Hi, yesterday we talked about how truth and math
is not just something that we are told. It is
something we can discover, something we can see.
And once we see it, we're convinced. But once
we begin to see truth clearly, sometimes we are
shocked. Because sometimes the truth we discover
is not what we expected. This is absolutely one
of the most delightful things about math. Even
when we follow the logic carefully, even when
we ask good questions, the answer can still surprise
us. A simple example is with prime numbers. Prime
numbers are numbers like 2, 3, 5, 7, numbers
that are only divisible by 1 and themselves.
If you start listing them in order, they begin
to spread out. They appear less often. And it's
natural to wonder, do they eventually run out?
Do we reach a point where there are no more primes?
Well, over 2 ,000 years ago, Euclid asked that
same question. What he discovered was that no
matter how many prime numbers you find, there
is always another one. always another, then another.
When you think about it, this is amazing because
how do you prove that a list of numbers is infinite
if you cannot list an infinite set of numbers?
But just because we cannot list the infinite
set, we can prove that the set is infinite. And
the proof is surprisingly simple. It is such
a thing of beauty that we will discuss in detail
on a later day. For now, the key idea is that
the proof shows that even if you tried to gather
all the prime numbers together, you could always
construct a new number that leads you to another
one. And when you see it, really see it, it changes
how you think. Because the truth is bigger than
you expected. And I think that is the part of
what makes truth so compelling. It does not just
confirm what we already believe. Sometimes it
stretches us. Sometimes it expands what we thought
was possible. And when that happens, we have
a choice. We can hold on to what feels familiar,
or we can stay open. to what we are discovering.
Because truth is not always comfortable, but
it is worth following. Truth does not just answer
our questions, it reshapes them. So my question
today is, am I open to discovering what is true,
even when it stretches what I expect?