Final Exam Extra Credit Opportunity

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1 Final Exam Extra Credit Opportunity For extra credit, counted toward your final exam grade, you can write a 3-5 page paper on (i) Chapter II, Conceptions in Antiquity, (ii) Chapter V, Newton and Leibniz, and (iii) One other chapter of your choice, from The History of the Calculus and its Conceptual Development, by Carl Boyer. To receive credit, the paper should demonstrate a robust understanding of both the history and the mathematics involved. Here are some questions that you might have in mind while reading: 1. What is the role of definition in the development of mathematics? 2. Compare and contrast the ancient Greek method of exhaustion, the seventeenth-century idea of infinitessimals, and our contemporary notion of limits. 3. What are Zeno s paradoxes? How would the method of exhaustion, infinitessimals, and limits address these paradoxes? 4. Read a Socratic dialogue (see below). How might Socrates have challenged contemporary, seventeenthcentury, and modern-day mathematicians on their treatment of the infinite. 5. What was the controversy between Newton and Leibniz? 6. Given that modern calculus does not use infinitessimals, why do you think Leibniz s notation is still popular today? Notes 1. Conceptions in Antiquity... [I]n Egyptian arithmetic... with the exception of 2 3, all rational fractions were expressed as sums of unit fractions. (page 15) A unit fraction is a fraction of the form 1 n with n N.... [T]he problem of incommensurability... (18) 1

2 The ancient Greeks did not share our modern notion of real numbers. To them, it did not make sense to speak of the length of an abstract line segment. Rather, they spoke of the ratio of lengths between two line segments. Thus, they would say that the length of this line: is five times the length of this line: But they considered it meaningless to say that the line: has length five. Put differently, they believed in the relative lengths between two lines, but not the absolute length of one line. Not surprisingly, they held similar beliefs about the areas of shapes in the plane, and the volumes of solids in space. In fact, the ancient Greeks defined numbers to be ratios of line segments. The number 2 1 2, for example, was defined to be the ratio of the length of to the length of Notice that zero and negative numbers were not included in this definition. The Greeks held a further belief: namely, that any number (by their definition) can be expressed as the ratio of the lengths of two lines, each of which is itself an integral multiple of some third line segment. For example, is defined to be the ratio of the length of to the length of where each line is, respectively, seven and five times the length of the third length This belief, though intuitively compelling, is incorrect. Consider the square: It can be shown that the ratio of the length of the diagonal 2

3 to the length of the side cannot be expressed as the ratio of any integral multiple of a given length to any other integral multiple of that same length. In our modern terminology, we would say that there are no two integers m, n Z for which the ratio of the length of the diagonal to the length of the side is equal to m n. We call such numbers, which are not the ratio of any two integers, irrational numbers. The corresponding line segments are sometimes called incommensurable. The Greeks actually discovered, fairly early on, that the diagonal and the side of a square are incommensurable. This created a small crisis in mathematics. This incommensurability is established as follows. For simplicity, we will use our modern notation and terminology. Consider the square with side-length equal to 1, and let α be the length of the diagonal. Our diagonal length goal is to show that side length = α is not the ratio of any two integers. The Pythagorean theorem (i.e. a 2 + b 2 = c 2, for a right triangle with hypotenuse c and side lengths a and b) tells us that α 2 = = 2 Suppose for the moment that α is in fact the ratio of two integers, say α = n m for n, m Z. We may additionally assume that m and n are not both even since, if they were both even, then we could factor out 2 from the numerator and denominator until one of them is odd. Then ( n ) 2 = 2 m and so n 2 = 2m 2 Since 2m 2 is even, it follows that n 2 is even. This in turn implies that n must be even (why?). Since n is even, there is a p Z with n = 2p. Consequently, Dividing both sides by 2 yields 2m 2 = n 2 = (2p) 2 = 4p 2 m 2 = 2p 2 from which it follows that m is even. But this contradicts our original assumption that m and n are not both even! Therefore, we must reject our initial assumption, and conclude that there cannot exist any integers m, n Z for which α = m n. This confusion of the abstract and the concrete... was characteristic of the whole Pythagorean school. (18) The Pythagoreans constituted a highly spiritual school, active on the island Samos in the Aegean Sea. They believed that the universe is, in a rather literal sense, entirely mathematical. Among other things, they held that the number 10 is sacred, and that there is some kind of counter-earth which revolves around the sun. Their central figure, Pythagoras, may or may not have existed. It is said that it was a Pythagorean, named Hippasus, who discovered the incommensurability of the diagonal of a square to its side. The story goes on to say that the Pythagoreans were so horrified at this discovery that they took him out to sea and drowned him. This account is almost certainly false. 3

4 ... Ionian hylozoistic interpretations had made little headway. (18) The term hylozoism refers to the belief that either all matter is alive, or that it otherwise possesses some kind of a life-force. For more information, see the brief and freely available essay Hylozoism: A Chapter in the Early History of Science, by William Hammond.... [T]he assertion that... it was the authority of Aristotle which held back for two thousand years a transformation which the Academy sought to complete. (27) Aristotle was a student of Plato, in turn a student of Socrates. These three figures are so monumental in the development of Western philosophy that all prior philosophical schools are usually termed pre-socratic. Socrates himself wrote nothing; rather, he wandered about Athens challenging his interlocutors to give basic definitions (e.g. of truth, justice, courage,... ) and generally annoying people. Our knowledge of him reaches us through the writings of Plato. See, for example, the Apology, Crito, Phædo. Plato, on the other hand, not only published but also founded the Academy, a prototype of a modernday university. Tradition has it that above the door of the Academy were inscribed the words Let no one ignorant of geometry enter here, though the historical accuracy of this claim is in doubt. Aristotle was a prolific writer. Unlike his mentor, he took an interest not just in abstract philosophy but in the natural world as a whole. Much of his Organon ( instrument ), a collection six tracts on the study of logic, is still widely relevant today. He eventually went on to establish the Lyceum, a second Athenian school to rival the Academy. 3. Newton and Leibniz... [W]e shall not discuss the shamefully bitter controversy as to the priority and independence of the inventions by Newton and Leibniz. (188) For a more contemporary account of this controversy, see where... [H]e used the infinitely small, both geometrically and analytically... and extended its applicability by the use of the binomial theorem. (191) The binomial theorem asserts that, for any a, b R and any n N, we have is called the binomial coefficient of n and k. (a + b) n = ( ) n = k n k=0 ( ) n a n k b k k n! k! (n k)!... [H]e sent a letter to Leibniz... in which he gave in the form of an anagram a statement of the fundamental problem of his calculus. (196) 4

5 At the time, it was common for European scholars to send anagrams of their discoveries to their competitors. Naturally, the recipient would be unable to decipher the message, which was intended to be kept secret. The point of this was that, should this competitor eventually discover the secret on their own, the original scholar could send the solution of the anagram and, in so doing, establish that they made the discovery first.... [T]he quadrature depended upon the sum of the ordinates... for infinitesimal intervals in the abscissas. (203) The words quadrature, ordinates, and abscissas are old-fashioned synonyms for integration, vertical-axis values (on a graph), and horizontal-axis values, respectively. 5

Greece In 700 BC, Greece consisted of a collection of independent city-states covering a large area including modern day Greece, Turkey, and a multitu

Chapter 3 Greeks Greece In 700 BC, Greece consisted of a collection of independent city-states covering a large area including modern day Greece, Turkey, and a multitude of Mediterranean islands. The Greeks