Hi, imagine holding a meter stick and looking
at the evenly spaced centimeter marks. It feels
obvious that every distance between 0 and 100
centimeters exists somewhere on that stick. Not
just whole numbers like 1 or 57, but fractions.
Even irrational numbers like pi or square root
of 2. Somehow, infinitely many numbers fit onto
a single stick. Now imagine sitting in a race
car, accelerating from 0 to 100 miles per hour.
At some instance, did the car go exactly 70 miles
per hour? What about 70 .5? Or 70 .00001? Did
it pass through every possible speed between
0 and 100? Or did it somehow skip some values?
And digging deeper, What does it really mean
to say a car is traveling 70 miles per hour at
an instant? A snapshot of time has no duration,
but motion seems to require duration. So how
can something move during an instant that contains
no time? Okay, for most of us, these questions
do not affect everyday life. We still drive cars.
We still trust speedometers. Life moves on. But
mathematicians refuse to simply dismiss these
tensions. They stay with them. And one direct
benefit we have from wrestling with paradoxes
is the concept of calculus, the mathematics of
motion and change. Math usually becomes more
interesting once we kick things into motion.
A golf ball is a special type of sphere, but
hit it with a club? and suddenly we find a parabola.
Calculus and math are full of paradoxes, moments
where reality seems to pull in two different
directions at once. Of course, paradox is not
just a math thing. It's a human thing. In relationships,
two thoughtful people can experience the same
situation completely different. In life, many
of us would gladly choose success over struggle.
Yet, when I look back on my life, I often realize
I learned the most during difficult seasons rather
than easy ones. Why is that? Well, author Brene
Brown says that one of the greatest challenges
of life is learning not to run from paradox,
but to remain inside it long enough for deeper
understanding to emerge. But that's difficult.
We naturally move towards certainty because certainty
feels safe. We want quick conclusions, simple
categories, clear sides, right or wrong, strong
or weak, certain or uncertain. But life rarely
stays inside those neat boxes. Sometimes we feel
both confident and afraid. Sometimes grief and
gratitude exist together. Sometimes we feel both
broken and hopeful at the same time. And perhaps
one of the dangers of certainty is that our understanding
of reality becomes too small. I know I often
want to be right. But over time, I've discovered
that there are moments when love matters more
than winning an argument. Moments where reconciliation
matters more than proving my point. Paradox stretches
us because it forces us to consider that reality
may be larger than our current categories can
hold. So, this week we will explore paradoxes
that involve motion, infinity, patterns, truth,
and the strange unity hidden inside mathematics
itself. Not because math removes mystery, but
because it teaches us how to thoughtfully live
within it. And one thing that I've experienced
is that as I remain open and inside the tension
of paradox, that it is part of the journey towards
reasonable hope.