Axiom 1: Human life has value, saving a life is thus positive utility.

Assumption 1: We know nothing of the people tied to the trolley -> All humans are probabilistically equal.

Thus, the utilitarian presses the button and saves 3 at the cost of 1.

And even if you had some suspicions, once you honestly factor in your certainty about predictive value of your suspicions, it multiplies down to nothing - so in practice, an uniform prior is a pretty good choice.

Yes, it's true that all this stuff is not computable in practice, that it's too complex - but it's not really an obstacle! Everything in life is like that. When you're cooking and a recipe calls for 200g of an ingredient, you're fine with anything between 180g or 220g, and don't care how much of it is water. A restaurant may need to do 200g ± 5g. A chemical plant may want to do 200g ± 0.5g, after first demoisturizing. Nobody tries to hit 200g to ± 1 dalton, because that would be ridiculously complex to do, and require to control for phenomena that most people don't even know exist.

We've been trading precision for compute time ever since humans first figured out that things can be measured. The solution is what it's always been: calculating using heuristics and approximations, while keeping tracks of the bounds of those heuristics and approximations.

Does it? You seem to be marginalizing across the entire population, but it's eminently obvious that most humans do _not_ see a human and erase all aspects of their being and then consider them to be one human in a uniformly distributed pool. Humans make snap judgements based on hundreds of subconscious factors all the time. Most humans look at a person and immediately bucket them into a cohort using some mix of physical attributes and prior beliefs of these attributes. Once we're in a cohort, relative probabilities can change. In this particular example, what happens if one of the 3 happens to be wearing military fatigues? What if they have a well-known authoritarian political symbol on their clothing?

Anyway, lots of intelligent philosophers have argued against utilitarianism and have brought up flaws in utilitarianism, such as Utility Monsters, and then there's the issue of even being able to impose a utility-based order between disparate things such as "tasty candy" and "rocks underneath my feet". Feel free to read those on the internet.

That indeed happens, but going by the gut isn't a bullet-proof moral philosophy. Snap judgements are something we can evaluate and consciously calibrate when we're not in the heat of the moment.

> lots of intelligent philosophers have argued against utilitarianism and have brought up flaws in utilitarianism

Yeah, I read some of those, many are really solid objections. I'm not arguing that utilitarianism is the be-all, end-all moral philosophy - just that it's a surprisingly good heuristic that can be pushed quite far before it starts to go "off the rails". And primarily, I'm arguing that working with approximations and heuristics is something we know how to do - not being able to compute a perfectly precise answer isn't a valid argument against a method.

Presumably by this logic, you'd have no opinion on whether a midwife should save a newborn baby's life, as it's unknowable whether the baby will one day orchestrate a genocide?

Regardless, there are lots of criticisms of utilitarianism, and utilitarianism is far from the ascendant moral philosophy, so feel free to find others online.

The very low chance of a future genocide does not weigh highly in this decision.