Looks Can Be Deceiving
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.
