Hi. From my observation, there are essentially
two different mindsets. One mindset is fixed,
where problems get bigger and possibilities get
smaller. The other is a growth mindset, where
we ask, what is a bigger reality that I'm not
yet seeing? This week is for that second mindset.
And our topic for the week is infinity. On one
hand, infinity feels vague. What even is infinity?
But on the other hand, it feels straightforward.
The positive integers go on forever. One, two,
three, four, dot, dot, dot. Given any positive
integer, we can always create the next integer
simply by adding one. And because we can always
add one more, the integers continue without bound.
That idea feels intuitive. We can also look at
subset of integers, like the odd numbers. The
odd numbers are also infinite because we always
know how to create the next odd number. Simply
add two. Last week, we studied prime numbers.
Prime numbers are another special set of integers.
But unlike the odd numbers, we do not have a
simple pattern for generating the next prime
number. In fact, prime numbers become more spread
out as numbers get larger. So this raises a natural
question. Is there a largest prime number? Or
do the primes always continue forever? This question
is much more difficult because knowing that,
say, 107 is prime tells us almost nothing about
what the next prime number will be. So how could
we possibly know whether the primes are infinite.
Well, more than 2 ,000 years ago, Euclid proved
that there are infinitely many prime numbers.
And this opened the door to a new kind of infinity.
Not an infinity based on a simple repeating pattern
like adding 1 or adding 2, but an infinity hidden
inside something unpredictable. Suddenly, The
positive integers no longer look like one simple
collection. Hidden structure began to emerge.
It's a little like looking at Earth from far
away. From a distance, it appears as one sphere.
But as we move closer, we begin to notice oceans
and continents, mountains and deserts. Structure
hidden within what first appeared simple. And
doing mathematics is often like that. We take
familiar things, like numbers, and learn to see
them in deeper ways. And this week, we're going
to do that with infinity. We will discover that
prime numbers are only the tip of the infinity
iceberg. And what I find beautiful is that exploring
this iceberg has been a deeply human journey.
Human history is filled with many stories of
war, division, destruction, and the way we tear
things apart. But this week, we're going to study
a fresh side of human history. A story of people
across civilizations and centuries building something
together. Greek and Indian mathematicians. Islamic
scholars. The list goes on. Different cultures.
different languages, different centuries. Despite
these differences, step by step, humanity uncovered
deeper truths about infinity and slowly built
a shared structure of understanding together.
To me, that is one of the most remarkable and
beautiful parts of mathematics. So I am super
excited. to explore a few pieces of that larger
infinity story with you this week. So, are you
ready to grow your mind?