Hi, so far this week we've talked about submitting
ourselves to logic, allowing our intuition to
be corrected, and admitting that we make mistakes.
So today I want to talk about another kind of
humility. The humility to recognize that other
people may see something that you do not. Now
one of the beautiful things about math is that
there is often more than one correct way to solve
a problem. So two students can arrive at the
same answer, but using completely different approaches.
One might use algebra and another might use a
graphical argument, for example. Someone else
might recognize a clever substitution that everyone
else missed. Now, sometimes one method is more
efficient and sometimes another method might
give deeper insight. And sometimes a method that
seems longer. actually teaches you more about
why the result is true. And that is one reason
why I really enjoy looking at other people's
solutions. Not because mine is necessarily wrong,
but because theirs might reveal an idea that
I had not considered. So when we're first learning
math, it's easy to become attached to our way
of doing things. After all, it is the method
that makes sense to us. So when we see someone
solve the problem in a different way, our first
reaction can be, why would they do it that way?
So maybe a better question is, what do they see
that I do not? And that question can change everything
because it turns another person's solution from
something to maybe criticize into something to
learn from. And this is one of the reasons mathematics
has progressed so rapidly. Every generation builds
on the insights of those who came before. Different
mathematicians bring different perspectives.
They bring different tools and different ways
of thinking. No single person sees everything.
And of course that is okay. Humility reminds
us that our understanding is always incomplete.
There is almost always another viewpoint that
can deepen our understanding or simply reveal
a simpler way to think about a problem. And this
is a lesson I've had to learn the hard way. And
of course, not every approach is correct. Mathematics
still demands logical consistency. Different
methods must ultimately agree with the underlying
truth. But when multiple correct approaches exist,
we do not have to defend our favor as if it's
the only valid path. Instead, we can appreciate
the richness that comes from seeing the same
truth from different angles. So, when someone
creates a solution that is different from yours,
here are some questions you can ask yourselves.
Why do they choose that approach? And what does
their method make easier to see? And also, what
can I learn from it? Humility is not just admitting
when we're wrong. Sometimes it is recognizing
that even when we are right, we have plenty that
we can learn.