Today in the library I came across this little book – An Introduction to Mathematics – by Whitehead, 1911. Yes, the same Alfred North Whitehead, coauthor of the Principia Mathematica along with Bertrand Russell. Its a wonderful read, this little book. It doesn’t teach mathematics but it explains much of its spirit. And explains it wonderfully.

The book starts out with a description of the mathematics that I feel is apt for the state of appreciation of programming language theory today. Today, I feel one has to do little to defend the general utility or appreciation for mathematics in comparison to what I suppose it was like in Whitehead’s times. However any real appreciation for language theory is moot, not just among the general populace but even to the vast majority of programmers and other computing related people. So I quote Whitehead (and for my interests in language theory, please substitute mathematics with “language theory” as appropriate):

The study of mathematics is apt to commence in disappointment. The important applications of the science, the theoretical interest of its ideas, and the logical rigour of its methods, all generate the expectation of a speedy introduction to the processes of interest. We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet’s father, this great science eludes the efforts of our mental weapons to grasp it – ” ‘Tis here, ’tis there, ’tis gone” – and what we do see does not suggest the same excuse for illusiveness as sufficed for the ghost, that it is too noble for our gross method.

Thus begins the introduction. It’s a neat little book with all sorts of quotes and very tangible parallels applicable to language theory. For instance, anyone who has struggled with an abstract machine, a new calculus or formalizing a new language construct would know the importance of discovering the right syntax. Whitehead says:

The interesting point to notice is the admirable illustration which this numeral system affords to the importance of a good notation. By relieving the brain of all unnecessary work, a good notation sets it free to focus on more advanced problems, and in effect increases the mental power of the race. Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility.

Interestingly enough, the concerns of Whitehead and the metaphors he uses to motivate his mathematics do not apply to language theory merely by coincidence. There is a sense in which the mathematics of Russell and Whitehead was actually an early form of language theory – in fact some of its very foundations.

If you like this sort of thing, you can download a scanned copy of the book from here:

http://www.archive.org/details/introductiontoma00whitiala