Hi. Don't you just love it when you struggle
understanding something in math for a long time
and finally the light bulb turns on? This happened
to me this week. I've known for a couple of years
that mathematicians state that the set of integers
and set of even numbers are the same size. Now
that does not seem possible since the even numbers
are only half of the integers. But finally this
week, I looked at it in a different way. I considered
what happens if you take the positive integers
and multiply them by 2. The result is the set
of even integers. Multiplying numbers in a set
by a constant 2 should not change the number
of items in a set. Another slightly different
perspective is to refer to the integers as yards.
So 1 is 1 yard, 2 is 2 yards, etc. Then we take
this pattern to infinity. But what if we take
those same distances and measure feet? Then we
have 3 feet, 6 feet, 9 feet, dot, dot, dot. We're
not changing the size of the set, just scaling
the measurement of the set. Thus, scaling the
integers should not change the number of items
in a set. So the light bulb came on and it all
made sense. I like math not only because understanding
enhances my knowledge and challenges my thinking
as to what is true, but it also challenges me
to be patient and give these paradox time to
sink in. I have to think about them from different
angles, consider different perspectives. But
once the light bulb turns on, I feel like I joined
this special club of math people. that have also
wrestled with these paradoxes and have seen it
through. Understanding a paradox is different
than just solving a difficult problem. To understand
paradoxes, we have to let go of one intuition,
like the even set of numbers is the same size
as the evens and the odds, and adopt a new intuition.
Eugenia Cheng takes this a step further and says
that math is not always identifying what is true
or not true, but identifying in what sense something
is true. This seems to be a good muscle to not
only understand paradoxes, but to grow in healthier
relationships. Sometimes other people's perspectives
seem difficult to understand, almost like a paradox.
I find it is easy to quickly dismiss their perspective
as wrong, but my math training, encourages me
to stay in the conversation longer and understand
in what sense they are right and in what sense
do I need to expand my thinking. Paradoxes are
not only fun puzzles to solve, but they remind
me to invite new perspectives.