Download Greek Deductive Reasoning in Mathematics: A Cultural & Historical View and more Study notes Mathematics in PDF only on Docsity!
Birth of the Mathematical Spirit
The mathematics that existed before Greek times has already been characterized as a collection of empirical conclusions. Its formulas were the accretion of ages of experience much as many medical practices and remedies are today. Though experience is no doubt a good teacher in many situations it would be a most inefficient way of obtaining knowledge.
Fortunately, there is a method of reasoning that does guarantee the certainty of the conclusions it produces. The method is known as deduction. Let us consider some examples. If we accept the facts that all apples are perishable and that the object before us is an apple, we must conclude that this object is perishable. As another example, if all good people are charitable and if I am good, then I must be charitable. And if I am not charitable I am not good. Again, we may argue deductively from the premises that all poets are intelligent and that no intelligent people deride mathematics, to the inevitable conclusion that no poet derides mathematics.
It does not matter, in so far as the reasoning is concerned, whether we agree with the premises. What is pertinent is that if we accept the premises we must accept the conclusion. Unfortunately, many people confuse the acceptability or truth of a conclusion with the validity of the reasoning that leads to this conclusion. From the premises that all intelligent beings are humans and that readers of this book are human beings, we might conclude that all readers of this book are intelligent. The conclusion is undoubtedly true but the purported deductive reasoning is invalid because the conclusion does not necessarily follow from the premises. A moment's reflection shows that even though all intelligent beings are humans, there may be human beings who are not intelligent, and nothing in the premises tells us to which group of human beings the readers of this book belong.
Deductive reasoning, then, consists of those ways of deriving new statements from accepted facts that compel the acceptance of the derived statements. We shall not pursue at this point the question of why it is that we experience this mental conviction. What is important now is that man has this method of arriving at new conclusions and that these conclusions are unquestionable if the facts we start with are also unquestionable.
Deduction, as a method of obtaining conclusions, has many advantages over trial and error or reasoning by induction and analogy. The outstanding advantage is the one we have already mentioned, namely, that the conclusions are unquestionable if the premises are. Truth, if it can be obtained at all, must come from certainties and not from doubtful or approximate inferences. Second, in contrast to experimentation, deduction can be carried on without the use or loss of expensive equipment. Before the bridge is built and before the long-range gun is fired, deductive reasoning can be applied to decide the outcome. Sometimes deduction has the advantage of being the only available method. The calculation of astronomical distances cannot be carried out by applying a yardstick. Moreover, whereas experience confines us to tiny portions of time and space, deductive reasoning may range over countless universes and aeons.
With all of its advantages, deductive reasoning does not supersede experience, induction, or reasoning by analogy. It is true that 100 per cent certainty can be attached to the conclusions of deduction when the premises can be vouched for 100 per cent. But such unquestionable premises are not necessarily available. No one, unfortunately, has been able to vouchsafe the premises from which a cure for cancer could be deduced. For practical purposes, moreover, the certainty deduction grants is sometimes superfluous. A high degree of probability may suffice. For centuries
the Egyptians used mathematical formulas drawn from experience. Had they waited for deductive proof the pyramids at Giza would not be squatting in the desert today.
Each of these various ways of obtaining knowledge, then, has its advantages and disadvantages. Despite this fact, the Greeks insisted that all mathematical conclusions be established only by deductive reasoning. By their insistence on this method, the Greeks were discarding all rules, formulas, and procedures that had been obtained by experience, induction, or any other non- deductive method and that had been accepted in the body of mathematics for thousands of years preceding their civilization. It would seem, then, that the Greeks were destroying rather than building; but let us withhold judgment for the present.
Why did the Greeks insist on the exclusive use of deductive proof in mathematics? Why abandon such expedient and fruitful ways of obtaining knowledge as induction, experience, and analogy? The answer can be found in the nature of their mentality and society.
The Greeks were gifted philosophers. Their love of reason and their delight in mental activity distinguished them from other peoples. The educated Athenians were as much devoted to philosophy as our smart-set is to night-clubbing; and pre-Christian fifth-century Athens was as deeply concerned with the problems of life and death, immortality, the nature of the soul, and the distinction between good and evil as twentieth-century America is with material progress.
Philosophers do not reason, as do scientists, on the basis of personally conducted experimentation or observation. Rather their reasoning centers about abstract concepts and broad generalizations. It is difficult, after all, to experiment with souls in order to arrive at truths about them. The natural tool of philosophers is deductive reasoning, and hence the Greeks gave preference to this method when they turned to mathematics.
Philosophers are, moreover, concerned with truths, the few, immaterial wisps of eternity that can be sifted from the bewildering maze of experiences, observations, and sensations. Certainty is the indispensable element of truth. To the Greeks, therefore, the mathematical knowledge accumulated by the Egyptians and Babylonians was a house of sand. It crumbled to the touch. The Greeks sought a palace built of ageless, indestructible marble.
The Greek preference for deduction was, surprisingly, a facet of the Hellenic love for beauty. Just as the music lover hears music as structure, interval, and counterpoint, so the Greek saw beauty as order, consistency, completeness, and definiteness. Beauty was an intellectual as well as an emotional experience. Indeed, the Greek sought the rational element in every emotional experience. In a famous eulogy Pericles praises the Athenians who died in battle at Samos not merely because they were courageous and patriotic, but because reason sanctioned their deeds. To people who identified beauty and reason, deductive arguments naturally appealed because they are planned, consistent, and complete, while conviction in the conclusions offers the beauty of truth. It is no wonder, then, that the Greeks regarded mathematics as an art, as architecture is and though its principles may be used to build warehouses.
