The Inference
Pre-Flight
So we've randomized. We do the experiment. We may see something interesting. How do we know this is not due to pure chance? Due to some other funky stuff going on?
Mister Fisher
You treat the Potential Outcomes $Y^a$ as fixed. The Randomness comes from the Assignment $A$.
Mister Neyman
Here we focus on the average causal effect size. Fisher was a smoke test: "Is anything there?" This is a "How much is there and how confident are you?"