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, a² = ab. (2) Subtracting b² from both sides, a²
- b² = ab - b² . (3) Factoring both sides, (a + b)(a - b) = b(a - b). (4)
Dividing both sides by (a - b), a + b = b. (5) If now we let a = 1 = b, we
conclude, from step (5), that 2 = 1. Or we can subtract b from both sides and
conclude that a, which can be taken as any number, must be equal to zero. Or we
can substitute b for a and conclude that any number is double itself. Our result
can thus be interpreted in a number of ways, all equally ridiculous. The paradox
arises from a disguised breach of the arithmetical prohibition on division by
zero, occurring at Step (5). Namely, since a = b, dividing both sides by (a - b)
is dividing by zero, which renders the equation meaningless. As Northrop goes on
to show, the same trick can be used to prove, for example, that any two unequal
numbers are equal, or that all positive whole numbers are equal (Internet 4).
Paradox 4: Squares and Rectangles The area of the square, shown above, is 8 x 8
= 64 units². The square is cut in the four parts A, B, C, and D, which are
rearranged into the rectangle shown below. This rectangle, however, has an area
of 13 x 5 = 65 units². This can lead to the potential of making 65 units² of
gold out of only 64 units². How can you justify this transformation in area and
the creation of matter? The picture of the rectangle is deceptive! The line XY
shown in the picture of the rectangle (see above) is not a line at all. The
parts XU and VY have a gradient of 2 / 5 = 0.4, and the parts XV and UY have a
gradient of 3 / 8 = 0.375. So, in fact, XUYV is a parallelogram with an area of
1, not a line! Paradox 5: Where Is The Missing Dollar? Three people check into a
hotel. They pay $30 to the manager and go to their room. The manager remembers
that the room rate is $25 and gives $5 to the bellboy to return. On the way to
the room, the bellboy reasons that $5 would be difficult to share among three
people so he pockets $2 and gives $1 to each person. Now each person paid $10
and got back $1. So they each paid $9, totaling $27. The bellboy has $2,
bringing the total up to $29. Where is the missing $1? The correct response to
this question is that since all three people paid $9 each, we are looking at a
total of $27. The manager has $25 for the room while the bellboy has $2 for
himself. The bellboy’s $2 should be added to the manager’s $25 or subtracted
from the tenant’s $27, not added to the tenant’s $27. The existence of a
paradox is proof that either, at least one of the propositions are false, or the
logic used to arrive at the paradox is false, at which point you do not really
have a paradox. As stated previously, there really is no such thing as a
paradox, for its own existence proves that the assumptions it is based on are
wrong. (Internet 5) Searching For Answer’s A survey was composed in order to
demonstrate the extent of comprehension that the general public has in terms of
paradoxes. Ten individuals, whom of which ranged from the ages 16-42, answered
the questionnaires. The survey consisted of five paradoxes that were randomly
chosen, each individual was given an opportunity to choose from one of three
responses (yes, no, or uncertain) for each paradox. The survey showed that 32%
responded yes, 16% responded no and 46% responded uncertain to the ten questions
that were asked. These results justify that the individuals, who answered yes to
most of the questions, were tricked by false propositions. These individuals
ignored common sense and allowed themselves to be deceived. Moreover, the
majority of individuals who answered no to most of the questions were aware that
the paradoxes were somewhat misleading. However, they were unable to explain any
further. Also, the questions that were answered with an uncertain apparently
left the individuals pondering. Survey Results In Chart Form Number of answers
which fall in each category Individual Yes No Uncertain Person #1 4 0 1 Person
#2 3 1 1 Person #3 3 1 1 Person #4 2 2 1 Person #5 1 2 2 Person #6 2 0 3 Person
#7 3 0 2 Person #8 0 1 4 Person #9 0 0 5 Person #10 1 1 3 Total 19/50 8/50 23/50
Percentages 38% 16% 46% Survey Results Represented On A Pie Graph Conclusion
Paradoxes Survey Question 1: Acceptance Percentages for College A and College B
College A College 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 Do the women have reason to claim
sexual discrimination against the university? a) Yes. Explain: b) No. Explain:
c) Uncertain Question 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? (2) a) Yes. Explain: b) No.
Explain: c) Uncertain Question 3: Does Two Equal One? Assume that a = b. (1)
Multiplying both sides by a, a² = ab. (2) Subtracting b² from both sides, a²
- b² = ab - b². (3) Factoring both sides, (a + b)(a - b) = b(a - b). (4)
Dividing both sides by (a - b), a + b = b. (5) Let a=1=b, 2=1. Do you agree? a)
Yes. Explain: b) No. Explain: c) Uncertain. Question 4: Squares and Rectangles
The area of the square, shown above, is 8 x 8 = 64 units². The square is cut in
the four parts A, B, C, and D, which are rearranged into the rectangle shown
below. This rectangle, however, has an area of 13 x 5 = 65 units². This can
lead to the potential of making 65 units² of gold out of only 64 units². Is
this a valid transformation in area? a) Yes. Explain: b) No. Explain: c)
Uncertain Question 5 The Missing Dollar Three people check into a hotel. They
pay $30 to the manager and go to their room. The manager remembers that the room
rate is $25 and gives $5 to the bellboy to return. On the way to the room, the
bellboy reasons that $5 would be difficult to share among three people so he
pockets $2 and gives $1 to each person. Now each person paid $10 and got back
$1. So they each paid $9, totaling $27. The bellboy has $2, bringing the total
up to $29. Is a dollar missing? a) Yes. Explain: b) No. Explain: c) Uncertain.
Work Cited (Internet 1) http://www.colchsfe.ac.uk/