Chapter Content
Okay, so, you know, we all kind of crave certainty, right? In this crazy world. And you might think, "Where can I find it?" Well, a lot of people point to mathematics, this, like, fabled land where everything's, you know, perfectly proven. This island of pure logic, where the antidote to uncertainty is just, like, 100% faith-free proof.
But, uh, that's, like, totally wrong.
I mean, sure, some mathematicians probably dream of that, you know? This, like, enchanted island. But, as a mathematician myself, I can tell you, there's just no such thing as a faith-free proof. Really. The whole idea, the whole concept of mathematical proof, it's built on a huge, vast sea of faith. It's, like, mathematics totally depends on beliefs that can't be proven, can't even be imagined sometimes, and are, in a lot of cases, just plain, like, preternatural.
And, uh, it's kind of risky to say this out loud, actually. There was this one guy, you know, back in, well, some time ago, who went onto this online forum called Stack Exchange. It's supposed to be a place for people interested in, like, the fundamental nature of knowledge and reality and all that. Anyway, he asked this simple question to the math community: "Does mathematical proof require faith?"
He was an engineer, and he thought it would, you know, spark a good discussion. But he said he was basically, like, "run out of town." People got really mad at him!
So, does it? Does mathematical proof require faith?
Well, let's go back in time, way back, to Athens and Alexandria. And let's meet these two guys, Aristotle and Euclid. They're, like, the Madonna and Prince of their day, you know, just going by single names.
Aristotle, he's just invented the rules of logic, this, like, super strict new way of thinking. It's, um, best shown with something called a syllogism. It goes like this: all ravens are black, Edgar is a raven, therefore, Edgar is black.
Okay, so that first statement, "all ravens are black," that's an axiom. It's an assumption. It's a stated belief. Hopefully, a smart one. But it can't be proven. You just gotta have faith in it, you know? If you believe it, then you'll see something important about Edgar. Remember, like, believing is seeing.
And Euclid, he's, like, super impressed with Aristotle's logic. He's using it to figure out everything about plane geometry, the study of shapes on a flat surface.
And to do that, Euclid has to assume, like, thirty-three axioms! Beliefs he can't prove. There are, like, twenty-three definitions, five postulates, and five common notions.
The common notions are things like, "Things that equal the same thing also equal one another." Or, "If equals are added to equals, then the wholes are equal." They're, like, really obvious, right?
But think about it. Why? Why do these axioms seem so obvious to you?
It's because they're, like, logical things you can easily check yourself by counting marbles or beads. They're, like, trivial truths that only need, like, IQ-based faith to believe.
But some of Euclid's other axioms, they're not so obvious, not so logical, not so trivial. They're actually, like, super weird and require, like, SQ-based faith to believe.
Like, Euclid says a point is something with no width, no depth, and no length. It's, like, both something and nothing! Just like the quantum vacuum.
A point, it doesn't make sense. It's not logical. You can't even see it or imagine it.
Try to picture a point. I dare you. You can't do it any more than you can picture something that's both alive and dead, black and white, true and false.
But Euclid's definition of a point, it's not nonsense. You can't just ignore it. Why? Because it's a profound, like, translogical axiom that he's using to prove the whole thing, the whole of plane geometry.
Is your head spinning yet?
Think about this. A Euclidean point, it's something your eyes can't see, your IQ can't understand, your imagination can't even grasp. But your brain, the right hemisphere, it can kind of see it sightlessly, wordlessly. And with its help, you can draw, like, a crude version of it with a pencil.
And, Euclid's religious-like faith in these axioms, these things that aren't logical and can't be seen, it turned out to be super smart. He revolutionized math.
For, like, over two thousand years, we've used Euclid's geometry to build bridges and skyscrapers, to lay out floor plans, and to, like, send rockets to the moon. More geometry textbooks have been sold than any other book except for the Bible, supposedly.
So, Euclidean geometry, it shows that logic built on enlightened faith can be super powerful. But is there a limit to logic's power? Or is it, like, all-powerful, like some people think?
And the answer is, no. It has limits. Big ones.
It started with this German logician, Gottlob Frege. He wanted to do for arithmetic what Euclid did for geometry.
