The Information - James Gleick [78]
De Morgan knew more about the scholastic traditions of the subject, but Boole was the more original and free-thinking mathematician. By post, for years, they exchanged ideas about converting language, or truth, into algebraic symbols. X could mean “cow” and Y “horse.” That might be one cow, or a member of the set of all cows. (The same?) In the algebraic fashion the symbols were to be manipulated. XY could be “name of everything which is both X and Y” while X,Y stood in for “name of everything which is either X or Y.”♦ Simple enough—but language is not simple and complications reared up. “Now some Zs are not Xs, the ZYs,”♦ wrote De Morgan at one point. “But they are nonexistent. You may say that nonexistents are not Xs. A nonexistent horse is not even a horse; and (a fortiori?) not a cow.”
He added wistfully, “I do not despair of seeing you give meaning to this new kind of negative quantity.” He did not post this and he did not throw it away.
Boole thought of his system as a mathematics without numbers. “It is simply a fact,”♦ he wrote, “that the ultimate laws of logic—those alone on which it is possible to construct a science of logic—are mathematical in their form and expression, although not belonging to the mathematics of quantity.” The only numbers allowed, he proposed, were zero and one. It was all or nothing: “The respective interpretation of the symbols 0 and 1 in the system of logic are Nothing and Universe.”♦ Until now logic had belonged to philosophy. Boole was claiming possession on behalf of mathematics. In doing so, he devised a new form of encoding. Its code book paired two types of symbolism, each abstracted far from the world of things. On one side was a set of characters drawn from the formalism of mathematics: p’s and q’s, +’s and –’s, braces and brackets. On the other were operations, propositions, relations ordinarily expressed in the fuzzy and mutable speech of everyday life: words about truth and falsity, membership in classes, premises and conclusions. There were “particles”: if, either, or. These were the elements of Boole’s credo:
That Language is an instrument of human reason, and not merely a medium for the expression of thought.
The elements of which all language consists are signs or symbols.
Words are signs. Sometimes they are said to represent things; sometimes the operations by which the mind combines together the simple notions of things into complex conceptions.
Words … are not the only signs which we are capable of employing. Arbitrary marks, which speak only to the eye, and arbitrary sounds or actions … are equally of the nature of signs.♦
The encoding, the conversion from one modality to the other, served a purpose. In the case of Morse code, the purpose was to turn everyday language into a form suitable for near-instantaneous transmission across miles of copper wire. In the case of symbolic logic, the new form was suitable for manipulation by a calculus. The symbols were like little capsules, protecting their delicate cargo from the wind and fog of everyday communication. How much safer to write:
1 − x = y(1 − z) + z(1 − y) + (1 − y)(1 − z)
than the real-language proposition for which, in a typical Boolean example, it stood:
Unclean beasts are all which divide the hoof without chewing the cud, all which chew the cud without dividing the hoof, and all which neither divide the hoof nor chew the cud.♦
The safety came in no small part from draining the words of meaning. Signs and symbols were not just placeholders; they were operators, like the gears and levers in a machine. Language, after all, is an instrument.
It was seen distinctly now as an instrument with two separate functions: expression and thought.