# Ultrafilters, Ultraproducts, and Hypernaturals 1: Introduction

This is the series I wish would have existed a month ago when I first started learning about ultrafilters and ultraproducts. First of all, what’s the motivation for learning about these ultra-objects? I imagine that if you’re here, you probably already have some degree of interest in ultrafilters, ultraproducts, or nonstandard models of arithmetic. But I’ll see if I can bolster that interest a little bit.

The most exciting application to me personally is that ultraproducts give you a recipe for constructing new mathematical structures out of familiar ones. Out of some model M one can construct a a new model *M, which typically has a much richer and stranger structure, but is nonetheless elementarily equivalent to M (this is called Łoś’s theorem). Elementary equivalence means that all the expressive resources of first-order logic are insufficient to tell M and *M apart: they agree on all first-order sentences.

For example, the ultraproduct of ℝ (the real numbers) is *ℝ, the hyperreal numbers. The hyperreals contain an enormous supply of infinitesimal quantities clustered around every real, as well as infinitely large quantities. And all the usual operations on ℝ, like addition and multiplication, are already nicely defined in *ℝ, meaning that these infinitesimals and infinities have a well-defined algebraic structure. And Łoś’s theorem tells us that ℝ and *ℝ are elementary equivalent: any first-order sentence true of the reals is also true of the hyperreals.

Another example: the ultraproduct of ℕ (the natural numbers) is *ℕ, the hypernaturals. The hypernaturals don’t contain any infinitesimals, but they do contain an uncountable infinity of infinite numbers. And since *N and N are elementarily equivalent, *ℕ is a model of true arithmetic! This is super exciting to me. True arithmetic is this unimaginably complicated set of sentences; there’s no Turing machine that decides which sentences reside in true arithmetic, nor is there a Turing machine with a halting oracle that does the job, nor even a Turing machine with a halting oracle for Turing machines with halting oracles! The computational complexity of true arithmetic is the limit of this sequence of progressively harder halting problems: you can only decide the set if you have an oracle for the halting problem, plus an oracle for the halting problem for TMs with oracles for the halting problem, plus an oracle for the halting problem for TMs with oracles for the halting problem for TMs with oracles for the halting problem, and so on. This level of complexity is known as 0(ω).

So true arithmetic is this impossibly uncomputable set of sentences. We know trivially that ℕ is a model of true arithmetic, because that’s how true arithmetic is defined { φ | ℕ ⊨ φ }. And we also know that true arithmetic has nonstandard models, because it’s a first-order theory and no first order theory can categorically define ℕ (I think the easiest way to see why this is true is to apply the Löwenheim-Skolem theorem, but – self-promotion alert – there’s also this very nice and simple proof from first principles). And ultraproducts allow us to construct an explicit example of one of these nonstandard models! You can actually write down some of these nonstandard numbers (they look like infinite sequences of natural numbers) and discover their strange properties. For example, (2, 3, 5, 7, 11, …) is a nonstandard prime number, (1, 2, 4, 8, 16, …) is infinitely even, and (0!, 1!, 2!, 3!, 4!, …) is a multiple of every standard natural number. We’ll dive into all of this soon enough.

Ultrafilters and ultraproducts also have applications outside of logic. A fairly basic result about ultrafilters (that every ultrafilter over a finite set contains a singleton) is equivalent to Arrow’s impossibility theorem in voting theory (that any voting system with unanimity, independence of irrelevant alternatives (IIR), and finitely many voters contains a dictator). And the existence of a singleton-free ultrafilter over an infinite set (a free ultrafilter) shows that with infinitely many voters, there’s a non-dictatorial voting system with unanimity and IIR. There’s a pretty good description of those results here, but finish reading my post first!

Nick Bostrom in an old paper titled Infinite Ethics describes how to apply ultrafilters to resolve some of the issues that arise when trying to apply utilitarian ethics in a universe with infinitely many experiencers. Suppose that the current universe contains an infinite number of experiencers, each of whom is having an experience with a valence of +1. By pressing a button, you can immediately transition to a universe where everybody is now experiencing a valence of +2. Clearly pressing the button is a utilitarian good. But 1+1+1+… = ∞ = 2+2+2+…. So it looks like a standard account of utility as real numbers says that the utility of the +2 world is the same as the utility of the +1 world (both just ∞). But if we treat utilities as hyperreal numbers instead of ordinary real numbers, then we can do the comparison and get the right answer. It’s worth noting that this approach doesn’t fix all the problems raised by an infinite universe; for instance, if pressing the button only increases a finite number of experiencers from valence +1 to valence +2, then the net utility of the universe is unchanged, even with hyperreal utilities. (This corresponds to a property that we’ll see soon, which is that a free ultrafilter contains no finite sets.)

Okay, introduction over. Hopefully you’re now excitedly asking yourself the question “So what are these ultrafilters and ultraproducts anyway??” The story begins in the next post!

Hypernaturals Simplified

# Some half-baked thoughts on moral arbitrariness

I find that there’s often a crucial assumption implicit in discussions of abortion ethics. It comes up at mentions of when personhood arises in the development from zygote to fetus to baby. One person claims that some particular moment is the threshold at which personhood arises. The other points out that zooming in to that moment and looking at extremely nearby moments, we see no particular reason to privilege one over the others. This arbitrariness is taken to be a fatal blow for the account of personhood.

This raises an interesting question. Could the fundamental moral laws be arbitrary? By analogy, think about the laws of physics. The laws of physics contain certain parameters like the gravitational constant G and me/mp, the ratio of the mass of an electron to the mass of a proton, whose values are likely arbitrary to some degree. Even taking into account fine-tuning for life, it’s probable that the fine-tuning isn’t infinitely precise and there’ll be some level of arbitrariness in the 1000th decimal place value of G.