Another explanation of the Greek preference for deduction is found in the organization of their society. The philosophers, mathematicians, and artists were members of the highest social class. This upper stratum either completely disdained commercial pursuits and manual work or regarded them as unfortunate necessities. Work injured the body and took time from intellectual and social activities and the duties of citizenship.
To the Egyptians, for example, a straight line was quite literally no more than either a stretched rope or a line traced in sand. A rectangle was a fence bounding a field.
With the Greeks not only was the concept of number consciously. recognized but also they developed arithmetica, the higher arithmetic or theory of numbers; at the same time mere computation, which they called logistica and which involved hardly any appreciation of abstractions, was deprecated as a skill in much the same way as we look down upon typing today. Similarly in geometry, the words point, line, triangle, and the like became mental concepts merely suggested by physical objects but differing from them as the concept of wealth differs from land, buildings, and jewelry and as the concept of time differs from a measure of the passage of the sun across the sky.
The Greeks eliminated the physical substance from mathematical concepts and left mere husks. They removed the Cheshire cat and left the grin. Why did they do it? Surely it is far more difficult to think about abstractions than about concrete things. One advantage is immediately apparent - the gain in generality. A theorem proved about the abstract triangle applies to the figure formed by three match sticks, the triangular boundary of a piece of land, and the triangle formed by the Earth, sun, and moon at any instant.
The Greeks preferred the abstract concept because it was, to them, permanent, idea; and perfect whereas physical objects are short-lived, imperfect, and corruptible. The physical world was unimportant except in so far as it suggested an ideal one; man was more important than men. The strong preference for abstractions will be evident from a brief glance at the leading doctrine of Greece's greatest philosopher.
Plato was born in Athens about 428 B.C. of a distinguished and active Greek family, at a time when that city was at the height of her power. While still a youth he met Socrates and later supported him in the defense of the aristocracy's leadership of Athens. When the democratic party took power, Socrates was sentenced to drink poison and Plato became persona non grata in Athens. Convinced that there was no place in politics for a man of conscience - of course, politics was different in those days - he decided to leave the city. After traveling extensively in Egypt and visiting the Pythagoreans in lower Italy, he returned to Athens about 387 B.C. where he founded his academy for philosophy and scientific research. Plato devoted the latter forty of his eighty years of life to teaching, writing, and the making of mathematicians. His pupils, friends, and followers were the greatest men of his age and of many succeeding generations, and among them could be found every noteworthy mathematician of the fourth century B.C.
There is Plato maintained, the world of matter, the Earth and the objects on it, which we perceive through our senses. There is also the world of spirit, of divine manifestations, and of ideas such as Beauty, Justice, Intelligence, Goodness, Perfection, and the State. These abstractions were to Plato as the Godhead is to the mystic, the Nirvana to the Buddhist, and the spirit of God to the Christian. Whereas our senses grasp the passing and the concrete, only the mind can attain the contemplation of these eternal ideas. It is the duty of every intelligent man to use his mind toward this end, for these ideas alone, and not the daily affairs of man, are worthy of attention. These idealizations, which are the core of Plato's philosophy, are on exactly the same mental level as the abstract concepts of mathematics. To learn how to think about the one is to learn how to think about the other. Plato seized upon this relationship.
In order to pass from a knowledge of the world of matter to the world of ideas, he said, man must prepare himself. Light from the highest realities reside in the divine sphere, blinds the person who is not trained to face it. He is, to use Plato's own famous figure, like one who lives
continually in the deep shadows of a cave and is suddenly brought out into the sunlight. To make the transition from darkness to light, mathematics is the ideal means. On the one hand, it belongs to the world of the senses, for mathematical knowledge pertains to objects on this Earth. It is, after all, the representation of properties of matter. On the other hand, considered solely as idealization, solely as an intellectual pursuit, mathematics is indeed distinct from the physical objects it describes. Moreover, in the making of proofs, physical meanings must be shut out. Hence mathematical thinking prepares the mind to consider higher forms of thought. It purifies the mind by drawing it away from the contemplation of the sensible and perishable to the eternal. The path to salvation, then, to the understanding of Truth, Beauty, and Goodness, led through mathematics. This study was an initiation into the Mind of God. In Plato's words, '. ..geometry will draw the soul towards truth, and create the spirit of philosophy, ...' For geometry is concerned not with material things but with points, lines, triangles, squares, and so on, as objects of pure thought.
when it has shape. It is no wonder, then, that geometry, the study of forms, was the special concern of the Greeks.
Finally, it was the solution of a vital mathematical problem that drove the Greek mathematicians into the camp of the geometers. Irrational numbers. We have already spoken of the fact that the Babylonian civilization, as well as earlier ones, used integers and fractions. The Babylonians were familiar also with a third type of number which arose though the application of a theorem on right triangles.
When most people describe the Greek contributions to modern civilization, they talk in terms of art, philosophy, and literature. No doubt the Greeks deserve the highest praise for what they bequeathed to us in these fields. Greek philosophy is as alive and significant today as it was then. Greek architecture and sculpture, especially the latter, are more beautiful to the average educated person of the twentieth century than the creations of his own age. Greek plays still appear on Broadway. Nevertheless, the contribution of the Greeks that did most to determine the character of present-day civilization was their mathematics. By altering the nature of the subject in the manner we have related, they were able to proffer their supreme gift. This we proceed to examine.
