Hi, mathematics has taught me that when something
feels impossible to imagine, the limitation may
belong to my intuition rather than to reality.
A wonderful example is infinity. For a long time,
I used to think of infinity as simply endlessness,
something unimaginably large. that keeps going
and going and going forever. If two collections
were both infinite, I assume that they must be
the same size. After all, how could anything
be larger than something that never ends? Well,
mathematics offers a surprising answer. Some
infinities are larger than others. Consider the
counting numbers 1, 2, 3, 4, and so on. And now,
Consider only the even numbers, 2, 4, 6, 8, and
so on. Now, it seems that there should be fewer
even numbers because they make up only part of
the counting numbers. But every counting number
can be paired with exactly one even number. 1
with 2, 2 with 4, 3 with 6, and so on. Neither
list ever runs out. So in the mathematical sense,
These two infinite collections have the same
size. Wow. Okay, the rational numbers. These
are the numbers that can be written as fractions.
They, of course, are also infinite. Between any
two rational numbers, we can always find another.
They seem to be everywhere along the number line.
Remarkably, however, they can still be arranged
into a list. Their infinity is the same size
as our counting numbers. Okay, then we encounter
the irrational numbers, numbers such as square
root of 2 and pi. These cannot be expressed as
ratios of whole numbers. Now together, the rational
and irrational numbers, they form the real number
line. But the real numbers cannot be placed into
a complete list, even an infinite one. So their
infinity is larger. This is one of those mathematical
discoveries that makes me feel it's very strange,
but it's also very spacious. The number line
appears so simple when we draw it, and yet hidden
within it are levels of infinity that ordinary
intuition would never predict. Ideas like this
have changed me because they have made me less
confident that the limits of my imagination are
the limits of what is possible. It is formed
by experience with finite objects in an everyday
world. It is not always prepared for infinity.
Mathematics lets me move beyond that first intuition.
It gives me tools for approaching ideas that
I cannot picture and language for distinctions
I would not otherwise know how to make. I may
never be able to visualize infinity, but mathematics
has taught me that understanding does not always
begin with being able to imagine the answer.
Sometimes understanding begins when I allow my
imagination to be changed.