If the laws of physics can be arbitrary, why not the laws of morality? Perhaps there’s just one arbitrary point at which moral personhood emerges, and there’s not much motivation for that point over any other. How strange you consider this to be will likely depend on what your meta-ethical theory is. Trivially, if you don’t think there are moral facts at all, then this puzzle never even arises for you. If you think there are moral facts, but there’s somehow socially or biologically determined, then it’s not so puzzling that there would be arbitrariness in the moral facts. But if you’re a moral realist that believes in an objectively true set of laws governing morality, then this view starts to look strange.

Among moral objectivists, it seems to me like anti-Humeans would not be okay with arbitrariness in the laws of morality. In meta-ethics, anti-Humeans are those who believe that moral facts are intrinsically motivating. This doesn’t mesh well with arbitrariness. If the moral laws are arbitrary, then why should I follow them rather than a neighboring set of laws that work just as well? Almost by definition, arbitrariness in the moral laws implies a lack of motivation, both motivation for the letter of the laws and motivation to live by the laws. On the other hand, if one takes a Humean stance on meta-ethics, perhaps arbitrariness is not so puzzling.

Moral arbitrariness might also be troubling to divine command theorists, who believe that the moral rules are set by God. There’s something that seems quite strange about saying that God’s commands are arbitrary to some extent (though to be fair, I say this from a very atheistic perspective, so perhaps my intuitions differ from theists here). But if this feels strange, then why shouldn’t it feel just as strange to say that the laws of the physical universe are arbitrary? Presumably God also decided on the precise values of all the physical parameters, and there seems to be arbitrariness there. Is there something particularly troubling about the idea that God’s choice of moral laws is arbitary?

Moral arbitrariness seems like an inevitable consequence of most, maybe all, moral systems. A rights-based approach has to deal with tradeoffs between different rights: how severe a breach of bodily autonomy is severe enough that it’s better to violate a person’s right to life? Any binary account of personhood seems bound to end up drawing the line at some arbitrary point. And gradualist accounts of personhood come with their own type of arbitrariness: why should the curve of increasing personhood with time look like precisely this, rather than some other very similar curve? Virtue theoretic approaches talk about virtues arising in a happy medium between two vices (e.g. bravery arising between cowardice and foolhardiness), but where is the precise middle point? If one were to completely codify virtue ethics, they would have to say precisely what level of riskiness is bravery, and when it tips over into foolhardiness. But there will always be thought experiments that place you just barely on either side of this threshold and reveal that there is no apparent moral difference between one side and the other.

Perhaps the framework that has the least trouble with moral arbitrariness is consequentialism. Something like utilitarianism says that the threshold for when you should choose Act 1 over Act 2 is exactly when the expected net utility produced by Act 1 exceeds the expected net utility produced by Act 2 (where utility is something like “happiness minus sadness”). Unfortunately, I think that this approach runs in to problems as well. Happiness is not one-dimensional, and neither is suffering. How do you make different types of happiness commensurable? How many sips of hot chocolate are equivalent to a roller-coaster ride? How many minutes in front of a fire on a cold night are equivalent to the moment of insight when you solve a tough mathematical problem? I find it hard to imagine that non-arbitrary answers to these types of questions exist.

If it’s true that most all moral frameworks contain fundamental arbitrariness, as I believe it is, then I think that this turns into a powerful argument against many types of moral realism. If you’re an anti-Humean, then you have to either deny the arbitrariness or explain why arbitrary moral laws would be intrinsically motivating to us. If you think that God created the moral laws, then you have to reckon with the apparent arbitrariness of those laws. Presumably God always makes the optimal choice when one exists, but what does God do when faced with a choice where there is no optimum?

# Immoral or Inconsistent?

In front of you sits a button. If you press this button, an innocent person will be subjected to the worst possible forms of torture constantly for the next year. (There will be no “resistance” built up over time – the torture will be just as bad on day 364 as it was on day 1.) If you don’t press it, N people will receive a slight pin-prick in their arm, just sharp enough to be noticeably unpleasant.

Clearly if N is something small, like, say, 10, you should choose the pin-pricks rather than the torture. But what about for very very large N? Is there any N so large that you’d switch to the torture option? And if so, what is it?

I’m going to take it as axiomatic that no value of N is high enough that it’s worth inflicting the year long torture. Even if you were faced with the choice between year long torture for one and momentary pin-prick for a literal infinity of people, the right choice would be the pin-pricks.

I feel safe in taking this as axiomatic for three main reasons. Firstly, it’s the obvious answer to the average person, who hasn’t spent lots of time thinking about normative ethics. (This is just in my experience of posing the problem to lay-people.)

Secondly, even among those that choose the torture, most of them do so reluctantly, citing an allegiance to some particular moral framework. Many of them say that they think that it’s technically what they should do, but that in reality they probably wouldn’t, or would at least be extremely hesitant to.

Thirdly, my own moral intuitions deliver a clear and unambiguous judgement on this case. I would 100% choose an infinity of people having the pin-pricks, and wouldn’t feel the tiniest bit of guilt about it. My intuition here is at the same level of obviousness as something like “Causing unhappiness is worse than causing happiness” or “Satisfying somebody’s preferences is a moral good, so long as those preferences don’t harm anybody else.”

With all that said, there are some major problems with this view. If you already know what these problems are, then the rest of this post will probably not be very interesting to you. But if you feel convinced that the torture option is unambiguously worse, and don’t see what the problem could be with saying that, then read on!

First of all, pain is on a spectrum. This spectrum is not continuous, but nonetheless there’s a progression of pains from “slight pin-prick” to “year long torture” such that each step is just barely noticeably worse than the previous one.

