I don't understand why, in mathematical discourse, addition and multiplication are so often regarded as duals of each other, considering that, for example, $\forall x, y, z\in \mathbb{Z}$ (say),
$ x \times (y + z) = (x\times y) + (x \times z) $
but $x + (y \times z) \neq (x + y) \times (x + z).$
Any clarification would be much appreciated!
Edit: I'm shocked by the comments. We must live on different planets. Mentions of the multiplication/addition duality are so frequent and ubiquitous in my experience with mathematical texts, that I don't bother keeping a record of them, so it will be impossible for me to document my post's claims to anyone's satisfaction. Of course, I have no doubt that (once I've recovered from the shock) I'll find examples of said practice, but no reasonable number of examples could do justice to its pervasiveness.
Edit 2: OK, here's a quick and easy one: I have often seen some variant of the word "sum" (e.g. "direct sum", "disjoint sum") used as an alternative name for the categorical coproduct. This word choice obviously implies a multiplication/addition duality. I'll think of some more...
Edit 3: Recently, in a math paper, I came across a pair of perfectly dual definitions, where the names for the two terms being defined were of the form "multiplicative <X>" and "additive <X>". Also, it is highly significant that the author of these definitions does not bother to justify this highly suggestive choice of names. This tells me that the author felt that the duality of multiplication and addition would be obvious to his intended audience.
Edit 4: The most recent instance that I came across of this pervasive notion was, one or two days ago, in one of TheCatsters' videos. In this video, Cheng says something like "...the dual [of the product], which is, of course, the sum." Finding the exact chapter-and-verse will take some time, since I've watched many of those videos recently, and I can't think of any way to find the exact reference other than watching them all over again.