Log xy = Log x + Log y

In their (rich, original, ground-breaking) writings on plurals, Alex Oliver and Timothy Smiley more than once say that “mathematical practice” shows that addition is allowed to take plural terms as arguments. Thus, with a trivial change of variables, in their  ‘A modest logic of plurals’ (JPL, 2006) they write

One might think that [‘+’] can only take singular terms as arguments, … Mathematical practice shows that this is quite wrong, as witness the right-hand side of this equation (Hardy, 1925, p. 408)

Log xy = Log x + Log y

bearing in mind that Log z is an infinitely many-valued function.

But this is surely a mis-step. In fact Hardy himself realises that — given that the Log terms are many-valued — he’d better explain the extended use of the equation notation here (that is the novelty). So he does. Hardy tells us that we are to read this as “Every value of either side is one of the values of the other side”. So already, the surface equation is frankly offered as syntactic sugar for something more structured. Fair enough: we do this sort of thing all the time. And how are to read Hardy’s snappy explication? Obviously, if you want the gory details, he intends this — what else?

For any (particular!) a such that a is a value of Log xy, there is a b such that b is a value of Log x and a c such that c is a value of Log y, and a = b + c. And for any b such that b is a value of Log x and c such that c is a value of Log y, there is an a such that a is a value of Log xy and a = b + c.

(In a slogan: you do the selections from the many values before the addition.) Which reveals that, when the wraps are off, the addition functor here is  straightforwardly applied to singular terms, just as you’d expect.

Writing Log xy = Log x + Log y is a handy extension of equation notation to plural terms produced by many-valued functions (though like Hardy, careful writers on complex analysis take it that it does need explaining). But once this sort of extension is explained, we don’t need to discern any further accompanying novelty in the use of addition. No?

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