Second of all, for each step along this progression, there’s a trade-off to be made. For instance, a pin-prick for one person is better than a very slightly worse pin-prick for one person. And a pin-prick for each of one million people is worse than a slightly more painful pin-prick for one person. So there’s some number of people N that will cause you to change your choice from “pin-pricks for N people” to “slightly worse pin-prick for one person.”

Let’s formalize this a little bit. We’ll call our progression of pains p1, p2, p3, …, pn, where p1 is the pain of a slight pin-prick and pn is the pain of yearlong torture. And we’ll use the symbol < to mean “is less bad than”. What we’ve just said is that for each k, (pk for one person) < (pk+1 for one person) AND (pk for one million people) > (pk+1 for one person). (The choice of million really doesn’t matter here, all that we need is that there’s some number for which the slighter pain becomes worse. One million is probably high enough to do the job.)

Now, if (pk for N) < (pk+1 for 1), then surely (pk for 2N) < (pk+1 for 2). The second is just the first one, but two times! If the tradeoff was worth it the first time, then the exact same tradeoff should be worth it the second time. But now what this gives us is the following:

p1 for 1,000,000n > p2 for 1,000,000n-1 > … > pn-1 for 1,000,000 > pn for 1

In other words, if we are willing to trade off at each step along the progression, then we are forced on pain of inconsistency to trade off between the slight pin prick and the yearlong torture! I.e. there has to be some number (namely 1000000n) for which we choose the torture for 1 over the pin-prick for that number of people.

Writing this all out:

1. There’s a progression of pains p1, p2, …, pn-1, pn such that for each k, (pk for one million people) > (pk+1 for one person).
2. If (p for n) > (q for m), then (p for k⋅n) > (q for k⋅m).
3. > is transitive.
4. Therefore, (p1 for 1,000,000n) > (pn for 1)

If you accept the first three premises, then you must accept the fourth. And the problem is, all three premises seem very hard to deny!

For premise 1: It seems just as clearly immoral to choose to inflict a pain on a million people rather than inflict a slightly worse pain on one person as it does to choose to inflict torture on one rather than a pin-prick on an arbitrarily large number of people. Or to make the point stronger, no sane moral system would say that it’s worse to inflict a pain on one person than a very slightly less bad pain on an arbitrarily large number of people. We have to allow some tradeoff between barely different pains.

Premise 2: if you are offered a choice between two options and choose the first, and then before you gain any more information you are offered the exact same choice, with no connection between the consequences of the first choice and the consequences of the second choice, then it seems to me that you’re bound by consistency to choose the first once more. And if you’re offered that choice k times, then you should take the first every time.

(Let me anticipate a potential “counter-example” to this. Suppose you have the choice to either get a million dollars for yourself or for charity. Then you are given that same choice a second time. It’s surely not inconsistent to choose the million for yourself the first time and the million for charity the second time. I agree, it’s not inconsistent! But this example is different than what we’re talking about in Premise 2, because the choice you make the second time is not the same as the choice you made the first. Why? Because a million dollars to a millionaire is not the same as a million dollars to you or me. In other words, the goodness/badness of the consequences of the second choice are dependent on the first choice. In our torture/pin-prick example, this is not the case; the consequences of the second choice are being enacted on an entirely different group of people, and are therefore independent of the first choice.)

Premise 3: Maybe this seems like the most abstract of the three premises, and hence potentially the most easy to deny. But the problem with denying premise 3 is that it introduces behavioral inconsistency. If you think that moral badness is not transitive, then you think there’s an A, B, and C such that you’d choose A over B, B over C, but not A over C. But if you choose A over B and B over C, then you have in effect chosen A over C, while denying that you would do so. In other words, a moral system that denies transitivity of badness cannot be a consistent guide of action, as it will tell you not to choose A over C, while also telling you to take actions that are exactly equivalent to choosing A over C.

And so we’re left with the conclusion that pin-pricks for 1,000,000n people is worse than torture for one person for a year.

Okay, but what’s the take-away of this? The take-away is that no consistent moral system can agree with all of the following judgements:

• Barely distinguishable pains are morally tradeoffable.
• There’s a progression of pains from “momentary pin-prick” to “torture for a year”, where each step is just barely distinguishably worse from the last.
• Torturing one person for a year is worse than inflicting momentary pin-pricks on any number of people.

You must either reject one of these, or accept an inconsistent morality.

This seems like a big problem for anybody that’s really trying to take morality seriously. The trilemma tells us in essence that there is no perfect consistent moral framework. No matter how long you reflect on morality and how hard you work at figuring out what moral principles you should endorse, you will always have to choose at least one of these three statements to reject. And whichever you reject, you’ll end up with a moral theory that is unacceptable (i.e. a moral theory which commits you to immoral courses of action).

# Six Case Studies in Consequentialist Reasoning

Consequentialism is a family of moral theories that say that an act is moral or immoral based on its consequences. If an act has overall good consequences then it is moral, and if it has bad consequences then it is immoral. What precisely counts as a “good” or “bad” consequence is what distinguishes one consequentialist theory from another. For instance, act utilitarians say that the only morally relevant feature of the consequences of our actions is the aggregate happiness and suffering produced, while preference utilitarians say that the relevant feature of the consequences is the number and strength of desires satisfied. Another form of consequentialism might strike a balance between aggregate happiness and social equality.

What all these different consequentialist theories have in common is that the ultimate criteria being used to evaluate the moral status of an action is only a function of the consequences of that action, as opposed to, say, the intentions behind the action, or whether the action is an instance of a universalizable Kantian rule.

In this essay, we’ll explore some puzzles in consequentialist theories that force us to take a more nuanced and subtle view of consequentialism. These puzzles are all adapted from Derek Parfit’s Reasons and Persons, with very minor changes.

First, we’ll consider a simple puzzle regarding how exactly to evaluate the consequences of one’s actions, when one is part of a collective that jointly accomplishes some good.

