Thursday, December 29, 2011

136: Are Coincidences Just Coincidences?

In past episodes of this podcast, I've poked fun at the human tendency to try to spot deep significance in coincidences.  We know from statistics that unlikely things are very likely to happen on occasion, due to all the millions of events occurring in the world.  For example, someone in the near future will listen to a Math Mutation podcast only days after dreaming they would listen to a Math Mutation podcast.  Does this mean they had psychic powers and predicted it?  Or does it just mean that if they are the kind of person who would listen to math podcasts, chances are very high that they would both have dreams featuring podcasts, and come across this one in iTunes?   Hopefully listeners to this podcast would choose the second explanation, but you never know.  Somehow we are hard-wired to notice coincidences and act on them.  And one point that I have not given enough podcast time to is that, on many occasions, this tendency to notice coincidences and patterns is actually a *good* thing.  In fact, noticing such patterns and questioning whether there might be an underlying cause has been a key basis of nearly all progress in math and science.  Let's look at a few examples.

Most of us learned about the periodic table of the elements in high school.  This is a table that lists the basic chemical elements in rows, according to their atomic number, and has the miraculous-seeming property that elements in each column engage in similar types of chemical reactions.  For example, the leftmost column is mainly highly reactive or 'alkali' metals, while the rightmost column is noble gases.  In modern times, we know this is mainly due to the elements in each column having a similarly built outer electron shell.  But how was this table discovered?  It began when early chemists, such as Dobereiner in the early 1800s, saw that many elements could be grouped into triads based on their properties.  They also noticed that in each triad, when arranged by atomic weight, the middle element was roughly the average weight of the first and the third.  Could this pattern have just been a coincidence?  The thought that this lucky coincidence might have deeper significance began the path to the periodic table and modern chemistry.

Another example is a geometric coincidence whose explanation is well-known today: the shapes of the continents.  Ever since the writings of Ortelius in the late 1500s, people had noticed that the continents kind of look like they should fit together, most notably in the cases of Africa and South America.  They wondered if there was some deeper explanation, perhaps the continents being torn apart by earthquakes or volcanoes, and then drifting away.  Many later scientists expanded on the concept, most notably Alfred Wegener in 1912.  For a long time this was considered a fringe theory-- even in the 1950s, papers were published debunking the idea.  But when the theory of plate tectonics was developed in the 1960s, the mechanism was finally understood, and the idea of continental drift was proven correct.

There are also other examples of numerical patterns that truly are just coincidences, but can be harnessed and used effectively in science and technology.  For example, if you square pi, the ratio between the circumference and diameter of a circle, you find that pi squared is very close to 10.  This made it very convenient to design slide rules with scales based on pi instead of the square root of 10, enabling precise engineering calculations while still maintaining a relationship to our base-10 number system. Another example that is common in the computer world is the fact that 2 to the 10th power is very close to 1000, enabling the slightly inaccurate but very convenient use of the metric prefixes kilo-, mega-, tera-, etc to be used for computer memory which comes in multiples of powers of 2, instead of having to generate all-new terminology.

And while we're on this topic, we also shouldn't neglect examples of seeming patterns or coincidences that led otherwise brilliant scientists down dead ends.  One classic example is the one I mentioned back in podcast 26.  Kepler, one of the founders of modern astronomy, thought he had hit on something when he noticed that the planetary orbits could be inscribed approcimately within giant versions of the five regular polyhedra.  He spent a lot of time and energy on this theory, which today we consider just as silly as psychic dream phenomena.   But by continually looking for coincidences and patterns in the sky, he made enough real scientific discoveries that today we forgive him.

So, while continuing to make fun of New Agers who derive all sorts of significance from patterns that are mere statistical coincidences, keep this in mind:  the very same human mental foibles that cause us
to fall for these things are the ones that have enabled our amazing progress in the sciences and mathematics.

And this has been your math mutation for today.

• Mathematical Coincidence at Wikipedia
• Periodic Table at Wikipedia
• Continental Drift at Wikipedia
• Kepler at Wikipedia