The basic assumptions are random mutation plus differential gene survival via natural selection (non-random) and drift (random).
It appears to me that randman means, by "uniformitarianism", the idea that physical laws were the same in the past. IMHO he thinks they weren't, or at least may not have been. Never mind the evidence we have that they
were the same in the past, and never mind that he has no evidence that they were
different in the past.
The premise (I don't like "assumption" with its implication of "untested") that physical laws were the same in the past is an underlying premise when applying the ToE to the fossil record.
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