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Eliezer_Yudkowsky comments on Are imaginary and complex numbers of decibans meaningful? - Less Wrong
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Comments (15)
No, in accordance with whatchamacallit's law.
If you end up with complex probabilities, you won't be able to plug them into an expected utility formula to get a preference ordering. This has always been the knockdown argument for quantitatively scaled real-number subjective probabilities in my book. Even if underlying physics turns out to use complex-numbered reality fluid, I don't see how I can make choices if my degree of anticipation for something happening to me is not a real number - I don't know of any complex analogue of the von Neumann-Morgenstern theorem which yields actual decision outputs.
To put it simply, complex numbers lack an order compatible with the algebraic structure: just that makes them unsuitable for decision theoretic criteria.
After the first dozen responses, I'm currently thinking of writing something along the lines: "While the unusual math of noncommutative probabilities allows for complex probabilities, which have applications in quantum superpositions and eigenstates, there is little likelihood of any practical application involving (the protocol). A (protocol) statement may be written with a complex number for its confidence, but a (protocol) reader or interpreter need only concern itself with the real portion of that number."
Either that, or just stating 'real numbers only'.
(PS: I've never written anything which has even a chance at being an Internet Draft, let alone an RFC; but if the tag: URI made it in, nym: just might pass muster, too - and I would welcome any and all advice.)