No Universally Compelling Arguments in Math or Science

Last week, I started a thread on the widespread sentiment that people don’t understand the metaethics sequence. One of the things that surprised me most in the thread was this exchange:

Commenter: “I happen to (mostly) agree that there aren’t universally compelling arguments, but I still wish there were. The metaethics sequence failed to talk me out of valuing this.”

Me: “But you realize that Eliezer is arguing that there aren’t universally compelling arguments in any domain, including mathematics or science? So if that doesn’t threaten the objectivity of mathematics or science, why should that threaten the objectivity of morality?”

Commenter: “Waah? Of course there are universally compelling arguments in math and science.”

Now, I realize this is just one commenter. But the most-upvoted comment in the thread also perceived “no universally compelling arguments” as a major source of confusion, suggesting that it was perceived as conflicting with morality not being arbitrary. And today, someone mentioned having “no universally compelling arguments” cited at them as a decisive refutation of moral realism.

After the exchange quoted above, I went back and read the original No Universally Compelling Arguments post, and realized that while it had been obvious to me when I read it that Eliezer meant it to apply to everything, math and science included, it was rather short on concrete examples, perhaps in violation of Eliezer’s own advice. The concrete examples can be found in the sequences, though… just not in that particular post.

First, I recommend reading The Design Space of Minds-In-General if you haven’t already. TLDR; the space of minds in general ginormous and includes some downright weird minds. The space of human minds is a teeny tiny dot in the larger space (in case this isn’t clear, the diagram in that post isn’t remotely drawn to scale). Now with that out of the way...

There are minds in the space of minds-in-general that do not recognize modus ponens.

Modus ponens is the rule of inference that says that if you have a statement of the form “If A then B”, and also have “A”, then you can derive “B”. It’s a fundamental part of logic. But there are possible mind that reject it. A brilliant illustration of this point can be found in Lewis Carroll’s dialog “What the Tortoise Said to Achilles” (for those not in the know, Carroll was a mathematician; Alice in Wonderland is secretly full of math jokes).

Eliezer covers the dialog in his post Created Already In Motion, but here’s the short version: In Carroll’s dialog, the tortoise asks Achilles to imagine someone rejecting a particular instance of modus ponens (drawn from Euclid’s Elements, though that isn’t important). The Tortoise suggests that such a person might be persuaded by adding an additional premise, and Achilles goes along with it—foolishly, because this quickly leads to an infinite regress when the Tortoise suggests that someone might reject the new argument in spite of accepting the premises (which leads to another round of trying to patch the argument, and then..)

“What the Tortoise Said to Achilles” is one of the reasons I tend to think of the so-called “problem of induction” as a pseudo-problem. The “problem of induction” is often defined as the problem of how to justify induction, but it seems to make just as much senses to ask how to justify deduction. But speaking of induction...

There are minds in the space of minds-in-general that reason counter-inductively.

To quote Eliezer:

There are possible minds in mind design space who have anti-Occamian and anti-Laplacian priors; they believe that simpler theories are less likely to be correct, and that the more often something happens, the less likely it is to happen again.

And when you ask these strange beings why they keep using priors that never seem to work in real life… they reply, “Because it’s never worked for us before!”

If this bothers you, well, I refer you back to Lewis’ Carroll’s dialog. There are also minds in the mind design space that ignore the standard laws of logic, and are furthermore totally unbothered by (what we would regard as) the absurdities produced by doing so. Oh, but if you thought that was bad, consider this...

There are minds in the space of minds-in-general that use a maximum entropy prior, and never learn anything.

Here’s Eliezer again discussing a problem where you have to predict whether a ball drawn out of an urn will be red or white, based on the color of the balls that have been previously drawn out of the urn:

Suppose that your prior information about the urn is that a monkey tosses balls into the urn, selecting red balls with 14 probability and white balls with 34 probability, each ball selected independently. The urn contains 10 balls, and we sample without replacement. (E. T. Jaynes called this the “binomial monkey prior”.) Now suppose that on the first three rounds, you see three red balls. What is the probability of seeing a red ball on the fourth round?

First, we calculate the prior probability that the monkey tossed 0 red balls and 10 white balls into the urn; then the prior probability that the monkey tossed 1 red ball and 9 white balls into the urn; and so on. Then we take our evidence (three red balls, sampled without replacement) and calculate the likelihood of seeing that evidence, conditioned on each of the possible urn contents. Then we update and normalize the posterior probability of the possible remaining urn contents. Then we average over the probability of drawing a red ball from each possible urn, weighted by that urn’s posterior probability. And the answer is… (scribbles frantically for quite some time)… 1/​4!

Of course it’s 14. We specified that each ball was independently tossed into the urn, with a known 14 probability of being red. Imagine that the monkey is tossing the balls to you, one by one; if it tosses you a red ball on one round, that doesn’t change the probability that it tosses you a red ball on the next round. When we withdraw one ball from the urn, it doesn’t tell us anything about the other balls in the urn.

If you start out with a maximum-entropy prior, then you never learn anything, ever, no matter how much evidence you observe. You do not even learn anything wrong—you always remain as ignorant as you began.

You may think, while minds such as I’ve been describing are possible in theory, they’re unlikely to evolve anywhere in the universe, and probably they wouldn’t survive long if programmed as an AI. And you’d probably be right about that. On the other hand, it’s not hard to imagine minds that are generally able to get along well in the world, but irredeemably crazy on particular questions. Sometimes, it’s tempting to suspect some humans of being this way, and even if that isn’t literally true of any humans, it’s not hard to imagine as just a more extreme form of existing human tendencies. See e.g. Robin Hanson on near vs. far mode, and imagine a mind that will literally never leave far mode on certain questions, regardless of the circumstances.

It used to disturb me to think that there might be, say, young earth creationists in the world who couldn’t be persuaded to give up their young earth creationism by any evidence or arguments, no matter how long they lived. Yet I’ve realized that, while there may or may not be actual human young earth creationists like that (it’s an empirical question), there are certainly possible minds in the space of mind designs like that. And when I think about that fact, I’m forced to shrug my shoulders and say, “oh well” and leave it at that.

That means I can understand why people would be bothered by a lack of universally compelling arguments for their moral views… but you shouldn’t be any more bothered by that than by the lack of universally compelling arguments against young earth creationism. And if you don’t think the lack of universally compelling arguments is a reason to think there’s no objective truth about the age of the earth, you shouldn’t think it’s a reason to think there’s no objective truth about morality.

(Note: this may end up being just the first in a series of posts on the metaethics sequence. People are welcome to discuss what I should cover in subsequent posts in the comments.)

Added: Based on initial comments, I wonder if some people who describe themselves as being bothered the lack of universally compelling arguments would more accurately describe themselves as being bothered by the orthogonality thesis.