question why the idea that mathematical facts must be assured by logical ...
Lancelot Hogben, in Mathematics for the Million, suggests that the need for proof
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Math 489
Test 1 Solutions
L. Hogben 2 / 1 5 / 9 5
2/03 Note regarding historical questions (1,2,3): The solutions to #1 is longer than you would have time for on a test and considerably more complete than is necessary to receive 95% or higher. Note #3 asks for only one of the two answers given. Although you should be able to answer these questions, you will not necessarily be familiar with sources cited in #1 or all the details given in #2.
1. Discuss possible explanations for the Greek Mystery, i.e., the question why the idea that mathematical facts must be assured by logical demonstration rather than laboratory experimentation, arose only once, in ancient Greece. Which explanation do you consider most plausible? Why? The Greek Mystery is the question why the idea that mathematical facts must be assured by logical demonstration rather than laboratory experimentation arose only once, in ancient Greece. Any explanation of this mystery must explain both why this happened in ancient Greece, and also why it did not happen in other cultures. It must also explain it in a way that does not merely raise another question. Howard Eves offers several possible explanations in Great Moments: 1) The peculiar mental bias of the Greeks toward philosophical inquiries. 2) The Greek love of beauty, as manifest in their art, writing and architecture. 3) The slave based nature of the Greek economy, freeing a privileged class for intellectual endeavors. 4) The sweeping economic and political changes occurring at this time: the iron age, the alphabet, coinage, geographic discoveries. Eves seems to find each explanation individually unsatisfactory, he finds all taken together satisfactory. I do not. (1) and (2) beg the question. If mathematics came from love of art, what caused love of art here and not in other civilizations? (3) is directly contradicted by Eves' assertion in Cultural Connection IV, Introduction to the History of Mathematics, that the decline of mathematics during the Roman Empire was caused in part by the fact that it was based on slave labor and there was therefore no incentive for labor saving devices. This also ignores the Greeks' clear appreciation of the sheer beauty of mathematics. (4) Strikes me as plausible; however, taken alone it does not explain the "only" part, i.e., proof arose only in ancient Greece. It is related to the next idea. Lancelot Hogben, in Mathematics for the Million, suggests that the need for proof arose in ancient Greece because "barbarians" simultaneously encountered several developed cultures, and had to decide what to accept from each. The idea of proof developed as way to determine what should be accepted from each culture. The Greeks were the "barbarians," encountering the well developed cultures of the Babylonians, Egyptians, Phoenicians and Jews. Notice that this form of contact is different than two developed cultures encountering each other, or a new group encountering only one developed civilization. Furthermore, this is the only time and place this happened in the ancient world. This is the explanation I find most convincing.
2. Discuss the importance of unsolved problems in mathematics. Give examples of famous problems and mathematics that grew out of attempts to solve the problem (the problem may be solved now; the attempt need not be successful). Unsolved problems can give birth to whole new branches of mathematics in attempts to solve them. The three famous construction problems of antiquity, duplicating the cube, trisecting the angle, and squaring the circle provide many examples of this. It is first necessary to clarify the problems themselves. To duplicate the cube means given a cube of volume V, construct a cube of volume 2V. It did not mean to make an identical copy. To trisect the angle meant to be able to cut any angle into three congruent pieces, not just to do it for special angles, such as a right angle. To square the circle means give a circle, construct a square having the same area. As the problems were posed in antiquity, it was required to carry out the constructions by straightedge and compass. Hippocrates of Chios reduced the problem of duplicating the cube to the construction of the second mean proportional, i.e., give a and b construct x and y such that a:x = x:y = y:b. Meneachmus (ca. 350 B.C.) invented conic sections for the purpose of solving this problem. Diocles invented the cissoid curve to solve this. (Many mechanical solutions were also invented, but are not relevant here.) Various curves were also invented for trisecting the angle and squaring the circle, including the quadratrix of Hippias and the spiral of Archimedes. In the 19th century it was proved that solution of these problems with straightedge and compass is impossible, but the mathematics created in the attempt to solve them, e.g., conic sections, lives on. Although we have not yet studied it, Fermat's Last "Theorem" is another good example. Algebraic number theory was developed for the purpose of proving this result, although it has now been proved (?) by other means, elliptic curves.
