I have noticed that when I give talks about Fortress and start talking about the importance of algebraic properties for parallel programming, I often see many pairs of eyes glaze over, if only a bit. It occurs to me that maybe not everyone is conversant or comfortable with the terminology of modern albegra (maybe they had a bad experience with the New Math back in the 1960s, a fate I barely escaped), and this may obscure the essential underlying ideas, which after all are not that difficult, and perhaps even obvious to the average programmer when explained in everyday terms.
It's a good goal. Using technical terms often obscures the real point one is trying to make. Worse, technical terms are often dress up a claim to sound like it says much more than it does. For all kinds of reasons, it is better to well-known informal terms whenever they will work.
Nonetheless, I am not so sure about changing terms like commutative and associative in the Fortress library. It looks like a case where the hard part is not in the terminology, but in the underlying theory itself. Once a programmer understands the theory well enough to work with the library, they'll almost certainly know the standard formal terms anyway.
A similar issue come up for Scala, where writers of very flexible libraries end up working with rather complicated static types. In such cases, there is no getting around understanding how the Scala type system works. The Scala collections library is deep mojo.
That doesn't mean the complexity must leak out to users of the library, however. In both cases, the designers of core libraries must think hard, because they are working with deep abstractions. If the designers do well, then users of these libraries will find everything working effortlessly.
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