How are addition and multiplication duals of each other?

Question

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.

Answer 1

The cases you cite in the question all appear to descend, via more or less twisted paths, from Boolean algebra, where the OR operation (often written additively) and the AND operation (often written multiplicatively) are indeed duals of each other and each distribute over the other.

For example, via the Curry-Howard isomorphism, these concepts find their way from logic into applied lambda calculi, where the dual to a product type, which is naturally interpreted as a variant or disjoint union type, is called a sum type for this reason (and possibly also due to an analogy with cardinal arithmetic). From there, since cartesian closed categories are models of lambda calculus, category-theoretic constructions dual to products become "sums". There may be a more direct connection here, but as a computer scientist that's the direction I know of.

In most of these cases, it appears that the the main examples of "products" look more or less like embellished Cartesian products, whereas the corresponding "sums" look more varied.

Answer 2

It's the product and coproduct in an arbitrary category $C$ that are dual to each other, and this is because their definitions are categorically dual: in the dual category $C^{op}$, the product is the coproduct in $C$ and vice versa.

In a particular category $C$, though, the product and coproduct may not have dual properties, and this should merely be taken as evidence that $C$ is very different from $C^{op}$. For example, in the category of sets, the product is the Cartesian product and the coproduct is the disjoint union. Decategorifying Cartesian product and disjoint union on finite sets, we then obtain multiplication and addition on non-negative integers (and this, as far as I know, is the main reason why we call products products and occasionally call coproducts sums). As you correctly observe, product distributes over coproduct but coproduct does not distribute over product in this situation, and all this means is that $\text{Set}$ is not equivalent to $\text{Set}^{op}$.

Three more examples:

• In the category $\text{CRing}$ of commutative rings, coproduct distributes over product but not the other way around. This is one indication that $\text{CRing}^{op}$ behaves similarly to $\text{Set}$ (or at least more similarly than $\text{CRing}$ does), and this is one abstract way to justify scheme theory.

• In the category $\text{Ab}$ of abelian groups, finite products and finite coproducts agree, and neither operation distributes over the other. ($\text{Ab}^{op}$ turns out to be equivalent to the category of compact abelian groups through Pontrjagin duality.)

• In a poset regarded as a category in the usual way ($a \le b$ means that there is a single arrow $a \to b$), the product is the infimum (if it exists) and the coproduct is the supremum (if it exists); in particular, the two-element product is the minimum and the two-element coproduct is the maximum. A poset is a lattice if finite products and coproducts exist and a distributive lattice if either distributes over the other (the two conditions turn out to be equivalent for lattices). In lattice theory, therefore, it is very natural to regard "addition" (supremum) and "multiplication" (infimum) as dual.