3. Defend or refute the statement, "Archimedes is the inventor of integral calculus." Defense: The finding of areas and volumes is one of the key ideas of integral calculus, and Riemann sums form the basis for the definition of the definite integral. Archimedes was able to find areas and volumes better than most modern students of calculus, using a method based on Riemann sums. Archimedes used Eudoxus' Method of Exhaustion to rigorously prove formulas derived by his own method described in the treatise M e t h o d . The Method of Exhaustion involves assuming a ratio is greater than it is and doing a construction to derive a contradiction. Archimedes' technique described in M e t h o d essentially uses Riemann sums. In the example we examined in class, Archimedes told us to balance slices of a sphere and a cone against slices a cylinder. As he sliced these shapes he was implicitly using Riemann sums. Because he did not have concepts of infinity or limits, he could not make this method rigorous, but the essential ideas were all there. Refutation: Archimedes developed many of the techniques used in integral calculus, but he cannot be considered the inventor of it, because he lacked the essential unifying concept. Archimedes used Eudoxus' Method of Exhaustion to rigorously prove formulas derived by his own method described in the treatise M e t h o d . The Method of Exhaustion involves assuming a ratio is greater than it is and doing a construction to derive a contradiction. Archimedes' technique described in M e t h o d essentially uses Riemann sums. Although Archimedes had the idea of Riemann sums, he lacked the concepts of infinity and limit, so he could not make his method rigorous. More importantly, he lacked the Fundamental Theorem of Calculus, which is essential to the understanding of integration. Even Archimedes' Method was ad hoc: It gave no general way of finding area or volume. In the example we examined, Archimedes told us to balance a sphere and a cone against a cylinder. How were these found?
4. Proposition 14 of Book II of Euclid' E l e m e n t s provides the construction of a square that is equal in area to a given rectangle. Let ABCD be the given rectangle with BC = b and DC = a, BE be collinear with AB and BE = b, AE be the diameter of a semicircle, F be the intersection of the semicircle with BC extended, and BFGH be a square, as shown:
Prove the area of BFGH equals the area of ABCD. Draw line segments AF and EF as shown above. Angle AFE is a right angle because it is an angle inscribing a semicircle. angle BAF + angle AFB = right angle = angle BFE + angle AFB, so angle BAF = angle BFE. In the same way, angle AFB = angle BEF. Thus the triangles AFE, ABF, and FBE are similar. Therefore, BE x b x = , i.e., = , or ab = x2 . Thus the area of the rectangle ABCD x a x AB equals the area of the square FGHB.
5. State the proof that 2 is irrational. Identify at which step it breaks down if you try to use the same method to show 4 is irrational. Suppose 2 is rational. Then it can be expressed as a fraction in lowest a terms, 2 = , where a and b are relatively prime positive integers. b a 2 = with a and b relatively prime 1) Assumed: b 2 b = a 2) Multiply both sides by b: 3) Square both sides: 2b 2 = a2 4) 2b2 is even, so a2 is even 5) a must be even or odd. If a is odd, odd x odd is odd, but a2 is even. Thus a is even. 6) a is even means a = 2c. 7) From (3) and (6): 2b 2 = (2c)2 = 4c2 8) Divide both sides by 2: b 2 = 2c2 9) 2c2 is even, so b2 is even 10) b must be even or odd. If b is odd, odd x odd is odd, but b2 is even. Thus b is even. 11) Both a and b are even, so 2|a and 2|b, contradicting a and b are relatively prime. The proof breaks down at step 5. Step 4 becomes 4b2 = a2 . Step 5 gets rewritten as a is of form 4k, 4k+1, 4k+2, or 4k+3. In step 5, a = 4k+2 gives a2 = 16k2 + 16k + 4, which is clearly divisible by 4.
6. Proposition 13 of Archimedes' On the Sphere and Cylinder states: The surface area of any right circular cylinder, excluding its bases, is equal to the area of a circle whose radius is the mean proportional between the side of the cylinder and the diameter of the base of the cylinder. Translate this statement into modern symbolic notation, and prove it.
Let the cylinder have height h and radius r, as shown above. Then its surface area excluding bases is the circumference times height, 2 π r h (as can be seen by cutting off the bases and cutting through the side vertically). The diameter is 2r. If x is the mean proportional between 2r and h, then 2r:x = x:h, or 2rh = x2 . The area of the circle of radius x is π x 2 = π 2rh = 2π rh, the surface are of the open cylinder.