Case 1: There are 100 miners stuck in a mineshaft with flood waters rising. These men can be brought to the surface in a lift raised by weights on long levers. The leverage is such that just four people can stand on a platform and provide sufficient weight to raise the lift and save the lives of the hundred men. But if any fewer than four people stand on the platform, it will not be enough to raise the lift. As it happens, you and three other people happen to be standing there. The four of you stand on the platform, raising the lift and saving the lives of the hundred men.

The question for us to consider is, how many lives did you save by standing on the platform? The answer to this question matters, because to be a good consequentialist, each individual needs to be able to compare their contribution here to the contribution they might make by going elsewhere. As a first thought, we might say that you saved 100 lives by standing on the platform. But the other three people were in the same position as you, and it seems a little strange to say that all four of you saved 100 lives each (since there weren’t 400 lives saved total). So perhaps we want to say that each of you saved one quarter of the total: 25 lives each.

Parfit calls this the Share-of-the-Total View. We can characterize this view as saying that in general, if you are part of a collective of N people who jointly save M lives, then your share of lives saved is M/N.

There are some big problems with this view. To see this, let’s amend Case 1 slightly by adding an opportunity cost.

Case 2: Just as before, there are 100 miners stuck in a mineshaft with flood waters rising, and they can be saved by four or more people standing on a platform. This time though, you and four other people happen to be standing there. The other four are going to stand on the platform no matter what you do. Your choice is either to stand on the platform, or to go elsewhere to save 10 lives. What should you do?

The correct answer here is obviously that you should leave to save the 10 lives. The 100 miners will be saved whether you stay or leave, and the 10 lives will be lost if you stick around. But let’s consider what the Share-of-the-Total View says. According to this view, if you stand on the platform, your share of the lives saved is 100/5 = 20. And if you leave to go elsewhere, you only save 10 lives. So you save more lives by staying and standing on the platform!

This is a reductio of the Share-of-the-Total View. We must revise this view to get a sensible consequentialist theory. Parfit’s suggestion is that we say that when you join others who are doing good, the good that you do is not just your own share of the total benefit. You should also add to your share the change that you caused in the shares of the benefits produced by each other by joining. On their own, the four would each have a share of 25 lives. So by joining, you have a share of 20 lives, minus the 5 lives that have been reduced from the share of each of the other four. In other words, by joining, you have saved 20 – 5(4) lives, in other words, 0 lives. And of course, this is the right answer, because you have done nothing at all by stepping onto the platform!

Applying our revised view to Case 1, we see that if you hadn’t stepped onto the platform, zero lives would be saved. By stepping onto the platform, 100 lives are saved. So your share of those lives is 25, plus 25 lives for each of the others that would have had zero without you. So your share is actually 100 lives! The same applies to the others, so in our revised view, each of the four is responsible for saving all 100 lives. Perhaps on reflection this is not so unintuitive; after all, it’s true for each of them that if they change their behavior, 100 lives are lost.

Case 3: Just as in Case 2, there are 100 miners stuck in a mineshaft. You and four others are standing on the platform while the miners are slowly being raised up. Each of you know of an opportunity to save 10 lives elsewhere (a different 10 lives for each of you), but to successfully save the lives you have to leave immediately, before the miners are rescued. The five of you have to make your decision right away, without communicating with each other.

We might think that if each of the five of you reasons as before, each of you will go off and save the other 10 lives (as by staying, they see that they are saving zero lives). In the end, 50 lives will be saved and 100 lost. This is not good! But in fact, it’s not totally clear that this is the fault of our revised view. The problem here is lack of information. If each of the five knew what the other four planned on doing, then they would make the best decision (if all four planned to stay then the fifth would leave, and if one of the other four planned to leave then the fifth would stay). As things stand, perhaps the best outcome would be that all five stay on the platform (losing the opportunity to save 10 extra lives, but ensuring the safety of the 100). If they can use a randomized strategy, then the optimal strategy is to each stay on the platform with probability 97.2848% (saving an expected 100.66 lives)

Let’s move on to another type of scenario.

Case 4: X and Y simultaneously shoot and kill me. Either shot, by itself, would have killed.

The consequence of X’s action is not that I die, because if X had not shot, I would have died by Y’s bullet. And the same goes for Y. So if we’re evaluating the morality of X or Y’s action based on its consequences, it seems that we have to say that neither one did anything immoral. But of course, the two of them collectively did do something immoral by killing me. What this tells us that the consequentialist’s creed cannot be “an act is immoral if its consequences are bad”, as an act can also be immoral if it is part of a set of acts whose collective consequences are bad.

Inheriting immorality from group membership has some problems, though. X and Y collectively did something immoral. But what about the group X, Y, and Barack Obama, who was napping at home when this happened? The collective consequences of their actions were bad as well. So did Obama do something immoral too? No. We need to restrict our claim to the following:

“When some group together harm or benefit other people, this group is the smallest group of whom it is true that, if they had all acted differently, the other people would not have been harmed, or benefited.” -Parfit

A final scenario involves the morality of actions that produce imperceptible consequences.

Case 5: One million torturers stand in front of one million buttons. Each button, if pushed, induces a tiny stretch in each of a million racks, each of which has a victim on it. The stretch induced by a single press of the button is so minuscule that it is imperceptible. But the stretch induced by a million button presses produces terrible pain in all the victims.

Clearly we want to say that each torturer is acting immorally. But the problem is that the consequences of each individual torturer’s action are imperceptible! It’s only when enough of the torturers press the button that the consequence becomes perceptible. So what we seem to be saying is that it’s possible to act immorally, even though your action produces no perceptible change in anybody’s conscious experience, if your action is part of a collection of actions that together produce negative changes in conscious experiences.

This is already unintuitive. But we can make it even worse.

