proposed a number of paradoxes describing supposed difficulties with
the division of time and space into infinitely small increments.
Here's an example you've probably heard in many forms before: Suppose
you want to cross a road. Before you get across the road, you must
get halfway across the road. But before you cross the second half, you
must get halfway across that. And so on: there are an infinite
number of halfway points as you get closer and closer to the other
side, so therefore you can never reach it. The total distance you
need to walk is 1/2 + 1/4 + 1/8 + ..., with an infinite total number
of terms in the series.
Some philosophers still talk about this paradox, but in a
mathematical sense, it's pretty well resolved. One concept that Zeno
may not have been aware of, though you probably were taught at some
point in high school algebra, is that an infinite series of
geometrically decreasing elements can indeed add up to an easily
calculated finite sum. As a quick refresher, let S equal the sum
discussed above, 1/2 + 1/4 + 1/8 +... If you multiply S by 1/2, you
will see that you have the same series, except with the first term,
1/2, missing. So S = 1/2 + 1/2S, and solving for S, we get S = 1.
Of course, the fact that we can indeed cross roads also proves to
refute Zeno's paradox, and in fact is not something that we find very
exciting. But this ability to sum up an infinite series can have
interesting consequences when we look at another area: population
sizes. As you have probably heard, many modern countries have
fertility rates below replacement: that means that the number of
people is declining every generation. Let's take Russia as an
example. Their estimated fertility rate in 2007 is 1.39 children per
couple, which means that for every person in the current generation,
there will be about 0.7 people in the next generation. That means
that the ratio of the total number of children that will ever be born
in Russia, compared to the current population, is given by the series
1 + .7 + .7 squared + .7 cubed ..., which by the same method we used
before, sums up to about 3.33.
This is pretty scary if you think about it-- it means that even
after an infinite number of years, at the current fertility rates, the
total number of people that will *ever* have existed in Russia is a
little over 3 times the current population. It's actually a bit less
than that, of course, since after the terms become smaller than whole
numbers, you can't really keep summing up fractional children. Now we
can understand those wacky stories about Russian baby-producing
contests that have been in the media recently. On the other hand,
there's nothing unique about Russia: this issue of declining
population is striking large portions of the Western world. Perhaps
soon other Western countries will emulate Russia's survival strategy,
and baby-producing will be the top event at the 2012 Olympics.
And this has been your math mutation for today.