Hi. More than 2 ,000 years ago, Euclid organized
geometry around a small collection of starting
assumptions called postulates. Most of these
seemed simple. A straight line can be drawn between
any two points. We can draw a circle with any
center at any radius. But one assumption was
different. Eventually, we named it the parallel
postulate. In short, it says that if you have
a line and a point outside that line, exactly
one line can pass through the point without ever
intersecting the original line. Now, unlike Hewlett's
other assumptions, this one seemed complicated.
It did not feel as obvious or even fundamental.
So, mathematicians wondered whether it was really
necessary. Perhaps the parallel postulate was
not an assumption at all. Perhaps we could prove
it from the other postulates. So, for roughly
2 ,000 years, mathematicians tried. Again and
again, they constructed proofs. Again and again,
their reasoning failed. Now, by the usual definition,
they failed. But then, something remarkable happened.
Mathematicians began asking a different question.
What if the parallel postulate could not be proven
because it was truly independent of the others?
And what if they replaced it with something else?
Suppose there were no parallel lines through
the point. Or, suppose there were many. Now,
at first, these possibilities seemed absurd.
Surely, changing the parallel postulate would
produce a geometry filled with contradictions.
But, it did not. Instead, mathematicians discovered
entirely new geometries. For example, on a curved
surface, the angles of a triangle do not necessarily
sum to 180 degrees. Parallel lines may behave
very differently from what we learned in school.
The geometry of a flat surface is only one geometry
among several possibilities. Eventually, these
new ways of thinking gave Albert Einstein the
mathematical language he needed to describe curve
space and the theory of relativity. Wow! So what's
the big idea here? 2 ,000 years of unsuccessful
proofs open the door to a larger understanding
of reality. Those mathematicians did not solve
the problem that they originally wanted to solve,
but they discovered that their question was pointing
towards something far greater. And I find that
encouraging, because many of my small ideas or
small problems often are part of something bigger
than I ever imagined. if I'm open to see it.
Many of my failures and struggles were not dead
ends, but they were invitations to question the
path itself and to discover an entirely new world
of possibilities. So, how about you? What problem
are you trying to solve that you are ready to
quit? problem important and if it is it is likely
worth persisting