Case 6: Consider the final torturer of the million. At the time that he pushes his button, the victims are all in terrible agony, and his press doesn’t make their pain any perceptibly worse. Now, imagine that instead of there being 999,999 other torturers, there are zero. There is just the one torturer, and the victims have all awoken this morning in immense pain, caused by nobody in particular. The torturer presses the button, causing no perceptible change in the victims’ conditions. Has the torturer done something wrong?

It seems like we have to say the same thing about the torturer in Case 6 as we did in Case 5. The only change is that Nature has done the rest of the harm instead of other human beings, but this can’t matter for the morality of the torturer’s action. But if we believe this, then the scope of our moral concerns is greatly expanded, to a point that seems nonsensical. My temptation here is to say “all the worse for consequentialism, then!” and move to a theory that inherently values intentions, but I am curious if there is a way to make a consequentialist theory workable in light of these problems.

# Sapiens: How Shared Myths Change the World

I recently read Yuval Noah Harari’s book Sapiens and loved it. In additional to fascinating and disturbing details about the evolutionary history of Homo sapiens and a wonderful account of human history, he has a really interesting way of talking about the cognitive abilities that make humans distinct from other species. I’ll dive right into this latter topic in this post.

Imagine two people in a prisoner’s dilemma. To try to make it relevant to our ancestral environment, let’s say that they are strangers running into one another, and each see that the other has some resources. There are four possible outcomes. First, they could both cooperate and team up to catch some food that neither would be able to get on their own, and then share the food. Second, they could both defect, attacking each other and both walking away badly injured. And third and fourth, one could cooperate while the other defects, corresponding to one of them stabbing the other in the back and taking their resources. (Let’s suppose that each of the two are currently holding resources of more value than they could obtain by teaming up and hunting.)

Now, the problem is that on standard accounts of rational decision making, the decision that maximizes expected reward for each individual is to defect. That’s bad! The best outcome for everybody is that the two team up and share the loot, and neither walks away injured!

You might just respond “Well, who cares about what our theory of rational decision making says? Humans aren’t rational.” We’ll come back to this in a bit. But for now I’ll say that the problem is not just that our theory of rationality says that we should defect. It’s that this line of reasoning implies that cooperating is an unstable strategy. Imagine a society fully populated with cooperators. Now suppose an individual appears with a mutation that causes them to defect. This defector outperforms the cooperators, because they get to keep stabbing people in the back and stealing their loot and never have to worry about anybody doing the same to them. The result is then that the “gene for defecting” (speaking very metaphorically at this point; the behavior doesn’t necessarily have to be transmitted genetically) spreads like a virus through the population, eventually transforming our society of cooperators to a society of defectors. And everybody’s worse off.

One the other hand, imagine a society full of defectors. What if a cooperator is born into this society? Well, they pretty much right away get stabbed in the back and die out. So a society of defectors stays a society of defectors, and a society of cooperators degenerates into a society of defectors. The technical way of speaking about this is to say that in prisoner’s dilemmas, cooperation is not a Nash equilibrium – a strategy that is stable against mutations when universally adopted. The only Nash equilibrium is universal defection.

Okay, so this is all bad news. We have good game theoretic reasons to expect society to degenerate into a bunch of people stabbing each other in the back. But mysteriously, the record of history has humans coming together to form larger and larger cooperative institutions. What Yuval Noah Harari and many others argue is that the distinctively human force that saves us from these game theoretic traps and creates civilizations is the power of shared myths.

For instance, suppose that the two strangers happened to share a belief in a powerful all-knowing God that punishes defectors in the afterlife and rewards cooperators. Think about how this shifts the reasoning. Now each person thinks “Even if I successfully defect and loot this other person’s resources, I still will have hell to pay in the afterlife. It’s just not worth it to risk incurring God’s wrath! I’ll cooperate.” And thus we get a cooperative equilibrium!

Still you might object “Okay, but what if an atheist is born into this society of God-fearing cooperative people? They’ll begin defecting and successfully spread through the population, right? And then so much for your cooperative equilibrium.”

The superbly powerful thing about these shared myths is the way in which they can restructure society around them. So for instance, it would make sense for a society with the cooperator-punishing God myth to develop social norms around punishing defectors. The mythical punishment becomes an actual real-world punishment by the myth’s adherents. And this is enough to tilt the game-theoretic balance even for atheists.

The point being: The spreading of a powerful shared myth can shift the game theoretic structure of the world, altering the landscape of possible social structures. What’s more, such myths can increase the overall fitness of a society. And we need not rely on group selection arguments here; the presence of the shared myth increases the fitness of every individual.

A deeper point is that the specific way in which the landscape is altered depends on the details of the shared myth. So if we contrast the God myth above to a God that punishes defectors but also punishes mortals who punish defectors, we lose the stability property that we sought. The suggestion being: different ideas alter the game theoretic balance of the world in different ways, and sometimes subtle differences can be hugely important.

Another take-away from this simple example is that shared myths can become embodied within us, both in our behavior and in our physiology. Thus we come back to the “humans aren’t rational” point: The cooperator equilibrium becomes more stable if the God myth somehow becomes hardwired into our brains. These ideas take hold of us and shape us in their image.

Let’s go further into this. In our sophisticated secular society, it’s not too controversial to refer to the belief in all-good and all-knowing gods as a myth. But Yuval Noah Harari goes further. To him, the concept of the shared myth goes much deeper than just our ideas about the supernatural. In fact, most of our native way of viewing the world consists of a network of shared myths and stories that we tell one another.

