One of my favorite things about mathematics is
that it is incredibly creative. When people think
of math, they often imagine a rigid system where
every rule has already been decided. But that's
not actually how mathematics begins. Mathematicians
invented notation. We've created the definitions
and we've decided what assumptions or axioms
we're going to start with. So in that sense,
mathematics is an act of creativity. For example,
someone, specifically think Leonard Euler, had
to decide that we would use the symbol pi for
the ratio of a circle's circumference to its
diameter. Someone had to define what a vector
is. Someone had to formalize the idea of a group,
or a field, or even a matrix. These were not
discovered lying around in nature. They were
human inventions designed to help us describe
patterns more clearly. But here is the part that
absolutely fascinates me. Once we've chosen our
definitions and assumptions, our freedom ends.
And at that point, we do not get to decide what
comes next. Logic takes over. We do not get to
vote on the theorems we would like to be true.
We do not get to ignore conclusions that make
us uncomfortable. We simply follow the logical
consequences of the choices that we have already
made. And this is why in the history of mathematics,
we have uncovered so many surprising discoveries.
We've talked about them, irrational numbers,
non -Euclidean geometry. and different sizes
of infinity, all challenge our intuition. And
many mathematicians initially resisted these
ideas because they felt wrong. But logic did
not care how anyone felt. If the reasoning was
sound, then the conclusions had to be accepted.
And I think that is one of the most humbling
lessons that math teaches us. Our intuition is
valuable. It often points us towards interesting
questions, but intuition is not our final authority.
Truth is. So as we work through problems in our
work, and as you work through your problems,
you will probably have moments where you're convinced
an answer must be right, and then only to discover
you have a mistake in your reasoning. I know
I've certainly had many of those moments. They
are frustrating, but they're also one of the
ways mathematics shapes us. It teaches us to
ask a different question. It's not what do I
want to be true, but given these assumptions,
what must be true? And that's a very different
way of thinking. And it is the foundation for
this week's theme on humility. To me, humility
is not thinking less of myself, but it is recognizing
that once the assumptions are set, all of us,
the students, the professors, the professional
mathematicians alike, we all stand under the
same authority, the authority of logic. Wonderful.
Well, tomorrow, we're going to look... at one
of the biggest reasons that humility is necessary
in mathematics. Our intuition is often wrong,
and that is perfectly okay.