Flying around the world isn’t correctly Euclidean

The Abel Prize – the “Nobel prize of mathematics” – was awarded this year to a man who proved that the shortest distance between two points isn’t always a straight line.  

Most of us were taught in geometry that it was.  Rather like 2+2 always equals 4, the straight line of Euclidean geometry had the hard immutability of always, irrevocably being self-evident fact.  One might even say, of always being true.

But what Mikhail Gromov proved is that the shortest distance between two points isn’t a straight line if you aren’t on a flat plane.  What pilots have known for a long time is that if the points are located on a round three-dimensional object like Earth, the shortest distance between two points is a curved line.  It’s why, if you are flying between London and San Francisco, the flight heads north from London before heading south.  It’s not to avoid other planes on a busy airline highway.  It’s because it’s the shortest way to go. 

If you have a globe of the Earth which is big enough, you can test for yourself whether this and various other Euclidean “facts” are actually valid in a curved geometry.  

This is an illustration of the relativity of facts in science.  All scientific facts exist in a context.  Sometimes it takes no more than weeks, sometimes it take centuries to discover a different context in which an “absolute” fact is actually relative.  It’s the facts that took centuries to limit their context that gave us the impression for so long that some scientific facts are absolute.

But we now know from Einstein’s theory of relativity, for instance, that Newton’s theory of gravity doesn’t apply equally throughout the universe, and that time and space are not absolute.  We now know from quantum mechanics that before and after, up and down, and even existence and non-existence don’t appear to operate on the particle level the way they do in the world in which most of us spend our lives.

If you still are not convinced that all scientific facts are facts only within a specified context, think of the following puzzle that was offered to me as an adolescent: 

How many examples can you think of when 1 + 1 do not equal 2?  There are many.


About Terry Sissons

Terry Sissons is the author of The Big Bang to Now: All of Time in Six Chunks, and this blog is a dialogue about the universe and what’s happened in the last 13 billions years.
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