Is God a Mathematician_ - Mario Livio [86]
Overall, Boole managed to mathematically tame the logical connectives and, or, if…then, and not, which are currently at the very core of computer operations and various switching circuits. Consequently, he is regarded by many as one of the “prophets” who brought about the digital age. Still, due to its pioneering nature, Boole’s algebra was not perfect. First, Boole made his writings somewhat ambiguous and difficult to comprehend by using a notation that was too close to that of ordinary algebra. Second, Boole confused the distinction between propositions (e.g., “Aristotle is mortal”), propositional functions or predicates (e.g., “x is mortal”), and quantified statements (e.g., “for all x, x is mortal”). Finally, Frege and Russell were later to claim that algebra stems from logic. One could argue, therefore, that it made more sense to construct algebra on the basis of logic rather than the other way around.
There was another aspect of Boole’s work, however, that was about to become very fruitful. This was the realization of how closely related logic and the concept of classes or sets were. Recall that Boole’s algebra applied equally well to classes and to logical propositions. Indeed, when all the members of one set X are also members of set Y (X is a subset of Y), this fact can be expressed as a logical implication of the form “if X then Y.” For instance, the fact that the set of all horses is a subset of the set of all four-legged animals can be rewritten as the logical statement “If X is a horse then it is a four-legged animal.”
Boole’s algebra of logic was subsequently expanded and improved upon by a number of researchers, but the person who fully exploited the similarity between sets and logic, and who took the entire concept to a whole new level, was Gottlob Frege (figure 48).
Friedrich Ludwig Gottlob Frege was born at Wismar, Germany, where both his father and his mother were, at different times, the principals at a girls’ high school. He studied mathematics, physics, chemistry, and philosophy, first at the University of Jena and then for an additional two years at the University of Göttingen. After completing his education, he started lecturing at Jena in 1874, and he continued to teach mathematics there throughout his entire professional career. In spite of a heavy teaching load, Frege managed to publish his first revolutionary work in logic in 1879. The publication was entitled Concept-Script, A Formal Language for Pure Thought Modeled on that of Arithmetic (it is commonly known as the Begriffsschrift). In this work, Frege developed an original, logical language, which he later amplified in his two-volume Grundgesetze der Arithmetic (Basic Laws of Arithmetic). Frege’s plan in logic was on one hand very focused, but on the other extraordinarily ambitious. While he primarily concentrated on arithmetic, he wanted to show that even such familiar concepts as the natural numbers, 1, 2, 3,…, could be reduced to logical constructs. Consequently, Frege believed that one could prove all the truths of arithmetic from a few axioms in logic. In other words, according to Frege, even statements such as 1 + 1 = 2 were not empirical truths, based on observation, but rather they could be derived from a set of logical axioms. Frege’s Begriffsschrift has been so influential that the contemporary logician Willard Van Orman Quine (1908–2000) once wrote: “Logic is an old subject,