[MUSIC PLAYING] Did humans just make mathematics, as in are prime numbers just a figment of our collective imagination?

Or do they really exist separate from our human minds?

To start, here's a fairly simple math statement.

The Goldbach conjecture says with every even number is the sum of two primes.

A prime number is one that's only divisible by itself and 1, like 2, 3, 5, 7, 11, 13, 17, and so on.

For example, 6 is not prime because it's divisible by 2 and 3.

So the Goldbach conjecture says that every even number is a prime number plus another prime.

For example, 4 is 2 plus 2.

8 is 3 plus 5.

20 is 7 plus 13, and so on.

Prime numbers are defined by and principally about multiplication and division.

But the Goldbach conjecture is about what happens when you add them.

That's part of what's weird about the Goldbach conjecture and mathematics in general.

Humans pick how to define mathematical objects.

But then those objects take on a life of their own and have properties and behaviors we didn't intend.

This throws into question whether or not math exists independently of the human mind.

In the case of the Goldbach conjecture, we picked the definition of prime number, and then their relationship with even numbers just sort of sprung up.

In this way, mathematics can feel like you wrote the first page of a book, and then you're figuring out the rest.

Nobody had to write the next page.

It just sort of unfolds, like you're tapping into a story that was already written, which is strange.

And what gets even stranger about the Goldbach conjecture is that in the nearly 300 years since the conjecture was formulated, nobody has ever proved it.

Computers have tested even numbers up to this huge number, and they've always been able to find two primes that add up to the even number.

But mathematicians don't have a proof that it holds for all even numbers.

And this points to something really important about math.

Math isn't like science where you gain enough evidence and declare something true.

You have to have a precise logical proof.

In this sense, mathematics is about perfect knowledge.

Individual mathematicians may make logical mistakes, but whole mathematical theories aren't disproven the way they are in science.

Mathematical knowledge just accumulates rather than revises itself.

But what is all this knowledge about?

Well, it's about mathematical objects, like prime numbers.

And numbers are weird objects.

In fact, there's a serious debate about whether numbers even exist.

You can't see or touch them, which makes them seem sort of made up.

But unlike most fictions, numbers never change.

You can make up new rules for checkers, but 2 plus 3 is always going to be 5.

And besides, if numbers don't really exist, how come we can use them to construct skyscrapers that don't fall over?

Let's move away from prime numbers and try to answer these questions with another mathematical object, the circle.

In school, you probably learned that pi is the ratio of the circumference to the diameter of a circle.

This statement is not about a real circle.

Real circles have defects.

And while the number pi is infinitely precise, any real measurement has a limited amount of precision.

So actually, pi is the ratio of the circumference to the diameter of a perfect circle.

This fact about the definition of pi seems simple enough, but there are actually two very different ways to interpret it.

The first option is called realism.

A realist would say that there is some fixed, idealized circle, and all mathematicians throughout all of space and time are talking about the properties of that circle.

Pi is the ratio of the circumference to the diameter of that circle.

Realists think that the objects that mathematicians study are real, really real, and exist independently of the humans that think about them.

So a perfect circle exists, just like a water molecule or the planet Jupiter exist.

One version of this is called Platonism, as in the ancient Greek philosopher Plato and his idea of platonic forms.

There's a form of a circle.

And all circular things in the physical world embody that form.

But mathematicians aren't studying physical circles.

They're studying the platonic form of a circle, just as chemists study molecules and astrophysicists study planets.

Here's what's great about realism.

It aligns with the way people feel while doing math.

It positions mathematicians as investigators of sorts.

They're like explorers in a sea of numbers and shapes and ideas.

It's how actual mathematicians talk.

But Platonism leads to a big problem.

Where does our knowledge of these forms come from?

Actual explorers learn by physical interactions.

When Neil Armstrong famously described the surface of the Moon, he was relying on his senses like touch and sight.

Photons were bouncing off the Moon into his eyeballs.

But when Goldbach saw the connection between even numbers and primes, it's hard to believe he was literally interacting with the Platonic forms of the numbers.

If numbers or a perfect circle exist, how do we access them to learn about them?

This question is irritating enough that many philosophers of math have taken the opposite approach, antirealsim.

Let's go back to our fact about pi.

Pi is the ratio of the circumference to the diameter of a circle.

The antirealist would say that mathematicians just made up the rules for being a circle and are drawing out the consequences of those rules.

One version of this is called Formalism, which sort of equate mathematics with a game, like chess.

Mathematicians are playing out the rules of the game.

It's dependent on the human minds that created it.

What's great about Formalism is that it solves the problem of Platonism.

We know things about math because we made up the rules.

We don't have to look anywhere else.

The problem in treating math like it's completely made up is what mathematician Eugene Wigner described as the "unreasonable effectiveness of mathematics in the natural sciences."

Math is robust and an amazingly precise language to describe everything, from quantum mechanics to the stock market.

It's hard to imagine that a game so accurately describes reality.

Knowing the rules of chess doesn't help scientists launch rockets into space, but knowing math does.

That's part of why it's so difficult to understand what math is.

Mathematician Reuben Hersh wrote that "The working mathematician is a Platonist on weekdays, a Formalist on weekends."

At the end of the day, most mathematicians, this one included, are content just mapping it.

Mathematicians love their subject.

It's beautiful and mysterious, abstract yet concretely descriptive, whatever it is.

Which camp do you fall into?

Let us know in the comments, and we'll see you next week on "Infinite Series."

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