We already have a science of memory: "mnemology". This is named after Richard Semon's fine concept of the "mneme" - which was great, but ultimately lost out to the "gene". Unfortunately, mnemology does not seem to be a well-known subject area. There are also "memory studies" and "memory theory" - which don't quite seem to be the same thing. The science associated with memory seems to be a bit fragmented. Animal memory seems to have been assimilated into psychology and cognitive science - and not given a name of its own. The topic of "genetic memory" has been assimilated into genetics.
If you look for a science of copying, there doesn't seem to be much out there - except for evolutionary theory, biology, genetics and "epigenetics" [sic]. There is information theory - perhaps that is the nearest thing.
A science of copying would probably have two foundations: the concept of mutual information - from Shannon information theory - and the concept of causality. I've gone into this in more detail in my article: What are inheritance and copying?
Since one way for patterns to persist is via copying, we could have the science of memory including the science of copying. That way we could still call it the "science of heredity", "science of persistence", or "genetics". Persisting without copying is a topic unto itself - but perhaps one that is not worthy enough one to have its own separate field.
This whole situation seems rather unfortunate. The sciences of persistence and memory seem fragmented, while the science of copying barely seems to exist - outside of biology.
Perhaps my perspective on this is a bit different from other scientists. For me, storage and copying are fundamental operations. They are certainly fundamental operations inside computers. If you think about it, they are pretty fundamental operators in the rest of the world as well. To have a science of memory and a science of copying seems like a natural way to carve the world up to me. However, what we currently have seems to be a long way from this.
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