twiffy
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There is likely a broader-scoped discussion on this topic that I haven't read, so please point me to such a thread if my comment is addressed -- but it seems to me that there is a simpler resolution to this issue (as well as an obvious limitation to this way of thinking), namely that there's an almost immediate stage (in the context of highly-abstract hypotheticals) where probability assessment breaks down completely.
For example, there are an uncountably-infinite number of different parent universes we could have. There are even an uncountably-infinite number of possible laws of physics that could govern our universe. And it's literally impossible to have all these... (read more)
As a tangent, I think it's relatively clear both how physicists tend to think differently from mathematicians, and how they came up with path-integration-like techniques in QFT. In both math and physics, researchers will come up with an idea based on intuition, and then verify the idea appropriately. In math the correct notion of verification is proof; in physics it's experimentation (with proof an acceptable second). This method of verification has a cognitive feedback loop to how the researcher's intuition works. In particular physicists have intuition that's based on physical intuition and (generally) a thoroughly imprecise understanding of math, so that from this perspective, using integral-like techniques without any established mathematical underpinnings is intuitively completely plausible. Mathematicians would shirk away from this almost immediately as their intuition would hit the brick wall of "no theoretical foundation".