Ridiculous math things which Ethics shouldn't depend on but does

There is a scene in Gulliver's Travels where the protagonist calls up the ghosts of all the philosophers since Aristotle, and the ghosts all admit that Aristotle was way better than them at everything. Especially Descartes – Jonathan Swift wants to make very clear that Aristotle is a way better philosopher than Descartes, and that all of Descartes's ideas are stupid. (I think this was supposed to prove a point in some long-forgotten religious dispute.)

If I ever become a prominent philosopher and we develop the technology to call up ghosts in order to win points in literary holy wars (I will let the reader decide which of those two conditions is more likely), please reincarnate me to talk ethics with Aristotle. Basically all the problems I'm worried about deal with mathematical concepts which weren't developed until around a century ago, and I'm excited to hear whether a virtuous person would accept Zorn's Lemma.

Today I want to share two mathematical assumptions which are so esoteric that even most mathematicians don't bother worrying about them. Despite that, they actually critically influence what we think about ethics.

The Axiom of Choice

The Axiom of Choice is everyone's favorite example of something which seems like an innocuous assumption but isn't. (The Axiom of Choice is the axiom of choice for such situations, if you will.) Here's Wikipedia's informal description:
The axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin.
Seems pretty reasonable right? Unfortunately, it leads to a series of paradoxes like that any ball can be doubled into two balls, both of which have the same size as the first.

In many cases, a weaker assumption known as the "axiom of dependent choice" suffices and has the advantage of not leading to any (known) paradoxes. Sadly, this doesn't work for ethics.

Consider the two following reasonable assumptions:

  1. Weak Pareto: if we can make someone better off and no one worse off, we should.
  2. Intergenerational Equality: we should value the welfare of every generation equally.

Theorem (proven by Zame): we cannot prove the existence of an ethical system which satisfies both Weak Pareto and Intergenerational Equality without using the axiom of choice (i.e. the axiom of dependent choice doesn't work).

Sorry grandma, but unless you can make that ball double in size we're gonna have to start means-testing Medicare

Hyperreal numbers

The observant reader will note that the previous theorem showed only that we could prove the existence of a "good" ethical system if we use the axiom of choice, it didn't say anything about us actually being able to find it. To get that we have to enter the exciting world of hyperreal numbers!

The founding fathers weren't as impressed with Thomas Jefferson's original nonconstructive proof that the Bill of Rights could, in theory, be created

I recently asked my girlfriend whether she would prefer:
  1. Having one unit of happiness every day, for the rest of eternity, or
  2. Having two units of happiness every day, for the rest of eternity
She told me that the answer was obvious: she's a total utilitarian and in the first circumstance she would have one unit of happiness for an infinite amount of time, i.e. one infinity's worth of happiness. But in the second case she would have two units for an infinite amount of time, i.e. two infinities of happiness. And clearly two infinities are bigger than one.

My guess is that how reasonable you think this statement is will depend in a U-shaped way on how much math you've learned:



To the average Joe, it's incredibly obvious that two infinities are bigger than one. More advanced readers will note that the above utility series don't converge, so it's not even meaningful to talk about one series being bigger than another. But those who've dealt with the bizarre world of nonstandard analysis know that notions like "convergence" and "limit" are conspiracies propagated by high school calculus teachers to hide the truth about infinitesimals. In fact, there is a perfectly well-defined sense in which two infinities are bigger than one, and the number system which this gives rise to is known as the "hyperreal numbers."

From an ethical standpoint, here are the relevant things you need to know:

Theorem (proven by Basu and Mitra): if we use only our normal "real" numbers, then we can't construct an ethical system which obeys the above Weak Pareto and Intergenerational Equality assumptions.
Theorem (proven by Pivato): we can find such a system if we use the hyperreal numbers.

To any TV producers reading this: the success of the hyperreal approach over the "standard calculus" approach would make me an excellent soft-news-show guest. While most stations can drum up some old crotchety guy complaining about how schools are corrupting the minds of today's youths, only I can actually prove that calculus teaches kids to be unethical.

Conclusion / Apologies / Further Reading

As far as the laws of mathematics refer to reality, they are not certain; as far as they are certain, they do not refer to reality. - Einstein
It goes without saying that I've heavily simplified the arguments I've cited, and any mistakes are mine. If you are interested in using logical reasoning to improve the world, then you should check out Effective Altruism. If you are more of a "nonconstructive altruist" then you can do a Google scholar search for "sustainable development" or read the papers cited below to learn more.

And most importantly: if you are student who is being punished for misbehaving in a calculus class, please 1) tell your teacher the Basu-Mitra-Pivato result about how calculus causes people to disrespect their elders and 2) film their reaction and put it on YouTube. (Now that's effective altruism!)

  • Basu, Kaushik, and Tapan Mitra. "Aggregating infinite utility streams with intergenerational equity: the impossibility of being Paretian." Econometrica 71.5 (2003): 1557-1563.
  • Pivato, Marcus. "Sustainable preferences via nondiscounted, hyperreal intergenerational welfare functions." (2008).
  • ZAME, WILLIAM R. "Can intergenerational equity be operationalized?."Theoretical Economics 2 (2007): 187-202.

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