Is God a Mathematician_ - Mario Livio [82]
Logic and Mathematics
Traditionally, logic dealt with the relationships between concepts and propositions and with the processes by which valid inferences could be distilled from those relationships. As a simple example, inferences of the general form “every X is a Y; some Z’s are X’s; therefore some Z’s are Y’s” are constructed so as to automatically ensure the truth of the conclusion, as long as the premises are true. For instance, “every biographer is an author; some politicians are biographers; therefore some politicians are authors” produces a true conclusion. On the other hand, inferences of the general form “every X is a Y; some Z’s are Y’s; therefore, some Z’s are X’s” are not valid, since one can find examples where in spite of the premises being true, the conclusion is false. For example: “every man is a mammal; some horned animals are mammals; therefore, some horned animals are men.”
As long as some rules are being followed, the validity of an argument does not depend on the subjects of the statements. For instance:
Either the butler murdered the millionaire or his daughter killed him;
His daughter did not kill him;
Therefore the butler murdered him.
produces a valid deduction. The soundness of this argument does not rely at all on our opinion of the butler or on the relationship between the millionaire and his daughter. The validity here is ensured by the fact that propositions of the general form “if either p or q, and not q, then p” yield logical truth.
You may have noticed that in the first two examples X, Y, and Z play roles very similar to those of the variables in mathematical equations—they mark the place where expressions can be inserted, in the same way that numerical values are inserted for variables in algebra. Similarly, the truth in the inference “if either p or q, and not q, then p” is reminiscent of the axioms in Euclid’s geometry. Still, nearly two millennia of contemplation of logic had to pass before mathematicians took this analogy to heart.
The first person to have attempted to combine the two disciplines of logic and mathematics into one “universal mathematics” was the German mathematician and rationalist philosopher Gottfried Wilhelm Leibniz (1646–1716). Leibniz, whose formal training was in law, did most of his work on mathematics, physics, and philosophy in his spare time. During his lifetime, he was best known for formulating independently of (and almost simultaneously with) Newton the foundations of calculus (and for the ensuing bitter dispute between them on priority). In an essay conceived almost entirely at age sixteen, Leibniz envisaged a universal language of reasoning, or characteristica universalis, which he regarded as the ultimate thinking tool. His plan was to represent simple notions and ideas by symbols, more complex ones by appropriate combinations of those basic signs. Leibniz hoped to be able to literally compute the truth of any statement, in any scientific discipline, by mere algebraic operations. He prophesied that with the proper logical calculus, debates in philosophy would be resolved by calculation. Unfortunately, Leibniz did not get very far in actually developing his algebra of logic. In addition to the general principle of an “alphabet of thought,” his two main contributions have been a clear statement about when we should view two things as equal and the somewhat obvious recognition that no statement can be true and false at the same time. Consequently, even though Leibniz’s ideas were scintillating, they went almost entirely unnoticed.
Logic became more in vogue again in the middle of the nineteenth century, and the sudden surge in interest produced important works, first by Augustus De Morgan (1806–71) and later by George Boole (1815–64), Gottlob Frege (1848–1925), and Giuseppe Peano (1858–1932).
De Morgan was an incredibly prolific writer who published literally thousands of articles and books on a variety of topics in mathematics, the history of mathematics, and philosophy. His more unusual work included an almanac of full