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Laura Creighton's avatar

One of the more annoying things about Utilitarians, in general, is that they argue 'we've got the math on our side'. This irritates people who have studied basic Set Theory, who know that this isn't so. It is not as if mathematics is the best foundation for moral sentiments, but Utilitarianism has a strong appeal among people who pride their own rationalism above and beyond all else.

Infinite sequences are a source of strange paradoxes. Most of them are not actually contradictory but merely indicative of a mistaken intuition about the nature of infinity and the notion of a set.

"What is larger," wondered Galileo Galilei in _Two New Sciences_, published in 1638, "the set of all positive numbers (1,2,3,4 ...) or the set of all positive squares (1,4,9,16 ...)?" (He wasn't the first to do this sort of wondering, of course, but it's a convenient starting point, i.e. there are links.)

For some people the answer is obvious. The set of all squares are contained in the set of all numbers, therefore the set of all numbers must be larger. But others reason that because every number is the root of some square, the set of all numbers is equal to the set of all squares. Paradox, no?

Galileo concluded that the totality of all numbers is infinite, that the number of squares is infinite, and that the number of their roots is infinite; neither is the number of squares less than the totality of all the numbers, nor the latter greater than the former; and finally the attributes "equal," "greater," and "less," are not applicable to infinite, but only to finite, quantities.

See 'Galileo's Paradox' https://en.wikipedia.org/wiki/Galileo%27s_paradox .

We haven't changed our minds much about this in the mathematical world. We've become more rigourous in our thinking, and have invented fancy notation -- typographical conventions -- to talk about them, but the last big thing in such comparisons was the idea of 'Cardinality of infinite sets', thank you Georg Cantor. The Cardinality of a set is how many items are in it. If the set is finite, then you just count the elements. Cantor wanted to make it possible to do certain comparisons with infinite sets. I could explain a whole lot of math here, which would bore most of the readership to tears, but people interested in this stuff can find it all over the internet. If you come from a country where set theory is taught in high school, you will have already learned this.

The bottom line is that the set of all numbers and the set of all squares are both the same size, the size being 'countable infinity' or 'aleph-null' in the jargon. Aha, you conclude. So where is the mistaken intuition that creates these paradoxes? The mistaken intuition is that you can compare infinite sets and conclude things like 'the set of all numbers is twice the size of the set of all even numbers'.

Which brings us to the Utilitarian's favourite hobby horse, trolley problems. If you consider each human being on the track as 'an infinite set of potentials', not just metaphorically, but mathematically too, then you can no longer conclude that killing 1 person is better than killing 4. They've all got the same cardinality. (No, I cannot prove this one. But for a thought experiment, we can assume it.)

And this is, after all, what the non-utilitarian moral philosophers have been insisting for all this time. People are not fungible. Non-utilitarians still have to make the tough moral decisions about whether you let one person die to save four, but we don't get to hide behind a shallow mistaken intuition all the while singing the 'we're superior because we have the math on our side' song, as loudly as we can.

P.S. I think that this dreadful state where we all end up living in capsule-hotel accommodations feeling just a hair above misery, with only one duty, to fill the world with people in the same state is simply a restatement of Hilbert's Paradox of the Grand Hotel --

https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

where the hotel has a particularly lousy rating for hospitality on Trip Advisor.

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walruss's avatar

I still think self-dealing is the biggest practical problem with utilitarianism. While it's true that every moral system gets abused, utilitarianism is uniquely susceptible.

Imagine I decide to eat more healthily. I have two options: Banish cream-filled Hostess cupcakes from my life entirely, or acknowledge that cream-filled Hostess cupcakes are a sometimes food that should be eaten extremely rarely.

Now probably I would increase my amount of life enjoyment if I could usually not eat a cream-filled Hostess cupcake, but occasionally, on a special occasion, have one cream-filled Hostess cupcake. So the "math" says to go with option 2.

I do so. But oh no! Now every time I come across a cream-filled Hostess cupcake I have to do a mental analysis. Is this one of these rare occasions when I am permitted a cupcake? And hey, look at all this psych research that says I will *not* make that decision based on my calculating mind, but on the hind-brain that thinks it might die if I don't have that cupcake. So I eat the cupcake and tell myself that I've been dealing with a lot of stress lately and that this will help me cope with that stress, and probably I'll be more effective in my diet if I don't make myself miserable through denial, and etc. etc.

Or I go with option one, say "sorry, I don't eat cream-filled Hostess cupcakes" and miss out on a tiny bit of pleasure, but gain the advantage of this decision not being a decision.

For me the big lie in utilitarianism is less that you might reach a repugnant conclusion. As you point out the whole purpose of the project is to increase human flourishing in human terms, so if our logic chain hits a point of not doing that, at the point of action I think 99.9% of sane people will revise the chain.

And it's not that I don't think there is a right and wrong answer to moral questions: I absolutely am okay decreasing human happiness and flourishing today for drastically increased human happiness and flourishing in the future.

The big lie is that we are mentally capable of handling every single moral choice through calculation. We absolutely are not.

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