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JerL's avatar

On ideal vs non ideal:

Imagine two different characterizations of some mathematical object as the infinite limit of some process, say two different infinite sums, or an infinite sum vs an infinite product, or the argmax of a function found by exhaustive search through an infinite search space vs a fixed point of a dynamical system computed iteratively.

On the one hand, it would be absurd for the proponents of say an infinite product representation of a positive number to think that they've "refuted" an infinite sum representation because one of its partial sums is negative.

But on the other hand, depending on the task at hand, noting that a sum converges slowly, or that an exhaustive search has no optimality guarantees if truncated at finite steps, are genuinely valid criticisms of those representations even if, in some sense, they are correctly calculated approximations of the true ideal object.

Which is to say, I think there's a middle ground where we try and understand objective features of the finite approximations and how they compare to each other and to the ideal object they approach.

I gather from your writing that most deontologists aren't past the stage of pointing out, "the 1000th partial sum of your series for Pi is (negative/rational/algebraic) but everyone knows that Pi is (positive/irrational/transcendental)! Refuted!" so I think it's fair to stress the distinction between ideal and non ideal.

But I still think there is a better argument that has some of the same form that I don't think is addressed by the distinction, something like the claim that "consequentialism converges very, very slowly and requires intractably many terms to compute before you get results that can be achieved with just the first few terms of other theories" (note I don't claim this is true, but I think it's a better version of the sort of thing deontologists are claiming).

Appealing to the necessity of a better decision theory strikes me as similar to someone using the arctan series for Pi appealing to the need for better calculators. Maybe, but maybe it's a problem that you've chosen a slowly converging sum, and for any given strength of calculator you could do better (at certain tasks) by just picking a different representation.

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Matthias's avatar

I wholeheartedly agree with your analysis and hope it reaches a wider audience. In my view, ethics is a field where many different questions are often mixed together. If you're interested, I’ve written a paper on this issue (specifically in the context of teaching ethics).

https://ojs.ub.rub.de/index.php/JDPh/article/view/10811/10995

Section 2 may be particularly relevant to your work, especially my analysis of the notion of "rightness" in the context of ethics.

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