He started with six axioms, and he thought he could prove all of arithmetic, starting with 1+1=2. He worked for years, and finally, he published the first part of this big thing he was working on.
Nine years later, he finished the second part. But just as he was about to send it to the publisher, he got some bad news from Bertrand Russell, this other mathematician.
Russell found a mistake in the first part. Not a typo, a, like, fatal flaw.
Russell found a problem with Frege's logic that went back to his fifth axiom, which is about how groups, or sets, work. Sets are important in math, so you have to be careful how you define them.
Like, the group of all living people on Earth is the set that contains all and only those people on Earth who are not dead. Simple, right?
But imagine a village of only clean-shaven men. And the village's only barber says, "I shave all and only the men who don't shave themselves."
So, what can you say about the set of all men who shave themselves? Does the barber shave himself?
Think about it.
If you say yes, then the barber's wrong, because he only shaves men who don't shave themselves.
If you say no, then the barber's wrong again, because he shaves all the men who don't shave themselves.
So, there's no logical answer. Logic just fails completely.
Here's another way to think about it: is this headline true or false? "This statement is false."
If it's true, it's false. If it's false, it's true!
Logic just goes crazy and sucks us into this never-ending loop.
That's the flaw Russell found in Frege's work. Frege was shocked.
He said, "Your discovery has surprised me beyond words...it has rocked the ground on which I meant to build arithmetic."
Frege published the second part anyway, but he added this sad warning. He never published the third part, and he basically gave up on using logic to prove arithmetic.
All mathematicians were upset by this. By 1925, David Hilbert, this other mathematician, said that the paradoxes were just, like, "intolerable." He said, "If mathematical thinking is defective, where are we to find truth and certitude?"
So, Hilbert told his colleagues not to give up hope. The paradoxes, he said, were just because of bad axioms. The solution was to choose better axioms, more carefully.
Mathematicians were all for it. Russell and Whitehead, these other guys, they published this thing called Principia Mathematica, which was supposed to put math back on solid, logical ground.
But then Kurt Gödel came along. In 1931, he found a big problem, not just in Principia, but in logic itself. A problem that couldn't be fixed. Ever.
Gödel's big thing is called the incompleteness theorems. Here's a kind of technical way to explain them:
If you have any system of logic that's strong enough to describe all of arithmetic, then it'll either be incomplete, meaning it can't prove everything that's true, or it'll be inconsistent, meaning it has paradoxes and it's completely unreliable.
So, whenever you try to think logically about a complicated thing, one of two things will happen:
One, you'll say something that's true, but you can't prove it. Logic will fail you because it's not strong enough. As one logician said, "There is more to truth than can be caught by proof." Truth is bigger than proof.
Or two, you'll prove something is true using logic, but it's actually not true. Your logic seems good, but it has hidden paradoxes.
That was Frege's problem. And now, Gödel showed, it was Russell's and Whitehead's problem too.
Russell was crushed. He'd always been an atheist, he was always bashing Christianity. He put his faith in logic and math. But Gödel had destroyed that faith.
Russell said he wanted certainty like people want religious faith. He thought he could find it in math. But after years of work, he realized he couldn't make mathematical knowledge completely certain.
Today, Gödel's theorems affect a lot more than just math.
First, they make it less likely there'll be a "theory of everything" in physics. Einstein tried to find one, but he failed. Gödel's theorems say that any logically consistent theory of everything is as impossible as the Tooth Fairy.
Second, Gödel's theorems make it possible for "God exists" to be true but unprovable. Truth is bigger than proof.
If logic can't even handle arithmetic, it definitely can't settle an argument about God.
Third, Gödel's theorems say that the Isle of Mathematics floats in a sea of faith. Mathematicians have to believe in things they can't prove.
No matter how smart they are, no mathematician can make a logical argument without first believing in some assumptions.
At worst, the assumptions will be bad and lead to disaster, like with Frege and Russell. At best, they'll be good and lead to great things, like with Euclid and Gödel.
Either way, Gödel's theorems prove, using logic, that math is built on faith. Like any religion.
As one mathematician said, "If a 'religion' is defined to be a system of ideas that contains unprovable statements, then Gödel taught us that mathematics is not only a religion, it is the only religion that can prove itself to be one."
