Does math have a major flaw? - Jacqueline Doan and Alex Kazachek
Consider this mathematician,
with her standard-issue infinitely sharp
knife and a perfect ball.
She frantically slices and distributes
the ball into an infinite number of boxes.
She then recombines the parts
into five precise sections.
Gently moving and rotating
these sections around,
seemingly impossibly, she recombines them
to form two identical, flawless,
and complete copies
of the original ball.
This is a result known in mathematics
as the Banach-Tarski paradox.
The paradox here is not
in the logic or the proof—
which are, like the balls, flawless—
but instead in the tension
between mathematics
and our own experience of reality.
And in this tension lives some beautiful
and fundamental truths
about what mathematics actually is.
We’ll come back to that in a moment,
but first,
we need to examine the foundation
of every mathematical system: axioms.
Every mathematical system
is built and advanced
by using logic to reach new conclusions.
But logic can’t be applied to nothing;
we have to start with some basic
statements, called axioms,
that we declare to be true,
and make deductions from there.
Often these match our intuition
for how the world works—
for instance, that adding zero to a number
has no effect is an axiom.
If the goal of mathematics is to build
a house, axioms form its foundation—
the first thing that’s laid down,
that supports everything else.
Where things get interesting is that
by laying a slightly different foundation,
you can get a vastly different
but equally sound structure.
For example, when Euclid laid
his foundations for geometry,
one of his axioms implied that given
a line and a point off the line,
only one parallel line exists
going through that point.
But later mathematicians,
wanting to see if geometry was
still possible without this axiom,
produced spherical
and hyperbolic geometry.
Each valid, logically sound,
and useful in different contexts.
One axiom common in modern mathematics
is the Axiom of Choice.
It typically comes into play in proofs
that require choosing elements from sets—
which we’ll grossly simplify
to marbles in boxes.
For our choices to be valid,
they need to be consistent,
meaning if we approach a box,
choose a marble,
and then go back in time and choose again,
we'd know how to find the same marble.
If we have a finite number of boxes,
that’s easy.
It’s even straightforward
when there are infinite boxes
if each contains a marble that’s readily
distinguishable from the others.
It’s when there are infinite boxes
with indistinguishable marbles
that we have trouble.
But in these scenarios,
the Axiom of Choice lets us summon
a mysterious omniscient chooser
that will always select the same marbles—
without us having to know anything
about how those choices are made.
Our stab-happy mathematician,
following Banach and Tarski’s proof,
reaches a step in constructing
the five sections
where she has infinitely many boxes
filled with indistinguishable parts.
So she needs the Axiom of Choice
to make their construction possible.
If the Axiom of Choice can lead
to such a counterintuitive result,
should we just reject it?
Mathematicians today say no,
because it’s load-bearing for a lot
of important results in mathematics.
Fields like measure theory
and functional analysis,
which are crucial
for statistics and physics,
are built upon the Axiom of Choice.
While it leads to some
impractical results,
it also leads to extremely practical ones.
Fortunately, just as Euclidean geometry
exists alongside hyperbolic geometry,
mathematics with the Axiom of Choice
coexists with mathematics without it.
The question for many mathematicians
isn’t whether the Axiom of Choice,
or for that matter any given axiom,
is right or not,
but whether it’s right
for what you’re trying to do.
The fate of the Banach-Tarski paradox
lies in this choice.
This is the freedom mathematics gives us.
Not only is it a way to model
our physical universe
using the axioms we intuit
from our daily experiences,
but a way to venture into abstract
mathematical universes
and explore arcane geometries and laws
unlike anything we can ever experience.
If we ever meet aliens, axioms which seem
absurd and incomprehensible to us
might be everyday common sense to them.
To investigate, we might start by handing
them an infinitely sharp knife
and a perfect ball,
and see what they do.
