical, something numinous, something like a divine spark. And on our best theories of what things are like if there is no God, there is nothing like that.

So:

1. There are affronts to human dignity.

2. If there are affronts to human dignity, there is human dignity.

3. If there is human dignity, there is a God.

4. So, there is a God.

This argument is very close to the one I made here, but manages to avoid some rabbit-holes.