Chapter 12: The Principle of Computational Equivalence

Section 10: Intelligence in the Universe

Mathematical notation

While it is usually recognized that ordinary human languages depend greatly on history and context, it is sometimes believed that mathematical notation is somehow more universal. But although it so happens that essentially the same mathematical notation is in practice used all around the world by speakers of every ordinary language, I do not believe that it is in any way unique or inevitable, and in fact I think it shows most of the same issues of dependence on history and context as any ordinary language.

As a first example, consider the case of numbers. One can always just use n copies of the same symbol to represent an integer n—and indeed this idea seems historically to have arisen independently quite a few times. But as soon as one tries to set up a more compact notation there inevitably seem to be many possibilities. And so for example the Greek and Roman number systems were quite different from current Hindu-Arabic base-10 positional notation. Particularly from working with computers it is often now assumed that base-2 positional notation is somehow the most natural and fundamental. But as pages 560 and 916 show, there are many other quite different ways to represent numbers, each with different levels of convenience for different purposes. And it is fairly easy to see how a different historical progression might have ended up making another one of these seem the most natural.

The idea of labelling entities in geometrical diagrams by letters existed in Babylonian and Greek times. But perhaps because until after the 1200s numbers were usually also represented by letters, algebraic notation with letters for variables did not arise until the late 1500s. The idea of having notation for operators emerged in the early 1600s, and by the end of the 1600s, notably with the work of Gottfried Leibniz, essentially all the basic notation now used in algebra and calculus had been established. Most of it was ultimately based on shortenings and idealizations of ordinary language, an important early motivation just being to avoid dependence on particular ordinary languages. Notation for mathematical logic began to emerge in the 1880s, notably with the work of Giuseppe Peano, and by the 1930s it was widely used as the basis for notation in pure mathematics.

In its basic structure of operators, operands, and so on, mathematical notation has always been fairly systematic—and is close to being a context-free language. (In many ways it is like a simple idealization of ordinary language, with operators being like verbs, operands nouns, and so on.) And while traditional mathematical notation suffers from some inconsistencies and ambiguities, it was possible in developing Mathematica StandardForm to set up something very close that can be interpreted uniquely in all cases.

Mathematical notation works well for things like ordinary formulas that involve a comparatively small number of basic operations. But there has been no direct generalization for more general constructs and computations. And indeed my goal in designing Mathematica was precisely to provide a uniform notation for these (see page 852). Yet to make this work I had to use names derived from ordinary language to specify the primitives I defined.

From Stephen Wolfram: A New Kind of Science [citation]