After all, the universe is just physics. We’re atoms bumping into one another. There are no particles of fairness or human rights, no quantum fields for human meaning or karmic debts. These are all shared myths. Economic systems consist of mostly shared stories that we tell each other, stories about how much a dollar bill is worth and what the stock price of Amazon is. None of these things are really out there in the world. They are in our brains, and they are there for an important reason: they open up the possibility for societal structures that would otherwise be completely impossible. Imagine having a global trade network without the shared myth of the value of money. Or a group of millions of humans living packed together in a city that didn’t all on some level believe in the myths of human value and respect.

Just think about this for a minute. Humans have this remarkable ability to radically change our way of interacting with one another and our environments by just changing the stories that we tell one another. We are able to do this because of two features of our brains. First, we are extraordinarily creative. We can come up with ideas like money and God and law and democracy and whole-heartedly believe in them, to the point that we are willing to sacrifice our lives for them. Second, we are able to communicate these ideas to one another. This allows the ideas to spread and become shared myths. And most remarkably, all of these ideas (capitalism and communism, democracy and fascism) are running on essentially the same hardware! In Harari’s words:

While the behaviour patterns of archaic humans remained fixed for tens of thousands of years, Sapiens could transform their social structures, the nature of their interpersonal relations, their economic activities and a host of other behaviours within a decade or two. Consider a resident of Berlin, born in 1900 and living to the ripe age of one hundred. She spent her childhood in the Hohenzollern Empire of Wilhelm II; her adult years in the Weimar Republic, the Nazi Third Reich and Communist East Germany; and she died a citizen of a democratic and reunited Germany. She had managed to be a part of five very different sociopolitical systems, though her DNA remained exactly the same.

# Against moral realism

Here’s my primary problem with moral realism: I can’t think of any acceptable epistemic framework that would give us a way to justifiably update our beliefs in the objective truth of moral claims. I.e. I can’t think of any reasonable account of how we could have justified beliefs in objectively true moral principles.

Here’s a sketch of a plausible-seeming account of epistemology. Broad-strokes, there are two sources of justified belief: deduction and induction.

Deduction refers to the process by which we define some axioms and then see what logically follows from them. So, for instance, the axioms of Peano Arithmetic entail the theorem that 1+1=2 – or, in Peano’s language, S(0) + S(0) = S(S(0)). The central reason why reasoning by deduction is reliable is that the truths established are true by definition – they are made true by the way we have constructed our terms, and are thus true in every possible world.

Induction is scientific reasoning – it is the process of taking prior beliefs, observing evidence, and then updating these beliefs (via Bayes’ rule, for instance). The central reason why induction is reliable comes from the notion of causal entanglement. When we make an observation and update our beliefs based upon this observation, the brain state “believes X” has become causally entangled with the truth of the the statement X. So, for instance, if I observe a positive result on a pregnancy test, then my belief in the statement “I am pregnant” has become causally entangled with the truth of the statement “I am pregnant.” It is exactly this that justifies our use of induction in reasoning about the world.

Now, where do moral claims fall? They are not derived from deductive reasoning… that is, we cannot just arbitrarily define right and wrong however we like, and then derive morality from these definitions.

And they are also not truths that can be established through inductive reasoning; after all, objective moral truths are not the types of things that have any causal effects on the world.

In other words, even if there are objective moral truths, we would have no way of forming justified beliefs about this. To my mind, this is a pretty devastating situation for a moral realist. Think about it like this: a moral realist who doesn’t think that moral truths have causal power over the world must accept that all of their beliefs about morality are completely causally independent of their truth. If we imagine keeping all the descriptive truths about the world fixed, and only altering the normative truths, then none of the moral realist’s moral beliefs would change.

So how do they know that they’re in the world where their moral beliefs actually do align with the moral reality? Can they point to any reason why their moral beliefs are more likely to be true than any other moral statements? As far as I can tell, no, they can’t!

Now, you might just object to the particular epistemology I’ve offered up, and suggest some new principle by which we can become acquainted with moral truth. This is the path of many professional philosophers I have talked to.

But every attempt that I’ve heard of for doing this begs the question or resorts to just gesturing at really deeply held intuitions of objectivity. If you talk to philosophers, you’ll hear appeals to a mysterious cognitive ability to reflect on concepts and “detect their intrinsic properties”, even if these properties have no way of interacting with the world, or elaborate descriptions of the nature of “self-evident truths.”

(Which reminds me of this meme)

None of this deals with the central issue in moral epistemology, as I see it. This central issue is: How can a moral realist think that their beliefs about morality are any more likely to be true than any random choice of a moral framework?

# Constructing the world

In this six and a half hour lecture series by David Chalmers, he describes the concept of a minimal set of statements from which all other truths are a priori “scrutable” (meaning, basically, in-principle knowable or derivable).

What are the types of statements in this minimal set required to construct the world? Chalmers offers up four categories, and abbreviates this theory PQIT.

# P

P is the set of physical facts (for instance, everything that would be accessible to a Laplacean demon). It can be thought of as essentially the initial conditions of the universe and the laws governing their changes over time.

# Q

Q is the set of facts about qualitative experience. We can see Chalmers’ rejection of physicalism here, as he doesn’t think that Q is eclipsed within P. Example of a type of statement that cannot be derived from P without Q: “There is a beige region in the bottom right of my visual field.”

# I

Here’s a true statement: “I’m in the United States.” Could this be derivable from P and Q? Presumably not; we need another set of indexical truths that allows us to have “self-locating” beliefs and to engage in anthropic reasoning.

# T

Suppose that P, Q, and I really are able to capture all the true statements there are to be captured. Well then, the statement “P, Q, and I really are able to capture all the true statements there are to be captured” is a true statement, and it is presumably not captured by P, Q, and I! In other words, we need some final negative statements that tell us that what we have is enough, and that there are no more truths out there. These “that’s all”-type statements are put into the set T.

