[FOM] 605: Integer and Real Functions

[Mitchell Spector]

In response to Bill Taylor <W.Taylor at math.canterbury.ac.nz on whether, as I had suggested, string 
concatenation is additive, and more generally, on what it means for a binary operation to be 
"additive" in nature:



1.  The distributivity of (elementwise) string concatenation over set union that Bill mentions isn't 
relevant, since _every_ binary operation is distributive over set union in the same sense:

Let # be any binary operation on a set S, and let P(S) denote the power set of S.  Define a binary 
operation #' on P(S) by setting

X #' Y = { x # y | x belongs to X, and y belongs to Y}.

Then X #' (Y union Z)  = (X #' Y) union (X #' Z).


Since this is true for any binary operation #, it says nothing about whether # (or #') is additive 
or multiplicative in nature.



2. Algebraic properties such as the distributive law, although elegant, aren't fundamental or 
essential -- in those cases when they happen to be true, they're derived properties.

In fact, in the example of the ordinal numbers, multiplication isn't distributive over addition. 
(It's left-distributive, but not right-distributive.)



3. String concatenation over an alphabet of size 1 is isomorphic to addition of natural numbers 
(and, if you allow strings of well-ordered transfinite length, it's isomorphic to addition of 
ordinal numbers).

In contrast, there is no way, as far as I can see, to interpret any sort of numerical multiplication 
as string concatenation.



4.  Here's the actual principle involved here:

In essence, a binary operation is "additive in nature" if it just involves taking its two arguments 
and "putting them together", without modifying or processing them in any way.  (Maybe we're keeping 
track of the order of things, and maybe we're not, but that level of specificity doesn't matter.)

Needless to say, this is a philosophical definition, not one that you can give a rigorous 
mathematical interpretation of, but I think its meaning is apparent.

It clearly applies to addition of natural numbers, addition of ordinal numbers, addition of cardinal 
numbers, and string concatenation.  It applies to addition of real numbers if you think of reals as 
lengths (directed lengths would accommodate negative numbers).  It applies to addition of complex 
numbers if you think of complex numbers as vectors.  It applies to addition of vectors themselves as 
well.  It applies to the set union operation.  (This is what I meant when I described string 
concatenation as "quintessentially additive".)

It cannot be interpreted to apply, as far as I can see, to any indisputably "multiplicative" operation.

This perspective explains why addition is generally viewed as the simplest non-trivial binary 
operation -- the other operations involve modifying or processing their arguments, or using them in 
some relatively complicated way.


It's true that ring theorists and other algebraists have found it productive to look in general at 
binary operations that happen to satisfy some of the same algebraic laws as numerical addition and 
multiplication.  That's because much can be deduced just from the fact that those algebraic 
properties hold.

But this doesn't imply that those algebraic properties are fundamental to the nature of addition and 
multiplication.  Those properties are just very useful derived facts (when they happen to be true, 
of course).



5.  One final remark: This viewpoint suggests, perhaps surprisingly, that composition of functions 
is an "additive" operation.  I haven't yet given any thought as to what this might tell us.



Overall, a foundational analysis, rather than an algebraic approach, is needed if we want to 
understand the nature of fundamental mathematical objects.

Mitchell Spector