What's Linearity For ?

In physics (and geometry) one has occasion to add things and to scale them. There are restrictions on which things are amenable to addition; but addition generally produces a thing of the same kind as the things added. Multiplication can produce an answer of a different kind from the things multiplied (as, for instance, length×length = area); and one has to take some care to work out how (if at all) one may multiply certain kinds of things (e.g. vectors).

Both addition and multiplication are associative binary operators: addition is invariably commutative (a+b = b+a) and cancellable (a+c = b+c always implies a=b) but multiplication is less reliable (at least in some of its forms).

Consider a collection, D, of things which may be added to one another, producing answers in D: so, for any c, e in D, c+e is a member of D. One can view any a in D as 1×a; a+a as 2×a, ..., a+a+a+a+a+a+a as 7×a and so on: (1+n)×a = a + n×a. In this way one may induce a natural scaling by the positive integers, applicable to any additive domain. With a little effort one may show that (n+m)×a = n×a + m×a and n×(a+b) = n×a + n×b for arbitrary a, b in D and positive integers n, m.

One may find that, for every c in D and natural n, there is exactly one b in D for which n×b = c: in such a case, one may wish to write b as c/n; scaling this up by m, we obtain m×c/n and it makes sense to think of this as (m/n)×c. We thus expand our available scalings from the positive integers to the positive rationals (fractions).

If D contains an additive identity, zero in D for which zero+c = c for every c in D, one may naturally extend this multiplication to include 0×a = zero. Likewise, if D supports subtraction we may extend our scaling to work with all integers (or all rationals, if we also have the divisibility above).

One may, in various ways, augment the rationals to obtain further numbers, such as π and the square root of 7. Multiplying a number by a member of D always produces a member of D and we think of the effect as a change of scale, or scaling. Consequently, numbers are called scalars. While the things they scale may also be numbers, they need not be: but scalars can only be added to scalars. One generally selects one collection of scalars, whether it be the naturals, the surreals or the complex numbers, and discusses an assortment of additive domains on which the given scalars act as scalings. All other kinds of multiplication are, ultimately, built up from consideration of scaling.