⁂⁂⁂

So this is a basic sketch of Chalmer’s construction. I like that we can use these tags like PQIT or PT or QIT as a sort of philosophical zip-code indicating the core features of a person’s philosophical worldview. I also want to think about developing this further. What other possible types of statements are there out there that may not be captured in PQIT? Here is a suggestion for a more complete taxonomy:

p    microphysics
P    macrophysics (by which I mean all of science besides fundamental physics)
Q    consciousness
R    normative rationality
E
normative ethics
C    counterfactuals
L
mathematical / logical truths
I     indexicals
T    “that’s-all” statements

I’ve split P into big-P (macrophysics) and little-p (microphysics) to account for the disagreements about emergence and reductionism. Normativity here is broad enough to include both normative epistemic statements (e.g. “You should increase your credence in the next coin toss landing H after observing it land H one hundred times in a row”) and ethical statements. The others are fairly self-explanatory.

The most ontologically extravagant philosophical worldview would then be characterized as pPQRECLIT.

My philosophical address is pRLIT (with the addendum that I think C comes from p, and am really confused about Q). What’s yours?

# Moving Naturalism Forward: Eliminating the macroscopic

Sean Carroll, one of my favorite physicists and armchair philosophers, hosted a fantastic conference on philosophical naturalism and science, and did the world a great favor by recording the whole thing and posting it online. It was a three-day long discussion on topics like the nature of reality, emergence, morality, free will, meaning, and consciousness. Here are the videos for the first two discussion sections, and the rest can be found by following Youtube links.

Having watched through the entire thing, I have updated a few of my beliefs, plan to rework some of my conceptual schema, and am puzzled about a few things.

A few of my reflections and take-aways:

1. I am much more convinced than before that there is a good case to be made for compatibilism about free will.
2. I think there is a set of interesting and challenging issues around the concept of representation and intentionality (about-ness) that I need to look into.
3. I am more comfortable with intense reductionism claims, like “All fact about the macroscopic world are entailed by the fundamental laws of physics.”
4. I am really interested in hearing Dan Dennett talk more about grounding morality, because what he said was starting to make a lot of sense to me.
5. I am confused about the majority attitude in the room that there’s not any really serious reason to take an eliminativist stance about macroscopic objects.
6. I want to find more details about the argument that Simon DeDeo was making for the undecidability of questions about the relationship between macroscopic theories and microscopic theories (!!!).
7. There’s a good way to express the distinction between the type of design human architects engage in and the type of design that natural selection produces, which is about foresight and representations of reasons. I’m not going to say more about this, and will just refer you to the videos.
8. There are reasons to suspect that animal intelligence and capacity to suffer are inversely correlated (that is, the more intelligent an animal, the less capacity to suffer it likely has). This really flips some of our moral judgements on their head. (You must deliver a painful electric shock to either a human or to a bird. Which one will you choose?)

Let me say a little more about number 5.

I think that questions about whether macroscopic objects like chairs or plants really REALLY exist, or whether there are really only just fermions and bosons are ultimately just questions about how we should use the word “exist.” In the language of our common sense intuitions, obviously chairs exist, and if you claim otherwise, you’re just playing complicated semantic games. I get this argument, and I don’t want to be that person that clings to bizarre philosophical theses that rest on a strange choice of definitions.

But at the same time, I see a deep problem with relying on our commonsense intuitions about the existence of the macro world. This is that as soon as we start optimizing for consistency, even a teeny tiny bit, these macroscopic concepts fall to pieces.

For example, here is a trilemma (three statements that can’t all be correct):

1. The thing I am sitting on is a chair.
2. If you subtract a single atom from a chair, it is still a chair.
3. Empty space is not a chair.

These seem to me to be some of the most obvious things we could say about chairs. And yet they are subtly incoherent!

Number 1 is really shorthand for something like “there are chairs.” And the reason why the second premise is correct is that denying it requires that there be a chair such that if you remove a single atom, it is no longer a chair. I take it to be obvious that such things don’t exist. But accepting the first two requires us to admit that as we keep shedding atoms from a chair, it stays a chair, even down to the very last atom. (By the way, some philosophers do actually deny number 2. They take a stance called epistemicism, which says that concepts like “chair” and “heap” are actually precise and unambiguous, and there exists a precise point at which a chair becomes a non-chair. This is the type of thing that makes me giggle nervously when reflecting on the adequacy of philosophy as a field.)

As I’ve pointed out in the past, these kinds of arguments can be applied to basically everything in the macroscopic world. They wreak havoc on our common sense intuitions and, to my mind, demand rejection of the entire macroscopic world. And of course, they don’t apply to the microscopic world. “If X is an electron, and you change its electric charge a tiny bit, is it still an electron?” No! Electrons are physical substances with precise and well-defined properties, and if something doesn’t have these properties, it is not an electron! So the Standard Model is safe from this class of arguments.

Anyway, this is all just to make the case that upon close examination, our commonsense intuitions about the macroscopic world turn out to be subtly incoherent. What this means is that we can’t make true statements like “There are two cars in the garage”. Why? Just start removing atoms from the cars until you get to a completely empty garage. Since no single-atom change can make the relevant difference to “car-ness”, at each stage, you’ll still have two cars!

As soon as you start taking these macroscopic concepts seriously, you find yourself stuck in a ditch. This, to me, is an incredibly powerful argument for eliminativism, and I was surprised to find that arguments like these weren’t stressed at the conference. This makes me wonder if this argument is as powerful as I think.

# Defining racism

How would you define racism?

I’ve been thinking about this lately in light of some of the scandal around research into race and IQ. It’s a harder question than I initially thought; many of the definitions that pop to mind end up being either too strong or too weak. The term also functions differently in different contexts (e.g. personal racism, institutional racism, racist policies). In this post, I’m specifically talking about personal racism – that term we use to refer to the beliefs and attitudes of those like Nazis or Ku Klux Klan members (at the extreme end).

