Yesterday, we talked about one of the most remarkable
features of mathematics. We have the freedom
to choose our starting assumptions. But once
those assumptions are in place, we no longer
get to choose the conclusions. Logic leads the
way. Well, today, I want to show you just how
surprising that journey can be. For many years,
mathematicians believed that proving the Pythagorean
theorem, required a particular trig identity,
specifically sine squared plus cosine squared
equals one. The identity seems so fundamental
that it almost feels eternally connected to the
theorem itself. In fact, many believed there
simply was not another way. In fact, I have a
book that is on the authority of the Pythagorean
theorem. that specifically states that this trig
identity must be assumed. Then, in 2022, mathematicians
were humbled because two high school students
from Ohio found a proof. Check this. While working
on a school project, they developed a proof of
the Pythagorean theorem. that avoided using this
identity altogether. And the proof was carefully
examined by professional mathematicians, and
ultimately it was even published in a peer -reviewed
journal. Think about how remarkable that is.
The Pythagorean theorem has been studied for
over 2 ,000 years. There's hundreds of proofs
that have been discovered. Yet two students,
Simply by asking, what if we try a different
approach, found something that many experts did
not expect. And to me, that is one of the beautiful
things that math teaches us. Even when we are
convinced we understand something completely,
there may still be another perspective waiting
to be discovered. Now, this does not mean experts
are unreliable. In fact, it's quite the opposite.
Mathematics progresses because mathematicians
are willing to let logic correct even their strongest
expectations. Humility does not mean doubting
everything. It means holding our conclusions
with enough openness that if logic points somewhere
unexpected, we are willing to follow. Let me
end with one of my most favorite quotes from
Paul Lockhart in his book, The Mending of Broken
Bones, which is a modern guide to classical algebra.
So here's what Paul says. We thought it was obvious
that any two lengths must be commensurable. Not.
We assumed that every number was algebraically
or geometrically describable. Not. We thought
we were the lords and masters of mathematical
reality. Not even close. We get to play there,
and we certainly get to invent and discover and
explore. But we most definitely do not own the
place. In fact, my favorite feature of mathematics
is that it continually blows my mind and teaches
me humility. To be utterly and absolutely wrong
is one of the greatest gifts one can receive.
And here's how he ends it. Luckily, mathematics
is extremely generous in this regard.