The Closing of the Western Mind: The Rise of Faith and the Fall of Reason Read Online Free PDF Page B

condition of being human. Similarly, anyone who met Socrates would have agreed that he was a man. From these two assumptions could be drawn the conclusion: “Therefore Socrates is mortal.” Aristotle went further, replacing the subjects of the assumptions with letters, so that it follows if all As are B, and C is an A, then C is B. One can substitute any suitable premises to create a valid conclusion. Aristotle goes on to explore the cases where the logic does not work. “A dog has four feet” and “A cat has four feet” are both reasonable assumptions to make from one’s experience of dogs and cats in everyday life, but it does not follow that a cat is a dog, and the student in logic has to work out why this is so. “All fish are silver; a goldfish is a fish; therefore a goldfish is silver” cannot be sustained because the example of a living goldfish would itself show that the premise that “All fish are silver” is not true.
    Aristotle’s syllogisms can take us only so far; their premises have to be empirically correct and relate to each other in such a way that a conclusion can be drawn from their comparison. They provide the basis for deductive argument, an argument in which a specific piece of knowledge can be drawn from knowledge already given. The development of the use of deductive proof was perhaps the greatest of the Greeks’ intellectual achievements. Deductive argument had, in fact, already been used in mathematics by the Greeks before Aristotle systematized it. In an astonishing breach of conventional thinking, the Greeks conceived of abstract geometrical models from which theorems could be drawn. While the Babylonians knew that in any actual right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides, Pythagoras’ theorem generalizes to show that this must be true in any conceivable right-angled triangle, a major development both mathematically and philosophically. A deductive proof in geometry needs to begin with some incontrovertible statements, or postulates as the mathematician Euclid (writing c. 300 B.C.) named them. Euclid’s postulates included the assertion that it is possible to draw a straight line from any point to any other point and that all right angles are equal to each other. His famous fifth postulate stipulated the conditions under which two straight lines will meet at some indefinite point. (It was the only one recognized as unprovable even in his own day and eventually succumbed to the analysis of mathematicians in the nineteenth century.) Euclid also recognized what he termed “common notions,” truths that are applicable to all sciences, not merely mathematics, such as “If equals be added to equals, the wholes are equal.” These postulates and “common notions” might seem self-evident, but in his
Elements,
one of the outstanding textbooks in history, Euclid was able to draw no less than 467 proofs from ten of them, while a later mathematician, Apollonius of Perga, was to show 487 in his
Conic Sections.
As Robert Osserman has put it in his
Poetry of the Universe:
    In a world full of irrational beliefs and shaky speculations, the statements found in
The Elements
were proven true beyond a shadow of a doubt . . . The astonishing fact is that after two thousand years, nobody has ever found an actual “mistake” in
The Elements—
that is to say a statement that did not follow logically from the given assumptions. 13
    Later mathematicians, such as the great Archimedes (see below, p. 43), were to develop new branches and areas of mathematics from these foundations.
    Dealing with the natural world is a much more complex business. It seems to be in a constant state of change—the weather changes, plants grow, wars happen, men die. As Heraclitus had observed, all is in a process of flux. Yet if an underlying order can be assumed and isolated, then some progress can be made. Such progress assumes that the gods do not disturb the