Looks Can Be Deceiving Essays (1861 words) - Logic, Philosophy

Looks Can Be Deceiving Paradoxes are sometimes composed of contradictory ideas presented together, ultimately leading to an unworkable situation. Paradoxes, however, are not simply ambiguous questions. Paradoxes are the essence of the inherent complexity of systems (Internet 1). Each paradox must be analyzed and clearly understood before it can be explained. Since mathematics is, in a sense, a universal language, certain paradoxes and contradictions have arisen that have troubled mathematicians, dating from ancient times to the present. Some are false paradoxes; that is, they do not present actual contradictions, and are merely slick logic tricks. Others have shaken the very foundations of mathematics ? requiring brilliant, creative mathematical thinking to resolve. Others remain unresolved to this day, but are assumed to be solvable. One recurring theme concerning paradoxes is that each of them can be solved to some degree of satisfaction, but are never completely conclusive. In other words, new answers will likely replace older ones, in an attempt to solidify the answer and clarify the problem. A paradox can be defined as an unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises. This essay provides an introduction to a range of paradoxes and their possible solutions. In addition, a questionnaire was composed in order to demonstrate the extent of knowledge that the general population has pertaining to paradoxes. Paradoxes are useful things, despite their mind-boggling appearance. Generally, however, most paradoxes can be "solved" by searching for specific properties that they may contain. Therefore, if you try to describe a situation and you end up with a paradox (contradictory outcome), it usually means that the theory is wrong, or the theory or the definitions break down along the way. Also, it is possible that the situation cannot possibly occur, or the question may simply be meaningless for some other reason. Any of these possibilities are relevant, and if you exhaust all the possible interpretations, one of them should prove to be incorrect (Internet 1). The following type of paradox is called Simpson's Paradox. This paradox involves an apparent contradiction, because when the data are presented one way, one particular conclusion is inferred. However, when the same data are presented in another form, the opposite conclusion results. Paradox 1: Acceptance Percentages for College A and College Chart 1 Section A Section B Accepted Rejected Total Percent Accepted Accepted Rejected Total Percent Passing Women 400 250 650 61% 50 300 350 14% Men 50 25 75 67% 125 300 425 29% Total 450 275 725 175 600 775 As is evident in Chart 1, when the data are presented in two separate tables, it looks as if men are accepted more often than women, because in each case (College A and College B), men are accepted at a higher ratio than women. However, when the same data are combined into one table (Chart 2), a contradicting result is implied. Acceptance Percentage Totals for the University Chart 2 Accepted Rejected Total Percent Accepted Women 450 550 1000 45% Men 175 325 500 35% Total 625 875 1500 This table shows women actually having a higher overall acceptance rate than men. This is an example of Simpson's Paradox because it involves misleading data. Obviously, the presentation of the data is very important, and can lead to incorrect assumptions if the data are not used properly (Internet 2). Paradox 2: An Arrow in Flight One can imagine an arrow in flight, toward a target. For the arrow to reach the target, the arrow must first travel half of the overall distance from the starting point to the target. Next, the arrow must travel half of the remaining distance. For example, if the starting distance was 10m, the arrow first travels 5m, then 2.5m. If one extends this concept further, one can imagine the resulting distances getting smaller and smaller. Will the arrow ever reach the target? (Internet 3) The answer is, of course, yes the arrow will reach the target. Our common sense tells us so. But, mathematically, this fact can be proven because the sum of an infinite series can be a finite number. The question contains a premise, which implies that the infinite series will result in an infinite number. Thus, 1/2 + 1/4 + 1/8 + ... = 1 and the arrow hits the target (Internet 3). Paradox 3: Two Equals One? Assume that a = b. (1) Multiplying both sides by a, a? = ab. (2) Subtracting b? from both sides, a? - b? = ab - b? . (3) Factoring both sides, (a + b)(a - b) =