I’m going to walk through a few possible definitions. This will be fairly stream-of-consciousness, so I apologize if it’s not incredibly profound or well-structured.

Definition 1 Racism is the belief in the existence of inherent differences between the races.

‘Inherent’ is important, because we don’t want to say that somebody is racist for acknowledging differences that can ultimately be traced back to causes like societal oppression. The problem with this definition is that, well, there are inherent differences between the races.

The Chinese are significantly shorter than the Dutch. Raising a Chinese person in a Dutch household won’t do much to equalize this difference. What’s important, it seems, is not the belief in the existence of inherent differences, but instead the belief in the existence of inherent inferiorities and superiorities. So let’s try again.

Definition 2 Racism is the belief in the existence of inherent racial differences that are normatively significant.

This is pretty much the dictionary definition of the term “racism”. While it’s better, there are still some serious problems. Let’s say that somebody discovered that the Slavs are more inherently prone to violence than, say, Arabs. Suppose that somebody ran across this fact, and that this person also held the ethical view that violent tendencies are normatively important. That is, they think that peaceful people are ethically superior to violent people.

If they combine this factual belief with this seemingly reasonable normative belief, they’ll end up being branded as a racist, by our second definition. This is clearly undesirable… given that the word ‘racism’ is highly normatively loaded, we don’t want it to be the case that somebody is racist for believing true things. In other words, we probably don’t want our definition of racism to ever allow it to be the right attitude to take, or even a reasonable attitude to take.

Maybe the missing step is the generalization of attitudes about Slavs and Arabs to individuals. This is a sentiment that I’ve heard fairly often… racism is about applying generalizations about groups to individuals (for instance, racial profiling). Let’s formalize this:

Definition 3 Racism is about forming normative judgments about individuals’ characteristics on the basis of beliefs about normative group-level differences.

This sounds nice and all, but… you know what another term for “applying facts about groups to individuals” is? Good statistical reasoning.

If you live in a town composed of two distinct populations, the Hebbeberans and the Klabaskians, and you know that Klabaskians are on average twenty times more likely than Hebbeberans to be fatally allergic to cod, then you should be more cautious with serving your extra special cod sandwich to a Klabaskian friend than to a Hebbeberan.

Facts about populations do give you evidence about individuals within those populations, and the mere acknowledgement of this evidence is not racist, for the same reason that rationality is not racist.

So if we don’t want to call rationality racist, then maybe our way out of this is to identify racism with irrationality.

Definition 4 Racism is the holding of irrational beliefs about normative racial differences.

Say you meet somebody from Malawi (a region with an extremely low average IQ). Your first rational instinct might be to not expect too much from them in the way of cognitive abilities. But now you learn that they’re a theoretical physicist who’s recently been nominated for a Nobel prize for their work in quantum information theory. If the average IQ of Malawians is still factoring in at all to your belief about this person’s intelligence, then you’re being racist.

I like this definition a lot better than our previous ones. It combines the belief in racial superiority with irrationality. On the other hand, it has problems as well. One major issue is that there are plenty of cases of benign irrationality, where somebody is just a bad statistical reasoner, but not motivated by any racial hatred. Maybe they over-updated on some piece of information, because they failed to take into account an important base-rate.

Well, the base-rate fallacy is one of the most common cognitive biases out there. Surely this isn’t enough to make them a racist? What we want is to capture the non-benign brand of irrational normative beliefs about race – those that are motivated by hatred or prejudice.

Definition 5 Racism is the holding of irrational normative beliefs about racial differences, motivated by racial hatred or prejudice.

I think this does the best at avoiding making the category too large, but it may be too strong and keep out some plausible cases of racism. I’d like to hear suggestions for improvements on this definition, but for now I’ll leave it there. One potential take-away is that the word ‘racism’ is a nasty combination of highly negatively charged and ambiguous, and that such words are best treated with caution, especially when applied them to edge cases.

# The Scourge of Our Time

Human life must be respected and protected absolutely from the moment of conception. From the first moment of his existence, a human being must be recognized as having the rights of a person – among which is the inviolable right of every innocent being to life.

Since it must be treated from conception as a person, the embryo must be defended in its integrity, cared for, and healed, as far as possible, like any other human being.

Catechism of the Catholic Church, #2270, 2274

In this paper, Toby Ord advances a strong reductio ad absurdum of the standard pro-life position that life begins at conception. I’ve heard versions of this argument before, but hadn’t seen it laid out so clearly.

Here’s the argument:

1. The majority (~62%) of embryos die within a few weeks of conception (mostly from failure to implant in the lining of the uterus wall). A mother of three children could be expected to also have had five spontaneous abortions.
2. The Catholic Church promotes the premise that an embryo at conception has the same moral worth as a developed human. On this view, more than 60% of the world population dies in their first month of life, making this a more deadly condition than anything else in human history. Saving even 5% of embryos would save more lives than a cure for cancer.

3. Given the 200 million lives per year at stake, those that think life begins at conception should be directing massive amounts of resources towards ending spontaneous abortion and see it as the Scourge of our time.

Here are two graphs of the US survival curve: first, as we ordinarily see it, and second, as the pro-lifer is obligated to see it:

This is of course a really hard bullet for the pro-life camp to bite. If you’re like me, you see spontaneous abortions as morally neutral. Most of the time they happen before a pregnancy has been detected, leaving the mother unaware that anything even happened. It’s hard then to make a distinction between the enormous amount of spontaneous abortions naturally occurring and the comparatively minuscule number of intentional abortions.

I have previously had mixed feelings about abortion (after all, if our moral decision making ultimately comes down to trying to maximize some complicated expected value, it will likely be blind to whether is a real living being or just a “potential” living being), but this argument pretty much clinches the deal for me.