Even though mathematicians liked Aristotle's logic and Euclid's geometry, they wondered, is that it? Is there only one logic? Is there only one geometry?
And the answer is no. There are lots of logics and lots of geometries. They all depend on different beliefs.
Aristotle believed that something is either true or false, there's nothing in between. But there are other ways to think about it. For example:
Three-valued logic says something can be true, false, or unknown.
Four-valued logic says something can be true, false, true and false, or unknown.
Fuzzy logic says something can have an infinite number of truth values, anywhere from zero to 100 percent true.
Fuzzy logic is used in things like antilock brakes. The computer has to decide how hard to hit the brakes by weighing the truth values of lots of different things.
The discovery of all these different kinds of logic has made Aristotle's logic less important. One mathematician said that Aristotle's logic "is inadequate for mathematics. It was already inadequate for the mathematics of his day."
So, there's a lot of talk about critical thinking, about teaching students to be critical thinkers. I agree with that.
But you have to understand that critical thinking now means more than just logical thinking.
Aristotle's logic is just one way to think wisely. It's the Model T of critical thinking.
Same thing with Euclid's geometry.
Euclid said that parallel lines never cross, even if you extend them forever.
But that's only true for flat surfaces, not curved surfaces. That's led to lots of other kinds of geometry.
Spherical geometries are for round surfaces, like Earth. Lines of longitude are like parallel lines, but they bend toward each other and meet at the poles.
Hyperbolic geometries are for saddle-shaped surfaces. On those surfaces, parallel lines move away from each other.
Riemannian geometries are for surfaces with four, five, six, or more dimensions. These surfaces can be flat, spherical, or hyperbolic.
We can't see or imagine multidimensional Riemannian surfaces, just like we can't imagine Euclid's point. They're amazing products of human intelligence.
And like Euclid's point, Riemannian surfaces are actually useful. Einstein used a 4D Riemannian surface in general relativity to describe gravity. It has three space dimensions and one time dimension.
Experiments have shown that Einstein's theory is true. That means that the Riemannian surfaces it uses are the creation of enlightened faith.
Einstein was surprised that math, something that's just a product of human thought, is so good at describing the real world.
One time, I was at a physics seminar with this mathematician named Eugene Wigner.
Wigner didn't know about my ideas, but he realized that math seems to be powered not just by logic, but by something beyond that.
He said that "the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and ... there is no rational explanation for it."
There was a time when mathematical proofs were short enough to check by hand. The proofs I did in high school geometry were like that, and my teacher was super strict about it. I didn't realize it then, but that was the real start of my scientific training.
But in high-level math, those days are gone.
It didn't happen all at once, but by the 1990s, it was clear that math wasn't as clear as it used to be.
One journalist wrote that mathematicians used to measure progress by what they could prove. Now, he said, doubts have finally infected math.
He wrote that because a mathematician named Andrew Wiles claimed to have proven this old math problem. The proof was hundreds of pages long, so it was hard to check. And when people finally did, they found a mistake.
Wiles had to go back and fix it.
A year later, he said he'd fixed it. People were skeptical. But after checking the proof carefully, other mathematicians agreed that it was correct.
That proof showed that math was entering a new age, an age of growing uncertainty.
Since then, it's only gotten worse. Proofs are longer and harder to check, and more of them are being done with computers. That's making math even more uncertain.
One mathematician said that we're now in an age where the big statements of math are so complex that we might never know for sure if they're true or false. That puts us in the same boat as all the other scientists.
As of this writing, the longest mathematical proof in history was made a few years ago by some people and a supercomputer. It's 200 terabytes long, which is like the entire US Library of Congress. It would take ten billion years just to read it, and even longer to check each step.
So, the Sea of Faith that the Isle of Mathematics floats on is much deeper than anyone thought. Aristotle and Euclid didn't see any of this coming.
But that's only bad news for people who thought that faith-free proof was possible. It's not. It never has been.
The Sea of Faith on which math floats is full of unprovable, unimaginable beliefs.
Beliefs that defy mere IQ.
Beliefs that describe the real world with amazing accuracy.
Beliefs you have to accept if you want to see their astonishing truths.