Hi. It seems natural that tying and untying a
shoe would have been discovered at the same time.
One action creates the knot, the other reverses
it. Mathematics is filled with similar pairs
of doing and undoing. We add 3 plus 4 equals
7. Then subtraction undoes the addition. 7 minus
4 returns to 3. Multiplication is repeated addition.
4 plus 4 plus 4. equals 12. And division undoes
multiplication. 12 divided by 3 equals 4. Mathematics
keeps building these layers. Repeated multiplication
becomes exponentiation. 4 times 4 times 4 equals
4 cubed, which equals 64. And then logarithms
undo exponentiation. At least, that's how we
think about them today. But here's our paradox.
Exponentials and logarithms were not originally
discovered together. In fact, they emerged from
completely different worlds. In the early 1600s,
John Napier invented logarithms as a practical
tool to simplify massive calculations used in
astronomy and navigation. His goal was efficiency.
Logarithms transformed multiplication into addition,
making difficult calculations far easier. Meanwhile,
a completely different story was unfolding with
future actuaries in mind. Mathematicians studying
compound interest and continuous growth kept
encountering a strange number that was near 2
.71828... Later, we gave this number the letter
E. And this number emerged naturally from growth
itself. One discovery came from simplifying calculations.
The other came from studying growth and change.
Different people, different problems, different
centuries. At first, these ideas seemed unrelated.
And then... Euler entered the story. Euler discovered
something truly amazing, especially at the time.
Log rhythms and exponential growth were deeply
connected. They were inverses of each other.
The same hidden structure had been sitting underneath
both ideas all along. I just wonder about the
aha moment Euler had in this discovery. suddenly
mathematics became more unified than anyone had
imagined. And that is one of the reoccurring
paradoxes of math. Truth often appears in scattered
pieces before we recognize the deeper pattern
connecting them. Life works a bit this way too.
Sometimes we may see how things are connected
quickly, but many times the connection requires
time. But if we take time to notice, we can discover
that some experiences that seem completely unrelated
later reveal hidden connections. A difficult
season later produces compassion. Failure creates
wisdom. Loss deepens gratitude. At the time,
these pieces may feel disconnected. Only later
do we begin to see the larger story. Adam Grant
talks about the importance of rethinking, being
willing to let new understanding reshape old
assumptions. Mathematics trains us to do exactly
that. It repeatedly reminds us that reality may
be more connected than our first impressions
suggest, that beneath the scattered pieces, there
may be hidden coherence waiting to be discovered.
And perhaps hope grows when we learn not to cling
too tightly to our current understanding. Because
reality may still be revealing deeper connections
that our eyes cannot yet see.