Hi !
Congrats on sharing your first post here !
Sorry for the unpolished bullet points, I’m in a bit if a hurry right now and would probably forget later, but I think it may still be worth it to point out a few things :
The way value is defined seems not clearly stated but implicitly guessable to me here. If you look at counterfactual value, the instant value in the blowout scenario could and would probably be negative. It seems to me that you consider value in a non-counterfactual way that attributes zero on the instant scale to nothing existing at all and zero on the cumulative scale to nothing having ever existed. Is that indeed what you have in mind ?
About the math : as long as you are integrating a continuous positive function of a real variable on a segment, you’ll get a positive increasing continuously differentiable function. This may help you reflect on the uncertainty you mention at the end regarding noise.
I agree with you that CV is more intuitive—I sometimes think theoretically about maximizing or minimizing (just a matter of convention) some integral over spacetime spanning the whole universe for space and its whole period of past, present and future existence for time but that’s impractical. However, I think the way you present your example actually makes things look unintuitive. A party where everyone dies at the end is given a positive value but no, I still wouldn’t attend given the chance. My take is, if you think in terms of counterfactual cumulative value it would be negative, and if you set the zero to nothing having ever existed, your scenario may be positive, but probably less than the scenario that would have happened otherwise.
Thanks for your answer !
I thought IV was assumed continuous based on your drawing. Still, I’d be surprised—and I would love to know about it—if you could find an function with a discontinuous integral and does not seem unfit to correctly model IV to me—both out of interest for the mathematical part and of curiosity about what functions we respectively think can correctly model IV.
I think that piecewise continuity and local boundedness are already enough to ensure continuity and almost-everywhere continuous differentiability of the integral. I personally don’t think that functions that don’t match these hypotheses are reasonable candidates for IV, but I would allow IV to take any sign. What are your thoughts on this ?