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Dubious Math in Infinite Jest Let me say, first of all, that I am a huge fan of David
Foster Wallace in general and Infinite Jest in particular. On
my first reading of IJ, I noticed a few mathematical errors but thought
little of them. After reading the essay
Derivative Sport in Torndao Alley in A
Supposedly Funny Thing I'll Never Do Again, though, I became curious about
why a writer with a clear aptitude for math would include such mistakes in his
opus. Therefore, during my second
reading of IJ, I made a note of the errors that I noticed. As it turned out, their number was smaller
than I had imagined. Consequently, I
lost interest in this topic until reading DFW's review of a pair of
mathematical novels in a scientific journal.
The broad knowledge of math demonstrated by DFW in this article
rekindled my curiousity about the errors in IJ, and I decided to
document them, in case others might be interested. My list is actually quite short - only four mistakes, two of which might well be
typographical - and I offer no theories about why they appear. One of the errors is attributable to the
omniscient narrator, while the other
three are spoken by Mike Pemulis. Both, we can assume, are competent
mathematicians. Before I outline the errors, I must
make a logistical note. Because I am
writing this using a simple text editor, I am somewhat symbolically
challenged. Exponentiation will be
denoted using a caret (e.g., 2^3 = 8).
Also, I will not even attempt to depict an integral sign or the usual
mathematical symbol for combinations, a pair of elongated parenthesis. I hope this makes the explanations no less
understandable. The first, and perhaps the most
interesting error that I noted, appears on page 259 of IJ. The narrator states that the odds of a 108
game tennis match ending in a 54-match-all tie are 1 in 2^27. This is incorrect by about seven orders of
magnitude; in fact, such an outcome is much more likely than the narrator
suggests. The problem of determining
the odds is an exercise in probabilty and combinatorics, a fancy mathematical
term for systematic counting. By the
laws of probability, the odds of a 108 game match ending in a tie are (the number of outcomes resulting
in a 54-54 tie) / (the total number of possible outcomes). The denominator in this expression
is easy to calculate. It is simply
2^108 (an example will follow). The
numerator can be found using the concept of combinations. That is, the correct anwer is the total
number of ways team A can be assigned exactly 54 victories out of the 108
matches. (Of course, if team A wins
exactly 54 matches, so must team B.) In
combinatorics this is referred to as "the number of combinations of 108
things taken 54 at a time."
Although the derivation is well beyond the scope of this discussion, it
can be shown that the number of combinations of n things taken m at a
time is n! / ((n-m)! m!). Therefore, the numerator in the probabilty expression above
is 108! / (54! * 54!), and the correct odds are (108! / (54! *54!))/(2^108) or approximately 0.0766. This is considerably greater than 1/2^27, or about
0.00000000745. To illustrate this
result, consider the following enumerable example: What are the odds of a 4 game match ending in a 2-2 tie? Again, the method above predicts that odds
are (4! / (2! * 2!))/(2^4)
= (24 / (2 * 2))/(16) = 6/16. The answer suggested by the narrator of IJ would be
1/(2^1) = 1/2. To see which is correct,
we can list all 16 possible outcomes as follows: AAAA ABAA BAAA BBAA AAAB ABAB BAAB BBAB AABA ABBA BABA BBBA AABB ABBB BABB BBBB Clearly, there are six outcomes resulting in a 2-2 tie
(AABB, ABAB, ABBA, BAAB, BABA, and BBAA), so the odds are, indeed, 6/16 as the
method above predicts. The second significant instance of dubious math in IJ
occurs in a footnote on pages 1023 and 1024.
In this section Mike Pemulis describes to Hal how the Mean Value Theorem
for integrals can be used to distribute megatons of thermonuclear weapons among
Eschaton combatants. The Mean Value
Theorem for integrals states that, for a function f(x) that is
continunous on the interval from x = a
to x = b, the integral from a to
b of
f(x)dx = f(x')(b - a) for some value x' between a
and b. In effect, this theorem simply states that the area
underneath the curve described by the function f(x) from x = a to x = b is exactly the same as the area of a rectangle whose width
extends from a to b and whose height has a value of f(x') for some value x' between a and b. Now, while Pemulis' description of the
theorem is essential correct, the problem in this footnote is the way the
theorem is supposedly applied.
Specifically, the Mean Value Theorem for integrals is a theoretical tool
for proving the existence of this
particular x'. It does not, however, offer any method of finding the value of x'. Therefore, it is difficult to imagine how
the Mean Value Theorem for integrals could be employed in Pemulis' Eschaton
calculations. Incidently, the third,
minor error, which may very well be typographical, also occurs on page 1024 in
this footnote. Note that the abstract
statement of the Mean Value Theorem for integrals which appears in the text
(i.e., f(x)dx = f(x')(b - a) ) is missing the sign for the integral
from a to b. The final mathematical error that I noted occurs in a footnote
on page 1063. Again, Mike Pemulis is
lecturing Hal, but this time he is helping Hal
prepare for the college board exams.
Pemulis states that for the
function x^n, the derivative is nx + x^(n-1). In fact, the correct expression is nx^(n-1). This, too, may be
a typographical error. As I have said, I have no theories to explain the existence
of these errors. I would, however, be
interested in the thoughts of others. Mike Strong smstrong [insert anti spam 'at' here] localnet.com |
